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Fung, ShingPen Leung, and ShingChiu Leung ACKNOWLEDGMENTS I would never be able to adequately thank Dr. R. Raymond Issa, my chair, for making room for me to develop my research question, as well as helping me to choose the direction of my life. I want to thank him not only for his tremendous guidance, considerable patience and encouragement throughout my study, but also for his endless trust, respect, and understanding, which has forged me into a better person, not only with intelligence, but with responsibility. I am especially grateful to Dr. Wayne Archer, Dr. Ian Flood, Dr. Kevin Grosskopf and Dr. Robert Cox, for their discussions, suggestions, and encouragement during the development of this dissertation. It is a great honor to have them serve on my committee. I am in debt to Dottie Beaupied for her tremendous helps, especially during the dissertation submission process. I would also like to acknowledge the generous financial support from the University of Florida and the UF Alumni Association, from which I will carry the Gator spirit for the rest of my life. I am in debt to Andrew Weiss, who has been the best mentor in my real estate profession, and has also provided generous help in data collection for this study. Besides him, I was working with an amazing team in Parmenter Realty Partners, and especially thankful to Darryl Parmenter, Ed Miller, and Mark Reese, for their insightful advice on career choices and their tremendous helps at work. Special thanks go to all my folks when I was in UF, whose love and friendship became part of the happiest memory of my life. I am especially grateful to Yujiao Qiao, Yang Zhu, Hongyan Du, Dongluo Chen, Jon Anderson, and Hazar Dib, whose encouragements have me to complete this dissertation in time. I want to extend a special word of thanks to all my mentors in the past, Dongshi Xu, Fuchang Lai, Shensheng Xu, Shouqing Wang, and David Ling, whose wisdom and insight have profound influence on my character and personality. This work is dedicated to Lezhou Zhan, my husband and best friend, for his company throughout my life in good days and in bad ones; and to my beloved family: Lau Leung, my father; Sau Pik Fung, my mother; ShingChiu Leung, my little brother; and ShingPen Leung, my deceased brother. The honor goes to them, for their thirty years of nurture with endless love and care. TABLE OF CONTENTS page A CK N O W LED G M EN T S ................................................................. ........... ............. ..... L IS T O F T A B L E S ................................................................................. 9 LIST OF FIGURES .................................. .. .... ..... ................. 10 ABSTRAC T ................................................... ............... 13 1 IN TR O D U C T IO N ................................................................................ 15 B a c k g ro u n d ....................................................................................................................... 1 5 Statement of Research Problem ...................................................................................................16 G oal and Objectives ............................................. 18 R research Scope ..................18................................................ Significance and Contributions................................................................ ...... 19 O organization of D issertation .............................................................................. ...............19 2 REVIEW OF REAL ESTATE VALUATION ...................................... ....................20 C u rren t P ra ctic e ..........................................................................................................2 0 Distinguishing Acquisition and Development ............... .....................................20 Typical A acquisition V aluation Process ........................................ ....................... 21 Current Real Option Approach and Limitations .................................. ...............24 D decision Tree Analysis and Lim stations ........................................ ...... ............... 25 R eal O options in R eal E state ................................................................................. .......... 25 Theoretical M odels ...................... ......... .. .......... ............... 25 E m pirical T testing ................................................................ 3 1 The RER O A approaches ................. .................................... .... ........ ...............3 1 Summary ......... ......... ......... .................................. ......... 32 3 L IT E R A TU R E R E V IE W .......................................... ......... .................. ............................33 Traditional Discounted Cash Flow Approaches..... .................... ...............33 C capital B u dgeting T heory ............................................................................. ....................34 M market R isk and P private R isk ............................................................... .....................34 C capital A sset Pricing M odel........................................ ..... .................... ............... 34 D iscou nt R ate ................................................................3 5 Option Pricing Theory .................. .. ... .............................. ........... 36 Definition and Type of Options ......... ...................... ................ 36 BlackSholes Model and Stochastic Partial Differential Equations..............................38 L attic e s ................................................................4 2 M onte C arlo Sim ulation ......................................................................... ...................4 5 R eal O options A analysis A approaches ............................................... ............................. 46 Practical R eal Options M odel in Real Estate................................... .................................... 50 6 D decision T ree A naly sis........... ........................................................................ ...... ....... 53 S u m m ary ................... ...................5...................4.......... 4 METHODOLOGY ................................. ......... ....................... 55 R E R O M modeling P procedures ......................................................................... ...................55 Problem Framing ............... ......... .......................55 Approach Selection ............................. ............. ........... .... .. .............. 57 Risk Drivers Identification and Estimation..... .................... ...............57 Base Case Modeling ....... ......... ....... ........ ................. ............... 57 O option M modeling ................... ...... ............................ .. ........ .. .............58 Sensitivity A analyses .................................... .. ......... .............. .. 58 RERO M modeling Approaches .................................................................... ............... 58 T he C om bined A approach ........................................................................ .................. 59 T he Separated A approach ........................................................................ ...................6 1 R E R O M odeling T echniques......................................................................... ...................63 Rational for Using Binomial Lattices.................. ............... .....................63 M onte C arlo Sim ulation ......................................................................... ................... 64 Replicating Portfolio ..................................... .. .. .. ...... .. ............64 Binom ial Lattice w ith Jum p Process ........................................ .......................... 66 Investm ent with Private Uncertainty .................................................... ...... ......... 68 5 THE COM BINED APPROACH................................................... ..................................... 72 5 THE COMBINED APPROACH.......... ... .. ........ ...... ........72 C a se D e sc rip tio n ............................................................................................................... 7 2 Building V valuation .............................................. 73 Problem Framing ............... ......... .......................73 A p p ro ach S election n ................................................................................................... 7 4 Base case NPV calculation ...................................................................................................74 R isk D rivers M modeling ..............................................................78 O option M modeling ....................................................... 85 Sen sitivity A naly ses ................................................................9 1 S u m m ary ................... ...................9...................5.......... 6 TH E SEPA R A TED A PPR O A CH .................................................................................... 96 C a se D e sc rip tio n ............................................................................................................... 9 6 L a n d V a lu a tio n ................................................................................................................. 9 7 P ro b lem F ram in g ....................................................................................................9 7 Approach Selection ................................................ 98 Risk Drivers Identification and Estimation .......................................98 Base Case M modeling .................................................................... ......... 103 O option M modeling ...............................................................103 S en sitiv ity A n aly se s ................................................................................................ 10 8 S u m m ary ................... ...................1...................1.........4 7 7 CONCLUSIONS AND RECOMMENDATIONS.........................................................115 C o n c lu sio n s ............... ..................................................................................... .............. 1 1 5 Recom m endations for Future Research....................................................... ................... 116 L IST O F R E F E R E N C E S ..................................................................................... ...................118 B IO G R A PH IC A L SK E T C H .......................................................................... ....................... 122 8 LIST OF TABLES Table page 21 Comparison of Research Subjects, Model Variants, Contributions and Limitations. .......28 31 Type of Real Options. ............. .................................... ...................... 47 32 Financial Options versus Real Options.................. ........... .......................47 33 Correspondence between Financial and Real Options.................................................51 51 M major A ssum options for A rgus. ........................................ ............................................75 52 Correlation Between Random Variables. ........................................ ....... ............... 87 53 Statistical Summary of Monte Carlo Simulation Result.................................................87 54 Event Tree A ssum options. ......................................................................... .....................88 55 Sum m ary of Variable Effect on Option Value. .............................. ................................91 61 D evelopm ent A ssum options. ...................................................................... ...................97 62 Probabilities of Jump Diffusion and Binomial Processes..............................................107 LIST OF FIGURES Figure page 21 Real estate phases and major factors to consider...........................................................22 22 Current acquisition valuation process ........................................ .......................... 23 23 Real Options approaches for land valuation. ........................................ ............... 27 31 Payoff of call option and put option. ............................................................................ 37 32 C all option payoff exam ple ............................................................................... .............37 33 C all prem ium vs. security price. ............................................................. .....................41 34 Stock and option price in a onestep binomial tree..........................................................42 35 Stock and option prices in general twostep tree. ................................... ............... 44 36 M onte Carlo sim ulation output. ............................................... ............................... 45 41 Critical steps in RER O analysis .............. .............................................. ............... 56 42 Twostep binomial lattice with different dividend yields ............................................66 43 Binomial lattice with jump process ........................... .......................... 68 44 Quadranom ial lattice ........... .............................................................. .. 70 45 Decision analysis. ...................................... .. ..... ..... .. ........... 70 51 211 Perim eter site plan ......................... ............ ... .. ........ ......... 73 52 B ase case N PV calculation. ...................................................................... ...................76 53 Spreadsheet model for Monte Carlo simulation. .................................... .................79 54 Historical market and subject property rental rates. ......................... .........................80 55 Returns correlation between market and subject property..............................................80 56 Normal distribution fit for historical returns on rental............... ..... .. ............... 82 57 Historical market and subject property occupancy rates. ...............................................83 58 Occupancy changes correlation between the local real estate market and the subject p rop erty ............................................................... ................ 84 10 59 Normal distribution fit for historical occupancy rates. ................................................. 84 510 Snap shot of Monte Carlo simulation assumptions........................................................86 511 Monte Carlo Simulation Result of Forecasting Variable z. .............................................86 512 Normal distribution fit of forecasting variable z............................................87 513 Event tree present value without flexibility ..................................................................... 89 514 Present value w ith flexibility. ................................................ ................................ 90 515 Option value in relation with present value. ........................................... ............... 92 516 Option value in relation with replacement cost. ..................................... ............... 92 517 Option value in relation with present value and volatility..............................................93 518 Option value in relation with volatility and discount rate...................................... 94 519 Option value in relation with replacement cost and volatility. ........................................94 520 Option value in relation with present value and replacement cost. ..................................95 61 Historical market average rental rates and return volatility......................... ............99 62 Normal distribution fit for historical market rental returns. ............................................99 63 Gross rental rate movement and probabilities. ..................................... ............... 100 64 Building value movement and probabilities. ......................................... ............... 101 65 Historical construction cost for highrise office building.........................................102 66 Construction cost change rate and inflation rate ............... ...... .... .....................102 67 D evelopm ent cost assume options ......... ................................................... ............... 103 68 Payoff and probabilities without flexibility. ...........................................104 69 Payoff matrices for project values without flexibility ............................... ...............105 610 Decision payoff and probabilities with flexibility. ................................ ............... 106 611 Payoff matrices of project value with flexibility. ............. ...................... ...............107 612 Present value in relation with rental rate and occupancy rate...................................109 613 Option value in relation with rental rate and occupancy rate. ............. ................110 614 Present value in relation with rental rate and development cost.................................. 110 615 Option value in relation with rental rate and development cost. ............. ................ 11 616 Present value in relation with rental rate and Cap rate. .............................112 617 Option value in relation with rental rate and Cap rate. ................. ...... .................112 618 Option value in relation with rental rate and Cap rate in 3D ................................... 113 619 Present value in relation with volatility and Cap rate............... ... .....................113 620 Option value in relation with volatility and Cap rate................... ................... ................114 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REAL OPTIONS FRAMEWORK FOR ACQUISITION OF REAL ESTATE PROPERTIES WITH EXCESSIVE LAND BY NgaNa Leung August 2007 Chair: Raymond Issa Major: Design, Construction, and Planning Our study touches a field that few researchers explore: the valuation model for acquisition of a property with excessive land that can be potentially converted into a new development. Traditional valuation focuses mainly on the building improvement. With the drastic capitalization rate compression, however, it becomes critical to identify and explore any hidden value in an acquisition. One of such challenges is valuing a large partially vacant parcel that can be potentially converted into a new development. Valuation of these parcels is not straightforward. Traditional discounted cash flow approach (DCF) cannot take into account the uncertainty and development flexibility. Alternative approaches are real options analysis (ROA) and decision tree analysis (DTA). However, the "twin asset" assumption required by the ROA methodology is often violated, especially for assets with private risk and rare events. The use of the same discount rate throughout valuation period in the DTA approach, regardless of changing risk characteristics upon the execution of decision making, allows for arbitrage opportunity. Our proposed real estate with real options (RERO) model is a framework that combines DCF, ROA and DTA analyses to specifically value real estate acquisition with excessive infill land. This methodology not only overcomes the shortcoming of current DCF method, but also is superior to the pure ROA or DTA analysis. Focusing on applicability in practice, this framework is developed intuitively with simple mathematics whenever possible. The study also explores a few unconventional real options cases, all of which could have been very complicated if modeled using the partial differential equations common in the academy, including (1)jump diffusion process that does not go back to normal diffusion, (2) risk drivers that do not follow the multiplicative stochastic movement, (3) private risk that has no market equivalent and hence violating the nonarbitrage option pricing assumption. All of these are implemented simply through binomial lattice with Monte Carlo simulation or DTA. The RERO framework is applied to a real case in Atlanta. Valuation has two parts: (1) the improvement is modeled using a combined approach with Monte Carlo simulation, and (2) the incremental value using a separated decision approach with binomial lattice technique. The valuation result is very close to the actual closing price. Three conclusions can be drawn from this study: (1) acquisition and development has different characteristics and deserve different kinds of attention; (2) consideration of managerial flexibility can change investment decisions; and (3) many unconventional real option valuation problems can be resolved by binomial lattice and Monte Carlo simulations. The novelty of this study is the research subject: property acquisition with excessive land. From the methodology standing point, the RERO framework is developed with ease of applicability in mind. It bridges the gap between research and practice for real options applications in the real estate industry. CHAPTER 1 INTRODUCTION Background Our study touches a field that very few academicians have explored: the valuation model for acquisition of a property with excessive land that can potentially be converted into a new development. The three major schemes in real estate property investment are acquisition, development, and operation. Acquisition is the ownership transaction of land and improvement; development is the process of adding improvement to the land; and operation is the daily management of the property. A majority of researchers focus on development, perhaps due to its high uncertainty. Acquisition, on the other hand, has been ignored to a certain extent considering its volume and size of transactions. Acquisition has been regarded as relatively low risk, since it is an investment on a touchable real property, which has historical operating track records, and numerous location attributes that last for decades and centuries. In recent years, however, real estate capitalization rates (defined by dividing the acquisition cost by annual net operating income) have compressed dramatically, meaning real estate is far more expensive to acquire than ever before. It becomes critical to identify and explore any hidden value in an acquisition target in order to be competitive. The proposed acquisition model has two parts: firstly, valuation of the income producing part of the property, mainly the improvement; secondly, the incremental value, mainly the excessive land that, depending on the circumstance of where the property is located, may have no value or substantial upside value. The proposed real estate with real options (RERO) model is a framework that combines real options and decision tree analyses. This methodology not only overcomes the shortcoming of the current discounted cash flow method, but also is superior to the existing real options or decision tree analysis. Focusing on applicability in practice, this framework is developed intuitively using simple mathematics whenever possible. The improvement is modeled using a consolidated approach with Monte Carlo simulation, and the incremental value using a separated decision approach with binomial lattice technique. Statement of Research Problem The fundamental value of real estate is the income producing capability of the property, which depends on many factors such as the amount of rental income to collect, the operating and financing expenses, the level of risk of the cash flow, the appreciation or depreciation of property value, and the performance of alternative investment instruments in the financial market. Acquisition valuation is the projection of future earning capability of a property related to other alternative investments. Traditional valuation mainly focuses on the building improvement. With the drastic capitalization rate compression, however, it becomes critical to identify and explore any hidden value in an acquisition. One such challenge is valuing a large partially vacant parcel that can be potentially converted into a new development. The attachment of excessive land to a property is not uncommon. Some developments were initially planned in phases, but the later phases were never implemented due to economic downturn or undesirable outcome of earlier phases. The land planned for later project phases thus remains vacant for a long time. Some early developments were planned on large parcels to insure sufficient space of surface parking. When the region becomes well developed and the economy turns to be more favorable, the vacant land becomes valuable for dense urban infill. Valuation of these parcels, however, is not as straightforward as applying the traditional Discounted Cash Flow (DCF) approach, which discounts expected future cash flows at a certain discount rate to get the Net Present Value (NPV). In the case of infill land, without new development, all future cash flow will be 0; with certain assumptions of new development, it will generate a value. Intuitively, in a hot real estate market where demand for developable land is high, such as in the South Florida, those parcels are extremely valuable. But in a warm or cold real estate market, the best use of such parcels may remain undeveloped until the market matures. The uncertainty and development flexibility need to be taken into account. Whether or not the land would be developed, when, what type, and what size all matters during the property acquisition. Alternative approaches are Real Options Analysis (ROA) and Decision Tree Analysis (DTA). The ROA approach has evolved from the financial option pricing theory to value real assets. Put simply, by acquiring a property, the owner has the right, but not the obligation, to develop the excessive land to its full use at a certain point of time in the future. Therefore, the value of a property with excess land should be higher than one without. The ROA methodology has been used to evaluate vacant land and to explain factors that affect development decisions. However, the ROA methodology requires one important assumption, that stochastic changes in the underlying value of the real asset to be developed are spanned by existing tradable assets or a dynamic portfolio of tradable assets, the price of which is perfectly correlated with the real asset (Pindyck, 1991). This so called twin asset is hard to find, especially for assets with private risk and rare events. Secondly, a lot of real options are compound options, which are options on options, not simply on a single asset, and consequently more complicated to solve by the pure option pricing methodology alone. The DTA approach evolves from management science. It is a method to identify all alternative actions with respect to the possible random events in a hierarchical tree structure. The DTA approach is developed to handle interactions between random events and management decisions. However, a major limitation of the DTA method is its use of the same discount rate throughout the valuation period, regardless of changing risk characteristics upon the execution of decision making, and thus allows for arbitrage opportunity (Copeland and Antikarov, 2005). Recent studies have turned to the combination of option pricing methodology, decision analysis, and game theory to solve real options problems. An ideal new approach should be able to address the unique characteristics of acquisition valuation with infill land, to handle the management flexibility, to take into account rare events such as new amenities driving up real estate value. It also needs to be intuitively simple for practical implementation. Goal and Objectives To overcome the above mentioned disadvantages of the current DCF, ROA, and DTA methodologies, this study has developed a framework, namely the Real Estate with Real Option (RERO) framework, as a combination of all three methods to specifically value real estate acquisition with excessive infill land. The objectives of this study are to: * Develop a theoretical integrated framework to address real estate acquisition problems; * Study factors affecting real estate acquisition and development, as well as their characteristics and statistical distributions; * Test and validate the model by applying it to real cases. Research Scope The research subject is real estate acquisition, which includes the value of the structural improvement, and the incremental value represented by excess developable land. The definition of excess land is that in addition to the portion necessarily attached to the existing structural improvement; the excess portion that is large enough for new development and at the same time meets local regulation requirements. Development factors are outside of our scope. Potential users of the framework are real estate investors who need a tool to estimate the building value and the land value during property acquisition. The proposed valuation model addresses mainly the economic risk and uncertainty for acquisition and development. Significance and Contributions The novelty of our study is the research subject: property acquisition with excessive land. To our knowledge, this is a field that few researchers have addressed. From the methodology standing point, the RERO framework is developed with ease of applicability in mind. It bridges the gap between research and practice for real options applications in the real estate industry. Organization of Dissertation In Chapter 2 we review the characteristics of real estate acquisition, existing valuation approaches and their limitations, as well as what a new approach needs to achieve. In Chapter 3 we review the theory and technical details of the different approaches currently available, in preparation for developing the proposed framework. We introduce the RERO framework in Chapter 4, including valuation procedures, the combined and separated approaches, and some new techniques developed to specifically apply to the case studies followed. Chapter 5 and 6 are case studies of the combined approach and separated approach respectively. Collectively they illustrate how the RERO framework can be applied to a broad spectrum of scenarios in practice. In Chapter 7 we conclude the study and suggest future research directions. CHAPTER 2 REVIEW OF REAL ESTATE VALUATION This chapter discusses the current practice in acquisition valuation, alternative approaches and their limitations, followed by a review of real options in real estate. It also analyzes how the proposed RERO framework needs to resolve the practical problems unique to real estate acquisitions. Current Practice Distinguishing Acquisition and Development Analogous to the financial market, the three major schemes in the real estate investment market are different and interrelated: acquisition, development, and operation. Acquisition is similar to a lumpy investment in a well established company with, in many cases, 100% ownership interest. Development is similar to the seeding of a startup company and bringing it to Initial Public Offering. Operation is the income producing process in the daily management of the property. This explains why research on development problems may not directly apply to acquisition valuation problems. A real estate investment firm may have a different agenda for the infill land than a real estate developer. The business of real estate development is to acquire and accumulate a considerable land bank, wait for appropriate timing and market demand to build new properties, and realize profit by selling the new properties to institutional investors. The business of commercial real estate investment, on the other hand, is to acquire existing properties, manage and improve the properties to receive the operating income from leasing. As an investment vehicle, commercial real estates tend to be traded more frequently than vacant land. As buildings get older and functionally obsolete, they usually change hands from passive institutional investors to active valueadded investors for cosmetic and functional upgrade and tenantmix adjustment. The developers, however, acquire land from different sources and wait more patiently in a real estate cycle before putting up new products to capture the maximal gain. Short holding periods and different business interest makes the infill land less valuable to an investor than the vacant land to a developer. The major factors to consider during acquisition are quite different from those in the development and operation processes (Figure 21). During acquisition, the major factors are location, market condition, market rent, pricing of the building and the land. Development factors, such as impact fee and school zoning, are outside the scope. If the investor wins the bid, he goes through the due diligence and financing process before actually plans for development of the vacant land. Although our model consists of the building value and the land value assuming possible development, it is by no means to substitute for a detailed financial planning before the development breaks ground. Typical Acquisition Valuation Process A real estate investment company buys and manages properties to capture the cash flow from operation. Many of these companies specialize in one or a few product types, such as office, retail, industrial, or residential properties. To evaluate a property with infill land, the management needs to answer the following questions: * What is the building worth? * What is the market demand for space? * What is the likelihood that the company, after acquiring the property, will put up new buildings? * If the company does not intend to build new properties, what is the likelihood of the next buyer to put up new buildings? * What type and size of development can add value to the land, and thus add value to the acquisition? Acquisition Location, Market, Rents, Pricing of Property and land s No Win Bid? Yes Acquisition I Due Dilience Development Feasibility Study Zoning, Density, Incentives, Impact Fee, School zone I Operation Operation Rent, Expense, Tenant Improvement; Leasing Figure 21. Real estate phases and major factors to consider. The typical decision process followed in current practice to acquire a property (an office building for example) with infill land is shown in Figure 22. First, the building value and the land value are segregated. Building value is derived from the standard DCF projection. Depending on the investor's perspective towards the market, the land could have no value or some value. In a weak demand region, the land probably does not generate any additional income besides parking, thus it has little or no value to the investor. In a strong demand region the investor conducts further investigation on the suitable product type to develop. If the best product type to develop is one that the investor is familiar with, say an office tower, the investor will further evaluate the project and land worth through a development model. If the best product type is not one the investor is familiar with, say a residential condominium or an industrial building, the investor probably hesitates to get involved in the development alone. Potential Acquisition Step 1: Segregating land value from building Land Value Building Value Step 2: Market demand analysis Strong Demand Weak Demand Have Value No Value Step 3: Product type analysis Other Type Office Not to Build To Build 0 Step 4: Assigning land value Subjective Development '.' Model v __ Land Value Step 5: Summing total value Offer Price Figure 22. Current acquisition valuation process. The investor might either find a development partner or consider selling off the land to such an interested party. In either case, for the acquisition purpose the investor will simply assign a subjective value to the land. The offer price consisting of the building and the land value is derived and submitted to the broker. Current Real Option Approach and Limitations In the ROA approach, by acquiring the property the investor not only receives all cash flows generated from leasing of the existing building, but also has the right, but not the obligation, to develop the vacant land to its full use at a certain point of time in the future. Therefore, the value of a property with infill land should be higher than one without. However, the current ROA models are not without limitations. Firstly, valuation methods for vacant land may not be suitable for infill land due to their different characteristics in the following aspects: (1) the price of acquiring the land could be substantially lower; (2) the building type to be developed may be restricted by zoning regulation on current property; (3) the synergy effect could be substantial between the proposed building and the existing building; (4) The surface parking is an inseparable part of the existing property. Secondly, a real estate investment firm has a different agenda for the infill land than a real estate developer. Short holding periods and different business interests make the infill land less valuable to an investor than to a developer. Thirdly, the current theoretical models are on a higher level to address real estate as a whole, while investors need practical models to address individual cases. The current theoretical models are on an aggregate level to explain real estate value in general. They have rigid restrictions, and can only be applied to the simplest cases (Miller and Park, 2002). They also lack flexibility to change variables to model realistic assumptions for practical use. Real assets often possess unique location, physical and contractual characteristics, many of which are subjective and unquantifiable. Using the real option method alone may be insufficient. Last, the existing "omnipotent" real options models are mathematically correct but too complicated to be used. Trigeorgis (2005) and others have advocated approximate methods to simplify the calculation for practical applications. In summary, although the ROA approaches can overcome some of the drawbacks of DCF and provide better valuation for acquisition, the method itself is not fully developed to address the specific needs of acquisition valuation in practice. Decision Tree Analysis and Limitations Another available approach is the Decision Tree Analysis approach (DTA). DTA is a method to identify all alternative actions with respect to the possible random events in a hierarchical tree structure. It is developed to handle the interaction between random events and management decisions. However, a major limitation of the DTA method is its use of the same discount rate throughout the valuation period, regardless of changing risk characteristics upon the execution of decision making, and thus allows for arbitrage opportunity (Copeland and Antikarov, 2005). This means using DTA alone is not sufficient for the acquisition with infill land problem. Real Options in Real Estate Applications of ROA in the real estate industry can be classified into the following categories: Vacant land for development, property redevelopment, and leasing (Ott, 2002). This section summarizes some theoretical models as well as empirical studies. Theoretical Models Titman (1985) developed a simple binominal tree model to explain why a piece of land could be more valuable remaining vacant today and when is optimal to develop. This seminal work is frequently cited in later papers, which all use Partial Differential Equations (PDE) and fall into two major categories by methodology: the optimal development timing problem, and the game theoretical problem. The optimal timing problem is represented by Clarke and Reed (1988, optimal timing and density for residential development), Capozza and Helsley (1990, conversion from agricultural to urban land use), Williams (1991, optimal timing and density to develop, optimal timing to abandon), and Geltner et al. (1996, two land use choice). The game theoretical problem is represented by Williams (1993, competition on simultaneous development), Grenadier (1996, competition on simultaneous or sequential development), and Childs et al. (2001, inefficient market with noisy effect on value). Figure 23 shows the genealogical relationship among these models. Table 21 itemizes the research subject, model variant, contributions and limitations of each study. Besides land valuation, there are two types of real estate applications of the ROA that are closely related to our research: property redevelopment and operational research. Williams (1997), Childs at al. (1996), Cederborg and Ekeroth (2004) have researched on the redevelopment or renovation of real assets. They view existing buildings as assets that can be repetitively invested and improved, sometimes by changing functional attributes, e.g., switching from offices to apartments. Grenadier (1995, 2003), Adams, Booth and MacGregor (2001), Bellalah (2002), Grenadier and Wang (2005), Capozza and Sick (1991), among others have focused on options embedded in the commercial lease agreements, such as forward leases, escalation clauses, leases with options to renew or cancel, adjustable rate leases, purchase options, saleleasebacks, ground leases, etc. Acquisitions have not been thoroughly researched using the real options approach, though common in practice. As discussed earlier, acquisitions with excessive land differ from ground up development. They also differ from redevelopment, since they are not simple renovations of the existing buildings. They might include valuation of the leases as a source of cash flow for the potential development, but would require a much simpler valuation process on the leases. In summary, although acquisition valuation is close to the three subjects mentioned above, the approach is significantly different. A new approach needs to be able to address both the building value and the land value, if any, for potential development. 1985 1987 1990 1991 1993 1996 2001 Clarke & Reed Development Timing Competition / Game Figure 23. Real Options approaches for land valuation. I Table 21. Comparison of research subjects, model variants, contributions and limitations. Author / title Subject description "Urban Land Explain why land is Prices under more valuable Uncertainty" remaining vacant for (Titman, future development: 1985) increased uncertainty leads to a decrease in current development activity. "A Stochastic Examine the qualitative Analysis of effects of the Land different types of Development uncertainty on the Timing and timing and structural Property density of land Valuation" development on (Clarke and residential projects. Reed, 1987) Model type & variant One time period binomial model assuming rents have two state values. PDE to solve for optimal development timing and density assuming rents and development cost follows stochastic processes. Contribution / limitation Seminal work of ROA in real estate. Simple. Two policy implications: (1) Government incentives to stimulate construction activities may actually lead to a decrease if the extent and duration of the activity is uncertain. (2) Initiation of height restrictions may lead to an increase in development activity due to reduced uncertainty regarding the optimal height of the area. One time period model. Assume only two states, and that construction costs are certain. Limited to residential development. Two limited assumptions: (1) new construction is small so that rents and development costs are uninfluenced by the newly added construction. However, in reality development is lumpy and will affect market rents and vacancy rate. (2) Efficient market in which all agents have equal information about the future probability distributions of rentals and costs. However, in reality real estate leasing and sales information is not as transparent as that in the stocks market, but more predictable, at least in a short run. Table 21. Continued. Author / title "The Stochastic City" (Capozza and Helsley, 1990) "Real Estate Development as an Option" (Williams, 1991) "Insights on the Effect of Aland Use Choice" (Geltner et al. 1996) Subject description Examine the land value of conversion from agricultural to urban use based on spatial characteristic of real estate such as distance or commuting time to the CBD. Optimal time to develop, optimal development density, and optimal time to abandon a project. Examine whether the multipleuse zoning add value to land by analyzing the land use choice between two different use types. Model type & variant PDE model built on the traditional mono centric urban theory to study spatial implication of land conversion value, assuming household income, rents and land prices follow stochastic processes. PDE model to solve for optimal timing of abandoning a project, in addition to optimal development timing and density, assuming carrying cost, rents and development cost follows GBM, also assuming carrying cost is significantly high so that during some circumstance it is better to abandon the project than bearing the cost. PDE to solve for optimal choice between two land use types, assuming development cost, value of first land use, value of second land use follow stochastic processes. Contribution / limitation Uncertainty (1) delays the conversion of land from agricultural to urban use, (2) imparts an option value to agricultural land, (3) causes land at the boundary to sell for more than its opportunity cost in other uses, and (4) reduces equilibrium city size. Does not explain very well land value in the emerging suburb economic centers. Looks at the downside of a project: optimal time to abandon. This is a put option. Maximum feasible density is determined by zoning restrictions. Assumes perfectly competitive market and perpetual option. Land use type choice is a unique perspective in real estate. Assume construction unit cost is the same regardless of building type to be developed. Table 21. Continued. Author / title "Equilibrium and Options on Real Assets" (Williams, 1993) "The Strategic Exercise of Options: Development Cascades and Overbuilding in Real Estate Markets" (Grenadier, 1996) "Noise, Real Estate Markets, and Options on Real Assets: Theory" (Childs et al. 2001) Subject description Examine industry equilibrium of optimal exercise policy under competition: the impact of competition erodes the value of the option to wait and leads to investment at very near zero net present value thresholds. Explain why building booms in the face of declining demand and property values: fearing preemption by a competitor, developers proceed into a panic equilibrium in which all development occurs during a market downturn. Optimal valuation of noisy real asset in an incomplete information game Model type and variant PDE to solve for perfect Nash equilibrium with finite elasticity of demand and finite development capacities in a less than perfectly competitive environment. Threestage model to explain real estate boomandbust cycle: valuation of land, construction lag, and "sticky vacancy" in operation PDE, assume optimal value include three terms: forward value estimate, historical value estimate, and the term that corrects for convexity effects due to incomplete information Contribution/ limitation Among the first to consider the effect of competition. Exercising options to develop affects the aggregate supply of developed assets and market price, which preclude simultaneous exercise of the option among all developers. Extend the Williams model from symmetric and simultaneous equilibrium to either simultaneous or sequential development, and allows for preemptive equilibria. Powerful to explain boom andbust markets such as real estate. Assume individual firms are identical and have all information. Extend to include the price lagging effect in real estate, where estimate value is different from market value, i.e., in a less than perfect market. Empirical Testing A majority of the ROA empirical works in real estate has been in aggregate studies. Quigg (1993), Holland et al. (2000), Sivitanidou and Sivitanides (2000), Bulan et al. (2004) all use a large sample of real estate data to test the premium of land price over intrinsic value, whether irreversibility is an important factor for real estate investment, whether uncertainty delays construction, and whether competitions among developers decrease the option value of waiting. As Bulan et al. (2004) point out, however, since real options models apply to individual investment projects and predict that trigger prices are nonlinear, aggregate investment studies may obscure these relationships. Moreover, these empirical tests are limited to qualitative results, such as whether each variable in the ROA model has positive or negative effect on the overall option value. Few of the ROA empirical works has focused on individual case studies and its implication in practice. The RERO Approaches The RERO framework attempts to move beyond the realm of academic interest to be used quantitatively in practical problems of acquisition valuation, development decision making, and land policy analysis. The approach should be able to address the unique characteristics of acquisition valuation with infill land, to handle the management flexibility, to take into account rare events such as new amenities driving up real estate value. This calls for the combination of DCF, ROA and DTA methodologies. It also needs to be intuitively simple for practical implementation. To achieve this goal, the problem is divided into two subproblems: (1) valuation of the building structure and (2) valuation of the infill land. Valuation of the building structure represents a normal case of acquisition. On the other hand, valuation of the infill land represents the extra value stemmed from creative management, i.e., the ability to uncover the hidden value in real estate and realize it through active development. Real estate valuation is an art and science. The RERO framework is not built on rigid reasoning and restricted assumptions to be precise, rather it is developed as a tool to solve a broad spectrum of practical real options problems. Specifically, it explores a few unconventional real option cases, including (1)jump diffusion process that does not go back to normal diffusion, (2) risk drivers that do not follow the multiplicative stochastic movement, (3) private risk that has no market equivalent and hence violating the nonarbitrage option pricing assumption. The mathematical models for these kinds of unconventional problems could be very complicated, if written in PDE equations. To facilitate practical implementation, the RERO framework applies the binomial lattice with Monte Carlo simulations and decision analysis method. The RERO framework is a simple yet powerful tool, intuitive to the practitioners, yet mathematically correct and precise. Summary This chapter compares the difference between real estate acquisition and development, reviews current practice of real estate acquisition valuation, discusses the three alternative valuation approaches, DCF, ROA, DTA and their limitations. Built on the strengths of these three approaches, the RERO framework needs to address practical problems of acquisition valuation, development decision making, and land policy analysis. The next few chapters explore modeling details of how this concept should be implemented. CHAPTER 3 LITERATURE REVIEW In Chapter 2 several different valuation methodologies were discussed conceptually: the Discounted Cash Flow approaches (DCF), the Real Option Analysis approaches (ROA), the Decision Tree Analysis approaches (DTA), and the proposed Real Estate with Real Option approaches (RERO). In this chapter the technical modeling details of the first three approaches, as well as the capital budgeting theory in finance will be discussed. The RERO approaches that built on the existing three will be discussed in Chapter 4. Traditional Discounted Cash Flow Approaches The Discounted Cash Flow (DCF) approaches include payback period, Internal Rate of Return (IRR), Net Present Value (NPV), and other forms such as Adjust Present Value. In this study DCF refers to the NPV method alone. The principle of the NPV method is to discount all projected free cash flow back to year 0, to get the net present value of the project (Equation 31). The NPV must be greater than 0, or the IRR must be greater than the company's hurdle rate, in order to justify the investment (Mun, 2002). If NPV is greater than 0, the project is regarded as optimal to be executed immediately. NPV= /p (31) =0 (1+ k)' where NPVis the net present value of the project at Year 0, F, is the projected free cash flow (including income, cost and terminal value) in year i, k is the project discount rate. The DCF method is suitable to evaluate projects that are well structured, with predictable future cash flows. For projects involve large uncertainty of timing, cost and cash flows, such as a real estate development, using the DCF approaches are difficult in the following three aspects (Miller and Park 2002; Feinstein and Lander 2002): firstly, selecting a fixed and appropriate discount rate; secondly, taking into account new information and changing the plan accordingly; thirdly, determining the optimal timing to carry out the project. Capital Budgeting Theory In the DCF approach and in all other approaches, one of the most influential factors is the discount rate to be used. To better understand discount rate, a brief discussion of the capital budgeting will follow. Market Risk and Private Risk Stocks are risky. For any individual stock, however, a large part of its risk can be eliminated by holding it in a large welldiversified portfolio. A portfolio consisting of all stocks is called a market portfolio. In reality, it can be approximated by a large amount of well diversified stocks. The part of the risk of a stock that can be eliminated is called private risk, or diversifiable risk; while the part that cannot be eliminated is called market risk, or systematic risk (Brigham et al. 1999, pl78). The Capital Asset Pricing Model (CAPM) indicates that the relevant riskiness of any individual stock is its contribution to the riskiness of a welldiversified portfolio, or the market risk portion only, which is measured by its /f coefficient. Capital Asset Pricing Model If the market portfolio m is efficient, the required return rOs of any stock i is the riskfree interest rate r plus a risk premium, as shown in Equation 32. = r + f (rm r) (32) Where r is the riskfree return, rF is the expected market return, = ", where am is the covariance between the stock and the market, and a o is the variance of the market portfolio. f, is an important variable to measure the risk characteristics of the stock i. If f, is greater than 1, the stock is more volatile than the average stock market; and iffl, is less than 1, the stock is less volatile than the average stock market. The more volatile a stock is, the more risky it is, and consequently the higher the required return needs to be in order to justify the risk an investor takes. Discount Rate A firm's hurdle rate is usually its Weighted Average Cost of Capital (WACC). A large real estate investment firm is usually formed as a Real Estate Investment Trust (REIT), which does not pay income taxes, so long as 95% of its income from operation is distributed to the investors on an annual basis. The WACC k of a REIT is calculated by Equation 33. S D k = r +r D (33) V V where r, and rd are the cost of equity and debt respectively, S, D and V are the market values of equity, debt, and total asset respectively; S + D = V. Equation 33 can also be used to value an investment project, as if every project was a separate mini company. However, it is difficult to determine the cost of equity and debt for a project, since the equity of a startup project, for example, may not be publicly traded, and the risk characteristics of a project are quite different than that of the company as a whole. The capital budgeting theory indicates that finding the right discount rate is extremely difficult, if not impossible. Since every company has different risk characteristics, the required discount rate is different from company to company. Also every project within the same company has different risk characteristics, and the correct discount rate required to value a project may not be the same as the company's WACC. This makes both the DCF and the DTA approached difficult to value infill land with development potential, although for an existing building with operating history the DCF and DTA approaches may work fine. Option pricing theory, on the other hand, does not rely on the risk characteristics of a particular firm or project. Neither does it rely on the risk preference of an individual investor. It is discounted at the riskfree interest rate r. The reason is that "private risk is alleviated through portfolio diversification and market risk can be diminished through the option's replicating portfolio" (Miller 2002). For development project that involves a lot of uncertainty, this is a huge benefit over the traditional DCF method. Option Pricing Theory Definition and Type of Options An option gives the holder the right but not the obligation to do something (Hull, 2006). In the financial market, there are two basic types of options: call options and put options. A call option gives the holder the right to buy the underlying asset by a certain date for a certain price. A put option gives the holder the right to sell the underlying asset by a certain date for a certain price (Figure 31). Based on exercise dates, options can be classified into two major types: American options can be exercised at any time up to the expiration date. European options can be exercised only on the expiration date. Most options are of the American type. The value of a financial option is determined by the current price of the underlining asset So, the strike price at maturity date K, the riskfree interest rate r, maturity date T, return volatility of the underlining asset o, and sometimes the dividends expected during the life of the option (Hull, 2006). Returns on options are asymmetric, i.e., options will only be exercised to the benefit of the holders. For example, if a holder of a call option can buy the stock 3 months later for $100 per share, and if the spot price at maturity becomes $120 per share, he will exercise this option, then sell the stock immediately, and earn $20 per share. However, if the spot price becomes $83 per share at maturity, he can let the option expire without exercised, thus avoid losing $17 per share. He only losses the premium initially paid for the option (Figure 32). His payoff is the difference between the spot price at maturity St and the exercise price K, or 0, whichever is greater (Equation 34). Max(S, K,O) (34) Payoff 01 Premium K Call Option * Stock Price SPayoff 01 Premium \ Stock K Price Put Option Figure 31. Payoff of call option and put option. SPayoff K=$10 0 / Stock S=$83 S=$120 Price Call Option Example 120 20 100 83 0 Example Payoff Figure 32. Call option payoff example. Option pricing theory is to determine what premium, or option price, a holder should pay for such flexibility. The types of option pricing methodology include continuous and discrete time models (Miller and Park, 2002; Lander and Pinches, 1998). Continuoustime models include closedform equations and stochastic partial differential equations. Discretetime models are mostly lattice models and Monte Carlo simulation. BlackSholes Model and Stochastic Partial Differential Equations The most famous closedform equation is the BlackScholes model, although it can only be used to price European options. The BlackScholes (1973) pricing formula is developed under the following ideal assumptions: stock price change follows the Wiener process, distribution of return is lognormal, efficient market, constant shortterm interest rate, no dividend payment, no transaction costs, and short selling is possible. A Wiener process, also called a Geometric Brownian Motion (GBM), is a random process with a mean change of 0 and a variance rate of 1. The values of dz for any two different short intervals of time dt, are independent (Equation 35). dz = Esdt (35) Where e has a standardized normal distribution 0(0,1), and (/,u, U) denotes a probability distribution that is normally distributed with mean yu and standard deviation o. A generalized Wiener process for a variable S can be defined by Equation 36. dS = uSdt + cSdz (36) where S is the underlying asset whose value change follows the Wiener process; dS is the change of value S during an infinitesimal time interval dt. Ito's Lemma (Hull, 2006, p273) is a theorem of stochastic calculus that shows second order differential terms of a Wiener Process can be considered to be deterministic when integrated over a nonzero time period. Since the stock price S follows the Wiener process, an optionf(be it a call option or a put option) contingent on S follows the Ito's Lemma (Equation 37). Of Of 1 02f 2 + df = ( + + a2S2_)dt+ Sdz (37) d S +t 2 OS2 3S The principle of option pricing methodology is to construct a riskless portfolio to prevent arbitrage. This portfolio H is short one option and long 8f/8S shares of the underlying stock. When the stock price S changes, the Of/8S shares must change accordingly. Later from Equation 310 we will see this portfolio is riskless because it does not involves dz over the time interval dt. The portfolio 7 is written as Equation 38. S= f + S (38) as During the time interval dt, the change in value of the portfolio is represented in Equation 39. cd = df + dS (39) as 8S Substitute dS from Equation 36 and dffrom Equation 37 into Equation 39, Of 1 2f dn = ( 2S2)dt (310) at 2 S2 To prevent arbitrage, the portfolio earns riskfree interest r during the time interval dt. cd = rHdt (311) From Equation 38, Equation 310 and Equation 311, we have af 1 a2f2 af ( 1 2 CS2)dt = r(f + S)dt at 2 S2 as Which simplifies to + S+ a2S2 = rf (312) at as 2 S2 Equation (312) is the BlackScholes partial differential equation. Subjected to the following boundary conditions: f = Max(S K,0), when t Tin the case of a call option, and f = Max(K S,0), when t = Tin the case of a put option. Integrating Equation 312, the BlackScholes formula can be written as Equations 313 and 314 (Black and Scholes, 1973; Hull, 2006). c= SN(d) Ke rrN(d2) (313) p = Ke rrN(d2) SoN(d,) (314) where In(So/K)+(r +a2/2)T d, = UT In(S/K) + (r 2 /2)T, UJT c is the value of a European call option; p is the value of a European put option; So is the current price of the underlying asset; K is the strike price of the option at maturity; r is the riskfree interest rate; Tis the time to maturity; N(*) is the cumulative standard normal distribution function. The BlackScholes model can be divided into two parts: The first part, SoN(dd), derives the expected benefit from acquiring a stock right now. This is found by multiplying stock price So by the change in the call premium with respect to a change in the underlying stock price N(d1). The second part of the model, KerrN(d2), gives the present value of paying the exercise price on the expiration day. The fair market value of the call option is then calculated by taking the difference between these two parts. The boundary condition of a call option is best depicted in Figure 33. The solid black line defines the call option value. The green line with square markers defines the maximum value of the option. For nonarbitrage, the option should never be worth more than the stock price S, otherwise an arbitrageur can easily make a riskless profit by buying the stock and selling the call option. The blue line with triangle markers defines the minimum value of the option. The call option should be worth more than Max(So Ke r,0), otherwise an arbitrageur can buy an option, short sell a share of stock, invest the surplus at riskfree interest rate and earn a profit. The possible option values fall in the region defined by the green line and the blue line and vary depending on the underlying stock volatility, option time to maturity, and riskfree interest rate. Call Option vs. Stock Price u  30 0 0 30 60 90 120 15( Stock Price Lower Bound Option Value  Upper Bound Figure 33. Call premium vs. security price. Though the BlackScholes pricing model has a lot of restrictions and can only value European options, there are a lot of stochastic partial differential equations with boundary conditions that relax some restrictions to a certain extent and can be used to value more specific questions. The benefits of these analytic continuoustime models are that they are flexible to model different circumstances, and mathematically accurate (Miller and Park, 2002). The drawback is that the modeling requires sophisticated mathematical knowledge, sometimes the solution does not exist, and even if it does, the process itself could become as complicated as a blackbox for the practitioners to comprehend (Lander and Pinches, 1998). In the case when analytical solutions to the stochastic differential equations do not exist, they must be solved numerically by using finitedifference methods, or Monte Carlo simulations (Miller and Park, 2002). Lattices Lattices are a type of discrete time model, which includes binomial tree, trinomial tree, quadranomial tree, and other multinomial models. Lattices are the approximation of the continuous models. The results of these two methods are very close when the time interval is infinitely small. The most commonly used binomial lattice was developed by Cox et al. (1979), in which values of the underlying asset are assumed to follow a multiplicative binomial distribution. The model assumes the up and down parameters u and d, the volatility of the underlying asset o, and riskneutral probabilities and 1 p are constant (Figure 34). SSoo 1p SSo'd fd Figure 34. Stock and option price in a onestep binomial tree. An optionf(be it a call option or a put one) is valued by constructing a riskless portfolio H of a long position in 6 shares of stock and a short position in 1 option (Equation 315). H = Sos f In an up movement of the stock price, the value of the portfolio is = Su8 f, In a down movement of the stock price, the value of the portfolio is Hd = Sod fd The two are equal when Soud f, = Sod fd or when 3= f fd (316) So(u d) The portfolio is riskless and must earn the riskfree interest rate r. The present value of the portfolio is represented by Equation 317. H = (Sou f,)er (317) From Equation 315 and Equation 317, we have So, f = (Sou f)erT (318) Substitute 6 from Equation 316 into Equation 318, f =e rr[pf, + (1 p)fd] (319) where e d P= ud u =e" d= Equation 319 is a onestep binomial model, which can be generalized to twostep and multistep models. Figure 35 shows a twostep binominal lattice. During each time step, the (315) stock value either moves up to u or down to d of its previous value. Option value is derived by working backward fromf, and fd to calculated, fromfud andfdd to calculated, then from, and fd to calculatef(Equations 320, 321 and 322). So*u*u So*u A f fud So'd fd So*d*d fdd Figure 35. Stock and option prices in general twostep tree. f, = e rt[pf, + (1 P)fd] (320) fd = e t[pfd + (1 P)fdd] (321) f =e t[pf + (1 p)fd ] (322) Substituting from Equation 320 and Equation 321 into Equation 322, we get f = e 2[p2 f, + 2p(l p)fd + (1 )2 fdd] (323) where et d ud u =et d= In general, for a binomial lattice with n steps, the ith step (0 < i < n) option value is calculated by Equation 324. f = e t[pf,,, + (1 p)f +,d ] (324) Lattice, though still complicated, is more intuitive to the practitioners than continuous time models. It is especially useful to evaluate American options, since analytic solutions are almost nonexisting in the continuous models. The drawback is that using lattice by itself is hard to model compound options. However, combined with DTA, lattice is capable to deal with a lot of complicated situations, even more flexible than PDEs in many circumstances. Monte Carlo Simulation Originally named after the casinos in Monte Carlo, Monaco, Monte Carlo simulation is about games of chance. It is now widely used to simulate stochastic processes by sampling large quantity of random outcomes for the processes (Figure 36). Because of the repetition of algorithms and the large number of calculations involved, Monte Carlo simulation is computationally complex, yet easy to model and understand. Histogram Statistic Preferences I Options 250 Rate of Value Change (2000 Trials) 1 .  1.0 *OSr D 0,5 IL 0.1 0.2363 0.1263 0.0233 0.0737 0.1737 0.27=.f Type ITwoTail Infinity Certainty [(%)\ 1o0.0o Figure 36. Monte Carlo simulation output. In real options modeling, Monte Carlo simulation can be used where there are several underlying variables. The drawback is that it is difficult to work backward to determine option exercise strategy, since Monte Carlo simulation is forward looking. In the RERO model, it is used as an intermediate step to estimate volatility of the project stems from multiple risk drivers. Real Options Analysis Approaches First coined by Myers (1977), the ROA approaches are to apply financial option pricing theory and methodology to evaluate real assets (Miller and Park, 2002; Trigeorgis, 2005). In the financial market, a derivative is a security whose value changes depend on the value changes of some other underlying assets. In real asset valuation, the value of a project can be viewed as a derivative contingent upon input costs, output yield, time and uncertainty (Miller and Park, 2002), and therefore can be evaluated by applying the financial option pricing principles. By using ROA, investment decisions are viewed as real options or combinations of real options, such as options to defer, expand, switch, contract, or abandon, as shown in Table 31 (Trigeorgis, 1996; Yao and Jaafari, 2003). Also included in the table are examples in the real estate and construction industry. Contrary to DCF method, in the ROA context greater volatility is not always worse, since losses are limited to the initial investment, or option premium, but the option holder can capture greater upswings if things turn out to be favorable. ROA is applied most commonly in the industries of natural resource, manufacturing, energy, research and development, startup companies, and others (Lander and Pinches, 1998; Trigeorgis, 1996). Applications in the real estate and construction industries are still limited. Although ROA borrows the option pricing theory, the distinguish characteristics of real assets demand different valuation assumptions and methodologies from direct applications of the option pricing theory without any modification. Table 32 lists the major differences between financial options and real options (Mun, 2002). Table 31. Types of real options. Options Features Examples Defer To postpone construction till optimal timing Time to develop Stage To create a series of stages to allow for Phased development abandonment or expansion in later stages depending on outcomes of earlier stages Contract To contract the project to a third party in order to Franchise stores mitigate risk or to speed up market domination Expand To expand the project scale in favorable market Airport expansion conditions Abandon To abandon the project and prevent severe lost in Bankruptcy of a unfavorable market conditions project entity Switch To change the output mix or input mix in response Coalfired vs. gas input/output to changing market demand fired power plants Compound Option on option, where the value of an earlier Case study in Chapter option can be affected by the value of later 5 and 6 options. Most real world options are of this kind Table 32. Comparison between Financial Options and Real Options. Characteristics Financial options Real options Maturity Short, usually in months Long, usually in years Underlying asset Traded stocks, with comparable and Not traded project free cash flow, pricing information proprietary in nature, with no explicit market comparable Management Value does not change due to Value has to do with individual manipulation individual management assumptions management assumptions and or actions actions Competition and Irrelevant to pricing Direct drivers of value market effect One of the major differences between financial options and real options is how to handle private risk. The underlying assets of financial options are traded market assets, and market risk is the major source of risk among all financial options. Private risk can be treated simply as errors. The underlying assets of real options, however, are usually nontraded assets that do not have market equivalent. Private risks cannot be hedged. The other difference is the effect of management and competition. Financial options on the same underlying asset and the same maturity date are identical. They are widely held to be market efficient. A single transaction usually does not affect the pricing of financial options, neither does management or competition. Real options, on the other hand, are lumpy or oneofthekind in nature. Exercise of real options by management can have profound impact on the underlying asset value. Consequently, there are a lot of debates in the academic world about how real options should be correctly priced. Borison (2005) classified existing real options approaches into 5 categories: * The classic approach, * The subjective approach, * The Market Asset Disclaimer approach, * The revised classic approach, and * The integrated approach. Borison also discussed the underlying assumptions of these approaches, the conditions that are appropriate for their applications, and the mechanics in applying them. The classic approach assumes that the capital market is complete, and an identical twin asset or portfolio exits for every real asset under evaluation. It makes explicit use of noarbitrage argument, and applies directly the BlackShores formula. The subjective approach also assumes that the capital market is complete. However, it relies on subjective judgment for input, as opposed to data from traded markets. This makes it an inconsistent approach, and limits to qualitative result. The Market Asset Disclaimer (MAD) approach assumes that the capital market is not complete. It relies on the estimate value of the asset without flexibility as the "twin asset" for the purpose of calculating the option value of the flexibility. Data is drawn from traded markets when available, and subjective judgment when not. Proponents of this approach justified this step explicitly: the same, weaker assumptions that are used to justify the applications of DCF can be used to justify the applications of option pricing to flexible corporate investment (Copeland and Antikarov, 2001). The revised classic approach assumes that the capital market is partially complete. It attempts to divide the world into black and white: For investments that have market equivalents, it applies the classic approach using market data; for investments that do not have market equivalents, it applies decision analysis using subjective judgment. The integrated approach also assumes that the capital market is partially complete. However, it uses capital market data for market risk and subjective judgment for private risk in an integrated model. The major difference among these approaches is how private risk is handled. The classic approach ignores private risk completely and treats real options exactly like financial options that all risks can be diversified away by constructing a hypothetical traded twin asset or portfolio. The subjective approach handles private risk by substituting market data by subjective assessment. The revised classic approach admits the limitations of direct applications of option pricing theory to real options analyses and classifies investments into those either dominated by market risk or by private risk. It applies the option pricing model only to investments dominated by market risk, and applies decision analysis to those dominated by private risk. Although it is a better approach than the previous two, the revised classic approach forces all investments into black or white, and implements two totally different approaches. The MAD approach, on the other hand, admits the difficulty of handling private risk, thus does not rely on the existence of a traded replicating portfolio. Instead, it uses the project value itself without flexibility as the twin security, as if it were traded in the financial market. After all, the best correlation with the project is the project itself (Copeland and Antikarov, 2001). Trigeorgis (1996) also argued that the assumptions underlying the DCF approach are traded assets of comparable risk (same beta), and MAD assumptions are no stronger than those of DCF. Contrary to Borison's understanding, Copeland and Antikarov (2005) clarified that the MAD approach does not blindly use all subjective assumptions. Similar to the integrated approach, MAD also uses traded market data whenever available, and uses subjective assumptions only when market estimates are impossible. The MAD approach and the integrated approach are considered to treat private risk in the same way, the difference remains only technical: MAD relies on simulations to evaluate project volatility, and attempts to combine all risks into one variable, whenever possible; while the integrated approach relies on utility functions, and models market risks and private risks explicitly and separately. Neither is superior to the other, and the selection of approaches depends on project characteristics on a casebycase basis. For this reason, the proposed RERO approaches are built on the MAD and the integrated approaches. Practical Real Options Model in Real Estate Ghosh and Sirmans (1999) were among the first to address the applications of real options to the corporate real estate practitioners, by developing a lookup table for the options value, which is derived from an approximation of the BlackScholes formula. They used the correspondence in Table 33 between financial and real options in order to apply the Black Scholes formula directly to real options. However, they did not explain whether the time value of money r is a riskfree rate or risk adjusted discount rate, nor how the risk of project cash flows o is determined. Table 33. Correspondence between Financial and Real Options. Variable Financial options Real options So Stock price Present value of projects expected cash flows K Exercise/strike price Cost of investment T Time to expiry Length of time the decision can be deferred r Riskfree rate Time value of money a Standard deviation of stock Risk of project cash flows returns They also developed a threestep approach to calculate the option value: Step 1: Calculate NPVq from Equation 325. NPVq = (325) SK /(1 + r) Step 2: Calculate oJT Step 3: Read the value of the call option as a percentage of the value of the underlying asset from the table. For example, if the stock price S is $100, strike price K is $100, time to expiry Tis 1 year, time value of money r is 5%, standard deviation of annual return a is 20%, then NPV = S/[K/( + r)] = 100/[100/(1.05)] = 1.05 oT = 0.20 x 1 = 0.20 From the lookup table, C is 10.4% of the asset value, C = 0.104 x 100 = $10.40. They did not specify how the lookup table is computed, but by comparing the Black Scholes formula and their threestep approach, it is not difficult to find that they did some approximations in order to simplify the calculation. From the BlackScholes formula of Equation 313, C Kerr = N(dl) N(d2) (326) So So where ln(S /K)+(r +a2 /2)T oUJT ln(S /K)+(r 02/2)T d2 = = d, T . UJT K S S K )T is an approximation of Ker and S can substitute S (r + 2 / 2)T (1 + r) K /(1 + r)Y K is ignored due to the low impact on the overall value. With the approximation and substituting Equation 325 into Equation 326, we have C 1 N[In(NPV)] (327) = 1 N (327) So NPV, aU Equation 327 is the formula to develop the lookup table. The Ghosh and Sirmans model falls into the subjective approach category of Borison's classification (Borison, 2005). As discussed in the previous section, this approach uses subjective assessment of variables without justification of its appropriateness. At a first glance, this approach is intuitive, especially for practitioners who are comfortable with NPV but unfamiliar with ROA. However, this direct application of the BlackScholes model is not without its limitations. Firstly, it is restricted to European options, where timing of execution of the option is perfectly known in advance. Secondly, it assumes future cash flow is as deterministic as in the traditional NPV method, and allows for only one scenario analysis. It does not allow for stochastic and dynamic changes of the underlying variables, such as development cost and rental rate, does not solve for optimal development timing. Lastly, while there is a tradeoff between simplicity and accuracy, the value derived from the lookup table has 10% variance from that calculated from the BlackScholes model, which is deemed inaccurate in many circumstance. In summary, the model developed by Ghosh and Sirmans is a good attempt to build the understanding of management flexibility value of corporate real estate in practice, however, it lacks accuracy and depth of applicability in the real estate industry, which is what this study plans to overcome. Decision Tree Analysis First coined by Howard (1964, in Ng and Bjornsson, 2004), decision analysis is the discipline comprising the philosophy, theory, methodology, and practice necessary to address important decisions. Graphical representation of decision analysis problems commonly use influence diagrams and decision trees. DTA is a method to identify all alternative actions with respect to the possible random events in a hierarchical tree structure. It is developed to handle the interaction between random events and management decisions. Uncertainties are represented through probabilities and distributions. The attitude of a decision maker to risk is represented by utility functions. Unlike the DCF approaches, there are no objectively correct DTA models. An appropriate model depends on the preferences and beliefs of the decision maker and hence is subjective. A decision analysis includes the following typical steps: first, defining the scope of the analysis; second, setting up a decision basis, including generating alternatives, collecting information, and estimating risk preference; third, constructing a decision tree with decision and uncertainty nodes; and forth, analyzing sensitivity of factors that have the largest effects (Ng and Bjornsson, 2004). Decision analytic methods are used in a wide variety of fields, including business, environmental remediation, health care research and management, energy exploration, litigation and dispute resolution, etc. DTA relies on subjective assessment of probabilities and distributions. This method alone cannot prevent arbitrage opportunity. However, the combination of ROA and DTA can eliminate the shortcoming of both, and creates a much better approach. Summary In this chapter we reviews modeling details of the DCF, ROA, DTA approaches, as well as capital budgeting theory, ROA applications in real estate. Treatment of private risk differentiates these approaches from one another. In ROA methodologies alone, there are various approaches advocated and debated in the academic community. Due to the characteristics of real options, it is inappropriate and inaccurate to directly apply the option pricing formula without any modification. The correct real option methods must be able to handle private risk as well as market risk in a consistent way. Only the MAD and the integrated approaches are considered appropriate and are subject to further use. CHAPTER 4 METHODOLOGY The RERO framework consists of two approaches to value real estate acquisitions: the combined approach and the separated approach. This chapter introduces the key elements and steps of the RERO approaches. The next two chapters present case studies that implement the principles introduced in this chapter. As mentioned in the previous chapter, the Market Asset Disclaimer (MAD) and the integrated approaches in ROA were adopted for this study. RERO Modeling Procedures The RERO framework adopts real options and decision analysis methodologies. It consists of a series of processes to solve a decision tree backward. The event tree starts by laying out all possible events and corresponding cash flows. Starting at the end of the analysis, we work backward through the tree at each decision node to calculate the payoff of all possible actions, using replicating portfolio or risk neutral discounting, choosing the optimal action that generates the highest payoff at each node. Eventually the possible cash flows generated by these future events and actions are folded back to a present value. The following 6 steps are critical in performing the RERO analysis (Figure 41): * Problem framing; * Approach selection; * Risk drivers identification and estimation; * Base case modeling; * Option modeling; and * Sensitivity analyses. Problem Framing For real estate acquisition, the first task is to review the case qualitatively, and to determine whether the asset itself is a sound investment. An investment that seems good by the numbers may not necessarily turn out to be a good investment in the end. Location, neighborhood development, economy growth, property visibility, accessibility, physical conditions, ownership and occupancy history, management capability, all these are unique characteristics of real estate that are nonquantifiable. Comprehensive local business knowledge and experience is needed to determine whether a piece of land is worth acquiring. Figure 41. Critical steps in RERO analysis. After this critical screening, if a property is good enough to go through the hassle of quantitative analysis, the problem is framed into a model and the story is told in a mathematical way. The goal becomes how much it is worth. Management flexibility and strategic options, if any, should be identified to determine which approach to use. Approach Selection DCF can solve most simple and conventional acquisition problems. It is only when a case has strategic options that cannot be valued by DCF should the RERO approaches be used. Depending on the characteristics of a project, the first step is to determine whether to use the combined approach or the separated approach. The differences between the two approaches are discussed in later sections. Risk Drivers Identification and Estimation The next step is to identify the risk drivers. Uncertainties of real estate acquisitions and development include rental income, operating costs, capital expenditure, discount rate, cap rate, development cost, etc. These variables flow through the model to affect the project value. Risk drivers are those key variables that have the most profound impact on project value change. To estimate the volatility of each risk driver, objective methods such as time series forecast or regression analysis should be used, if historical or comparable data exists. Alternatively, subjective methods may be used, such as subjective guesses, growth rate assumptions, expert opinions, etc (Mun, 2002). Base Case Modeling The expected project value without flexibility is the base case for the subsequent option value analysis. The base case value acts as the "twin asset" that the real option approach is based on. Option Modeling From the problem framing step, some strategic options have been identified; from the approach selection step, the combined approach or the separated approach has been selected; from the risk driver identification and estimation approach, the key uncertainties have been identified and their volatilities quantified. Now in the option modeling step, a Monte Carlo simulation is run, an event tree is constructed, with managerial flexibilities incorporated in each node, option values are calculated, optimal decisions are made at each node, and the value are tracked from the end of the analysis back to the starting time of the analysis. This process may be run back and forth for several times to ensure all option values are calculated correctly and the corresponding rational decisions are made. Sensitivity Analyses Setting the project value with flexibility and/or option value as the dependent variables, each risk variable can be changed, and the trend of value changes in the dependent variables can be observed. This sensitivity analysis helps the user to see the whole picture and determine how each risk variable should be managed. It also helps in understanding how uncertainty could have otherwise altered decision making. RERO Modeling Approaches For different treatments of risk drivers, there are two types of RERO modeling approaches: the combined approach and the separated approach. The combined approach is used for valuation of an existing building with a historical operating track record. For uncertainties of infill land development, the separated approach is more suitable. MAD has two key assumptions: firstly, the present value of the underlying risky asset without flexibility is the best estimate of the project value with flexibility. Secondly, properly anticipated cash flows fluctuate randomly. The second theorem allows the user to combine any number of uncertainties into a spreadsheet, and to produce an estimate of the project NPV conditional on the set of random variables drawn from their underlying distributions by using Monte Carlo simulation techniques (Copeland and Antikarov, 2001, p219). This is the theoretical foundation of the combined approach. By using the combined approach, uncertainties are assumed to be able to be resolved continuously over time. This assumption generally holds for stabilized assets. However, many projects in real estate, such as infill land development, have major uncertainties that do not get resolved smoothly over time. Many rare events, e.g., permit approval, development activities in the neighborhood, a new mall, a new subway station, can significantly change the real estate value. For projects with any risk of such jumping effect, the actual event tree is asymmetric with changes in value occurring when a significant part of the uncertainty is resolved. The separated approach is used to isolate the risks with jump diffusion effect from those resolved continuously, and to model their interaction explicitly. In other words, the separated approach also assumes that the underlying project value without flexibility is the best estimate of the project value with flexibility, but it does not assume that the cash flows fluctuate randomly. Rather, it separates the risk drivers with jump effect from the others without, and models the jump effect explicitly. The Combined Approach The combined approach is most suitable for valuation with risks resolved continuously. This approach can be best applied to acquisition valuation of stabilized real estate assets. The process is to model the parameters of different uncertainties and to estimate their effect on the volatility of the project value using Monte Carlo simulation techniques. The effects of individual risk drivers are thus combined into the project volatility, which is used to generate a binomial event tree. Actions of managerial flexibility are added to solve for option value. The following variables are typical in a property acquisition model: rental rate, occupancy rate, rentable square footage, expense recovery, operating expenses, capital expenditure, tenant improvement, leasing commission, goingout cap rate, discount rate, etc. Among these variables, the most influential ones are rental rate, stabilized occupancy rate, goingout cap rate, and discount rate. Rentable square footage is usually fixed; expense recovery and operating expenses vary but in a controllable small range related to the rental rate change; capital expenditure, tenant improvement, and leasing commission are tricky in reality, but could be assumed to be fixed on an annual basis for a highend office building. Rental rate and stabilized occupancy rate will be used as the two major variables in the case analyses. Rental rate is set by the market, and directly impacts the property value. For valueadded type of investors, who intend to upgrade amenities and enhance occupancy, the stabilized occupancy rate is an important factor for revenue estimation. The discount rate, however, is subjective to each investor. In finance theory, the discount rate should reflect the level of risk of a project. In practice, however, for an individual investor, the discount rate is usually his weighted average cost of capital. Risk is mainly adjusted through the Cap rate rather than discount rate (Wheaton et al., 2001). The discount rate can therefore be regarded as fixed. The change of rental rate depends on many factors, such as macro economics, employment growth, market occupancy rate, new construction pipeline, net absorption rate, etc. The change of rental rate is assumed to follow the multiplicative stochastic process. Historical data of rental rates will be examined in the next chapter. Another factor that affects rental revenue is stabilized occupancy rate. For a building that is not fully leased, there might be upside potential to lease up the vacant space, depending on market demand. In a market with strong job growth, demand for office space is also strong. It is relatively easy to lease up the vacant space. Assuming that vacant space can be leased up, the incremental Net Operating Income (NOI) is substantial compared to the incremental revenue, since the incremental operating expense is minimal. In other words, whether a building is 50% occupied or 100% occupied, a majority of the operating expenses is fixed, the 50% leaseup can potentially triple the NOI. Note that a multitenant office building is seldom fully occupied, therefore stabilized occupancy rate usually is close to but never reaches 100%. A general vacancy factor is deducted from the fully leased revenue. The change of occupancy rate is assumed to follow the additive stochastic process. This process is similar to the multiplicative stochastic process with the only difference being that the up and down movements in the lattice are assumed to be additive rather than multiplicative (Copeland and Antikarov, 2001, p123). The Separated Approach The separated approach is more complicated than the combined approach and should be used only when needed. It is best used for projects with major private risks that do not get resolved continuously. The infill land valuation is an example in this study that can be better modeled using the separated approach. The following variables are typical in an infill land development model: rental rate, development cost, development timing, development scale, operating expenses, expense recovery, cap rate, discount rate, etc. Among these variables, the most uncertain ones are rental rate, development cost, and development timing. Development scale is regarded as a major economic factor, but not a major uncertainty in the context of our case study, due to approved permit of the development scale. Since the goal of most commercial developments is to maximize the investor's wealth, developments are usually built to the largest size allowed by zoning and legal restrictions. Unless the development involves zoning changes, development scale is predictable, and thus is not modeled as a risk driver. As discussed in the combined approach, operating expenses and expense recovery are in a controllable range, and the discount rate for a particular project is fixed to a specific investor. Cap rate is assumed to be fixed in the integrated approach for simplicity. Development costs include hard costs and soft costs, and can be subdivided into costs associated with land, structure, tenant improvement, leasing commission, legal, finance, taxes, insurance, marketing, etc. Hard costs are construction costs that include demolition, foundation, structure, mechanical and engineering systems, general conditions, bonds and insurance of construction, design and management fees, tenant improvement, etc. Soft costs are intangible costs that go to legal, survey, marketing, financing, taxes, leasing commissions, etc. Since every project is unique, development costs represent the major private risk that does not correlate with the traded financial market, and thus cannot be replicated by the so called traded twin asset. Rental rate is discussed in the combined approach during normal circumstance. What needs to be pointed out in addition is the jump diffusion process. A jump diffusion process is defined as a type of stochastic process that has large discrete movements (jumps, or shocks), rather than small continuous movements (Amin, 1993). As Wheaton et al. (2001) noted: "In reaction to positive shocks, returns initially increase, but eventually diminish with the arrival of new supply. Similarly, negative shocks lead to building conversions, loss of stock and an eventual recovery of returns." One of the distinguishing characteristics of real estate, compared to traded securities, is its inelasticity, or slow reaction to shocks. The jump diffusion can be ignored in the acquisition of a nearly fully occupied property, since rental rates cannot be changed until lease expirations, which could be years from the emergence of the shock. But jump diffusion could be a major uncertainty in development, since all rental square footage is newly available. Developers can ask for higher rental rates in markets with rising demand. Development timing is also important. Development timing is different from development duration. Given the size of a development project, the duration of construction is usually fixed, but when to start the project could have profound impact on the value, given the real estate cycle. One of the major disadvantages of DCF valuation is its inability to determine the optimal development timing. The RERO framework, on the other hand, can analyze all possible scenarios and indicate the best action at each point in time. It is extremely valuable for the investor to hold the option of when to start the development. Another important factor is development scale, or the size of development. In the case study, the permit for around 1 million square feet of mixused development has been approved. Consequently no assumption needs to be made for changing development scale. But in many cases, when rezoning is required in order to develop more density, development scale is an important factor and should be modeled in the decision tree as whether or not the rezoning requirement will be approved. RERO Modeling Techniques Rational for Using Binomial Lattices Copeland and Antikarov (2001, p222) made the assumption that change in asset prices follow Geometric Brownian Motion, based on Samuelson's proof that "properly anticipated prices fluctuate randomly." In other words, change in asset value follows a random walk even if the risk drivers do not. This means multiple risk drivers, so long as they evolve continuously, can be combined and reduced to a single uncertainty, namely the expected underlying asset value change over time. This provides the rationale for using a binomial lattice to calculate real option value. Monte Carlo Simulation Monte Carlo simulation randomly generates values for uncertain variables to simulate a reallife model. In the combined approach, Monte Carlo simulation can be used as an intermediate step to estimate volatility of the project, the value of which is depended on multiple risk drivers. For this study Risk Simulator is used. Other simulation software available are Crystal Ball and @ Risk. The steps followed in the combined approach are to: 1. Identify risk drivers; 2. Estimate the probability distribution of each risk driver using historical data or subjective estimates; 3. Build present value model; 4. Define input variables with the possible range of value and a probability distribution in an MS Excel spreadsheet equipped with Monte Carlo simulation tools; 5. Define correlations among the risk variables; 6. Define forecast variables., e.g., rate of return for the project; 7. Run the simulation a thousand times; 8. Read the outputs of the forecast variables and their volatility distributions; and 9. Use the outputs as input variables to build the event tree. Replicating Portfolio In most cases the project cash flows are discounted at the riskadjusted rate to get to the project NPV. The riskadjusted discount rate is higher than the riskfree discount rate, since it is adjusted up to accommodate higher risk of the project than that of the treasury bonds. In order to apply a binomial lattice that is developed based on riskneutral probabilities and riskfree discount rates, riskadjusted probabilities should be used together with riskadjusted discount rates. To calculate the value of the option, the replicating portfolio method is used, but not the discounting method, since the risk characteristics of the project change over time depending on the decision made, and consequently the riskadjusted discount rates also change over time (Copeland and Antikarov, 2001). The riskadjusted up movement factor u and down movement factor d are the same as those in the riskneutral binomial lattice (Equations 41 and 42). u = e (41) d = (42) where a is the project volatility, and t is the time in years of each step in the binomial tree. The replicating portfolio formula can be derived by the same method as the option price is derived from binomial lattice. Construct a portfolio that consists of n shares of stock S and b amount of value in riskfree bonds. After a period of time t, the value of the portfolio can go up or down. Let the value be equal to the option value at that time. nuS +bert = C, (43) ndS +bert = Cd (44) From Equations 43 and 44, derive Equations 45 and 46. C, C, n = d (45) S(u d) b = ud d (46) er (u d) Consequently, the value of the option is calculated by Equation 47. C nSb Cd uCd dC C = nS +b = C d (47) ud e' (u d) Binomial Lattice with Dividend Chapter 3 covers binomial lattice without dividend. In real estate, the net cash flows from operation are collected from the property and distributed to the investor, which is similar to the dividend distribution of a stock. The stock dividend is usually assumed to be distributed at a constant yield, since corporations plan and manage the distribution process. The net cash flows at the property level, on the other hand, are the actually amounts collected from the property, and hence vary from period to period. Denote 6, to the dividend yield at Step i for 0 < i < n, and using all other notions in Chapter 3, the asset value changes are depicted in Figure 42 for a two period lattice. O So*u*u SOououo(12) so So u(]461) So*u Soouod ) So So ( d ()2) Sood(J61)SO Soodd SoS*d*(1 62) So* d* d* (132) Figure 42. Twostep binomial lattice with different dividend yields. At Step 2, the three possible values are calculated using Equations 48, 49, and 410. C, = Max[Suu(1 ,2) K,0] (48) Cud = Max[Sud(1 ,) K,O] (49) Cdd = Max[Sdd(1 ,2) K,0] (410) To calculate the option value at Step 1, the dividend yield 62 needs to be added back to the option value, before discounting at the riskfree rate, which is shown in Equations 411 and 412. SpC. + (1 p)C. (411) (1 32 )e pCud + (1 p)Cdd Cd = (412) (1 2 )ert The same method is followed to calculate the option value at Step 0, as shown in Equation 413. S= pC + (1 p)Cd (413) C = (413) (1 31)er" In general, for a binomial lattice with n steps, the ith step (0 < i < n) call option value with dividend is calculated by Equation 414. C, = pC,+,u + (1 p)C,+,d (414) C( e (414) (1 +, z)e" Binomial Lattice with Jump Process Chapter 3 covers binomial lattice during normal circumstance that the underlying asset strictly follows the GBM movement. However, in reality, the asset movement could be a jump. For example, the zoning change from agricultural land to urban land, the establishment of new amenities in the neighborhood, the construction of new freeway exits, all can have a sudden and profound influence on the estate value in an area. These events seldom happen. But once occur, they will completely change the project payoff pattern. Hence, these jump diffusion effects cannot be priced using the binomial lattice developed by Cox et al. (1979). Amin (1993) developed a discrete time model to value options when the underlying process follows a jump diffusion process. Unlike the financial jump diffusion process that reverses back to normal value quickly, a jump diffusion process in real estate usually is irreversible, at lease not in a short period of time. That is, if a large scale development occurs that drives up the rental rate in a neighborhood, that rental rate is likely to remain at the same level for several years until a new event happens. In this study the Amin model was modified to accommodate this change. Based on the assumption that the jump risk is diversifiable, a oneperiod call option is priced in the Equation 415 (Figure 43). yS Cy x/ x ) uS 1X Cu S C dS CC (1X)(1p) dS (1X)(1p) Cd Figure 43. Binomial lattice with jump process. C =e t{AC, +(1 )[pC + (1 p)Cd]} (415) where A is the probability of the jump event according to the Poisson distribution, and defined by e "n x! (where n is the expected number of successes, and x is the number of successes per unit); y is the capital gain return on the underlying asset when the jump event occurs; Cy is the option value at the time the jump event occurs; p is the adjusted probability of an up movement, and defined by e"' A d p P 1 A ud Investment with Private Uncertainty As discussed in Chapter 3, many investments include private and market uncertainties. Market uncertainty can be replicated with market participation and therefore diversifiable. Private uncertainty cannot. For example, the development project value depends on both the market uncertainty of rental rate and the private uncertainty of development cost. The principle of pricing in such investment, if no correlation between the market risk and private risk exists, is to use riskneutral probability for the market uncertainty and actual probability for the private uncertainty, both discounted at riskfree rate (Luenberger, 1998; Copeland and Antikarov, 2001; Smith and McCardle, 1999). Although formulas for pricing uncertainties with correlation exist, the no correlation assumption usually holds. To implement this principle, there are two alternative methods: the quadranomial lattice and the decision analysis method. The first method is to implement a quadranomial lattice. Figure 43 shows a onestep quadranomial lattice. If an option C is contingent upon the value of two underlying assets S1 and S2, assuming no correlation between S1 and S2, then the value of C is priced as Equation 416. C= e (p,,Cl +p12C12 +p21C21 +p22C22) (416) where P11 = PP2 P12 = P(1 p2) P21 = (1 p )p2 P22 = (1p1)(1 p2) p, is the riskneutral probability if S, is market uncertainty, or the actual probability if S, is private uncertainty. For each uncertainty, it can have more than two bifurcations. For example, if S is a market risk with jump diffusion (three bifurcations), and S2 is a private risk with three bifurcations, then C could be priced with nine nodes with corresponding probabilities and discount at the riskfree rate. In theory, an option can be contingent upon more than two separated assets, but in practice, the complexity of implementation will soon become intimidating. This study thus focuses on a few key risk drivers and combine them into two kinds of separated uncertainties: market uncertainty and private uncertainty. Pi Si p dSS U2S2 d2S2 Figure 44. Quadranomial lattice. Another way is to implement decision analysis methodology (Smith and Nau, 1995). For example, if the two underlying risks for a development are cost and rental rate, it can be modeled as shown in Figure 45. The expected value at each node is calculated and discounted at the risk free rate. Equation 417 shows how the expected value E(PVo) can be calculated. E(PVo)= ip, [E(PV)] J1 where j is a scenario labeled from 1 to m, 1 < j < m; E(PVj) is the expected present value of scenario for all the years i, 1 < i < n. Cost (417) Rent Figure 45. Decision analysis. Summary This chapter discusses the 6steps RERO framework: problem framing; approach selection; risk drivers identification and estimation; base case modeling; option modeling; and sensitivity analysis. Two modeling approaches are introduced to deal with different risk characteristics: the combined approach for projects with risk drivers that get resolved continuously, and the separated approach for project either with risk drivers that follow the jump diffusion process or involving private risk. The modeling techniques that will be applied in the case studies are also introduced, including the rationale of using the binomial lattice, Monte Carlo simulation, replicating portfolio, binomial lattice with jump diffusion process, and investment with private risk. CHAPTER 5 THE COMBINED APPROACH Chapter 5 and 6 present case studies that implement the principles of RERO described in Chapter 4. The two chapters describe the valuation of two parts of one case: valuation of the building using the combined approach, and valuation of the infill land using the separated approach. Together, these two case studies demonstrate how the RERO framework can be applied to different scenarios in the real estate acquisition and development analysis. Case Description The case identified is 211 Perimeter in Atlanta. This property is located in the Central Perimeter submarket of Atlanta. Adjacent to the Perimeter Mall and a subway station, 211 Perimeter is located in one of the largest suburban office markets in Atlanta. The property has an office building of 226,000sf rentable area, and 13 acres total land. The current owner has got approvals for over 1 million square feet of mixeduse development on the 9.5 acres developable site, and has built a 6storey parking garage with the intention to get as much value as the regulations allow from development of the excessive land (Figure 51). Furthermore, the property is strategically located within a larger neighborhood redevelopment planning of 38 acres and nearly 3 million square feet mixeduse development, although the timing of the neighborhood development is unknown. The land obviously has some value, but development might not break ground immediately. The real estate market in Atlanta is a commodity market, which means developments are spread out with few restrictions. As 2005, the Central Perimeter office submarket was over built, with several old office buildings torn down for new residential developments. It would be interesting to know how current bidders should price the land in addition to the building. .. ....... Figure 51. 211 Perimeter site plan. Building Valuation In this chapter only the building is evaluated using the combined approach with Monte Carlo simulation. The land valuation will be investigated in the next chapter using the separated approach. The following are the 6 steps used to perform the RERO valuation: * Problem framing; * Approach selection; * Base case modeling; * Risk drivers identification and estimation; * Option modeling; and * Sensitivity analyses. Problem Framing The property is located in a premium office market, with superior quality and tenant mix. Its strategic location within a larger neighborhood redevelopment plan makes real estate price appreciation in the future extremely promising, although the timing is still unknown. In short, the 211 Perimeter project is a sound investment that deserves further valuation. After the preliminary qualitative analysis, this project appears acceptable for quantitative analyses. The 11floor office building consists of 226,000sf rentable area. Current occupancy rate is 85%, with 15% upside potential to lease up the space. Major tenants collectively occupy 68% of the rentable square footage, which is deemed to be a sign of solid cash flow over the future. One of the major decisions to make is about the chiller system upgrade. The existing chillers are still in working condition but are at their maximum capacity, and consume far more energy than new ones. Preliminary research shows that replacement of the existing chillers will cost $950,000, and will increase the net cash flow by 5% per year. If both rental rates and occupancy rates are good, replacement of the chillers can justify its cost, and add value to the property. Otherwise, the capital improvement may not break even, and keeping the existing chillers is more economical. Approach Selection The combined approach is selected since both the rental rate and occupancy rate are market driven, and can be combined into the Monte Carlo simulation. Base case NPV calculation The following variables are typical in the NPV valuation model: rental rate, occupancy rate, rentable square footage, expense recovery, operating expenses, capital expenditure, tenant improvement, leasing commission, goingout cap rate, discount rate. Table 51 shows the assumptions used in the base case NPV calculation. Figure 52 shows the cash flow output from Argus, a software package for real estate valuation. Table 51. Major assumptions for Argus. Average rental rate $17/sf Capital expenditure $75,000 Occupancy rate 85% Tenant improvement $18/sf Rentable sf 225,924 sf Leasing commission 6.0% Expense recovery $0 Goingout Cap rate 7.0% Operation expenses $7.75/sf Discount rate 9.0% From the Argus cash flow output, modifications are made so that the model can be used for Monte Carlo simulation using Risk Simulator. Rental rate and occupancy rate have been identified as the two major risk variables that need to be simulated. Annual average rental rate and annual average occupancy rate are calculated from the Argus output, which are used to derive annual net cash flow. Purchase price is assumed to be fixed, so that we can compare the project value with and without flexibility. Operating expenses and expense recoveries are controllable variables. Capital items, such as capital expenditure, tenant improvement, and leasing commission, are also controllable. Cap rate and discount rate are also assumed to be fixed. Ignoring the option of chiller replacement, the project NPV has two components: (1) Total acquisition cost, including purchase price and closing cost; (2) Present value of annual net cash flow from operation and present value of net residual value (gross sale proceeds net out selling cost). These two parts are also called cost and benefit. The option of chiller replacement will be modeled later. In real estate fundamental analysis, property value consists of residual value and net cash flow from operation. The residual value, or value when the project is sold, is the major part. It is determined by Net Operating Income (NOI) and Capitalization rate (Cap rate). NOI is the gross Schedule Of Prospective Cash Flow In Inflated Dollars for the Fiscal Year Beginning 10/1/2005 Year 1 Year 2 Year 3 For the Years Ending Potential Gross Revenue Base Rental Revenue Absorption & Turnover Vacancy Base Rent Abatements Scheduled Base Rental Revenue Expense Reimbursement Revenue Miscellaneous Conference Room Antenna Revenue Total Potential Gross Revenue General Vacancy Effective Gross Revenue Operating Expenses Cleaning Repairs and Maint. Utilities Grounds Security Parking/Fitness Center Management Administrative RE Taxes for Building Insurance NonRecoverable Total Operating Expenses Net Operating Income Leasing & Capital Costs Tenant Improvements Leasing Commissions Reserves 8F Corridor & Common Area Total Leasing & Capital Costs Cash Flow Before Debt Service Sep2006 Sep2007 Sep2008 Sep2009 $3,865,560 $3,933,779 $4,055,362 $4,187,044 (536,603) (120,821) (557,764) 2,771,193 2,400 6,000 44,874 2,824,467 (335,242) 3,477,716 2,472 6,180 46,220 3,532,588 2,824,467 3,532,588 181,060 (6,223) (17,347) (193,496) 3,855,643 26,417 2,546 6,365 47,607 3,938,578 (137,519) 3,801,059 4,169,697 52,555 2,623 6,556 49,035 4,280,466 (149,114) 4,131,352 184,366 216,739 222,801 243,591 250,899 258,426 266,178 318,091 326,111 42,095 128,504 8,182 116,706 193,246 43,358 355,126 365,464 44,659 45,998 132,359 136,330 140,420 8,427 106,141 8,680 133,037 8,941 144,597 199,043 205,015 211,165 418,558 431,115 444,048 457,370 62,129 45,185 1,757,347 1,067,120 64,277 46,747 1,792,843 1,739,745 119,610 1,818,406 33,160 396,647 33,889 75,000 261,659 805,461 35,060 2,250,113 (510,368) 66,233 48,169 1,916,462 1,884,597 11,118 9,094 36,127 56,339 1,828,258 68,197 49,598 1,980,729 2,150,623 Figure 52. Base case NPV calculation. Year 4 income from all sources (rental, storage, tenant reimbursement, antenna lease, etc) minus all operating expenses (common area maintenance, management fee, security, landscaping, insurance, real estate taxes, etc). For this reason, NOI is also regarded as the net income of the property. This is different from what the investor actually gets, which is called the Net Cash Flow. Net cash flow is calculated by taking out capital items from NOI. These capital items, such as capital improvement, tenant improvement, and leasing commission, are onetimeoff in nature. All these analyses are on an unleveraged beforetax basis, meaning debt financing and taxation are not considered. Figure 53 shows the modified Argus cash flow output for NPV calculation. For simulation simplicity, modifications of the Argus output are made so that the net operating income and net cash flow are calculated by Equation 51 and Equation 52. NOI Q SF Occ+ER OE (51) NCF = NOI TI LC CapX (52) where NOI is the net operating income; Q is the average rental rate; SF is the rentable square footage; Occ is the actual occupancy rate; ER is the expense recovery and other income; OE is the operating expenses; NCF is the net cash flow; TI is the tenant improvement; LC is the leasing commission; CapXis the capital expenditure. The residual value at sales is calculated by Equation 53. V = NOI SC (53) Cap where Vn is the net residual value at year n, and n is the holding period of the project; Cap is the goingout Cap rate; SC is the selling cost. The total benefit of the project PV,, which includes the present value of net cash flow NCF, and residual value Vn, can be calculated by Equation 54. NCF, V PV = ( + 1 (54) S(1+ k)' (1+ k)" where PVj is the project present value at Yearj, and 0 < j < n, where n is the holding period. When = 0, it is the present value at time 0, or PVo. NCF, is the net cash flow at Year i, k is the discount rate of the project. The NPV of the project is the present value of total cost PPo and total benefit PVo at time 0, as calculated by Equation 55. NPVo = PVo PPo (55) Risk Drivers Modeling Among the variables, those that have the most profound impact on the project NPV changes are rental rate and stabilized occupancy rate, both are market driven. Rental rates differ leasebylease, but for simplicity we take the average rental rate over the entire building. Stabilized occupancy rate is subjective based on management's estimates In this case the 15% vacant space is assumed to be leased up within 2 years, after which a general vacancy factor of 3% is taken out. Figure 54 shows the historical rental rates of the Central Perimeter Class A office market and the subject property in 15 years. The quarterly data is from CoStar. The change of rental rate is assumed to follow GBM. This means the logarithm of the rental rate Qi is normally distributed; and the return (also called the change of rental rate) qi follows a random walk. Using Equation 56, a rental return analysis was performed and the scatter chart was plotted as shown in Figure 55, with market return variables on Xaxis and corresponding subject property return variables on Yaxis. It shows negative correlation (0.1445), which indicates that the rental Rental Square Footage Base Rent psf Rental Adjustment Multiple Return Random Variable Occupancy Rate Occupancy Adjustment Variable Occupancy Random Variable For the Years Ending Potential Gross Revenue Base Rental Revenue Vacancy Abatements Scheduled Base Rental Revenue Other Reimbursement Revenue Total Potential Gross Revenue General Vacancy Effective Gross Revenue Operating Expenses Total Operating Expenses Net Operating Income Leasing & Capital Costs Total Leasing & Capital Costs Cash Flow Before Debt Service Net Sales Proceeds (Cap at 7%, 2% selling cost) Total Net Cash Flow Discount Rate PVo of Cash Flow PVI of Cash Flow PVo of Cash Flow Static 805,359 738,862 805,359 738,862 (510,762) 31,927,568 (429,898) (468,589) 24,653,941 26,872,795 (429,898) 24,653,941 Forecasting Variable Purchase and Closing Cost (3%) NPV of Project Figure 53. Spreadsheet model for Monte Carlo simulation. 225,924 $17.11 16.9121 1.0117 85.68% 0 8542 0 0026 Year 1 Sep2006 Year 0 Sep2005 $17.41 17.2087 1.0117 89.15% 0 8889 0 0026 Year 2 Sep2007 $3,933,337 (426,767) (29,248) 3,477,322 54,872 3,532,194 3,532,194 1,792,843 1,739,351 2,250,113 (510,762) $18.53 18.3157 1.0117 99.72% 0 9946 0 0026 Year 4 Sep2009 $4,186,372 (11,722) (5,623) 4,169,027 110,769 4,279,796 (149,114) 4,130,682 1,980,729 2,149,953 $17.95 17.7424 1.0117 99.91% 0 9965 0 0026 Year 3 Sep2008 $4,055,336 (3,650) (196,069) 3,855,617 82,935 3,938,552 (137,519) 3,801,033 1,916,462 1,884,571 56,339 1,828,232 30,099,336 $3,865,560 (553,548) (540,920) 2,771,091 53,274 2,824,365 2,824,365 1,757,347 1,067,018 261,659 805,359 1.0900 (24,205,000) 757,904 rate change of the subject property, 211 Perimeter, has very weak, if not negligible, correlation with the market. $26.00 $23.00 $20.00 $17.00 $14.00 / . \1 / r* , 1990 1992 1994 1996 1998 2000 2002 2004 3Q 3Q 3Q 3Q 3Q 3Q 3Q 3Q Market Subject Property Figure 54. Historical market and subject property rental rates. y6830.00% 20.00% y = 0.3276x 8E05 Subject Property R = 0.0209 0.0 * YA 7.00% 5.00% 3.00% 1.00% 1.00% 3.00% 5.00% 7.00% 10.00% * * 20.00% Market Figure 55. Returns correlation between market and subject property. q, = ln( ) (56) The seemingly controversial result of weak or no correlation between the rental return of the subject property and that of the market can be explained as due to two reasons: (1) Data reliability. CoStar started as a service portal mainly for commercial brokerage firms. In its early years data is derived from broker volunteer contributions. This would inevitably have led to data accuracy and timeliness issues. For example, from the first quarter of 1998 to the third quarter of 1999, the rental rates data of the subject property are missing, which are assumed to be $18.90/sf by the author for the purpose of data completeness. (2) The inelastic nature of real estate market. Compared to the financial market, the real estate market is lumpy and the performance is somewhat predictable, at least for the near term. Commercial lease terms are usually 3 to 7 years for office leases, 5 to 20 years for anchor retail leases, and even 100 years for ground leases. In most cases, the rental payment is set and documented in the contract throughout the terms. Market rental rate changes can only slowly affect individual property ask rates, since the landlord can change rental rate only when a lease negotiation happens, usually before the lease expires. However, market rates can directly affect the rates for new construction, since all spaces are newly available. Nevertheless, the data set from CoStar is the most comprehensive and consistent data available in the real estate industry. The characteristics of real estate require a different method than the one used to estimate stock volatility in the financial industry. Thus the correlation between the market and the subject property was ignored on purpose, and only the subject property rental rate data was used to estimate its volatility for acquisition. Risk Simulator is used to influence the distribution of the population from the available sample data. A lognormal distribution was chosen since the rental rates will never be negative. Due to the limitation of available data, the statistical significance of this distribution is low (P Value of 0.1625). Nevertheless, this is the most reasonable fit for the data. By fitting the sample data into a lognormal distribution (Figure 56), the following variables are determined: / is 0.0056 and a is 0.0548. Annualizing the quarterly o, and using Equation 57 and 58, mean of 18.0189 and standard deviation of 2.0218 for the return distribution are derived. To get the annual autocorrelation of the rental return, the quarterly return data is annualized by taking the average of the 4 quarters of each year, which turns out to be 0.0916. This autocorrelation of the samples is assumed to be the same as that of the population. X= e + 2 /2) (57) SD = e2(p ) e 2 (58) Statistical Summary Normal Distribution eoetical vs. Empirical Disribution Mu = 0.0056 1 ~Sigma = 0.0548 KolmogorovSmirnov Test Statistic Test Statistic: 0.3406 i:ii I PValue: 0.0000 Actual Theoretical IC0.0 Mean 0.0007 0.0056 Stdev 0.0548 0.0548 I I C0.I Skewness 0.7670 0.0000 Kurtosis 11.3812 0.0000 Figure 56. Normal distribution fit for historical returns on rental. CoStar also provides historical occupancy rates data for the market and subject property (Figure 57). Occupancy rate is assumed to follow the additive stochastic process. This means the change of occupancy rate o, between any two quarters is simply the difference of the occupancy rate 0, and O,_1 (Equation 59). From the scatter plot of the change of occupancy data shown in Figure 58, it can be concluded that the occupancy rate of the property also has very weak correlation with the market (0.1263). Thus, this correlation is also ignored on purpose and only the historical occupancy rates of the subject property will be relied on for forecasting. o, = 0, ,01 (59) Using RiskSimulator, the population u and a, the respective population mean and standard deviation of the normal distribution, are determined to be 0.0039 and 0.0471 respectively (Figure 59). Due to the limitation of available data, the statistical significance of this distribution is low (PValue of 0.00004). However, this low Pvalue might be a limitation of the software itself, i.e., its estimation of data in a small range is inaccurate. Nevertheless, this is the most reasonable fit for the data. To preserve accuracy, it was decided to keep the sample mean as the population mean (0.0026), and annualize the sample standard deviation as the 100.0% 80.0%  60.0% 40.0%  40.0%  20.0% 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 4Q 4Q 4Q 4Q 4Q 4Q 4Q 4Q 4Q 4Q Market Subject Property Figure 57. Historical market and subject property occupancy rates. 20.0% 10.0% y = 0.4502x + 0.0018 R2 = 0.016 P 10.0% 5.0% 0.( 5.0% 10.0%  10.0% 20.0%  2a0.0%  30. 0% Market Figure 58. Occupancy changes correlation between the local real estate market and the subject property. Figure 59. Normal distribution fit for historical occupancy rates. vs. Empirical Distrilbution Statistical Summary Normal Distribution Mu= 0.0039 Sigma = 0.0471 KolmogorovSmirnov Test Statistic Test Statistic: 0.2799 PValue: 0.0000 Actual Theoretical Mean 0.0026 0.0039 Stdev 0.0705 0.0471 Skewness 1.2849 0.0000 Kurtosis 15.0876 0.0000 ==NMI population volatility (0.1409). To calculate the auto correlation, the change of occupancy rate data is annualized by taking the average of the 4 quarters of each year, which turns out to be 0.1185. The correlation between rental return and change of occupancy rate is similar to the auto correlation of the two, which comes out to be 0.1575. For Monte Carlo simulation, the project volatility is the volatility of percentage changes in the value of the project from one time period to the next, defined by the forecasting variable z (Equation 510). This value is computed using the simulated present value of the project in Year 1 divided by the expected present value of the project in Year 0. In other words, PV1 is dynamic, while PVo is static. z = (510) PVo Option Modeling In the previous step the rental rate, occupancy rate, their respective volatilities, auto correlations, and the correlation between the two have been identified and quantified. With these variables, rental rates and occupancy rates for each year can be set as risk variables for the project value simulation. A total of 8 risk variables are defined and highlighted as shown in Figures 543, 510, and Table 52. The cash flows go through Equations 51 to 55 to generate annual net cash flow for the first 3 years. PVo and PVi are calculated based on the annual net cash flow. The forecasting variable z is defined in Equation 510. Setting PVo to be static and PV1 to be dynamic, and running the simulation for a 1000 times, the simulation result ofz is obtained as shown in Figure 511. Table 53 also shows the statistical summary of z, with a mean of 1.079 and standard a deviation of 0.3283. Name Enabled Cell Dynamic Simulation Range Minimum Maximum Distribudin Mean Standard Deviation Yr 4 Rertrn Yes $F$7 No infinity +Infinity Lognormal 0.0056 0.10968 Name Enabled Cell Dynamic Simulation Range Minimum Maximum Disribution Mean Standard Deviation Figure 510. Snap shot of Monte Carlo simulation assumptions. Figure 511. Monte Carlo Simulation Result of Forecasting Variable z. The statistical distribution fit for Variable z is then performed. By plotting the 1000 z values from the simulation output, as shown in Figure 512, it is determined that they are normally distributed with PValue of 0.8737. This result fits quite well with the theory Yr 1 Occ Yes $C$O1 No Infinity + infinity Normal 0.0026 0. 409 4.00 3.S0 3.00 F, R7 '7 F' 17 q7 1 11 1 'R  .00 .50 1O. '' 5'0 ; 1 31 ni 1 1i 17 3 SI,',. ..  1 ,3,', I I I IIrI I II S" ' ...., ,, ______7_ Table 52. Correlation between random variables. Yr 1 Yr 2 Yr 3 Yr 4 Yr 1 Yr 2 Yr 3 Yr 4 return return return return Occ Occ Occ Occ Yr 1 return 1.00 Yr 2 return 0.09 1.00 Yr 3 return 0.00 0.09 1.00 Yr 4 return 0.00 0.00 0.09 1.00 Yr 1 occ 0.16 0.00 0.00 0.00 1.00 Yr 20cc 0.00 0.16 0.00 0.00 0.12 1.00 Yr 3 Occ 0.00 0.00 0.16 0.00 0.00 0.12 1.00 Yr4 Occ 0.00 0.00 0.00 0.16 0.00 0.00 0.12 1.00 Table 53. Statistical summary of Monte Carlo simulation result. Description Value Number of data points 1000 Mean 1.0797 Median 1.0569 Standard deviation 0.3283 Variance 0.1078 Average deviation 0.2561 Maximum 2.4431 Minimum 0.0977 Range 2.3454 Skewness 0.3787 Kurtosis 0.7041 25% percentile 0.8722 75% percentile 1.2831 Error precision at 95% 0.0188 Statistical Summary Normal Distribution Teorellcal vs. Elliirial Disirlliilon Mu = 1.0738 Sigma = 0.3246 / \ KolmogorovSmirnov Test Statistic Test Statistic: 0.0275 PValue: 0.4326 S \ Actual Theoretical Mean 1.0797 1 0738 Stdev 0.3283 0 3246 'Skewness 0.3787 0.0000 Kurtosis 0.7041 0.0000 Figure 512. Normal distribution fit of forecasting variable z. developed by Samuelson and adopted by Copeland and Antikarov (2001), as discussed in Chapter 4, that changes in correctly expected asset prices follow Geometric Brownian Motion. From the Monte Carlo simulation, the mean u and the volatility a of forecasting variable z are calculated as 1.0797 and 0.3283 respectively. This means the expected average project return is 7.97% (1.0797 minus 1), and the volatility of the project is 30.4% (0.3283 divided by 1.0797). Using the assumptions in Table 54, with 30.4% volatility, and $24,963,000 PV derived from the base case analysis, a value tree is constructed as shown in Figure 513. Net cash flows are modeled as dynamic dividend yield times PV in the base case (Refer to Chapter 4 for details of binomial lattice with dividend). For example, in Year 1, the PV can go up to $34,664,000 with an up factor of 1.3886, the post dividend cash flow is therefore $33,638,000 (after taking out 2.96% yield from the $34,664,000 before dividend cash flow). Table 54. Event tree assumptions (Dollars in $1,00( Assumptions PV of asset value $24,963 Implementation cost $24,205 Maturity (years) 3.00 Riskfree discount rate (%) 5.00% Volatility (%) 32.83% Lattice steps 3 Option type Call NCF as percentage of PV Year 1 NCFi $805 PVi $27,210 Percentage 2.96% Intermediate computations Stepping time (dt) Up step size (up) Down step size (down) 2 ($511) $28,781 1.78% 1.0000 1.3886 0.7201 3 $1,828 $31,928 5.73% 46,7166,014 34,664 7,50 62,234 24,963 33,638 24,224 34,235 17,977 32,275 17,445 24,655 17,755 12,563 16,738 16,738 12,786 9,208 8,681 Figure 513. Event tree present value without flexibility (Number in $1,000). With the event tree of PV without flexibility, the chiller replacement option can now be modeled. An event tree of PV with flexibility is constructed (Figure 514). At the end nodes, the decision is whether to keep the existing chillers or replace them with new ones. For example, the value of Node A' is calculated as follow. Max (Replace, Keep) = Max (Present Value 1.05 Cost, Present Value) = (62234 1.05 950, 62234) = 64396 (Replace) At the intermediate nodes, the decision is about whether to leave the option open or to execute it immediately. To calculate the value of leaving the option open, the replicating portfolio method developed in Chapter 4 must be used, but not the discounting method, since riskadjusted probability and riskadjusted discount rate are used to construct the spread sheet and event tree. Equation 47 is the replication portfolio formula to be applied. 48,137 C A 68,175 35,486 8 A' 64,396 S34,4 ReplReplace 25,421 Open pen Opn 4,537 B 34,899 18,1 B' 32,939 173 26 Replace Open 1Open 17,755 12,563 16,738 16,738 12,786 Keep Open 9,208 8,681 Keep Figure 514. Present value with flexibility (Numbers in $1,000). For example, the value of keeping the option open at Node C' is 68175 34899 1.3886 x34889 0.7201 x68175 4 C = + = 48877 1.3886 0.7201 e0051 (1.3886 0.7201) Therefore, the value of node C' is Max (Replace, Open) = Max (47540*1.05950, 48877)= 48967 (Replace) The decision is to replace the chiller system immediately. Using Equation 414 to add back the implied net cash flow of negative $830,000, the before dividend present value is $48,137,000. Working backward the value at each node can be similarly calculated and the optimal action can be selected to maximize the present value, and eventually the maximum present value can be derived at time 0. The present value increases from $24,963,000 (without flexibility) to $25,421,000 (with flexibility), or an increase by $458,000. The NPV of the project is now $1,216,000. In other words, the option to replace the chillers system creates $458,000 value. If the building could be purchased at $24,205,000, the NPV increases to $1,216,000. Sensitivity Analyses Sensitivity analyses are conducted using option value as dependent variable, and present value, replacement cost, discount rate and volatility as independent variables. Table 55 summarizes the effect of each independent variable as well as their combined effects on the option value. Present value has positive effect on the option value (Figure 515). Replacement of the chiller system increases the annual net cash flow by 5%. And present value is positively related to net cash flow. Therefore, the higher the present value is, the higher the additional net cash flow would be when exercising the replacement option, and hence the higher the option value would be. Table 55. Summary of variable effect on option value. Present Replacement Discount rate Volatility value cost Present value Positive Uncertain Positive, most Positive, most Sensitive when in Sensitive when at themoney themoney Replacement Negative Uncertain, most Uncertain, most cost Sensitive when at Sensitive when at themoney themoney Discount rate Positive Positive, most Sensitive when at themoney Volatility Positive 1,000 3 800 > 600 S 400 o 200 10,000 20,000 30,000 Present Value Figure 515. Option value in relation with present value. As shown in Figure 516, the replacement cost has negative effect on the option value. The higher the replacement cost is, the less likely the replacement is breakeven, and hence the less likely the option would be exercised. 1,200 S 1,000 800 S 600 S 400 0 200 500 1,000 1,500 2,000 Replacement Cost Figure 516. Option value in relation with replacement cost. Volatility also has positive effect on the option value (Figure 517). The higher the volatility, the wider the present value spread becomes in later years, but the replacement option is only exercised in those scenarios with positive net cash flows. Therefore, the more uncertain the future cash flow is, the more valuable the option becomes. 1,000 800 600 6 400 200 10,000 20,000 30,000 Present Value 20% Volatility 33% Volatility 45% Volatility Figure 517. Option value in relation with present value and volatility. Riskfree interest rate has positive effect on the option value. But the effect is not significant. After examining the effect of each independent variable on the option value, combinations of each two independent variables can be looked at. The combination of present value and Risk free interest rate has positive effect on the option value. The two pairs of (1) present value and volatility (Figure 517), (2) volatility and riskfree rate (Figure 518) both exercise positive effect on option value, and are most sensitive when the option is atthemoney. The three pairs of (1) replacement cost and volatility (Figure 519), (2) replacement cost and riskfree rate, (3) present value and replacement cost (Figure 520) all display uncertain effect on the option value. This conclusion is best illustrated in Figure 520. The 3dimensional curve indicates that the higher the present value and the lower the replacement cost, the higher the option value. However, this effect is nonlinear. With higher present value and higher replacement cost, the option value may be higher or lower, depending on whether the option value is inthemoney. 600  550 500 450 400 350 300 10% 20% 30% 40% Volatility 3% RiskFree 5% RiskFree  7% RiskFree Figure 518. Option value in relation with volatility and discount rate. 1,200 1,000 800 600 400 200 50% 500 1,000 1,500 2,000 Replacement Cost 20% Volatility 33% Volatility  45% Volatility Figure 519. Option value in relation with replacement cost and volatility. 2,500 2,000 1,500 1,000 500 Cost Option Value ralue Figure 520. Option value in relation with present value and replacement cost. Summary This chapter applies the combined approach to determine the building value of the 211 Perimeter property in Atlanta. Rental rate and stabilized occupancy rate are identified as the two major risk drivers and their volatilities are estimated using historical data. The risk variables are combined in a spread sheet. Monte Carlo simulation is performed to estimate the project volatility. Event tree is constructed, in which the option to replace the chiller system is incorporated. The RERO approach indicates that the building is worth $25,421,000, and the value of managerial flexibility is worth $458,000. CHAPTER 6 THE SEPARATED APPROACH This chapter is the second part of the case study described in Chapter 5. In the previous chapter the RERO framework is applied to analyze the building structure and a managerial decision of chiller replacement. The combined approach with Monte Carlo simulation is used as the major methodology. This chapter, however, is about valuation of the infill land using the separated approach, with jump diffusion process and decision tree analysis techniques. Together, these two parts demonstrate how the RERO framework can be applied to different scenarios in the analysis of real estate acquisition and development. Case Description The previous chapter has full description of the case 211 Perimeter in Atlanta. This chapter only repeats the infill land portion. Besides the existing office building and the 6story garage, the current owner has got approvals for over 1 million square feet of mixeduse development on the 9.5 acres developable site. Furthermore, the property is strategically located within a larger neighborhood redevelopment planning of 38 acres and nearly 3 million square feet mixeduse development, although the timing of neighborhood development is unknown. The land obviously has some value, but development might not break ground immediately. The real estate market in Atlanta is a commodity market, which means, with little control of urban sprawl, developments are spread out easily as far as market demand exists. The Perimeter office submarket is currently overbuilt, with several old office buildings torn down for new residential developments. It would be interesting to know how current bidders should price the land in addition to the building. Land Valuation The value of the infill land (9.5 acres out of the 13 acres total) depends on the value and cost of the improvement should it be developed. The value of the improvement is determined by a function of its annual rental income and operating cost, just like the existing building. The cost of development includes hard costs and soft costs. Since every project is unique, development cost is assumed to be a private risk that does not correlate with the traded financial market. Problem Framing The addition of a 6story garage has freed the infill land from its original function as surface parking. With the 1 million square feet mixused development approval, the land can be sold for $4.75 million at anytime during the holding period. Its best value for the investor is being either developed or spinoff for $4.75 million. Table 61 shows the development assumptions. Assume the land allows for 1 million square feet to be built, gross rent is $24.5/sf, stabilized occupancy rate is 85%, operating expense is $8.5/sf, required cap rate is 8%, riskfree interest rate is 5%. Expected development cost is $227.5/sf Land carrying cost is assumed to be negligibly small compared to the development value. The land can be sold for $4.75 million at anytime. This can be viewed as the exercise price of a put option to the investor. Table 61. Development assumptions. Rentable sf 1,000,000 Site acres 9.50 Gross rent psf $24.50 Land $4.75 Occupancy rate 85.0% Value $154.06 Operating expenses psf $8.50 Cost $177.50 Net rent psf $12.33 Riskfree rate 5.0% Cap rate 8.0% In addition, management believes that the groundbreaking for the larger neighborhood redevelopment will have significant impact on the demand for new office space, and hence drive up rental rate of this development by 20%. This is a onetime event, but once the rental rate rises, it will remain at that level during the entire analysis period. Approach Selection The separated approach is selected because the impact when the rental rate jumps up by 20% is significant, and the chance is uncertain, depending on the timing of the neighborhood redevelopment. This is an example where one risk driver (the rental rate) does not get resolved smoothly, and must be modeled separately from the other risks. Risk Drivers Identification and Estimation The risk drivers are rental rates and development cost. Unlike the existing office building, the new building does not have a historical track record. For income, the building rental rate is assumed to have some premium over the average market rental rate. Changes in rental rate are assumed to follow the GBM movement, with a jumpdiffusion process corresponding to the groundbreaking of the neighborhood project. Figure 61 shows the historical market average rental returns for Class A office properties in the Central Perimeter submarket. Using the Risk Simulators, the quarterly lognormal returns are plotted into a normal fit as shown in Figure 62. Converted into annual data, the market rental rate volatility is 4.84%. As explained in Chapter 5, individual property is far more volatile than the market average. The management estimate doubles and becomes 9.68% per year for the infill land development project. The current gross rental rate is $21/sf for the average Class A building in the Central Perimeter submarket. According to management experience, a $3.50/sf premium for a brand new building can be secured. $30.00 6.00% $20.00 2 \ ,, Ii I 2.00% S$15.00 V / 2.00% $10.00 l 4.00% $5.00    6.00% $0.00 8.00% 1990 3Q 1992 3Q 1994 3Q 1996 3Q 1998 3Q 2000 3Q 2002 3Q 2004 3Q  Rental Return Figure 61. Historical market average rental rates and return volatility. Statistical Summary Normal Distribution Soretical vs. Empirical Distribrtion Mu = 0.0019 Sigma = 0.0242 20. KolmogorovSmirnov Test Statistic Test Statistic: 0.0691 1' PValue: 0.9282 1 i Actual Theoretical .. Mean 0.0024 0.0019 I ., Stdev 0.0242 0.0242 I 5 0.'5 1 Skewness 0.4240 0.0000 S Kurtosis 0.0318 0.0000 Figure 62. Normal distribution fit for historical market rental returns. Rental rate changes are assumed to follow the GBM movement. A Poisson distribution jumpdiffusion process corresponds to the groundbreaking of the neighborhood residential project, with 10% annual probability. The option value is calculated using Equation 415 developed in Chapter 4, where A is 10% and y is 1.2 (1 plus 20%). Figure 63 shows how to get rental rate changes from one period to the next period. At Year 0, gross rental rate is $24.50/sf. It could have three values in the next year: $29.40/sf (1.2 times $24.50/sf) if the neighborhood development breaks ground, $26.99/sf (up movement) or $22.24/sf (down movement) if the neighborhood development does not break ground, with probabilities of 0.10, 0.5895, and 0.3105 respectively. In year 2, it could have five values. If the neighborhood development breaks ground in Year 1, the rental rate $29.40/sf will follow the GBM movement with possible value of $32.39/sf or $26.69/sf, with probabilities of 0.7402 and 0.2598 respectively. If no development breaks ground in Year 1, the rental rates of $26.99/sf and $22.24/sf each follows the GBM with jump diffusion process and has three values, which combine into 5 possible values. In Year 3 the rental rates follow the same process and can have seven values. Notice, however, the probabilities to get to these values are different with and without the jump diffusion process. Sigma 9.68% a 1.1016 1 1( 'I':J t 1 d 0.9077 y 1.20 rf 5.:1: o p 0.7402 p 0.6550 1p 0.2598 (1p 0.5895 (11Xx1p) 0.3105 0.7402 S24.50 0.3105 0.3105 $32.39 $26.69 $29.73 S24.50 S20: 19 Year 0 Year Year 2 Figure 63. Gross rental rate movement and probabilities. $35.68 $29.a4 $24.23 S32 76 $26.99 $22.24 $18.33 yea 3 Taking out revenue lost from the 15% vacant space, $8.5/sf operating expense, and capping the net cash flow at 8% Cap rate, we can get the corresponding per square foot building value contingent upon the gross rental rate, stabilized occupancy, operating expenses, cap rate, and the likelihood of the neighborhood residential development (Figure 64). $272.85 0.7402 $7.87 $206.13 S2 06.13 0.25i9 5 $177.30 0.1 / 51.14 0.1 05895 $211.78 0.5895 20967 $180.52 Y .3105 18052 $154.06 O. > $154.06 035 $130.05 05895 / 30.05 0$108.24 0.3105 5 Year 0 Year 1 Year 2 Year 3 Figure 64. Building value movement and probabilities. There is no direct comparable data on development cost. Development cost includes hard and soft costs. For hard cost, the RS Means Building Cost Data manual (RS Means, 19982006) can be used. The cost per square foot data for highrise office buildings from Year 1998 to Year 2006 is shown in Figure 65. The historical data shows an upward trend, at a pace generally consistent with the inflation rate from inflatiodata.com (Figure 66). RS Means compiles market average data nation wide, which does not reflect the volatility of local markets. More over, there are no data about the soft cost. Each project is unique in some soft cost items, such as land acquisition cost, permit application cost, unexpected cost, etc. The best estimate would be from experienced managers. The development cost is assumed not to change with the financial market. It is a private risk that depends on the geological condition of the site, material and labor condition of the local market, etc. Management has estimated that with 50% probability the development cost would be $175/sf, with 20% to be $150/sf, and with 30% to be $200/sf, or 170.00 150.00 130.00 110.00 90.00 70.00 199 . _  __  _ 1999 2000 2001 2002 2003 2004 * Low Mid  High 2005 2006 Figure 65. Historical construction cost for highrise office building. 14.00% 12.00% 10.00% 8.00% 6.00% 4.00% 2.00% 0.00% 1999 0 m f 2000 2001 2002 2003  Cost Inflation Figure 66. Construction cost change rate and inflation rate. 2004 2005 2006 98 w r expected cost of $177.50/sf (Figure 67). Cost increases by 3% annually, consistent with the average inflation rate over the past 7 years. For simplicity, the buildable square footage is assumed to be the same as the rentable square footage. 0 150.00 $154.50 $159.14 $163.91 S177.50 175.00 $180.25 $185.66 $191.23 0.3 2.0 $S206.600 $212.18 $218.55 YeO 0 Year 1 Yea 2 Ye 3 Figure 67. Development cost assumptions. Base Case Modeling The expected PV without any flexibility is calculated as shown in Figure 68. It is better represented in matrices. Each table in Figure 69 is a matrix of possible PVs for a given year. Starting from Year 3, the possible outcomes of building values are listed in the first row, and the possible outcomes of development costs are listed in the first column. The values inside the rectangle are all possible combinations of costs and values. The same applies to the values for Year 2, Year 1 and Year 0. In Year 0, the expected value is calculated as the sum of the three values times the respective probabilities of their development cost. Option Modeling There are three possible kinds of decisions at each node: (1) to develop the land, (2) to keep the land asis, and (3) to sell it for $4.75 million. Figure 610 depicts the decisions and payoffs corresponding to the matrices in Figure 611. In this lattice, the notation below the value represents the optimal decision to be made: D for developing the land; K for keeping the option alive; and S for selling the land. $108.94 0.7402 $7874 $51.63 :2 22 0.1 0.2598 $18.17 ($12 77) 0.1 5895 /$77.87 0.5895 50.53 $26.02 03105 $16.61 4_06 0t 1 ($5_ > 0.3105 (24.45) 0.5895 (S33. 6'I 03105 "($50.89) 0.3105 (3105 75.45) $81.62 0.7402 $52.22 S 25.88 $14.90 0.2 0.1 0.2598 ($8.35) (4I i OS" .1 0 5895 $50.55 0.5895 // 24.01 0.5 0.27 03105 ($10.71) 2 .;44) I.2l': 94);' 0.1 ($31.60) 0.3105 "$5 i 05895/ (S6118) 0.3105 ($77.41) ($102271) 0.3 7 5431 0.7402 $25.69 f 12 ($12.42) 0.1 0.25987 ($3488) ($671.4 0 .1 895 $2323 0.5895 (S2.51) ($25 48) 03105 ( .. 45.94) 01 ($58 12.I 0 5895 0.31, ($75.95) 5895 ($88.50) 0.3105 ($10394) ($130.09) Year 0 Year 1 Ye 2 Year 3 Figure 68. Payoff and probabilities without flexibility (Dollars in $1,000,000). Development 163.91 Cost 191.23 218.55 Development 159.14 Cost 185.66 212.18 Development 154.50 Cost 180.25 206.00 T=3 Building Value 272.85 206.13 151.14 241.78 180.52 130.05 88.45 108.94 42.22 (12.77) 77.87 1661 (33.86) (75.45) 81.62 14.90 (40.08) 50.55 (10.71) (61.18) (102.77) 54.31 (12.42) (67.40) 23.23 (38 02) (88.50) (130.09) T=2 Building Value 237.87 177.30 209.67 154.06 108.24 78.74 18.17 50.53 (5.07) (50.89) 52.22 (8.35) 24.01 (31.60) (77.41) 25.69 (34.88) (2.51) (58.12) (103.94) 206.13 51.63 25.88 0.12 T=1 Building Value 180.52 130.05 26.02 (24.45) 0.27 (50.20) (25.48) (75.95) T=0 Building Value 154.06 Development 150.00 4.06 Cost 175.00 (20.94) (23.44) 200.00 (45.94) Figure 69. Payoff matrices for project values without flexibility (Numbers in $1,000,000). In Year 3, the decision will be either to develop the land or to sell it for $4.75 million, whichever generates the higher payoff. For example, the PV of Node A is calculated as follows: Max (Develop, Sell) = Max (Building Value Cost, Salvage Value) =(206.13 163.61, 4.75) = 42.22 (Develop) Working backward, in Year 2, the payoff is the greatest of the three: (1) the payoff of developing the land, which is the building value minus development cost; (2) the payoff of keeping the option open, i.e., the corresponding payoff in Year 3 discounted at riskfree interest rate using the binomial or jump diffusion probabilities calculated in Table 62; (3) the payoff of 0.7402 $6899 < _ 0.1 (K) 02598 B 01 .5895 0 5895 5 69 03105 (K) 01/ C $12.60 A 0.5895 0_3105 12 5 (K) 0.3105 0.7402 S4 94 (K) 0.2598 0.1 58E95 52S 47 0 3105 (K) 0.1 S5.59 4g05895 (K) 0.3105 0.7402 2?S 93 (K) 0.2598g S16. (K) 0.3105 U (s) $87.14 (K) $30_90 (K) $58.93 (K) $14.73 (K) (S) S61.16 (K) $11.66 (K) S.; .51 (K) $5.48 (K) 5175 (S) 0.3105 (K) for keepIt jg die :op:i .i'.e (D) for developing the land Year 0 Year 1 (S) for selling the land Year 2 Year 3 Figure 610. Decision payoff and probabilities with flexibility (Dollars in $1,000,000). $22.14 (K) C I'S 94 (D) A 42 22 (D) '175 (S) $77.87 (D) $16.61 (D) 4.75 $4.75 (S) $81 62 (D) $14.90 (D) 14.75 (s) $50.55 (D) (S) 5475 4'75 (S) $54.31 '1.75 (D) $4.75 (S) $23.23 (D) 1475 (S) 4175 (S) .4 75 (S) T=3 Buildig Value 272.85 206.13 151.14 108.94 81.62 54.31 42.22 14.90 4.75 4.75 4.75 4.75 T=2 Buiking Value 237.87 177.30 209.67 87.14 61.16 39.41 30.90 11.66 4.75 58.93 37.51 19.60 241.78 180.52 130.05 77.87 50.55 23.23 16.61 4.75 4.75 4.75 4.75 4.75 154.06 108.24 14.73 5.48 4.75 4.75 4.75 4.75 Buildig Value 206_13 18052 130_05 Development Cost 154 50 180.25 206.00 150.00 175.00 20000 154.06 35.91 21.99 2 22.14 1321 Figure 611. Payoff matrices of project value with flexibility (Numbers in $1,000,000). Table 62. Probabilities of jump diffusion and binomial processes. Jump diffusion No jump Jump Up Down Up Down X (l1X) (1X)(1p) p 1p 0.1000 0.5895 0.3105 0.7402 0.2598 the put option, which is to sell the land for $4.75 million. For the normal stochastic process, the payoff of keeping the option open at Node B, for example, is calculated using Equation 319 as follows: C = e [pC,, + (1 p)Cd =e e05 1 [0.7402 x 42.22 + 0.2598 x 4.75] =30.90 Development Cost 163.91 191.23 218.55 88.45 4.75 4.75 4.75 Developet Cost 159.14 185.66 212.18 68 99 45.94 28.93 45 69 28.47 16.14 1260 5.59 4.75 T=0 Deelopment Cost I I Consequently, the PV of Node B is calculated as follow. Max (Develop, Keep, Sell) = Max (Building Value Cost, Payoff of Keeping Option Open, Salvage Value) = (177.30 159.14, 30.90, 4.75) = 30.90 (Keep) For the jump diffusion, the payoff of keeping the option open at Node C, for example, is calculated using Equation 415 as follows: C = e rt{C, + (1 )[pC + (1 p)Cd]} = e 005" {0.1 x 42.22 + 0.5895 x 16.61+ 0.3105 x 4.75} = 14.73 Consequently, the PV of Node C is calculated as follows: Max (Develop, Keep, Sell) = Max (Building Value Cost, Payoff of Keeping Option Open, Salvage Value) = (154.06 159.14, 14.73, 4.75) = 14.73 (Keep) Working backward to Year 0, the PV of the project is expected PV of each cost scenario times its corresponding probability. The PV of Node D is calculated using Equation 417 as follows: E(PV) = p [E(PV)] = 0.2 x35.91+ 0.5 x 21.99 + 0.3 x13.21= 22.14 J1 The PV of the project increases from negative $23.44 million without flexibility to positive $22.14 million with the development and selloff flexibility. The option value is $22.14 ($23.44) = $45.57 million. Sensitivity Analyses Sensitivity analyses are conducted using gross rental rates, occupancy rates, volatility, Cap rates, and development cost as independent variables, and on two dependent variables: project value and option value. Project value is the PV of the project with the flexibility of deferred development, spinoff the land, and immediate development. Option value is the difference between PV with flexibility and PV without flexibility. Since the PV without flexibility also changes with variables, the project value and option value analyses have quite different results and implications. As shown in Figure 612 the rental rate has a positive effect on project value. Rental rate is directly linked to revenue. The higher the rental rate is, the higher the income the project will generate, and hence the higher the project value is. However, as shown in Figure 613 it has a negative effect on option value. This is because the higher the rental rate is, the more likely the project will be developed immediately, hence the option to wait or abandon the development by selling off the land is less worthy. In other words, higher rental rate not only increases the project value with flexibility, it also increases the value without flexibility at even higher pace. These two values cancel out each other, resulting in minimal option value. The combination of rental rate and occupancy rate has the same result: positive effect on the project value (Figure 612), and negative effect on the option value (Figure 613). Note that the option value is sensitive to stabilized occupancy rate when the option is atthemoney. 250 Z 200  S200 I 150 100 > 50 3 6 9 12 15 18 21 24 27 30 33 36 Gross Rent  60% Occ 85% Occ  100% Occ Figure 612. Present value in relation with rental rate and occupancy rate. 300 250 200 150 100 50 3 6 9 12 15 18 21 24 27 30 33 36 Gross Rent  60% Occ 85% Occ  100% Occ Figure 613. Option value in relation with rental rate and occupancy rate. Just opposite to the effect of rental rate, as shown in Figure 614, development cost has a negative effect on project value, but positive effect on option value (Figure 615), for the same reason as explained above. PV with Flexibility 200 180 160 140 120 100 80 60 40 20 Gross Rent 0 Development Cost 0 0" C' Figure 614. Present value in relation with rental rate and development cost. 400 350 300 250 200 Option Value 150 100 50 33 27 21 300 15 Gross Rent Development 200 9 tCost 1o0 Figure 615. Option value in relation with rental rate and development cost. As shown in Figure 616, cap rate has negative effect on project value. This is because cap rate is inversely related to property value. (Property value is determined by dividing net operating income by cap rate.) However, the effect of cap rate on option value is more profound. Figure 617 shows that at normal rental rate range ($11/sf to $3 1/sf), cap rate has a positive impact on the option value; however, in the low rental rate range ($0/sfto $11/sf), its impact is the opposite. Figure 618 illustrates how the combination of rental rate and cap rate results in different option value. Unlike most situations where a variable has monotonic impact on the option value, the shape of cap rate on option value is convex. For example, at $20/sf gross rent, the option value at 2% cap rate is $71 million, at 4% cap rate the option value drops to $47 million, and at 8% cap rate the option value comes back to $77 million. 250 r 250 200 150 1 00   _ 100 50  3 6 9 12 15 18 21 24 27 30 33 36 Gross Rent  6% Cap 8% Cap 10% Cap Figure 616. Present value in relation with rental rate and Cap rate. 300 250 S200 S150 100 50 3 6 9 12 15 18 21 24 27 30 33 36 Gross Rent 6% Cap 8% Cap 10/o Cap Figure 617. Option value in relation with rental rate and Cap rate. As shown in Figures 619 and 620 volatility has positive impact on both project value and option value. This finding is consistent with many observations in real options research (Titman, 1985; Williams, 1991; Quigg, 1993) that greater volatility increases option value, which is also the reason why the real options methodology should be applied to projects with high uncertainty. 900 800   700  "  700 600  Option Value 500 400 300 1 200 100 104 1p 3% 7% 9% Cap Rate Figure 618. Option value in relation with rental rate and Cap rate in 3D. 150 100 100 5 5 0 3% 6% 9% 12% 15% 18% 21% 24% 27% 30% 33% 36% Volatility  6% Cap 8% Cap 10% Cap Figure 619. Present value in relation with volatility and Cap rate. 100 80 > 60 S40 0 20 3% 6% 9% 12% 15% 18% 21% 24% 27% 30% 33% 36% Volatility 6% Cap 8% Cap  10% Cap Figure 620. Option value in relation with volatility and Cap rate. Summary This chapter applies the separated approach to value the infill land of the 211 Perimeter property in Atlanta. Rental rate and development cost are identified as the two major risk drivers. Rental rate is assumed to have jump diffusion effect due to the uncertainty of the larger neighborhood redevelopment project. Development cost is assumed to be a private risk with no corresponding traded twin asset and it is estimated subjectively based on management's experience. DTA methodology is applied and an event tree is constructed, in which three options are incorporated: the option to develop immediately, the option to delay development, and the option to sell the land. The RERO approach indicates that the land is worth $22,140,000 and the value of managerial flexibility is worth $45,570,000. In Chapter 5, the building is estimated to be worth $25 million; in this chapter, the land is estimated to be worth $22 million, totaling $47 million. This is very close to reality, because the property was actually sold for $43.5 million in 2005. CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS Conclusions Three main conclusions are drawn from this research: (1) acquisition and development has different characteristics and deserve different kinds of attention; (2) consideration of managerial flexibility can change investment decisions; and (3) many unconventional real option valuation problems can be realized by using binomial lattice and Monte Carlo simulations. Acquisition and development have different characteristics and thus deserve different kinds of valuation. The option value of acquisition is usually on a much lower scale than that of development, but by no means is it less significant. In the case studies, the option in the existing building is replacement of the chiller system. Its value is $496,000, or 52% of the replacement cost of $950,000. On the other hand, the option on the infill land is development timing and abandonment. The option value is as high as $45.65 million, but only 26% of the development cost of $177.5 million. Due to the scale of the valuations, it is better to have the option in the building and the options in the land valued separately. But the impact of management flexibility on acquisition and operation is as significant as, if not more than, that on development. The consideration of operating flexibility in acquisition is important. It adds competitive value to the bid for a property. In the case studies, the building is worth $25 million, and the land is worth $22 million, totaling $47 million. In other words, the infill land is worth almost as much as the building. This is very close to reality, because the property was actually sold for $43.5 million. Note that the present value of the development project without any flexibility is negative $23 million. With negative NPV, the project will not break ground. This means if management does not incorporate the flexibilities into the land valuation, the development is deemed worthless, and so is the land. The RERO framework explores a few unconventional real option cases, including (1)jump diffusion process that does not go back to normal diffusion, (2) risk drivers that do not follow the multiplicative stochastic movement, (3) private risk that has no market equivalent and hence violating the noarbitrage option pricing assumption. All of these can be implemented through a binomial lattice with Monte Carlo simulations or the DTA approach. The RERO framework is a simple yet powerful tool, intuitive to the practitioners, yet mathematically correct and precise. Recommendations for Future Research There are at least three directions that future research can go in: model perfection, game theory and phase investment. Model perfection is to improve the preciseness of outcome from the RERO models. Lattice is a discretetime method for option pricing. The smaller the time step, the closer the result will be to that calculated by continuoustime methods. At the same time, the development cost is assumed to have three values in our case study: the optimistic value, the most likely value, and the pessimistic value. More branches can be added to produce a more precise result. By dividing the lattice into more time steps, and breaking the development cost into more branches, a more precise result will be generated. A significant factor not considered in this study is competition. Without the consideration of competition, in most cases it is optimal to defer exercising an option until the end of the holding period. However, competition erodes the value of waiting, affects the value of option as competitors enter or exit the market place and changes the market dynamics (Williams 1993; Myerson, 1991). Should game theory be incorporated into the RERO framework, we predict the option value would be slightly lower, and hence even closer to the closing price. The other direction is stage investment and phased investment. Real estate development is a lengthy process, and it usually takes 2 to 3 years, if not longer. During this period, a lot of uncertainties can change the managerial strategies. Stage investment refers to dividing a real estate development project into different stages: planning, design, construction, sales, etc. This process can be valued similar to pharmaceutical research and development. Phased investment refers to dividing a large real estate development project into different phases, for example, Phase I retail corridor, Phase II residential condominium, Phase III office and hotel towers, etc. Decisions at later phases are contingent upon the outcome of earlier phases. However, there can be timing overlaps between two phases. 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"Equilibrium and Options on Real Assets," The Review ofFinancial Studies 6 (4), 825850. Williams, Joseph T. (1997). "Redevelopment of Real Assets," Real Estate Economics 25 (3), 387407. Yao, Junkui, and Ali Jaafari. (2003). "Combining Real Options and Decision Tree," The Journal of Structured and Project Finance 9 (Fall), 5370. BIOGRAPHICAL SKETCH NgaNa Leung earned her PhD degree in building construction from the University of Florida, Gainesville, FL. While earning this degree, she worked as an acquisition analyst for Parmenter Realty Partners in Miami, FL later for Acadia Realty Trust in White Plains, NY, and now for Antares Investment Partners in Greenwich, CT. She also holds a master of science degree in real estate from the University of Florida, a master's degree in building from the National University of Singapore, Singapore, and a bachelor's degree in architecture from Tongji University, Shanghai, China. NgaNa worked as an assistant project manager in the Environetics Design Group in Shanghai, China prior to coming to the US. At UF, she was supported by the Alumni Fellowship, the highest meritbased award for graduate students. After graduation NgaNa will continue her career in commercial real estate investment, including acquisitions, development, and management. PAGE 1 1 REAL OPTIONS FRAMEWORK FOR ACQUISITION OF REAL ESTATE PROPERTIES WITH EXCESSIVE LAND By NGANA LEUNG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 PAGE 2 2 2007 NgaNa Leung PAGE 3 3 To my husband, Lezhou Zhan, and my family, Lau Leung, SauPik Fung, ShingPen Leung, and ShingChiu Leung PAGE 4 4 ACKNOWLEDGMENTS I would never be able to adequately tha nk Dr. R. Raymond Issa, my chair, for making room for me to develop my res earch question, as well as helping me to choose the direction of my life. I want to thank him not only for hi s tremendous guidance, considerable patience and encouragement throughout my study, but also for his endless trust, respect, and understanding, which has forged me into a better person, not only with intelligence, but with responsibility. I am especially grateful to Dr. Wayne Arch er, Dr. Ian Flood, Dr. Kevin Grosskopf and Dr. Robert Cox, for their discussions, suggestions, and encouragement duri ng the development of this dissertation. It is a great honor to have them serve on my committee. I am in debt to Dottie Beaupied for her tremendous helps, especially during the dissertati on submission process. I would also like to acknowledge the generous financial support from the University of Florida and the UF Alumni Associ ation, from which I will carry the Gator spirit for the rest of my life. I am in debt to Andrew Weiss, who has been the best mentor in my real estate profession, and has also provided generous he lp in data collection for this study. Besides him, I was working with an amazing team in Parmenter Realty Partners and especially thankful to Darryl Parmenter, Ed Miller, and Mark Reese, for their insightfu l advice on career choices and their tremendous helps at work. Special thanks go to all my folks when I wa s in UF, whose love and friendship became part of the happiest memory of my life. I am especially grateful to Yujiao Qiao, Yang Zhu, Hongyan Du, Dongluo Chen, Jon Anderson, and Hazar Dib, whose encouragements have me to complete this dissertation in time. PAGE 5 5 I want to extend a special word of thanks to all my mentors in the past, Dongshi Xu, Fuchang Lai, Shensheng Xu, Shouqing Wang, and Da vid Ling, whose wisdom and insight have profound influence on my character and personality. This work is dedicated to Lezhou Zhan, my husband and best friend, for his company throughout my life in good days and in bad ones; and to my beloved family: Lau Leung, my father; Sau Pik Fung, my mother; ShingChiu Leung, my little brother; and ShingPen Leung, my deceased brother. The honor goes to them, for their thirty years of nurture with endless love and care. PAGE 6 6 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........9 LIST OF FIGURES................................................................................................................ .......10 ABSTRACT....................................................................................................................... ............13 1 INTRODUCTION..................................................................................................................15 Background..................................................................................................................... ........15 Statement of Research Problem..............................................................................................16 Goal and Objectives............................................................................................................ ....18 Research Scope................................................................................................................. ......18 Significance and Contributions...............................................................................................19 Organization of Dissertation...................................................................................................19 2 REVIEW OF REAL ESTATE VALUATION.......................................................................20 Current Practice............................................................................................................... .......20 Distinguishing Acquisition and Development.................................................................20 Typical Acquisition Valuation Process...........................................................................21 Current Real Option Approach and Limitations.............................................................24 Decision Tree Analysis and Limitations.........................................................................25 Real Options in Real Estate....................................................................................................25 Theoretical Models..........................................................................................................25 Empirical Testing............................................................................................................31 The RERO Approaches..........................................................................................................31 Summary........................................................................................................................ .........32 3 LITERATURE REVIEW.......................................................................................................33 Traditional Discounted Ca sh Flow Approaches.....................................................................33 Capital Budgeting Theory.......................................................................................................34 Market Risk and Private Risk..........................................................................................34 Capital Asset Pricing Model............................................................................................34 Discount Rate..................................................................................................................35 Option Pricing Theory.......................................................................................................... ..36 Definition and Type of Options.......................................................................................36 BlackSholes Model and Stochastic Partial Differential Equations................................38 Lattices....................................................................................................................... .....42 Monte Carlo Simulation..................................................................................................45 Real Options Analysis Approaches........................................................................................46 Practical Real Options Model in Real Estate..........................................................................50 PAGE 7 7 Decision Tree Analysis......................................................................................................... ..53 Summary........................................................................................................................ .........54 4 METHODOLOGY.................................................................................................................55 RERO Modeling Procedures..................................................................................................55 Problem Framing.............................................................................................................55 Approach Selection.........................................................................................................57 Risk Drivers Identification and Estimation.....................................................................57 Base Case Modeling........................................................................................................57 Option Modeling.............................................................................................................58 Sensitivity Analyses........................................................................................................58 RERO Modeling Approaches.................................................................................................58 The Combined Approach................................................................................................59 The Separated Approach.................................................................................................61 RERO Modeling Techniques..................................................................................................63 Rational for Using Binomial Lattices..............................................................................63 Monte Carlo Simulation..................................................................................................64 Replicating Portfolio.......................................................................................................64 Binomial Lattice with Jump Process...............................................................................66 Investment with Private Uncertainty...............................................................................68 Summary........................................................................................................................ .........71 5 THE COMBINED APPROACH............................................................................................72 Case Description............................................................................................................... ......72 Building Valuation............................................................................................................. .....73 Problem Framing.............................................................................................................73 Approach Selection.........................................................................................................74 Base case NPV calculation..............................................................................................74 Risk Drivers Modeling....................................................................................................78 Option Modeling.............................................................................................................85 Sensitivity Analyses........................................................................................................91 Summary........................................................................................................................ .........95 6 THE SEPARATED APPROACH..........................................................................................96 Case Description............................................................................................................... ......96 Land Valuation................................................................................................................. ......97 Problem Framing.............................................................................................................97 Approach Selection.........................................................................................................98 Risk Drivers Identification and Estimation.....................................................................98 Base Case Modeling......................................................................................................103 Option Modeling...........................................................................................................103 Sensitivity Analyses......................................................................................................108 Summary........................................................................................................................ .......114 PAGE 8 8 7 CONCLUSIONS AND RECOMMENDATIONS...............................................................115 Conclusions.................................................................................................................... .......115 Recommendations for Future Research................................................................................116 LIST OF REFERENCES.............................................................................................................118 BIOGRAPHICAL SKETCH.......................................................................................................122 PAGE 9 9 LIST OF TABLES Table page 21 Comparison of Research Subjects, Model Variants, Contributions and Limitations........28 31 Type of Real Options....................................................................................................... ..47 32 Financial Options versus Real Options..............................................................................47 33 Correspondence between Fina ncial and Real Options.......................................................51 51 Major Assumptions for Argus...........................................................................................75 52 Correlation Between Random Variables...........................................................................87 53 Statistical Summary of M onte Carlo Simulation Result....................................................87 54 Event Tree Assumptions....................................................................................................88 55 Summary of Variable Effect on Option Value..................................................................91 61 Development Assumptions................................................................................................97 62 Probabilities of Jump Diff usion and Binomial Processes................................................107 PAGE 10 10 LIST OF FIGURES Figure page 21 Real estate phases and major factors to consider...............................................................22 22 Current acquisition valuation process................................................................................23 23 Real Options approaches for land valuation......................................................................27 31 Payoff of call option and put option..................................................................................37 32 Call option payoff example................................................................................................37 33 Call premium vs. security price.........................................................................................41 34 Stock and option price in a onestep binomial tree............................................................42 35 Stock and option prices in general twostep tree...............................................................44 36 Monte Carlo si mulation output..........................................................................................45 41 Critical steps in RERO analysis.........................................................................................56 42 Twostep binomial lattice w ith different dividend yields..................................................66 43 Binomial lattice with jump process....................................................................................68 44 Quadranomial lattice....................................................................................................... ...70 45 Decision analysis.......................................................................................................... .....70 51 211 Perimeter site plan.................................................................................................... ...73 52 Base case NPV calculation................................................................................................76 53 Spreadsheet model for Monte Carlo simulation................................................................79 54 Historical market and su bject property rental rates...........................................................80 55 Returns correlation between market and subject property.................................................80 56 Normal distribution fit for historical retu rns on rental.......................................................82 57 Historical market and subj ect property occupancy rates...................................................83 58 Occupancy changes correlat ion between the local real es tate market and the subject property....................................................................................................................... .......84 PAGE 11 11 59 Normal distribution fit for historical occupancy rates.......................................................84 510 Snap shot of Monte Ca rlo simulation assumptions............................................................86 511 Monte Carlo Simulation Result of Forecasting Variable z ................................................86 512 Normal distribution fi t of forecasting variable z ................................................................87 513 Event tree present value without flexibility.......................................................................89 514 Present value with flexibility............................................................................................ .90 515 Option value in relati on with present value.......................................................................92 516 Option value in relation with replacement cost.................................................................92 517 Option value in relation with present value and volatility.................................................93 518 Option value in relation with volatility and discount rate..................................................94 519 Option value in relation with replacement cost and volatility...........................................94 520 Option value in relation with pr esent value and replacement cost....................................95 61 Historical market average rent al rates and return volatility...............................................99 62 Normal distribution fit for hist orical market rental returns...............................................99 63 Gross rental rate move ment and probabilities.................................................................100 64 Building value movement and probabilities....................................................................101 65 Historical construction cost for highrise office building................................................102 66 Construction cost change rate and inflation rate..............................................................102 67 Development cost assumptions........................................................................................103 68 Payoff and probabilitie s without flexibility.....................................................................104 69 Payoff matrices for project values without flexibility.....................................................105 610 Decision payoff and probabilities with flexibility...........................................................106 611 Payoff matrices of projec t value with flexibility.............................................................107 612 Present value in re lation with rental rate and occupancy rate..........................................109 613 Option value in relation with re ntal rate and occupancy rate..........................................110 PAGE 12 12 614 Present value in rela tion with rental rate and development cost......................................110 615 Option value in relation with re ntal rate and development cost......................................111 616 Present value in relation with rental rate and Cap rate....................................................112 617 Option value in relation with rental rate and Cap rate.....................................................112 618 Option value in relation with re ntal rate and Cap rate in 3D...........................................113 619 Present value in relation w ith volatility and Cap rate......................................................113 620 Option value in relation w ith volatility and Cap rate.......................................................114 PAGE 13 13 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REAL OPTIONS FRAMEWORK FOR ACQUISITION OF REAL ESTATE PROPERTIES WITH EXCESSIVE LAND BY NgaNa Leung August 2007 Chair: Raymond Issa Major: Design, Construction, and Planning Our study touches a field that few researchers explore: the valuation model for acquisition of a property with excessive land that can be po tentially converted into a new development. Traditional valuation focuses mainly on th e building improvement. With the drastic capitalization rate compression, however, it beco mes critical to identify and explore any hidden value in an acquisition. One of such challenges is valuing a large partially vacant parcel that can be potentially converted into a new development. Valuation of these parcels is not straight forward. Traditional discounted cash flow approach (DCF) cannot take into account the uncertainty and development flexibility. Alternative approaches are real options analysis (ROA) and d ecision tree analysis (DTA). However, the twin asset assumption require d by the ROA methodology is often violated, especially for assets w ith private risk and rare events. The use of the same discount rate throughout valuation period in the DTA approach, regardless of changing risk characteristics upon the execution of decision making, allows for arbitrage opportunity. Our proposed real estate with real options (RERO) model is a framework that combines DCF, ROA and DTA analyses to sp ecifically value real estate acquisition with excessive infill PAGE 14 14 land. This methodology not only overcomes the shor tcoming of current DCF method, but also is superior to the pure ROA or DTA analysis. Focusing on applicability in practice, this framework is developed intuitively with simple mathematic s whenever possible. The study also explores a few unconventional real options cases, all of whic h could have been very complicated if modeled using the partial differential equations common in the academy, including (1) jump diffusion process that does not go back to normal diffus ion, (2) risk drivers that do not follow the multiplicative stochastic movement, (3) private ri sk that has no market equivalent and hence violating the nonarbitrage option pricing assu mption. All of these are implemented simply through binomial lattice with M onte Carlo simulation or DTA. The RERO framework is applied to a real case in Atlanta. Valuation has two parts: (1) the improvement is modeled using a combined appr oach with Monte Carlo simulation, and (2) the incremental value using a separated decision ap proach with binomial lattice technique. The valuation result is very close to the actual closing price. Three conclusions can be drawn from this study: (1) acquisition and development has different characteristics and deserv e different kinds of attention; (2) consideration of managerial flexibility can change investment decisions; a nd (3) many unconventional real option valuation problems can be resolved by binomial lattice and Monte Carlo simulations. The novelty of this study is th e research subject: property acq uisition with excessive land. From the methodology standing point, the RERO framework is developed with ease of applicability in mind. It bri dges the gap between research and practice for real options applications in the real estate industry. PAGE 15 15 CHAPTER 1 INTRODUCTION Background Our study touches a field that very few academicians have explored: the valuation model for acquisition of a property with excessive land that can potentia lly be converted into a new development. The three major schemes in real estate prope rty investment are acquisition, development, and operation. Acquisition is the ownership tran saction of land and improvement; development is the process of adding improvement to the land; and operation is the da ily management of the property. A majority of researchers focus on developm ent, perhaps due to its high uncertainty. Acquisition, on the other hand, has been ignored to a certain exte nt considering its volume and size of transactions. Acquisition has been rega rded as relatively low risk, since it is an investment on a touchable real property, whic h has historical operati ng track records, and numerous location attributes that last for decades and centuries. In recent years, however, r eal estate capitalization ra tes (defined by dividing the acquisition cost by annual net ope rating income) have compressed dramatically, meaning real estate is far more expensive to acquire than ev er before. It becomes critical to identify and explore any hidden value in an acquisiti on target in order to be competitive. The proposed acquisition model has two parts: firstly, valuation of the income producing part of the property, mainly the improvement; secondly, the incremental value, mainly the excessive land that, depending on the circumstance of where the property is located, may have no value or substantial upside value. PAGE 16 16 The proposed real estate with real options (RERO) model is a framework that combines real options and decision tree analyses. This methodology not only overcomes the shortcoming of the current discounted cash flow method, but also is superior to the existing real options or decision tree analysis. Focusi ng on applicability in practice, this framework is developed intuitively using simple mathematics whenever possible. The improvement is modeled using a consolidated approach with Monte Carlo simula tion, and the incremental value using a separated decision approach with binomial lattice technique. Statement of Research Problem The fundamental value of real estate is th e income producing capability of the property, which depends on many factors such as the amount of rental income to collect, the operating and financing expenses, the level of ri sk of the cash flow, the apprecia tion or depreciation of property value, and the performance of alternative investment instruments in the financial market. Acquisition valuation is the project ion of future earning capability of a property related to other alternative investments. Trad itional valuation mainly focuses on the building improvement. With the drastic capitalization rate compression, however, it beco mes critical to identify and explore any hidden value in an acquisition. One such challenge is valuing a large partially vacant parcel that can be potentially converted into a new development. The attachment of excessive land to a prope rty is not uncommon. Some developments were initially planned in phases, but the later phases were ne ver implemented due to economic downturn or undesirable outcome of earlier phases. The land pl anned for later project phases thus remains vacant for a long time. Some early developments were planned on large parcels to insure sufficient space of surface parking. When the region becomes well developed and the economy turns to be more favorable, the vacant land becomes valuable for dense urban infill. PAGE 17 17 Valuation of these parcels, however, is not as straightforward as a pplying the traditional Discounted Cash Flow (DCF) approach, which disc ounts expected future cas h flows at a certain discount rate to get the Net Present Value (N PV). In the case of infill land, without new development, all future cash flow will be 0; wi th certain assumptions of new development, it will generate a value. Intuitively, in a hot real es tate market where demand for developable land is high, such as in the South Florida, those parcels are extremely valuable. But in a warm or cold real estate market, the best use of such parcels may remain undeveloped until the market matures. The uncertainty and de velopment flexibility n eed to be taken into account. Whether or not the land would be developed, when, what type, and what size all matters during the property acquisition. Alternative approaches are Real Options Analysis (ROA) and Decision Tree Analysis (DTA). The ROA approach has evolved from the financial option pricing theory to value real assets. Put simply, by acquiring a property, the owner has the right, but not the obligation, to develop the excessive land to its fu ll use at a certain point of time in the future. Therefore, the value of a property with excess land should be higher than one without. The ROA methodology has been used to evaluate vacant land and to expl ain factors that affect development decisions. However, the ROA methodology requires one importa nt assumption, that st ochastic changes in the underlying value of the real asset to be devel oped are spanned by existing tradable assets or a dynamic portfolio of tradable assets the price of which is perfectly correlated with the real asset (Pindyck, 1991). This so called twin asset is hard to fi nd, especially for assets with private risk and rare events. Secondly, a lot of real opti ons are compound options, which are options on options, not simply on a single a sset, and consequently more co mplicated to solve by the pure option pricing methodology alone. PAGE 18 18 The DTA approach evolves from management science. It is a method to identify all alternative actions with respect to the possible random events in a hierarchical tree structure. The DTA approach is developed to handle intera ctions between random events and management decisions. However, a major limitation of the DT A method is its use of the same discount rate throughout the valuation period, regardless of cha nging risk characteristics upon the execution of decision making, and thus allows for arbitrag e opportunity (Copeland and Antikarov, 2005). Recent studies have turned to the combination of option pricing methodology, decision analysis, and game theory to solv e real options problems. An ideal new approach should be able to address the unique characteristics of acquisi tion valuation with in fill land, to handle the management flexibility, to take into account rare events such as new amenities driving up real estate value. It also need s to be intuitively simple for practical implementation. Goal and Objectives To overcome the above mentioned disadvantag es of the current DCF, ROA, and DTA methodologies, this study has developed a framewor k, namely the Real Estate with Real Option (RERO) framework, as a combination of all thr ee methods to specifically value real estate acquisition with excessive infill land. The objectives of this study are to: Develop a theoretical integrated framework to address real estate acquisition problems; Study factors affecting real estate acquisition and devel opment, as well as their characteristics and statistical distributions; Test and validate the model by applying it to real cases. Research Scope The research subject is real estate acquisition, which includes the value of the structural improvement, and the incremental value represen ted by excess developable land. The definition of excess land is that in addition to the portion necessarily attached to the existing structural PAGE 19 19 improvement; the excess portion that is large eno ugh for new development and at the same time meets local regulation requirements. Developmen t factors are outside of our scope. Potential users of the framework are real estate investors who need a tool to estimate the building value and the land value during propert y acquisition. The proposed valu ation model addresses mainly the economic risk and uncertainty for acquisition and development. Significance and Contributions The novelty of our study is the research subj ect: property acquisition with excessive land. To our knowledge, this is a fiel d that few researchers have addressed. From the methodology standing point, the RERO framework is developed with ease of applicability in mind. It bridges the gap between research and practice for real opti ons applications in the real estate industry. Organization of Dissertation In Chapter 2 we review the characteristics of real estate acquis ition, existing valuation approaches and their limitations, as well as what a new approach needs to achieve. In Chapter 3 we review the theory and technical details of th e different approaches currently available, in preparation for developing the proposed framew ork. We introduce the RERO framework in Chapter 4, including valuation procedures, the co mbined and separated approaches, and some new techniques developed to specifically apply to the case studies followed. Chapter 5 and 6 are case studies of the combined approach and sepa rated approach respectively. Collectively they illustrate how the RERO framework can be applied to a broad spectrum of scenarios in practice. In Chapter 7 we conclude the study and s uggest future research directions. PAGE 20 20 CHAPTER 2 REVIEW OF REAL ESTATE VALUATION This chapter discusses the curr ent practice in acquisition valu ation, alternative approaches and their limitations, followed by a review of real options in real estate. It also analyzes how the proposed RERO framework needs to resolve th e practical problems unique to real estate acquisitions. Current Practice Distinguishing Acquisition and Development Analogous to the financial market, the three ma jor schemes in the real estate investment market are different and interrelated: acquisi tion, development, and operation. Acquisition is similar to a lumpy investment in a well es tablished company with, in many cases, 100% ownership interest. Development is similar to the seeding of a startup company and bringing it to Initial Public Offering. Oper ation is the income producing pro cess in the daily management of the property. This explains why research on development pr oblems may not directly apply to acquisition valuation problems. A real estate investment fi rm may have a different agenda for the infill land than a real estate developer. The business of real estate development is to acquire and accumulate a considerable land bank, wait for appropriate timing and market demand to build new properties, and realize profit by selling the ne w properties to institutional investors. The business of commercial real estate investment on the other hand, is to acquire existing properties, manage and improve th e properties to receiv e the operating income from leasing. As an investment vehicle, commercial real estates tend to be traded more frequently than vacant land. As buildings get older and functionally obsolete, they usua lly change hands from passive institutional investors to activ e valueadded investors for cosm etic and functional upgrade and PAGE 21 21 tenantmix adjustment. The developers, however acquire land from different sources and wait more patiently in a real estate cycle before put ting up new products to ca pture the maximal gain. Short holding periods and different business inte rest makes the infill land less valuable to an investor than the vacant land to a developer. The major factors to consider during acquis ition are quite different from those in the development and operation processes (Figure 21). During acquisition, the major factors are location, market condition, mark et rent, pricing of the building and the land. Development factors, such as impact fee and school zoning, are outside the scope. If th e investor wins the bid, he goes through the due diligence and financing pro cess before actually plans for development of the vacant land. Although our model consists of the building value and the land value assuming possible development, it is by no means to substitu te for a detailed financial planning before the development breaks ground. Typical Acquisition Valuation Process A real estate investment company buys and manages properties to capture the cash flow from operation. Many of these companies specia lize in one or a few product types, such as office, retail, industrial, or re sidential properties. To evaluate a property with infill land, the management needs to answer the following questions: What is the building worth? What is the market demand for space? What is the likelihood that the company, after acquiring the pr operty, will put up new buildings? If the company does not intend to build new pr operties, what is the likelihood of the next buyer to put up new buildings? What type and size of development can add va lue to the land, and thus add value to the acquisition? PAGE 22 22 Figure 21. Real estate phases and major factors to consider. Acquisition Feasibility Study Due Diligence End Public Relationship Design Construction Regulatory Schedule, Cost, Quality Control Permits, Approvals Equity, Loan, Title, Physical condition Zoning, Density, Incentives, Impact Fee, School zone Appearance, Plans, Structure, Building system Location, Market, Rents, Pricing of Property and land Community, Environmental Operation Rent, Expense, Tenant Improvement; Leasing Finance? Develop? Win Bid? N o Yes Yes Yes N o N o Disposition Sale price, Taxes, Loan repayment, Equity distribution Operate? Yes N o Acquisition Development Operation PAGE 23 23 The typical decision process followed in curren t practice to acquire a property (an office building for example) with infill land is show n in Figure 22. First, th e building value and the land value are segregated. Build ing value is derived from the standard DCF projection. Depending on the investors perspective towards the market, the land could have no value or some value. In a weak demand region, the land probably does not generate any additional income besides parking, thus it has little or no value to the investor. In a strong demand region the investor conducts further inve stigation on the suitabl e product type to develop. If the best product type to develop is one that the investor is familiar with, say an office tower, the investor will further evaluate the projec t and land worth through a development model. If the best product type is not one the investor is familia r with, say a residential condominium or an industrial building, the in vestor probably hesitates to get i nvolved in the development alone. Figure 22. Current acquisit ion valuation process. Step 1: Segregating land value from building Step 2: Market demand analysis Step 3: Product type analysis Step 4: Assigning land value Step 5: Summing total value Potential Acquisition Building Value Land Value No Value Have Value To Build Not to Build Strong Demand Weak Demand Office Other Type Offer Price Land Value 0 Subjective Development Model PAGE 24 24 The investor might either find a development partner or consider selling off the land to such an interested party. In either cas e, for the acquisition purpose the investor will simply assign a subjective value to the land. The offer price c onsisting of the building and the land value is derived and submitted to the broker. Current Real Option Approach and Limitations In the ROA approach, by acquiring the propert y the investor not onl y receives all cash flows generated from leasing of the existing building, but also has the right, but not the obligation, to develop the vacant land to its full use at a certain point of time in the future. Therefore, the value of a property with infi ll land should be higher than one without. However, the current ROA models are not with out limitations. Firstly, valuation methods for vacant land may not be suitable for infill land due to their different characteristics in the following aspects: (1) the price of acquiring the land could be substantially lower; (2) the building type to be developed may be restricted by zoning regulation on cu rrent property; (3) the synergy effect could be substantial between th e proposed building and the existing building; (4) The surface parking is an inseparable part of the existing property. Secondly, a real estate investment firm has a different agenda for the infill land than a real estate developer. Short holding periods and diffe rent business interests make the infill land less valuable to an investor than to a developer. Thirdly, the current theoretical models are on a higher level to addre ss real estate as a whole, while investors need practic al models to address individual cases. The current theoretical models are on an aggregate level to explain real estate value in genera l. They have rigid restrictions, and can only be applied to the simp lest cases (Miller and Park, 2002). They also lack flexibility to change variables to model rea listic assumptions for practical use. Real assets PAGE 25 25 often possess unique location, physical and cont ractual characteristics, many of which are subjective and unquantifiable. Using the real option method alone may be insufficient. Last, the existing omnipotent real options models are mathematically correct but too complicated to be used. Trigeorgis (2005) and others have advocated approximate methods to simplify the calculation for practical applications. In summary, although the ROA approaches can overcome some of the drawbacks of DCF and provide better valuation for acquisition, the method itself is not fully developed to address the specific needs of acquisi tion valuation in practice. Decision Tree Analysis and Limitations Another available approach is the Decision Tr ee Analysis approach (DTA). DTA is a method to identify all alternativ e actions with respect to the possible random events in a hierarchical tree structure. It is developed to handle the inte raction between random events and management decisions. However, a major limitation of the DTA met hod is its use of the same discount rate throughout the valuation period, regardless of cha nging risk characteristics upon the execution of decision making, and thus allows for arbitrag e opportunity (Copeland and Antikarov, 2005). This means using DTA alone is not sufficient for the acquisition with infill land problem. Real Options in Real Estate Applications of ROA in the real estate i ndustry can be classified into the following categories: Vacant land for development, property redevelopment, and leasing (Ott, 2002). This section summarizes some theoretical mode ls as well as empirical studies. Theoretical Models Titman (1985) developed a simple binominal tree model to explain why a piece of land could be more valuable remaining vacant today and when is optimal to develop. This seminal PAGE 26 26 work is frequently cited in later papers, whic h all use Partial Differen tial Equations (PDE) and fall into two major categories by methodology: th e optimal development timing problem, and the game theoretical problem. The optimal timing problem is represented by Clarke and Reed (1988, optimal timing and density for resident ial development), Capozza and Helsley (1990, conversion from agricultural to urban land us e), Williams (1991, optimal timing and density to develop, optimal timing to abandon), and Geltner et al. (1996, two land use choice). The game theoretical problem is represented by Williams (1993, competition on simultaneous development), Grenadier (1996, competition on si multaneous or sequential development), and Childs et al. (2001, inefficient market with noi sy effect on value). Figure 23 shows the genealogical relationship among these models. Ta ble 21 itemizes the research subject, model variant, contributions and limitations of each study. Besides land valuation, there are two types of re al estate applications of the ROA that are closely related to our research: property rede velopment and operational research. Williams (1997), Childs at al. (1996), Cederborg a nd Ekeroth (2004) have researched on the redevelopment or renovation of real assets. They view existing bu ildings as assets that can be repetitively invested and improved, sometimes by changing functional attributes, e.g., switching from offices to apartments. Grenadier ( 1995, 2003), Adams, Booth and MacGregor (2001), Bellalah (2002), Grenadier and Wang (2005), Capozza and Sick (1991), among others have focused on options embedded in the commercial lease agreements, such as forward leases, escalation clauses, leases with options to renew or cancel, adjustable rate leases, purchase options, saleleasebacks, ground leases, etc. Acquisitions have not been thoroughly research ed using the real opt ions approach, though common in practice. As discussed earlier, ac quisitions with excessive land differ from ground PAGE 27 27 up development. They also differ from redevelopm ent, since they are not simple renovations of the existing buildings. They might include valuat ion of the leases as a source of cash flow for the potential development, but would require a much simpler valuation process on the leases. In summary, although acquisition valua tion is close to the three s ubjects mentioned above, the approach is significantly different. A new approach needs to be ab le to address both the building value and the land valu e, if any, for potential development. Figure 23. Real Options approaches for land valuation. Titman Clarke & Reed Cappozza & Helsley Williams Geltner et al. Williams Grenadier Child et al. 1985 1987 1990 1991 1993 1996 2001 Competition / Game Development Timing PAGE 28 28 Table 21. Comparison of research subjects, m odel variants, contributions and limitations. Author / title Subject description Model type & variant Contribution / limitation "Urban Land Prices under Uncertainty" (Titman, 1985) Explain why land is more valuable remaining vacant for future development: increased uncertainty leads to a decrease in current development activity. One time period binomial model assuming rents have two state values. Seminal work of ROA in real estate. Simple. Two policy implications: (1) Government incentives to stimulate construction activities may actually lead to a decrease if the extent and duration of the activity is uncertain. (2) Initiation of height restrictions may lead to an increase in development activity due to reduced uncertainty regarding the optimal height of the area. One time period model. Assume only two states, and that construction costs are certain. "A Stochastic Analysis of Land Development Timing and Property Valuation" (Clarke and Reed, 1987) Examine the qualitative effects of the different types of uncertainty on the timing and structural density of land development on residential projects. PDE to solve for optimal development timing and density assuming rents and development cost follows stochastic processes. Limited to residential development. Two limited assumptions: (1) new construction is small so that rents and development costs are uninfluenced by the newly added construction. However, in reality development is lumpy and will affect market rents and vacancy rate. (2) Efficient market in which all agents have equal information about the future probability distributions of rentals and costs. However, in reality real estate leasing and sales information is not as transparent as that in the stocks market, but more predictable, at least in a short run. PAGE 29 29 Table 21. Continued. Author / title Subject description Model type & variant Contribution / limitation "The Stochastic City" (Capozza and Helsley, 1990) Examine the land value of conversion from agricultural to urban use based on spatial characteristic of real estate such as distance or commuting time to the CBD. PDE model built on the traditional monocentric urban theory to study spatial implication of land conversion value, assuming household income, rents and land prices follow stochastic processes. Uncertainty (1) delays the conversion of land from agricultural to urban use, (2) imparts an option value to agricultural land, (3) causes land at the boundary to sell for more than its opportunity cost in other uses, and (4) reduces equilibrium city size. Does not explain very well land value in the emerging suburb economic centers. "Real Estate Development as an Option" (Williams, 1991) Optimal time to develop, optimal development density, and optimal time to abandon a project. PDE model to solve for optimal timing of abandoning a project, in addition to optimal development timing and density, assuming carrying cost, rents and development cost follows GBM, also assuming carrying cost is significantly high so that during some circumstance it is better to abandon the project than bearing the cost. Looks at the downside of a project: optimal time to abandon. This is a put option. Maximum feasible density is determined by zoning restrictions. Assumes perfectly competitive market and perpetual option. "Insights on the Effect of Aland Use Choice" (Geltner et al. 1996) Examine whether the multipleuse zoning add value to land by analyzing the land use choice between two different use types. PDE to solve for optimal choice between two land use types, assuming development cost, value of first land use, value of second land use follow stochastic processes. Land use type choice is a unique perspective in real estate. Assume construction unit cost is the same regardless of building type to be developed. PAGE 30 30 Table 21. Continued. Author / title Subject de scription Model type and variant Contribution/ limitation "Equilibrium and Options on Real Assets" (Williams, 1993) Examine industry equilibrium of optimal exercise policy under competition: the impact of competition erodes the value of the option to wait and leads to investment at very near zero net present value thresholds. PDE to solve for perfect Nash equilibrium with finite elasticity of demand and finite development capacities in a less than perfectly competitive environment. Among the first to consider the effect of competition. Exercising options to develop affects the aggregate supply of developed assets and market price, which preclude simultaneous exercise of the option among all developers. "The Strategic Exercise of Options: Development Cascades and Overbuilding in Real Estate Markets" (Grenadier, 1996) Explain why building booms in the face of declining demand and property values: fearing preemption by a competitor, developers proceed into a panic equilibrium in which all development occurs during a market downturn. Threestage model to explain real estate boomandbust cycle: valuation of land, construction lag, and "sticky vacancy" in operation Extend the Williams model from symmetric and simultaneous equilibrium to either simultaneous or sequential development, and allows for preemptive equilibria. Powerful to explain boomandbust markets such as real estate. Assume individual firms are identical and have all information. "Noise, Real Estate Markets, and Options on Real Assets: Theory" (Childs et al. 2001) Optimal valuation of noisy real asset in an incomplete information game PDE, assume optimal value include three terms: forward value estimate, historical value estimate, and the term that corrects for convexity effects due to incomplete information Extend to include the price lagging effect in real estate, where estimate value is different from market value, i.e., in a less than perfect market. PAGE 31 31 Empirical Testing A majority of the ROA empirical works in real estate has been in aggregate studies. Quigg (1993), Holland et al. (2000), Sivitanidou and Siv itanides (2000), Bulan et al. (2004) all use a large sample of real estate data to test the premium of land price over intrinsic value, whether irreversibility is an important factor for real estate investment, whether uncertainty delays construction, and whether competitions among deve lopers decrease the option value of waiting. As Bulan et al. (2004) point out, however, sin ce real options models apply to individual investment projects and predict that trigger prices are nonlinear, aggregate investment studies may obscure these relationships. Mo reover, these empirical tests are limited to qualitative results, such as whether each variable in the ROA model has positive or negative effect on the overall option value. Few of the ROA empirical works has focused on individual case studies and its implication in practice. The RERO Approaches The RERO framework attempts to move beyond th e realm of academic interest to be used quantitatively in practical problems of acquisi tion valuation, development decision making, and land policy analysis. The approach should be able to address the unique characteristics of acquisition valuation with infill land, to handle the management fl exibility, to take into account rare events such as new amenities driving up real es tate value. This calls for the combination of DCF, ROA and DTA methodologies. It also needs to be intuitively simple for practical implementation. To achieve this goal, the problem is divided into two subproblems: (1) valuation of the building structure and (2) valuation of the in fill land. Valuation of the building structure represents a normal case of acquisition. On the other hand, valuation of th e infill land represents PAGE 32 32 the extra value stemmed from creative management i.e., the ability to uncover the hidden value in real estate and realize it through active development. Real estate valuation is an art and scien ce. The RERO framework is not built on rigid reasoning and restricted assumptions to be precise, rather it is developed as a tool to solve a broad spectrum of practical r eal options problems. Specificall y, it explores a few unconventional real option cases, including (1) jump diffusion pr ocess that does not go back to normal diffusion, (2) risk drivers that do not follow the multiplicati ve stochastic movement, (3) private risk that has no market equivalent and hence violating the nonarbitrag e option pricing assumption. The mathematical models for these kinds of unconven tional problems could be very complicated, if written in PDE equations. To facilitate pract ical implementation, the RERO framework applies the binomial lattice with Monte Carlo simulati ons and decision analysis method. The RERO framework is a simple yet powerful tool, intuitive to the practitioners, yet mathematically correct and precise. Summary This chapter compares the difference between real estate acquis ition and development, reviews current practice of real estate acquis ition valuation, discusses the three alternative valuation approaches, DCF, ROA, DTA and their limitations. Built on the strengths of these three approaches, the RERO framework needs to address practical problems of acquisition valuation, development decision making, and land policy analysis. The next few chapters explore modeling details of how this concept should be implemented. PAGE 33 33 CHAPTER 3 LITERATURE REVIEW In Chapter 2 several different valuation me thodologies were discussed conceptually: the Discounted Cash Flow approaches (DCF), the Real Option Analysis approaches (ROA), the Decision Tree Analysis approaches (DTA), and the proposed Real Estate with Real Option approaches (RERO). In this chapter the technical modeling details of the first three approaches, as well as the capital budgeting theory in finan ce will be discussed. The RERO approaches that built on the existing three will be discussed in Chapter 4. Traditional Discounted Cash Flow Approaches The Discounted Cash Flow (DCF) approaches include payback period, Internal Rate of Return (IRR), Net Present Value (NPV), and other fo rms such as Adjust Present Value. In this study DCF refers to the NPV method alone. The principle of the NPV me thod is to discount all projected free cash flow back to year 0, to get th e net present value of the project (Equation 31). The NPV must be greater than 0, or the IRR must be greater than the companys hurdle rate, in order to justify the investment (Mun, 2002). If NPV is greater than 0, the pr oject is regarded as optimal to be executed immediately. n i i ik F NPV0) 1 ( (31) where NPV is the net present value of the project at Year 0, Fi is the projected free cash flow (including income, cost and terminal value) in year i k is the project discount rate. The DCF method is suitable to evaluate projec ts that are well structured, with predictable future cash flows. For projects involve large uncertainty of timing, cost and cash flows, such as a real estate development, usi ng the DCF approaches are difficult in the following three aspects (Miller and Park 2002; Feinstei n and Lander 2002): firstly, sele cting a fixed and appropriate PAGE 34 34 discount rate; secondly, taking into account new information and ch anging the plan accordingly; thirdly, determining the optimal timing to carry out the project. Capital Budgeting Theory In the DCF approach and in all other approaches one of the most influential factors is the discount rate to be used. To better understand discount rate, a brief di scussion of the capital budgeting will follow. Market Risk and Private Risk Stocks are risky. For any i ndividual stock, however, a larg e part of its risk can be eliminated by holding it in a large welldiversified portfolio. A portfolio consisting of all stocks is called a market portfolio. In reality, it can be approxi mated by a large amount of welldiversified stocks. The part of th e risk of a stock that can be elim inated is called private risk, or diversifiable risk; while the part th at cannot be eliminated is called market risk, or systematic risk (Brigham et al. 1999, p178). The Capital Asset Pricing Model (CAPM) indicates that the relevant riskiness of any individu al stock is its contri bution to the riskiness of a welldiversified portfolio, or the market risk porti on only, which is measured by its coefficient. Capital Asset Pricing Model If the market portfolio m is efficient, the required return rs of any stock i is the riskfree interest rate r plus a risk premium, as shown in Equation 32. ) ( r r r rm i s (32) Where r is the riskfree return, mr is the expected market return, 2m im i where im is the covariance between the stock and the market, and 2 m is the variance of the market portfolio. PAGE 35 35 i is an important variable to measure the risk characteristics of the stock i If i is greater than 1, the stock is more volatile th an the average stock market; and if i is less than 1, the stock is less volatile than the average stock market. The more volatile a stock is, the more risky it is, and consequently the higher the requi red return needs to be in order to justify the risk an investor takes. Discount Rate A firms hurdle rate is usually its Weighted Average Cost of Capital (WACC). A large real estate investment firm is usually formed as a Real Estate Investment Trust (REIT), which does not pay income taxes, so long as 95% of its income from operation is distributed to the investors on an annual basis. The WACC k of a REIT is calculated by Equation 33. V D r V S r kd s (33) where rs and rd are the cost of equity and debt respectively, S, D and V are the market values of equity, debt, and total asset respectively; S + D = V. Equation 33 can also be used to value an i nvestment project, as if every project was a separate mini company. However, it is difficult to determine the cost of equity and debt for a project, since the equity of a startup project, for example, ma y not be publicly traded, and the risk characteristics of a projec t are quite different than that of the company as a whole. The capital budgeting theory indicates that fi nding the right discount rate is extremely difficult, if not impossible. Since every company has different risk charac teristics, the required discount rate is different from company to co mpany. Also every project within the same company has different risk characteristics, a nd the correct discount ra te required to value a project may not be the same as the company s WACC. This makes both the DCF and the DTA PAGE 36 36 approached difficult to value infill land with development potential, although for an existing building with operating history the DCF and DTA approaches may work fine. Option pricing theory, on the other hand, does not rely on the risk characteristics of a particular firm or project. Neith er does it rely on the risk preferen ce of an individual investor. It is discounted at the riskfree interest rate r. The reason is that private risk is alleviated through portfolio diversification and market risk can be diminished through the options replicating portfolio (Miller 2002). For deve lopment project that involves a lot of uncertainty, this is a huge benefit over the traditional DCF method. Option Pricing Theory Definition and Type of Options An option gives the holder the right but not the obligation to do something (Hull, 2006). In the financial market, there are two basic types of options: call options and put options. A call option gives the holder the right to buy the underlying asset by a certain date for a certain price. A put option gives the holder the right to sell the underl ying asset by a certain date for a certain price (Figure 31). Based on exer cise dates, options can be cla ssified into two major types: American options can be exercised at any time up to the expiration date. European options can be exercised only on the expiration date. Mo st options are of the American type. The value of a financial option is determined by the current price of the underlining asset S0, the strike price at maturity date K, the riskfree interest rate r, maturity date T, return volatility of the underlining asset and sometimes the dividends e xpected during the life of the option (Hull, 2006). Returns on options are asymme tric, i.e., options will only be exercised to the benefit of the holders. For example, if a holder of a call option can buy the stock 3 months later for $100 per share, and if the spot pri ce at maturity becomes $120 per share, he will exercise this option, then sell the stock immediately, and earn $20 per share. However, if the spot PAGE 37 37 price becomes $83 per share at maturity, he can let the option expire without exercised, thus avoid losing $17 per share. He only losses the pr emium initially paid for the option (Figure 32). His payoff is the difference between the spot price at maturity St and the exercise price K, or 0, whichever is greater (Equation 34). ) 0 (K S Maxt (34) Figure 31. Payoff of call option and put option. Figure 32. Call option payoff example. Option pricing theory is to determine what pr emium, or option price, a holder should pay for such flexibility. The types of option pric ing methodology include co ntinuousand discretetime models (Miller and Park, 2002; Lander a nd Pinches, 1998). Continuoustime models 0 Payoff Stock Price K Call Option Premium 0 Payoff Stock Price K Put Option Premium 0 Payoff Stock Price K=$100 Call Option Example S=$120 S=$83 100 120 20 83 0 Example Payoff PAGE 38 38 include closedform equations and stochastic partial differential e quations. Discretetime models are mostly lattice models and Monte Carlo simulation. BlackSholes Model and Stochastic Partial Differential Equations The most famous closedform equation is the BlackScholes model, although it can only be used to price European options. The BlackSc holes (1973) pricing formula is developed under the following ideal assumptions: stock price change follows the Wiener process, distribution of return is lognormal, efficient market, constant shortterm interest rate no dividend payment, no transaction costs, and short se lling is possible. A Wiener pr ocess, also called a Geometric Brownian Motion (GBM), is a rando m process with a mean change of 0 and a variance rate of 1. The values of dz for any two different short intervals of time dt, are independent (Equation 35). dt dz (35) Where has a standardized normal distribution ) 1 0 ( and ) ( denotes a probability distribution that is normally distributed with mean and standard deviation A generalized Wiener process for a variable S can be defined by Equation 36. Sdz Sdt dS (36) where S is the underlying asset whose value change follows the Wiener process; dS is the change of value S during an infinitesimal time interval dt Ito's Lemma (Hull, 2006, p273) is a theorem of stochastic calculus th at shows second order differential terms of a Wiener Process can be consid ered to be deterministic when integrated over a nonzero time period. Since the stock price S follows the Wiener process, an option f (be it a call option or a put option) contingent on S follows the Itos Lemma (Equation 37). Sdz S f dt S S f t f S S f df ) 2 1 (2 2 2 2 (37) PAGE 39 39 The principle of option pricing methodology is to construct a riskless portfolio to prevent arbitrage. This portfolio is short one option and long S f shares of the underlying stock. When the stock price S changes, the S f shares must change accordingly. Later from Equation 310 we will see this portfolio is riskless because it does not involves dz over the time interval dt The portfolio is written as Equation 38. S S f f (38) During the time interval dt, the change in value of the portfolio is represented in Equation 39. dS S f df d (39) Substitute dS from Equation 36 and df from Equation 37 into Equation 39, dt S S f t f d) 2 1 (2 2 2 2 (310) To prevent arbitrage, the portf olio earns riskfree interest r during the time interval dt. dt r d (311) From Equation 38, Equation 310 and Equation 311, we have dt S S f f r dt S S f t f) ( ) 2 1 (2 2 2 2 Which simplifies to rf S S f S S f t f 2 2 2 22 1 (312) Equation (312) is the BlackS choles partial differential e quation. Subjected to the following boundary conditions: ) 0 (K S Max f when t = T in the case of a call option, and PAGE 40 40 ) 0 (S K Max f when t = T in the case of a put option. Integrating Equation 312, the BlackScholes fo rmula can be written as Equations 313 and 314 (Black and Schol es, 1973; Hull, 2006). ) ( ) (2 1 0d N Ke d N S crT (313) ) ( ) (1 0 2d N S d N Ke prT (314) where T T r K S d ) 2 / ( ) / ln(2 0 1 ; T d T T r K S d 1 2 0 2) 2 / ( ) / ln( ; c is the value of a European call option; p is the value of a European put option; S0 is the current price of the underlying asset; K is the strike price of the option at maturity; r is the riskfree interest rate; T is the time to maturity; N() is the cumulative standard normal distribution function. The BlackScholes model can be divide d into two parts: The first part, S0N(d1) derives the expected benefit from acquiring a stock right now. This is found by multiplying stock price S0 by the change in the call premiu m with respect to a change in the underlying stock price N(d1) The second part of the model, KerTN(d2) gives the present va lue of paying the exercise price on the expiration day. The fair market value of the cal l option is then calculate d by taking the difference between these two parts. The boundary condition of a call option is best depi cted in Figure 33. The solid black line defines the call option value. The green line with square markers defines the maximum value of the option. For nonarbitrage, the option should never be wort h more than the stock price S otherwise an arbitrageu r can easily make a riskless profit by buying the stock and selling the call option. The blue line with tr iangle markers defines the minimum value of the option. The call PAGE 41 41 option should be worth more than ) 0 (0rTKe S Max otherwise an arbi trageur can buy an option, short sell a share of stock, invest the surplus at riskfree interest rate and earn a profit. The possible option values fall in the region defi ned by the green line and the blue line and vary depending on the underlying stock volatility, option time to maturity, and riskfree interest rate. Call Option vs. Stock Price 0 30 60 90 120 150 0306090120150 Stock PriceCall Optio n Lower Bound Option Value Upper Bound Figure 33. Call premium vs. security price. Though the BlackScholes prici ng model has a lot of restri ctions and can only value European options, there are a lot of stochastic partial differential equations with boundary conditions that relax some restrict ions to a certain extent and can be used to value more specific questions. The benefits of these analytic contin uoustime models are that they are flexible to model different circumstances, and mathemati cally accurate (Mille r and Park, 2002). The PAGE 42 42 drawback is that the modeli ng requires sophisticated mathem atical knowledge, sometimes the solution does not exist, and even if it does, the process itself could become as complicated as a blackbox for the practitioners to co mprehend (Lander and Pinches, 1998). In the case when analytical solutions to the stochastic differential equations do not exist, they must be solved numerically by using finite difference methods, or Monte Carlo simulations (Miller and Park, 2002). Lattices Lattices are a type of discrete time model, which includes binomial tree, trinomial tree, quadranomial tree, and other multinomial models. Lattices are the approximation of the continuous models. The results of these two methods are very close when the time interval is infinitely small. The most commonly used binomial lattice wa s developed by Cox et al. (1979), in which values of the underlying asset are assumed to fo llow a multiplicative binomial distribution. The model assumes the up and down parameters u and d the volatility of the underlying asset and riskneutral probabilities p and 1 p are constant (Figure 34). Figure 34. Stock and option pri ce in a onestep binomial tree. An option f (be it a call option or a put one) is va lued by constructing a riskless portfolio of a long position in shares of stock and a short pos ition in 1 option (Equation 315). S0 f S0d fd S0u fu p 1 p PAGE 43 43 f S 0 (315) In an up movement of the stock pric e, the value of the portfolio is u uf u S 0 In a down movement of the stock pric e, the value of the portfolio is d df d S 0 The two are equal when d uf d S f u S 0 0 or when ) (0d u S f fd u (316) The portfolio is riskless and must earn the riskfree interest rate r The present value of the portfolio is repres ented by Equation 317. rT ue f u S ) (0 (317) From Equation 315 and Equation 317, we have rT ue f u S f S ) (0 0 (318) Substitute from Equation 316 into Equation 318, ] ) 1 ( [d u rTf p pf e f (319) where d u d e prT Te u u d1 Equation 319 is a onestep binomial model, wh ich can be generalized to twostep and multistep models. Figure 35 shows a twostep binominal lattice. During each time step, the PAGE 44 44 stock value either moves up to u or down to d of its previous value. Option value is derived by working backward from fuu and fud to calculate fu, from fud and fdd to calculate fd, then from fu and fd to calculate f (Equations 320, 321 and 322). Figure 35. Stock and option pri ces in general twostep tree. ] ) 1 ( [ud uu rt uf p pf e f (320) ] ) 1 ( [dd ud rt df p pf e f (321) ] ) 1 ( [d u rtf p pf e f (322) Substituting from Equation 320 and Equation 321 into Equation 322, we get ] ) 1 ( ) 1 ( 2 [2 2 2 dd ud uu rtf p f p p f p e f (323) where d u d e prt te u u d1 In general, for a binomial lattice with n steps, the ith step (n i 0) option value is calculated by Equation 324. S0 f S0d fd S0u fu S0ud fud S0uu fuu S0dd fdd PAGE 45 45 ] ) 1 ( [, 1 1 d i u i rt if p pf e f (324) Lattice, though still complicated, is more intuit ive to the practitioners than continuous time models. It is especially useful to evaluate American options, si nce analytic solutions are almost nonexisting in the continuous models. The drawback is that using lattice by itself is hard to model compound options. However, combined with DT A, lattice is capable to deal with a lot of complicated situations, even more flex ible than PDEs in many circumstances. Monte Carlo Simulation Originally named after the casinos in Mont e Carlo, Monaco, Mont e Carlo simulation is about games of chance. It is now widely used to simulate stochastic processes by sampling large quantity of random outcomes for the processes (F igure 36). Because of the repetition of algorithms and the large number of calcula tions involved, Monte Carlo simulation is computationally complex, yet easy to model and understand. Figure 36. Monte Ca rlo simulation output. PAGE 46 46 In real options modeling, Monte Carlo simu lation can be used where there are several underlying variables. The drawback is that it is difficult to work backward to determine option exercise strategy, since Monte Ca rlo simulation is forward looking. In the RERO model, it is used as an intermediate step to estimate volatilit y of the project stems from multiple risk drivers. Real Options Analysis Approaches First coined by Myers (1977), the ROA approach es are to apply financial option pricing theory and methodology to evaluate real assets (Miller and Park, 2002; Trigeorgis, 2005). In the financial market, a derivative is a security whos e value changes depend on the value changes of some other underlying assets. In real asset valua tion, the value of a project can be viewed as a derivative contingent upon input costs, output yield, time and uncertainty (Miller and Park, 2002), and therefore can be evaluated by applyi ng the financial option pricing principles. By using ROA, investment deci sions are viewed as real options or combinations of real options, such as options to defer, expand, switc h, contract, or abandon, as shown in Table 31 (Trigeorgis, 1996; Yao and Jaafari, 2003). Also included in the ta ble are examples in the real estate and construction industry. Contrary to DCF method, in th e ROA context greater volatility is not always worse, since losses are limited to the initial investment, or option premium, but the option holder can capture greater ups wings if things turn out to be favorable. ROA is applied most commonly in the industries of natural re source, manufacturing, energy, research and development, startup companies, and others (Lander and Pinches, 1998; Trigeorgis, 1996). Applications in the real estate and cons truction industries are still limited. Although ROA borrows the option pricing theory, the distinguish charac teristics of real assets demand different valuation assumptions and methodologies from direct applications of the option pricing theory without any modification. Table 32 lists the major differences between financial options and real options (Mun, 2002). PAGE 47 47 Table 31. Types of real options. Options Features Examples Defer To postpone constructio n till optimal timing Time to develop Stage To create a series of stages to allow for abandonment or expansion in later stages depending on outcomes of earlier stages Phased development Contract To contract the project to a third party in order to mitigate risk or to speed up market domination Franchise stores Expand To expand the project scale in favorable market conditions Airport expansion Abandon To abandon the project and prevent severe lost in unfavorable market conditions Bankruptcy of a project entity Switch input/output To change the output mix or input mix in response to changing market demand Coalfired vs. gasfired power plants Compound Option on option, where the value of an earlier option can be affected by the value of later options. Most real world options are of this kind Case study in Chapter 5 and 6 Table 32. Comparison between Fi nancial Options and Real Options. Characteristics Financial options Real options Maturity Short, usually in months Long, usually in years Underlying asset Traded stocks, with comparables and pricing information Not traded project free cash flow, proprietary in nature, with no explicit market comparables Management manipulation Value does not change due to individual management assumptions or actions Value has to do with individual management assumptions and actions Competition and market effect Irrelevant to pricing Direct drivers of value One of the major differences between financia l options and real opti ons is how to handle private risk. The underlying assets of financial options are traded market assets, and market risk is the major source of risk among all financial op tions. Private risk can be treated simply as errors. The underlying assets of real options, however, are usually nontraded assets that do not PAGE 48 48 have market equivalent. Private risks cannot be hedged. The other difference is the effect of management and competition. Financial opti ons on the same underlying asset and the same maturity date are identical. They are widely he ld to be market efficient. A single transaction usually does not affect the pricing of financial options, neithe r does management or competition. Real options, on the other hand, are lumpy or oneofthekind in nature. Exer cise of real options by management can have profound imp act on the underlying asset value. Consequently, there are a lot of debates in the academic world about how real options should be correctly priced. Borison (2005) classi fied existing real opti ons approaches into 5 categories: The classic approach, The subjective approach, The Market Asset Disclaimer approach, The revised classic approach, and The integrated approach. Borison also discussed the underl ying assumptions of these ap proaches, the conditions that are appropriate for their applications, and the mechanics in applying them. The classic approach assumes that the capital market is complete, and an identical twin asset or portfolio exits for every real asset under evaluation. It ma kes explicit use of noarbitrage argument, and applies directly the BlackShores formula. The subjective approach also assumes that th e capital market is complete. However, it relies on subjective judgment for input, as opposed to data from traded markets. This makes it an inconsistent approach, and limits to qualitative result. The Market Asset Disclaimer (MAD) approach assumes that the capital market is not complete. It relies on the estimate value of the a sset without flexibility as the twin asset for the purpose of calculating the option va lue of the flexibility. Data is drawn from traded markets PAGE 49 49 when available, and subjective judgment when not. Proponents of this approach justified this step explicitly: the same, weaker assumptions that are used to justify the applications of DCF can be used to justify the applica tions of option pricing to flexib le corporate investment (Copeland and Antikarov, 2001). The revised classic approach assumes that the capital market is partially complete. It attempts to divide the world into black and white: For investments that have market equivalents, it applies the classic approach using market da ta; for investments that do not have market equivalents, it applies decision anal ysis using subjective judgment. The integrated approach also assumes that the capital market is partially complete. However, it uses capital market data for market risk and subjec tive judgment for private risk in an integrated model. The major difference among these approaches is how private risk is handled. The classic approach ignores private risk completely and treats real options exactly like financial options that all risks can be diversif ied away by constructing a hypothetical traded twin asset or portfolio. The subjective approach handles private ri sk by substituting market data by subjective assessment. The revised classic approach admits th e limitations of direct applications of option pricing theory to real options analyses and classifies invest ments into those either dominated by market risk or by private risk. It applies the option pricing mo del only to investments dominated by market risk, and applies decision analysis to those dominated by privat e risk. Although it is a better approach than the previous two, the revised classic appro ach forces all investments into black or white, and implements tw o totally different approaches. The MAD approach, on the other hand, admits th e difficulty of handli ng private risk, thus does not rely on the existence of a traded replica ting portfolio. Instead, it uses the project value PAGE 50 50 itself without flexibility as the twin security, as if it were traded in the financial market. After all, the best correlation with the project is the project itsel f (Copeland and Antikarov, 2001). Trigeorgis (1996) also argued that the assump tions underlying the DCF approach are traded assets of comparable risk (same beta), and MAD assumptions are no stronger than those of DCF. Contrary to Borisons unders tanding, Copeland and Antikar ov (2005) clarified that the MAD approach does not blindly use all subjectiv e assumptions. Similar to the integrated approach, MAD also uses traded market data whenever available, and uses subjective assumptions only when market estimates are impo ssible. The MAD approach and the integrated approach are considered to treat private risk in the same way, the difference remains only technical: MAD relies on simulations to evaluate project volatility, and attempts to combine all risks into one variable, whenever possible; wh ile the integrated approach relies on utility functions, and models market risks and private risks explicitly and separately. Neither is superior to the other, and the selection of approaches depend s on project characteristics on a casebycase basis. For this reason, the pr oposed RERO approaches are built on the MAD and the integrated approaches. Practical Real Options Model in Real Estate Ghosh and Sirmans (1999) were among the first to address the applicatio ns of real options to the corporate real estate pr actitioners, by developing a lookup table for the options value, which is derived from an approximation of the BlackScholes formula. They used the correspondence in Table 33 between financial a nd real options in orde r to apply the BlackScholes formula directly to real options. However, they did not explain whether the time value of money r is a riskfree rate or riskadjusted discount rate, nor how the risk of project cash flows is determined. PAGE 51 51 Table 33. Correspondence between Financial and Real Options. Variable Financial options Real options S0 Stock price Present value of proj ects expected cash flows K Exercise/strike price Cost of investment T Time to expiry Length of time the decision can be deferred r Riskfree rate Time value of money Standard deviation of stock returns Risk of project cash flows They also developed a threestep appr oach to calculate the option value: Step 1: Calculate NPVq from Equation 325. T qr K S NPV ) 1 /(0 (325) Step 2: Calculate T Step 3: Read the value of th e call option as a percentage of the value of the underlying asset from the table. For example, if the stock price S is $100, strike price K is $100, time to expiry T is 1 year, time value of money r is 5%, standard deviation of annual return is 20%, then 05 1 ] ) 05 1 /( 100 /[ 100 ] ) 1 /( /[1 T qr K S NPV 20 0 1 20 0 T From the lookup table, C is 10.4% of the asset value, 40 10 $ 100 104 0 C. They did not specify how the lookup table is computed, but by comparing the BlackScholes formula and their threestep approach, it is not difficult to find that they did some approximations in order to simplify the calculation. From the BlackScholes fo rmula of Equation 313, PAGE 52 52 ) ( ) (2 0 1 0d N S Ke d N S CrT (326) where T T r K S d ) 2 / ( ) / ln(2 0 1 ; T d T T r K S d 1 2 0 2) 2 / ( ) / ln( Tr K ) 1 ( is an approximation of rTKe, and Tr K S) 1 /(0 can substitute K S0, T r ) 2 / (2 is ignored due to the low impact on the overall value. With the approximation and substituting Equation 325 into Equation 326, we have T NPV N NPV S Cq q) ln( 1 10 (327) Equation 327 is the formula to develop the lookup table. The Ghosh and Sirmans model falls into the subjective approach cat egory of Borisons classification (Borison, 2005). As discussed in the previous s ection, this approach uses subjective assessment of variables without justification of its appropriateness. At a first glance, this approach is intuitive, especially for practitioners who are co mfortable with NPV but unfamiliar with ROA. However, this direct application of the BlackScholes model is not without its limitations. Firstly, it is restricted to European options, where timing of execution of the option is perfectly known in advance. Secondly, it assumes future cash flow is as deterministic as in the traditional NPV method, and allows for only one scenario analysis. It does not allow for stochastic and dynamic changes of the underlyi ng variables, such as development cost and rental rate, does not solve for optimal developm ent timing. Lastly, while there is a tradeoff between simplicity and accuracy the value derived from the lookup table has 10% variance PAGE 53 53 from that calculated from the BlackScholes model, which is deemed inaccurate in many circumstance. In summary, the model developed by Ghosh and Sirmans is a good attempt to build the understanding of management flexibility value of corporate real estate in practice, however, it lacks accuracy and depth of applicability in the real es tate industry, which is what this study plans to overcome. Decision Tree Analysis First coined by Howard (1964, in Ng and Bjornsson, 2004), decision analysis is the discipline comprising the philosophy, theory, me thodology, and practice necessary to address important decisions. Graphical representation of decision an alysis problems commonly use influence diagrams and decision trees. DTA is a method to identify all a lternative actions with respect to the possibl e random events in a hierarchical tree structure. It is developed to handle the interaction between random events and manage ment decisions. Uncertainties are represented through probabilities and distributio ns. The attitude of a decision maker to risk is represented by utility functions. Unlike the DCF approaches, there are no objecti vely correct DTA models. An appropriate model depends on the preferences a nd beliefs of the decision maker and hence is subjective. A decision analysis includes the following typical step s: first, defining the scope of the analysis; second, setting up a decision basis, including gene rating alternatives, collecting information, and estimating risk preference; third, constructing a decision tree with decision and uncertainty nodes; and forth, analyzing sensitivity of factors that have the largest effects (Ng and Bjornsson, 2004). Decision analytic methods are used in a wi de variety of fields, including business, environmental remediation, health care research and management, energy exploration, litigation and dispute resolution, etc. PAGE 54 54 DTA relies on subjective assessment of probabil ities and distributions. This method alone cannot prevent arbitrage opportunity. Howeve r, the combination of ROA and DTA can eliminate the shortcoming of both, and creates a much better approach. Summary In this chapter we reviews modeling details of the DCF, ROA, DTA approaches, as well as capital budgeting theory, ROA applica tions in real estate. Treatmen t of private risk differentiates these approaches from one another. In ROA methodologies alone, there are various approaches advocated and debated in the academ ic community. Due to the char acteristics of real options, it is inappropriate and inaccurate to directly apply the option pricing formula without any modification. The correct real option methods must be able to handle private risk as well as market risk in a consistent way. Only the M AD and the integrated approaches are considered appropriate and are subject to further use. PAGE 55 55 CHAPTER 4 METHODOLOGY The RERO framework consists of two approach es to value real estate acquisitions: the combined approach and the separated approach. This chapter introduces the key elements and steps of the RERO approaches. The next two ch apters present case studies that implement the principles introduced in this chapter. As mentioned in the previous chapter, the Market Asset Disclaimer (MAD) and the integrated approaches in ROA we re adopted for this study. RERO Modeling Procedures The RERO framework adopts real options and d ecision analysis methodologies. It consists of a series of processes to solve a decision tree backward. The event tree starts by laying out all possible events and corresponding cash flows. St arting at the end of the analysis, we work backward through the tree at ea ch decision node to calculate th e payoff of all possible actions, using replicating portfolio or ri sk neutral discounting, choosing the optimal action that generates the highest payoff at each node. Eventually the possible cash flows generated by these future events and actions are folded back to a presen t value. The following 6 steps are critical in performing the RERO analysis (Figure 41): Problem framing; Approach selection; Risk drivers identification and estimation; Base case modeling; Option modeling; and Sensitivity analyses. Problem Framing For real estate acquisition, the first task is to review the case qualitatively, and to determine whether the asset itself is a sound investment. An investme nt that seems good by the numbers PAGE 56 56 may not necessarily turn out to be a good i nvestment in the end. Location, neighborhood development, economy growth, property visibili ty, accessibility, physical conditions, ownership and occupancy history, management capability, all these are unique characteristics of real estate that are nonquantifiable. Comprehensive local bu siness knowledge and experience is needed to determine whether a piece of land is worth acquiring. Figure 41. Critical steps in RERO analysis. Problem Framing Approach Selection Risk Drivers Identification Base Case Modeling Sensitivity Analysis Option Modeling PAGE 57 57 After this critical screening, if a prope rty is good enough to go through the hassle of quantitative analysis, the problem is framed into a model and the story is told in a mathematical way. The goal becomes how much it is worth. Management flexibility a nd strategic options, if any, should be identified to de termine which approach to use. Approach Selection DCF can solve most simple and conventional acqui sition problems. It is only when a case has strategic options that cannot be valued by DCF should the RERO approaches be used. Depending on the characteristics of a project, the first step is to determine whether to use the combined approach or the separated approach. The differences between the two approaches are discussed in later sections. Risk Drivers Identification and Estimation The next step is to identify the risk drivers. Uncertainties of real estate acquisitions and development include rental income, operating cost s, capital expend iture, discount rate, cap rate, development cost, etc. These vari ables flow through the model to affect the project value. Risk drivers are those key variables that have th e most profound impact on project value change. To estimate the volatility of each risk driver, objective methods such as time series forecast or regression analysis should be used, if historical or comparable data exists. Alternatively, subjective methods may be used, such as subjec tive guesses, growth rate assumptions, expert opinions, etc (Mun, 2002). Base Case Modeling The expected project value without flexibility is the base case for the subsequent option value analysis. The base case value acts as the t win asset that the real option approach is based on. PAGE 58 58 Option Modeling From the problem framing step, some strategi c options have been identified; from the approach selection step, the combined approach or the separated approach has been selected; from the risk driver identification and estimation approach, the key uncertainties have been identified and their volatilities quantified. Now in the option modeling step, a Monte Carlo simulation is run, an event tree is constructed, w ith managerial flexibilities incorporated in each node, option values are calculated, optimal decisi ons are made at each node, and the value are tracked from the end of the analysis back to th e starting time of the analysis. This process may be run back and forth for several times to ensure all option values are cal culated correctly and the corresponding rational d ecisions are made. Sensitivity Analyses Setting the project value with flexibility and/ or option value as the dependent variables, each risk variable can be changed, and the trend of value changes in the dependent variables can be observed. This sensitivity analysis helps th e user to see the whole picture and determine how each risk variable should be mana ged. It also helps in understa nding how uncertainty could have otherwise altered decision making. RERO Modeling Approaches For different treatments of risk drivers, th ere are two types of RERO modeling approaches: the combined approach and the separated approach. The combined approach is used for valuation of an existing building with a historic al operating track record. For uncertainties of infill land development, the separated approach is more suitable. MAD has two key assumptions: firstly, the pr esent value of the underlying risky asset without flexibility is the best estimate of the project value with flexibility. Secondly, properly anticipated cash flows fluctuate randomly. The second theorem a llows the user to combine any PAGE 59 59 number of uncertainties into a spreadsheet, and to produce an estimate of the project NPV conditional on the set of random variables drawn from their underlying distributions by using Monte Carlo simulation techniques (Copeland and Antikarov, 2001, p219). This is the theoretical foundation of the combined approach. By using the combined approach, uncertainties are assumed to be able to be resolved continuously over time. This assumption generally holds for stabilized assets. However, many projects in real estate, such as infill land deve lopment, have major uncertainties that do not get resolved smoothly over time. Many rare events, e. g., permit approval, development activities in the neighborhood, a new mall, a new subway stati on, can significantly ch ange the real estate value. For projects with any risk of such jumpi ng effect, the actual event tree is asymmetric with changes in value occurring when a significant part of the uncertainty is resolved. The separated approach is used to isolate the risks with jump diffusion effect from t hose resolved continuously, and to model their interaction explicitly. In other words, the separated approach also assumes that the underlying project value without flexibility is the best estimate of the project value with flexibility, but it does not assume that the cash flows fluctuate randomly. Rather, it separates the risk drivers with jump effect from the others without, and models the jump effect explicitly. The Combined Approach The combined approach is most suitable for valuation with risks re solved continuously. This approach can be best applie d to acquisition valuation of stabilized real estate assets. The process is to model the parameters of different uncertainties and to estimate their effect on the volatility of the project value us ing Monte Carlo simulation technique s. The effects of individual risk drivers are thus combined into the project volatility, which is used to generate a binomial event tree. Actions of managerial flexibili ty are added to solve for option value. PAGE 60 60 The following variables are typical in a prope rty acquisition model: re ntal rate, occupancy rate, rentable square footage, expense recovery operating expenses, capit al expenditure, tenant improvement, leasing commission, goingout cap rate, discount rate etc. Among these variables, the most influential ones are rental rate, stab ilized occupancy rate, goi ngout cap rate, and discount rate. Rentable square footage is usually fixed; expens e recovery and operating expenses vary but in a controllable small range related to th e rental rate change; capital expenditure, tenant improvement, and leasing commission are tricky in reality, but could be assumed to be fixed on an annual basis for a highend office building. Rental rate and stabilized occupancy rate will be used as the two major variables in the case analyses. Rental rate is set by the market, and directly impacts the property value. For valueadded type of investors, who intend to upgrade amenities and enhance occupancy, the stabilized occupancy rate is an important f actor for revenue estimation. The discount rate, however, is subjective to each inve stor. In finance theory, the discount rate should reflect the level of risk of a project. In practice, however, for an individual investor, the discount rate is usually his weighted average cost of capital. Risk is mainly adju sted through the Cap rate rather than discount rate (Wheaton et al., 2001). The discount rate can therefore be regarded as fixed. The change of rental rate depends on many factors, such as macro economics, employment growth, market occupancy rate, new construction pi peline, net absorption ra te, etc. The change of rental rate is assumed to follow the multiplicative stochastic process. Historical data of rental rates will be examined in the next chapter. Another factor that affects rental revenue is stabilized occupancy rate For a building that is not fully leased, there might be upside poten tial to lease up the vacant space, depending on market demand. In a market with strong job growth demand for office space is also strong. It is PAGE 61 61 relatively easy to lease up the vacant space. Assuming that vacant space can be leased up, the incremental Net Operating Income (NOI) is subs tantial compared to the incremental revenue, since the incremental operating expense is minimal. In other words, whether a building is 50% occupied or 100% occupied, a majority of the ope rating expenses is fixe d, the 50% leaseup can potentially triple the NOI. Note that a multite nant office building is seldom fully occupied, therefore stabilized occupancy ra te usually is close to but never reaches 100%. A general vacancy factor is deducted from the fully leased revenue. The change of occupancy rate is assumed to follow the additive stochastic process. This process is similar to the multiplicative stochastic process with the only difference bei ng that the up and down mo vements in the lattice are assumed to be additive rather than multip licative (Copeland and Antikarov, 2001, p123). The Separated Approach The separated approach is more complicated than the combined approach and should be used only when needed. It is best used for pr ojects with major private risks that do not get resolved continuously. The infill land valuation is an example in this study that can be better modeled using the separated approach. The following variables are typical in an infill land development model: rental rate, development cost, development timing, development scale, operating expenses, expense recovery, cap rate, discount rate etc. Among these variables, the most uncertain ones are rental rate, development cost, and development timing. Development scale is regarded as a major economic factor, but not a major uncertainty in the context of our case study, due to approved permit of the development scale. Since the goa l of most commercial developments is to maximize the investors wealth, developments are usually built to the largest size allowed by zoning and legal restrictions. Unless the deve lopment involves zoning changes, development scale is predictable, and thus is not modeled as a risk driver. As discussed in the combined PAGE 62 62 approach, operating expenses and expense recove ry are in a controllabl e range, and the discount rate for a particular project is fixed to a specific investor. Cap rate is as sumed to be fixed in the integrated approach for simplicity. Development costs include hard costs and soft costs, and can be subdivided into costs associated with land, structure, tenant improveme nt, leasing commission, le gal, finance, taxes, insurance, marketing, etc. Hard costs are cons truction costs that include demolition, foundation, structure, mechanical and engineering system s, general conditions, bonds and insurance of construction, design and management fees, tenant improvement, etc. Soft costs are intangible costs that go to legal, survey, marketing, financ ing, taxes, leasing commissions, etc. Since every project is unique, develo pment costs represent the major privat e risk that does not correlate with the traded financial market, and thus cannot be re plicated by the so called traded twin asset. Rental rate is discussed in the combined approach during normal circumstance. What needs to be pointed out in addition is the jump diffusion process. A jump diffusion process is defined as a type of stochastic process that has large discrete movements (jumps, or shocks), rather than small continuous movements (Ami n, 1993). As Wheaton et al. (2001) noted: In reaction to positive shocks, returns initially increa se, but eventually diminish with the arrival of new supply. Similarly, negative shocks lead to building conversions, lo ss of stock and an eventual recovery of returns. One of the distinguishing characteris tics of real estate, compared to traded securities, is its inelasticity, or sl ow reaction to shocks. The jump diffusion can be ignored in the acquisition of a nearly fully o ccupied property, since rental rates cannot be changed until lease expirations, which could be y ears from the emergence of the shock. But jump diffusion could be a major uncertainty in de velopment, since all rental square footage is newly available. Developers can ask for highe r rental rates in markets with rising demand. PAGE 63 63 Development timing is also important. Devel opment timing is different from development duration. Given the size of a development project the duration of construc tion is usually fixed, but when to start the project could have profound imp act on the value, given the real estate cycle. One of the major disadvantages of DCF valuati on is its inability to determine the optimal development timing. The RERO framework, on the other hand, can analyze all possible scenarios and indicate the best action at each point in time. It is extremely valuable for the investor to hold the option of when to start the development. Another important factor is development scale, or the size of development. In the case study, the permit for around 1 million square feet of mixused development has been approved. Consequently no assumption needs to be made for changing development scale. But in many cases, when rezoning is required in order to de velop more density, development scale is an important factor and should be modeled in th e decision tree as whet her or not the rezoning requirement will be approved. RERO Modeling Techniques Rational for Using Binomial Lattices Copeland and Antikarov (2001, p222) made the a ssumption that change in asset prices follow Geometric Brownian Motion, based on Sa muelsons proof that properly anticipated prices fluctuate randomly. In other words, chan ge in asset value follows a random walk even if the risk drivers do not. This means multiple risk drivers, so long as they evolve continuously, can be combined and reduced to a single uncerta inty, namely the expected underlying asset value change over time. This provides the rationale for using a binomial lattice to calculate real option value. PAGE 64 64 Monte Carlo Simulation Monte Carlo simulation randomly generates valu es for uncertain variables to simulate a reallife model. In the combined approach Monte Carlo simulation can be used as an intermediate step to estimate volatility of the project, the value of which is depended on multiple risk drivers. For this study Risk Simulator is used. Other simulation software available are Crystal Ball and @ Risk. The steps followed in the combined approach are to: 1. Identify risk drivers; 2. Estimate the probability distribution of each risk driver using historical data or subjective estimates; 3. Build present value model; 4. Define input variables with the possible range of value and a probability distribution in an MS Excel spreadsheet equipped with Monte Carlo simulation tools; 5. Define correlations among the risk variables; 6. Define forecast variables., e.g., rate of return for the project; 7. Run the simulation a thousand times; 8. Read the outputs of the forecast variable s and their volatility distributions; and 9. Use the outputs as input variables to build the event tree. Replicating Portfolio In most cases the project cash flows are discount ed at the riskadjusted rate to get to the project NPV. The riskadjusted discount rate is higher than the riskfree discount rate, since it is adjusted up to accommodate higher risk of the project than that of the trea sury bonds. In order to apply a binomial lattice that is developed base d on riskneutral proba bilities and riskfree discount rates, riskadjusted proba bilities should be used togeth er with riskadjusted discount rates. To calculate the value of the option, the replicating portf olio method is used, but not the PAGE 65 65 discounting method, since the risk characterist ics of the project change over time depending on the decision made, and consequently the riska djusted discount rates al so change over time (Copeland and Antikarov, 2001). The riskadjusted up movement factor u and down movement factor d are the same as those in the riskneutr al binomial lattice (Equations 41 and 42). te u (41) u d 1 (42) where is the project volatility, and t is the time in years of each step in the binomial tree. The replicating portfolio formula can be derive d by the same method as the option price is derived from binomial lattice. Construct a po rtfolio that consists of n shares of stock S and b amount of value in riskfree bonds. After a period of time t the value of the portfolio can go up or down. Let the value be equal to the option value at that time. u rtC be nuS (43) d rtC be ndS (44) From Equations 43 and 44, de rive Equations 45 and 46. ) ( d u S C C nd u (45) ) (d u e dC uC brt u d (46) Consequently, the value of the optio n is calculated by Equation 47. ) (d u e dC uC d u C C b nS Crt u d d u (47) PAGE 66 66 Binomial Lattice with Dividend Chapter 3 covers binomial lattice without divide nd. In real estate, the net cash flows from operation are collected from the property and distri buted to the investor, which is similar to the dividend distribution of a stock. The stock dividend is usuall y assumed to be distributed at a constant yield, since corporations plan and mana ge the distribution process. The net cash flows at the property level, on the other hand, are the actually amounts collected from the property, and hence vary from period to period. Denote i to the dividend yield at Step i for n i 0, and using all other notions in Chapter 3, the asset va lue changes are depicted in Figure 42 for a twoperiod lattice. Figure 42. Twostep binomial lattic e with different dividend yields. At Step 2, the three possible values are cal culated using Equations 48, 49, and 410. ] 0 ) 1 ( [2 0K uu S Max Cuu (48) ] 0 ) 1 ( [2 0K ud S Max Cud (49) ] 0 ) 1 ( [2 0K dd S Max Cdd (410) To calculate the option value at Step 1, the dividend yield 2 needs to be added back to the option value, before discounting at the riskfree rate, which is s hown in Equations 411 and 412. S0 S0u S0u(11) S0d S0d(11) S0uu S0uu(12) S0ud S0ud(12) S0dd S0dd(12) PAGE 67 67 rt ud uu ue C p pC C) 1 ( ) 1 (2 (411) rt dd ud de C p pC C) 1 ( ) 1 (2 (412) The same method is followed to calculate the opt ion value at Step 0, as shown in Equation 413. rt d ue C p pC C) 1 ( ) 1 (1 (413) In general, for a binomial lattice with n steps, the i th step (n i 0) call option value with dividend is calculated by Equation 414. rt i d i u i ie C p pC C) 1 ( ) 1 (1 1 1 (414) Binomial Lattice with Jump Process Chapter 3 covers binomial lattice during nor mal circumstance that the underlying asset strictly follows the GBM movement. However, in reality, the asset movement could be a jump. For example, the zoning change from agricultura l land to urban land, the establishment of new amenities in the neighborhood, the construction of new freeway exits, all can have a sudden and profound influence on the esta te value in an area. These events seldom happen. But once occur, they will completely change the project payoff pattern. Hence, these jump diffusion effects cannot be priced using the binomial lattice developed by Cox et al. (1979). Amin (1993) developed a discrete time model to value opti ons when the underlying process follows a jump diffusion process. Unlike the financial jump diffusi on process that reverses back to normal value quickly, a jump diffusion process in real estate usually is irreversible, at lease not in a short period of time. That is, if a la rge scale development occurs that drives up the rental rate in a PAGE 68 68 neighborhood, that rental rate is likely to remain at the same le vel for several years until a new event happens. In this study the Amin model was modified to accommodate this change. Based on the assumption that the jump risk is diversifiable, a oneperiod call option is priced in the Equation 415 (Figure 43). Figure 43. Binomial lat tice with jump process. ]} ) ~ 1 ( ~ )[ 1 ( {d u y rtC p C p C e C (415) where is the probability of the jump event according to the Poisson distribution, and defined by ) ( x n e xx n (where n is the expected number of successes, and x is the number of successes per unit); y is the capital gain return on the underlying asset when the jump event occurs; Cy is the option value at the time the jump event occurs; p is the adjusted probability of an up movement, and defined by d u d y e prt 1 ~. Investment with Private Uncertainty As discussed in Chapter 3, many investments include private and market uncertainties. Market uncertainty can be replicated with mark et participation and therefore diversifiable. Private uncertainty cannot. For example, th e development project va lue depends on both the market uncertainty of rental rate and th e private uncertainty of development cost. C Cd Cu Cy (1)p (1)(1p ) S dS uS yS (1)p (1)(1p ) PAGE 69 69 The principle of pricing in such investment, if no correlation between the market risk and private risk exists, is to use riskneutral pr obability for the market uncertainty and actual probability for the private uncertainty, both di scounted at riskfree rate (Luenberger, 1998; Copeland and Antikarov, 2001; Smith and McCa rdle, 1999). Although formulas for pricing uncertainties with correlation exist, the no correlation assumption usually holds. To implement this principle, there are tw o alternative methods: the quadranomial lattice and the decision analysis method. The first method is to implement a quadranom ial lattice. Figure 43 shows a onestep quadranomial lattice. If an option C is contingent upon the value of two underlying assets S1 and S2, assuming no correlation between S1 and S2, then the value of C is priced as Equation 416. ) (22 22 21 21 12 12 11 11C p C p C p C p e Crt (416) where ) 1 )( 1 ( ) 1 ( ) 1 (2 1 22 2 1 21 2 1 12 2 1 11p p p p p p p p p p p p pi is the riskneutral probability if Si is market uncertainty, or the actual probability if Si is private uncertainty. For each uncertainty, it can have more than two bifurcations. For example, if S1 is a market risk with jump di ffusion (three bifurcations), and S2 is a private risk with three bifurcations, then C could be priced with nine node s with corresponding probabilities and discount at the riskfree rate. In theory, an option can be contingent upon more than two separated assets, but in practice, the comp lexity of implementation will soon become intimidating. This study thus focuses on a few key risk drivers and combine them into two kinds of separated uncertainties: market uncertainty and private uncertainty. PAGE 70 70 Figure 44. Quadranomial lattice. Another way is to implement decision an alysis methodology (Smith and Nau, 1995). For example, if the two underlying risks for a developmen t are cost and rental rate, it can be modeled as shown in Figure 45. The exp ected value at each node is calcul ated and discounted at the riskfree rate. Equation 417 shows how the expected value E(PV0) can be calculated. m j j jPV E p PV E1 0)] ( [ ) ( (417) where j is a scenario labeled from 1 to m m j 1; E(PVj) is the expected present value of scenario j for all the years i n i 1. Figure 45. Decision analysis. Jump Up Down Jump Up Down Jump Up Down Low Middle High Cost Rent C C22 C21 C12 C11 P21 p12 P22 p1 u1S1 d1S1 S1 p1 1p1 u2S2d2S2S2 p2 1p2 PAGE 71 71 Summary This chapter discusses the 6steps RERO framework: problem framing; approach selection; risk drivers identification and estima tion; base case modeli ng; option modeling; and sensitivity analysis. Two modeling approaches are introduced to deal with different risk characteristics: the combined approach for pr ojects with risk drivers that get resolved continuously, and the separated approach for project either with risk drivers that follow the jump diffusion process or involving private risk. The modeling techniques that will be applied in the case studies are also introduced, including the rati onale of using the binomial lattice, Monte Carlo simulation, replicating po rtfolio, binomial lattice with jump diffusion process, and investment with private risk. PAGE 72 72 CHAPTER 5 THE COMBINED APPROACH Chapter 5 and 6 present case studies that impl ement the principles of RERO described in Chapter 4. The two chapters describe the valua tion of two parts of one case: valuation of the building using the combined approach, and valuation of the infill land using the separated approach. Together, these two case studies demonstrate how the RERO framework can be applied to different scenarios in the real estate acquisition and development analysis. Case Description The case identified is 211 Perimeter in Atlant a. This property is located in the Central Perimeter submarket of Atlanta. Adjacent to the Perimeter Mall and a subway station, 211 Perimeter is located in one of the largest suburba n office markets in Atlanta. The property has an office building of 226,000sf rentable area, and 13 acres total land. The current owner has got approvals for over 1 million square feet of mi xeduse development on the 9.5 acres developable site, and has built a 6storey parking garage with the intention to get as much value as the regulations allow from development of the ex cessive land (Figure 51). Furthermore, the property is strategically located within a la rger neighborhood redevel opment planning of 38 acres and nearly 3 million square feet mixe duse development, although the timing of the neighborhood development is unknown. The land obviously has some value, but deve lopment might not break ground immediately. The real estate market in Atla nta is a commodity market, which means developments are spread out with few restrictions. As 2005, the Centra l Perimeter office submarket was over built, with several old office buildings torn down for new resi dential developments. It would be interesting to know how current bidders should price the land in addition to the building. PAGE 73 73 Figure 51. 211 Perimeter site plan. Building Valuation In this chapter only the building is valuated using the combined approach with Monte Carlo simulation. The land valuation will be inves tigated in the next chapter using the separated approach. The following are the 6 steps used to perform the RERO valuation: Problem framing; Approach selection; Base case modeling; Risk drivers identification and estimation; Option modeling; and Sensitivity analyses. Problem Framing The property is located in a premium office mark et, with superior quali ty and tenant mix. Its strategic location within a larger neighborhood redevelopment plan makes real estate price PAGE 74 74 appreciation in the future extremely promising, although the timing is still unknown. In short, the 211 Perimeter project is a sound investment that deserves further valuation. After the preliminary qualitativ e analysis, this project app ears acceptable for quantitative analyses. The 11floor office building consists of 226,000sf rentable area. Current occupancy rate is 85%, with 15% upside pot ential to lease up the space. Major tenants collectively occupy 68% of the rentable square footage, which is deemed to be a sign of solid cash flow over the future. One of the major decisions to make is about the chiller system upgr ade. The existing chillers are still in wo rking condition but are at their maximum capacity, and consume far more energy than new ones. Preliminary research show s that replacement of the existing chillers will cost $950,000, and will increase the net cash flow by 5% per year. If both rental rates and occupancy rates are good, replacement of the chill ers can justify its cost, and add value to the property. Otherwise, the capital improvement may not break even, and keeping the existing chillers is more economical. Approach Selection The combined approach is selected since both th e rental rate and occupancy rate are market driven, and can be combined into the Monte Carlo simulation. Base case NPV calculation The following variables are typical in the NP V valuation model: rental rate, occupancy rate, rentable square footage, expense recovery operating expenses, capit al expenditure, tenant improvement, leasing commission, goingout cap rate, discount rate. Table 51 shows the assumptions used in the base case NPV calculati on. Figure 52 shows the cash flow output from Argus, a software package for real estate valuation. PAGE 75 75 Table 51. Major assumptions for Argus. Average rental rate $17/sf Capital expenditure $75,000 Occupancy rate 85% Tenant improvement $18/sf Rentable sf 225,924 sf Leasing commission 6.0% Expense recovery $0 Goingout Cap rate 7.0% Operation expenses $7.75/sf Discount rate 9.0% From the Argus cash flow output, modifications are made so that the model can be used for Monte Carlo simulation using Risk Simulator. Rental rate and occupa ncy rate have been identified as the two major risk variables that need to be simulated. Annual average rental rate and annual average occupancy rate are calculat ed from the Argus output, which are used to derive annual net cash flow. Purc hase price is assumed to be fixed, so that we can compare the project value with and without flexibility. Operating expenses and expense recoveries are controllable variables. Capital items, such as capital expenditure, tenant improvement, and leasing commission, are also controllable. Cap ra te and discount rate ar e also assumed to be fixed. Ignoring the option of chiller replacement, th e project NPV has two components: (1) Total acquisition cost, including purchas e price and closing cost; (2) Pr esent value of annual net cash flow from operation and present value of net re sidual value (gross sale proceeds net out selling cost). These two parts are also called cost and benefit. The option of chil ler replacement will be modeled later. In real estate fundamental analysis, property value consists of residual value and net cash flow from operation. The residual value, or value when the project is sold, is the major part. It is determined by Net Operating Income (NOI) and Cap italization rate (Cap rate). NOI is the gross PAGE 76 76 Figure 52. Base case NPV calculation. PAGE 77 77 income from all sources (rental, storage, tenant reimbursement, antenna lease, etc) minus all operating expenses (common area maintenance, management fee, security, landscaping, insurance, real estate taxes, etc). For this reas on, NOI is also regarded as the net income of the property. This is different from what the inve stor actually gets, which is called the Net Cash Flow. Net cash flow is calculated by taking ou t capital items from NOI. These capital items, such as capital improvement, tenant improvement and leasing commission, are onetimeoff in nature. All these analyses are on an unleveraged beforetax basis, meaning debt financing and taxation are not considered. Figure 53 shows the modified Argus cash flow output for NPV calculation. For simulation simplicity, modifications of the Argus output are made so that the net operating income and net cash flow are calculated by Equation 51 and Equation 52. OE ER Occ SF Q NOI (51) CapX LC TI NOI NCF (52) where NOI is the net operating income; Q is the average rental rate; SF is the rentable square footage; Occ is the actual occupancy rate; ER is the expense recovery and other income; OE is the operating expenses; NCF is the net cash flow; TI is the tenant improvement; LC is the leasing commission; CapX is the capital expenditure. The residual value at sales is calculated by Equation 53. SC Cap NOI Vn n 1 (53) where Vn is the net residual value at year n and n is the holding peri od of the project; Cap is the goingout Cap rate; SC is the selling cost. PAGE 78 78 The total benefit of the project PVj, which includes the present value of net cash flow NCFi and residual value Vn, can be calculated by Equation 54. j n n n j i j i i jk V k NCF PV ) 1 ( ) 1 ( (54) where PVj is the project present value at Year j and n j 0, where n is the holding period. When j = 0 it is the present value at time 0, or PV0. NCFi is the net cash flow at Year i k is the discount rate of the project. The NPV of the project is the present value of total cost PP0 and total benefit PV0 at time 0, as calculated by Equation 55. 0 0 0PP PV NPV (55) Risk Drivers Modeling Among the variables, those that have th e most profound impact on the project NPV changes are rental rate and stabi lized occupancy rate, both are mark et driven. Rental rates differ leasebylease, but for simplicity we take the average rental rate ove r the entire building. Stabilized occupancy rate is subjective based on managements estimates In this case the 15% vacant space is assumed to be leased up within 2 years, after which a gene ral vacancy factor of 3% is taken out. Figure 54 shows the historical rental rates of the Central Perimeter Class A office market and the subject property in 15 years. The quarterly data is from CoStar. The change of rental rate is assumed to follow GBM. This means the logarithm of the rental rate Qi is normally distributed; and the return (also called the cha nge of rental rate) qi follows a random walk. Using Equation 56, a rental retu rn analysis was performed and th e scatter chart was plotted as shown in Figure 55, with market return variab les on Xaxis and corres ponding subject property return variables on Yaxis. It shows negative corr elation (0.1445), which indi cates that the rental PAGE 79 79 Figure 53. Spreadsheet model for Monte Carlo simulation. PAGE 80 80 rate change of the subject prope rty, 211 Perimeter, has very w eak, if not negligible, correlation with the market. $14.00 $17.00 $20.00 $23.00 $26.00 1990 3Q 1992 3Q 1994 3Q 1996 3Q 1998 3Q 2000 3Q 2002 3Q 2004 3Q Market Subject Property Figure 54. Historical market a nd subject property rental rates. Figure 55. Returns correlation betw een market and subject property. y = 0.3276x 8E05 R2 = 0.020920.00% 10.00% 0.00% 10.00% 20.00% 30.00% 7.00% 5.00%3.00% 1.00%1.00%3.00%5.00%7.00% Market Subject Property PAGE 81 81 ) ln(1 i i iQ Q q (56) The seemingly controversial resu lt of weak or no correlation be tween the rental return of the subject property and that of the market can be explained as due to two reasons: (1) Data reliability. CoStar started as a se rvice portal mainly for commercial brokerage firms. In its early years data is derived fr om broker volunteer contributions. This would inevitably have led to data accuracy and timelines s issues. For example, from the first quarter of 1998 to the third quarter of 1999, the rental rates data of the subject property are missing, which are assumed to be $18.90/sf by the author for the purpose of data completeness. (2) The inelastic nature of real estate market. Compared to the financial market, the real estate market is lumpy and the performance is some what predictable, at least for the near term. Commercial lease terms are usually 3 to 7 years fo r office leases, 5 to 20 years for anchor retail leases, and even 100 years for ground leases. In most cases, the rental payment is set and documented in the contract througho ut the terms. Market rental rate changes can only slowly affect individual property ask rates, since the land lord can change rental rate only when a lease negotiation happens, usually before the lease expires. However, mark et rates can directly affect the rates for new construction, since all spaces are newly available. Nevertheless, the data set from CoStar is the most comprehensive and consistent data available in the real estate indus try. The characteris tics of real estate require a different method than the one used to estimate stock volatility in the financial industry. Thus the correlation between the market and the subject property was ignored on purpose, and only the subject property rental rate data was used to estimate its vol atility for acquisition. PAGE 82 82 Risk Simulator is used to influence the di stribution of the populat ion from the available sample data. A lognormal distribution was chosen since the rental rates will never be negative. Due to the limitation of available data, the statisti cal significance of this distribution is low (PValue of 0.1625). Nevertheless, this is the most reasonable fit for the data. By fitting the sample data into a lognormal distribution (Figure 56), the following variables are determined: is 0.0056 and is 0.0548. Annualizing the quarterly and using Equation 57 and 58, mean of 18.0189 and standard deviation of 2.0218 for the re turn distribution are derived. To get the annual autocorrelation of the rental return, the qu arterly return data is annualized by taking the average of the 4 quarters of each year, which turn s out to be 0.0916. This autocorrelation of the samples is assumed to be the same as that of the population. ) 2 / (2 e X (57) 2 22 ) ( 2 e e SD (58) Figure 56. Normal distribution fi t for historical returns on rental. CoStar also provides historical occupancy rates data for th e market and subject property (Figure 57). Occupancy rate is assumed to follow the additive stochastic process. This means PAGE 83 83 the change of occupancy rate oi between any two quarters is simply the difference of the occupancy rate Oi and Oi1 (Equation 59). From the scatter plot of the change of occupancy data shown in Figure 58, it can be concluded that th e occupancy rate of the property also has very weak correlation with the market (0.1263). Thus, this correlation is also ignored on purpose and only the historical occupancy rates of the subj ect property will be relied on for forecasting. 1 i i iO O o (59) Using RiskSimulator, the population and the respective population mean and standard deviation of the normal distribu tion, are determined to be 0.0039 and 0.0471 respectively (Figure 59). Due to the limitation of available data, the sta tistical significance of this distribution is low (PValue of 0.00004). Howe ver, this low Pvalue might be a limitation of the software itself, i.e., its estimation of data in a small range is inaccurate. Nevertheless, this is the most reasonable fit for the data. To preser ve accuracy, it was decided to keep the sample mean as the population mean (0.0026), and annu alize the sample standa rd deviation as the 20.0% 40.0% 60.0% 80.0% 100.0% 1986 4Q 1988 4Q 1990 4Q 1992 4Q 1994 4Q 1996 4Q 1998 4Q 2000 4Q 2002 4Q 2004 4Q Market Subject Property Figure 57. Historical market a nd subject property occupancy rates. PAGE 84 84 y = 0.4502x + 0.0018 R2 = 0.01630.0% 20.0% 10.0% 0.0% 10.0% 20.0% 10.0%5.0%0.0%5.0%10.0% MarketSubject Property Figure 58. Occupancy changes corr elation between the local real estate market and the subject property. Figure 59. Normal distribution fit for historical occupancy rates. PAGE 85 85 population volatility (0.1409). To calculate the au to correlation, the change of occupancy rate data is annualized by taking the average of the 4 quarters of each year, which turns out to be 0.1185. The correlation between rental re turn and change of occupancy rate is similar to the autocorrelation of the two, which comes out to be 0.1575. For Monte Carlo simulation, the project volatilit y is the volatility of percentage changes in the value of the project from one time period to the next, defined by the forecasting variable z (Equation 510). This value is computed using the simulated present value of the project in Year 1 divided by the expected present value of the project in Year 0. In other words, PV1 is dynamic, while PV0 is static. 0 1PV PV z (510) Option Modeling In the previous step the rental rate, occupa ncy rate, their respective volatilities, autocorrelations, and the correlation be tween the two have been identifie d and quantified. With these variables, rental rates and occ upancy rates for each year can be set as risk variables for the project value simulation. A total of 8 risk va riables are defined and highlighted as shown in Figures 543, 510, and Table 52. The cash flows go through Equa tions 51 to 55 to generate annual net cash flow for the first 3 years. PV0 and PV1 are calculated based on the annual net cash flow. The forecasting variable z is defined in Equation 510. Setting PV0 to be static and PV1 to be dynamic, and running the simulation for a 1000 times, the simulation result of z is obtained as shown in Figure 511. Table 53 also shows the statistical summary of z with a mean of 1.079 and standard a deviation of 0.3283. PAGE 86 86 Figure 510. Snap shot of M onte Carlo simulation assumptions. Figure 511. Monte Carlo Simulati on Result of Forecasting Variable z The statistical distribution fit for Variable z is then performed. By plotting the 1000 z values from the simulation output, as shown in Figure 512, it is determined that they are normally distributed with PValue of 0.8737. Th is result fits quite we ll with the theory PAGE 87 87 Table 52. Correlation between random variables. Yr 1 return Yr 2 return Yr 3 return Yr 4 return Yr 1 Occ Yr 2 Occ Yr 3 Occ Yr 4 Occ Yr 1 return 1.00 Yr 2 return 0.09 1.00 Yr 3 return 0.00 0.09 1.00 Yr 4 return 0.00 0.00 0.09 1.00 Yr 1 occ 0.16 0.00 0.00 0.00 1.00 Yr 2 Occ 0.00 0.16 0.00 0.00 0.12 1.00 Yr 3 Occ 0.00 0.00 0.16 0.00 0.00 0.12 1.00 Yr 4 Occ 0.00 0.00 0.00 0.16 0.00 0.00 0.12 1.00 Table 53. Statistical summary of Monte Carlo simulation result. Description Value Number of data points 1000 Mean 1.0797 Median 1.0569 Standard deviation 0.3283 Variance 0.1078 Average deviation 0.2561 Maximum 2.4431 Minimum 0.0977 Range 2.3454 Skewness 0.3787 Kurtosis 0.7041 25% percentile 0.8722 75% percentile 1.2831 Error precision at 95% 0.0188 Figure 512. Normal distribution fit of forecasting variable z PAGE 88 88 developed by Samuelson and adopted by Copela nd and Antikarov (2001), as discussed in Chapter 4, that changes in correct ly expected asset prices follow Geometric Brownian Motion. From the Monte Carlo simulation, the mean and the volatility of forecasting variable z are calculated as 1.0797 and 0.3283 respectively. This means the expected average project return is 7.97% (1.0797 minus 1), and the volatil ity of the project is 30.4% (0.3283 divided by 1.0797). Using the assumptions in Table 54, with 30.4% volatility, and $24,963,000 PV derived from the base case analysis, a value tree is constr ucted as shown in Figure 513. Net cash flows are modeled as dynamic dividend yield times PV in the base case (Refer to Chapter 4 for details of binomial lattice with divi dend). For example, in Year 1, the PV can go up to $34,664,000 with an up factor of 1.3886, the post dividend cash flow is therefore $33,638,000 (after taking out 2.96% yield from the $34,664,000 before dividend cash flow). Table 54. Event tree assumptions (Dollars in $1,000). Assumptions Intermediate computations PV of asset value $24,963 Stepping time (dt) 1.0000 Implementation cost $24,205 Up step size (up) 1.3886 Maturity (years) 3.00 Down step size (down) 0.7201 Riskfree discount rate (%) 5.00% Volatility (%) 32.83% Lattice steps 3 Option type Call NCF as percentage of PV Year 1 2 3 NCFi $805 ($511) $1,828 PVi $27,210 $28,781 $31,928 Percentage 2.96% 1.78% 5.73% PAGE 89 89 Figure 513. Event tree present value w ithout flexibility (N umber in $1,000). With the event tree of PV without flexibility the chiller replacement option can now be modeled. An event tree of PV with flexibility is constructed (Fi gure 514). At the end nodes, the decision is whether to keep the existing chillers or replace them with new ones. For example, the value of Node A` is calculated as follow. Max (Replace, Keep) = Ma x (Present Value 1.05 Cost, Present Value) = (62234 1.05 950, 62234) = 64396 (Replace) At the intermediate nodes, the decision is a bout whether to leave the option open or to execute it immediately. To calculate the valu e of leaving the option open, the replicating portfolio method developed in Chapter 4 must be used, but not the discounting method, since riskadjusted probability and riska djusted discount rate are used to construct the spread sheet and event tree. Equation 47 is the replication portfolio formula to be applied. 34,664 33,638 17,977 17,445 12,563 12,786 9,208 8,681 17,755 16,738 34,235 32,275 66,014 62,234 46,710 47,540 24,224 24,655 24,963 PAGE 90 90 Figure 514. Present value with fl exibility (Numbers in $1,000). For example, the value of keepi ng the option open at Node C` is 48877 ) 7201 0 3886 1 ( 68175 7201 0 34889 3886 1 7201 0 3886 1 34899 681751 05 0 e C Therefore, the value of node C` is Max (Replace, Open) = Max (47540 *1.05950, 48877) = 48967 (Replace) The decision is to replace the chiller syst em immediately. Usi ng Equation 414 to add back the implied net cash flow of negative $830,000, the before dividend present value is $48,137,000. Working backward the value at eac h node can be similarly calculated and the optimal action can be selected to maximize th e present value, and eventually the maximum present value can be derived at time 0. Th e present value increases from $24,963,000 (without 35,486 34,461 18,124 17,593 12,563 12,786 9,208 8,681 17,755 16,738 34,899 32,939 68,175 64,396 48,137 48,967 24,537 24,967 25,421 Replace Replace Keep Keep Replace Open Open Open Open Open A A` B` B C C` PAGE 91 91 flexibility) to $25,421,000 (with flex ibility), or an increase by $458, 000. The NPV of the project is now $1,216,000. In other words, the option to replace the chillers system creates $458,000 value. If the building could be purchased at $24,205,000, the NPV increases to $1,216,000. Sensitivity Analyses Sensitivity analyses are conducted using option value as dependent variable, and present value, replacement cost, discount rate and vola tility as independent variables. Table 55 summarizes the effect of each independent variab le as well as their combined effects on the option value. Present value has positive effect on the opti on value (Figure 515). Replacement of the chiller system increases the annua l net cash flow by 5%. And pres ent value is pos itively related to net cash flow. Therefore, the higher the present value is, the highe r the additional net cash flow would be when exercising the replacement option, and hence the higher the option value would be. Table 55. Summary of vari able effect on option value. Present value Replacement cost Discount rate Volatility Present value Positive Uncertain Positive, most Sensitive when inthemoney Positive, most Sensitive when atthemoney Replacement cost Negative Uncertain, most Sensitive when atthemoney Uncertain, most Sensitive when atthemoney Discount rate Positive Positive, most Sensitive when atthemoney Volatility Positive PAGE 92 92 200 400 600 800 1,000 10,00020,00030,000 Present ValueOption Valu e Figure 515. Option value in relation with present value. As shown in Figure 516, the replacement cost has negative effect on the option value. The higher the replacement cost is the less likely the replacement is breakeven, and hence the less likely the option would be exercised. 200 400 600 800 1,000 1,200 5001,0001,5002,000 Replacement CostOption Valu e Figure 516. Option value in relation with replacement cost. Volatility also has positive effect on the option value (Figure 517). The higher the volatility, the wider the present value spread be comes in later years, but the replacement option is only exercised in those scenar ios with positive net cash flows. Therefore, the more uncertain the future cash flow is, the more valuable the option becomes. PAGE 93 93 200 400 600 800 1,000 10,00020,00030,000 Present ValueOption Valu e 20% Volatility 33% Volatility 45% Volatility Figure 517. Option value in relation with present value and volatility. Riskfree interest rate has positive effect on the option value. But the effect is not significant. After examining the effect of each independent variable on the option value, combinations of each two independent variables can be looked at The combination of present value and Riskfree interest rate has positive effect on the option value. The two pairs of (1) present va lue and volatility (Fi gure 517), (2) volatility and riskfree rate (Figure 518) both exercise positive effect on option value, and are most sensitive when the option is atthemoney. The three pairs of (1) replacement cost and volatility (Figure 519) (2) replacement cost and riskfree rate, (3) present value and repla cement cost (Figure 520) all display uncertain effect on the option value. This conclusion is best illustrated in Figure 520. The 3dimensional curve indicates that the higher the present valu e and the lower the replacement cost, the higher the option value. However, this effect is nonlinear. With higher present value and higher PAGE 94 94 replacement cost, the option value may be hi gher or lower, dependi ng on whether the option value is inthemoney. 300 350 400 450 500 550 600 0%10%20%30%40%50% VolatilityOption Valu e 3% RiskFree 5% RiskFree 7% RiskFree Figure 518. Option value in relation with volatility and discount rate. 200 400 600 800 1,000 1,200 5001,0001,5002,000 Replacement CostOption Valu e 20% Volatility 33% Volatility 45% Volatility Figure 519. Option value in relation with replacement cost and volatility. PAGE 95 95 200 1,200 2,200 2,000 12,000 22,000 32,000 42,000 500 1,000 1,500 2,000 2,500 Option Value Replacement Cost Present Value Figure 520. Option value in relation w ith present value and replacement cost. Summary This chapter applies the combined approach to determine the building value of the 211 Perimeter property in Atlanta. Re ntal rate and stabilized occupancy rate are identified as the two major risk drivers and their volatilities are estima ted using historical data. The risk variables are combined in a spread sheet. Monte Carlo si mulation is performed to estimate the project volatility. Event tree is constructed, in whic h the option to replace the chiller system is incorporated. The RERO appro ach indicates that the buildin g is worth $25,421,000, and the value of managerial flexibility is worth $458,000. PAGE 96 96 CHAPTER 6 THE SEPARATED APPROACH This chapter is the second part of the case st udy described in Chapter 5. In the previous chapter the RERO fr amework is applied to analyze the building structure and a managerial decision of chiller replacement. The combined approach with Monte Carlo simulation is used as the major methodology. This chapter, however, is about valuation of the infill land using the separated approach, with jump diffusion process and decision tree analysis techniques. Together, these two parts demons trate how the RERO framework can be applied to different scenarios in the analysis of real estate acquisition and development. Case Description The previous chapter has full description of the case 211 Perimeter in Atlanta. This chapter only repeats the infill land portion. Be sides the existing office building and the 6story garage, the current owner has got approvals for over 1 million square feet of mixeduse development on the 9.5 acres developable site. Fu rthermore, the property is strategically located within a larger neighborhood redevelopment planning of 38 acres and nearly 3 million square feet mixeduse development, although the timing of neighborhood deve lopment is unknown. The land obviously has some value, but deve lopment might not break ground immediately. The real estate market in Atla nta is a commodity market, which means, with little control of urban sprawl, developments are spread out easily as far as market demand exists. The Perimeter office submarket is currently overbuilt, with several old office buildings torn down for new residential developments. It would be interest ing to know how current bidders should price the land in addition to the building. PAGE 97 97 Land Valuation The value of the infill land (9.5 acres out of the 13 acres total) de pends on the value and cost of the improvement should it be developed. The value of the improvement is determined by a function of its annual rental in come and operating cost, just like the existing building. The cost of development includes hard cost s and soft costs. Since ever y project is unique, development cost is assumed to be a private risk that does no t correlate with the traded financial market. Problem Framing The addition of a 6story garage has freed th e infill land from its original function as surface parking. With the 1 million square feet mixused development approval, the land can be sold for $4.75 million at anytime during the holding period. Its best value for the investor is being either developed or spinoff for $4.75 million. Table 61 shows the development assumptions. Assume the land allows for 1 million square feet to be built, gross re nt is $24.5/sf, stabilized occupa ncy rate is 85%, operating expense is $8.5/sf, required cap rate is 8%, riskfree intere st rate is 5%. Expected development cost is $227.5/sf. Land carrying cost is assumed to be negligibly small compared to the development value. The land can be sold for $4.75 million at anytime. This can be viewed as the exercise price of a put option to the investor. Table 61. Development assumptions. Rentable sf 1,000,000 Site acres 9.50 Gross rent psf $24.50 Land $4.75 Occupancy rate 85.0% Value $154.06 Operating expenses psf $8.50 Cost $177.50 Net rent psf $12.33 Riskfree rate 5.0% Cap rate 8.0% PAGE 98 98 In addition, management believes that th e groundbreaking for the larger neighborhood redevelopment will have significant impact on the demand for new office space, and hence drive up rental rate of this development by 20%. This is a onetime event, bu t once the rental rate rises, it will remain at that level during the entire analysis period. Approach Selection The separated approach is selected because th e impact when the rental rate jumps up by 20% is significant, and the chance is uncerta in, depending on the timing of the neighborhood redevelopment. This is an exam ple where one risk driver (the re ntal rate) does not get resolved smoothly, and must be modeled sepa rately from the other risks. Risk Drivers Identification and Estimation The risk drivers are rental rates and developm ent cost. Unlike the existing office building, the new building does not have a historical track r ecord. For income, the building rental rate is assumed to have some premium over the average ma rket rental rate. Changes in rental rate are assumed to follow the GBM movement, with a jumpdiffusion process corresponding to the groundbreaking of the neighborhood proj ect. Figure 61 shows the historical market average rental returns for Class A office properties in th e Central Perimeter submarket. Using the Risk Simulator, the quarterly lognormal returns are plot ted into a normal fit as shown in Figure 62. Converted into annual data, the market rental rate volatility is 4.84%. As explained in Chapter 5, individual property is far more volatile than the market aver age. The management estimate doubles and becomes 9.68% per year for the infill land development project. The current gross rental rate is $21/sf for the average Cl ass A building in the Central Perimeter submarket. According to manageme nt experience, a $3.50/sf premium for a brand new building can be secured. PAGE 99 99 $0.00 $5.00 $10.00 $15.00 $20.00 $25.00 $30.00 1990 3Q1992 3Q1994 3Q1996 3Q1998 3Q2000 3Q2002 3Q2004 3Q 8.00% 6.00% 4.00% 2.00% 0.00% 2.00% 4.00% 6.00% Rental Return Figure 61. Historical market average rental rates and return volatility. Figure 62. Normal distri bution fit for historical market rental returns. Rental rate changes are assumed to follo w the GBM movement. A Poisson distribution jumpdiffusion process corresponds to the groundbreaking of the neighborhood residential project, with 10% annual probability. The op tion value is calculated using Equation 415 developed in Chapter 4, where is 10% and y is 1.2 (1 plus 20%). Figure 63 shows how to get rental rate change s from one period to the next period. At Year 0, gross rental rate is $24.50/ sf. It could have three values in the next year: $29.40/sf (1.2 PAGE 100 100 times $24.50/sf) if the neighborhood developmen t breaks ground, $26.99/sf (up movement) or $22.24/sf (down movement) if the neighborhood development does not break ground, with probabilities of 0.10, 0.5895, and 0.3105 respectively. In year 2, it could have five values. If the neighborhood development breaks ground in Year 1, the rental rate $29.40/sf will follow the GBM movement with possible va lue of $32.39/sf or $26.69/sf, w ith probabilities of 0.7402 and 0.2598 respectively. If no development breaks ground in Year 1, the rental rates of $26.99/sf and $22.24/sf each follows the GBM with jump diffusion process and has three values, which combine into 5 possible values. In Year 3 the re ntal rates follow the same process and can have seven values. Notice, however, the probabilities to get to these values are different with and without the jump diffusion process. Figure 63. Gross rental rate movement and probabilities. PAGE 101 101 Taking out revenue lost from the 15% vacant space, $8.5/sf operating expense, and capping the net cash flow at 8% Cap rate, we can get the corres ponding per square foot building value contingent upon the gross re ntal rate, stabilized occupanc y, operating expenses, cap rate, and the likelihood of the neighborhood re sidential development (Figure 64). Figure 64. Building value movement and probabilities. There is no direct comparable data on devel opment cost. Development cost includes hard and soft costs. For hard cost, the RS Mean s Building Cost Data manual (RS Means, 19982006) can be used. The cost per square foot data fo r highrise office buildings from Year 1998 to Year 2006 is shown in Figure 65. The historical data shows an upward trend, at a pace generally consistent with the inflation rate from inflati odata.com (Figure 66). RS Means compiles market average data nation wide, which does not reflect th e volatility of local mark ets. More over, there are no data about the soft cost. Each project is unique in some soft cost items, such as land acquisition cost, permit application cost, unexpected cost, etc. The best estimate would be from PAGE 102 102 experienced managers. The development cost is assumed not to change with the financial market. It is a private risk that depends on the geological condition of the site, material and labor condition of the local market, etc. Management has estimated that with 50% probability the development cost would be $175/sf, with 20% to be $150/sf, and with 30% to be $200/sf, or 70.00 90.00 110.00 130.00 150.00 170.00 199819992000200120022003200420052006$ psf Low Mid High Figure 65. Historical c onstruction cost for highrise office building. 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 19992000200120022003200420052006 Cost Inflation Figure 66. Construction cost chan ge rate and inflation rate. PAGE 103 103 expected cost of $177.50/sf (Fi gure 67). Cost increases by 3% annually, consistent with the average inflation rate over the past 7 years. For simplicity, the buildable square footage is assumed to be the same as the rentable square footage. Figure 67. Developmen t cost assumptions. Base Case Modeling The expected PV without any flex ibility is calculated as shown in Figure 68. It is better represented in matrices. Each table in Figure 69 is a matrix of possible PVs for a given year. Starting from Year 3, the possible outcomes of building values are listed in the first row, and the possible outcomes of development costs are listed in the first column. The values inside the rectangle are all possible combinations of costs and values. The same applies to the values for Year 2, Year 1 and Year 0. In Year 0, the expected value is ca lculated as the sum of the three values times the respective probabilities of their development cost. Option Modeling There are three possible kinds of decisions at each node: (1) to develop the land, (2) to keep the land asis, and (3) to sell it for $4.75 million. Figure 610 depicts the decisions and payoffs corresponding to the matrices in Figure 611. In this lattice, the notation below the value represents the optimal decision to be made: D for developing th e land; K for keeping the option alive; and S for selling the land. PAGE 104 104 Figure 68. Payoff and proba bilities without fl exibility (Dollars in $1,000,000). PAGE 105 105 Figure 69. Payoff matri ces for project values without fl exibility (Numbers in $1,000,000). In Year 3, the decision will be either to develop the land or to sell it for $4.75 million, whichever generates the higher payoff. For example, the PV of Node A is calculated as follows: Max (Develop, Sell) = Max (Buildi ng Value Cost, Salvage Value) = (206.13 163.61, 4.75) = 42.22 (Develop) Working backward, in Year 2, the payoff is the greatest of the three: (1) the payoff of developing the land, which is the building valu e minus development cost; (2) the payoff of keeping the option open, i.e., th e corresponding payoff in Year 3 di scounted at riskfree interest rate using the binomial or jump diffusion probabilitie s calculated in Table 62; (3) the payoff of PAGE 106 106 Figure 610. Decision payoff and probabilities with flexibility (Dollars in $1,000,000). PAGE 107 107 Figure 611. Payoff matrices of project value with flexib ility (Numbers in $1,000,000). Table 62. Probabilities of jump di ffusion and binomial processes. Jump diffusion No jump Jump Up Down Up Down (1) p ~ (1)(1p ~ ) p 1p 0.1000 0.5895 0.3105 0.7402 0.2598 the put option, which is to sell the land for $4.75 million. For the normal stochastic process, the payoff of keeping the option open at Node B, for example, is calculated using Equation 319 as follows: 90 30 ] 75 4 2598 0 22 42 7402 0 [ ] ) 1 ( [1 05 0 e C p pC e Cd u rt PAGE 108 108 Consequently, the PV of Node B is calculated as follow. Max (Develop, Keep, Sell) = Max (Building Value Cost, Payoff of Keeping Option Open, Salvage Value) = (177.30 159.14, 30.90, 4.75) = 30.90 (Keep) For the jump diffusion, the payoff of keeping the option open at Node C, for example, is calculated using Equation 415 as follows: 73 14 } 75 4 3105 0 61 16 5895 0 22 42 1 0 { ]} ) ~ 1 ( ~ )[ 1 ( {1 05 0 e C p C p C e Cd u y rt Consequently, the PV of Node C is calculated as follows: Max (Develop, Keep, Sell) = Max (Building Value Cost, Payoff of Keeping Option Open, Salvage Value) = (154.06 159.14, 14.73, 4.75) = 14.73 (Keep) Working backward to Year 0, the PV of the pr oject is expected PV of each cost scenario times its corresponding probability. The PV of Node D is calculated using Equation 417 as follows: 14 22 21 13 3 0 99 21 5 0 91 35 2 0 )] ( [ ) (1 0 m j j jPV E p PV E The PV of the project increases from negativ e $23.44 million without flexibility to positive $22.14 million with the development and se lloff flexibility. The option value is 57 45 $ ) 44 23 $ ( 14 22 $ million. Sensitivity Analyses Sensitivity analyses are conducted using gross re ntal rates, occupancy rates, volatility, Cap rates, and development cost as independent vari ables, and on two depende nt variables: project value and option value. Project value is the PV of the project with the flexibility of deferred development, spinoff the land, and immediate development. Option value is the difference PAGE 109 109 between PV with flexibility and PV without flexibility. Since th e PV without flexibility also changes with variables, the project value and op tion value analyses have quite different results and implications. As shown in Figure 612 the rental rate has a pos itive effect on project value. Rental rate is directly linked to revenue. Th e higher the rental rate is, the hi gher the income the project will generate, and hence the higher the project value is. However, as shown in Figure 613 it has a negative effect on option value. This is because the higher the rental rate is, the more likely the project will be developed imme diately, hence the option to wa it or abandon the development by selling off the land is less worthy. In other wo rds, higher rental rate not only increases the project value with flexibility, it also increases the value without fl exibility at even higher pace. These two values cancel out each other, resulting in minimal option value. The combination of rental rate and occupancy rate has the same result: positive effect on the project value (Figure 612), and negative effect on the option value (Figure 613). Note that the option value is sensitive to stabilized o ccupancy rate when the option is atthemoney. 50 100 150 200 250 369121518212427303336 Gross RentPV with Flexibili t 60% Occ 85% Occ 100% Occ Figure 612. Present value in relation with rental rate and occupancy rate. PAGE 110 110 50 100 150 200 250 300 369121518212427303336 Gross RentOption Valu e 60% Occ 85% Occ 100% Occ Figure 613. Option value in relation with rental rate and occupancy rate. Just opposite to the effect of rental rate, as shown in Figure 614, development cost has a negative effect on project value, but positive effect on option value (Figure 615), for the same reason as explained above. 3 9 15 21 27 33 100 200 300 20 40 60 80 100 120 140 160 180 200 PV with Flexibility Gross Rent Development Cost Figure 614. Present value in relation with rental rate and development cost. PAGE 111 111 3 9 15 21 27 33 100 200 300 50 100 150 200 250 300 350 400 Option Value Gross Rent Developmen t Cost Figure 615. Option value in relation w ith rental rate and development cost. As shown in Figure 616, cap rate has negative effect on project value. This is because cap rate is inversely related to property value. (Property value is dete rmined by dividing net operating income by cap rate.) However, the effect of cap rate on option value is more profound. Figure 617 shows that at normal rental rate ra nge ($11/sf to $31/sf), cap rate has a positive impact on the option value; however, in the low rent al rate range ($0/sf to $11/sf), its impact is the opposite. Figure 618 illustrates how the combin ation of rental rate and cap rate results in different option value. Unlike most situations where a variable has monotonic impact on the option value, the shape of cap rate on option valu e is convex. For example, at $20/sf gross rent, the option value at 2% cap rate is $71 million, at 4% cap rate the option value drops to $47 million, and at 8% cap rate the option value comes back to $77 million. PAGE 112 112 50 100 150 200 250 369121518212427303336 Gross RentPV with Flexibili t 6% Cap 8% Cap 10% Cap Figure 616. Present value in relati on with rental rate and Cap rate. 50 100 150 200 250 300 369121518212427303336 Gross RentOption Valu e 6% Cap 8% Cap 10% Cap Figure 617. Option value in relation with rental rate and Cap rate. As shown in Figures 619 and 620 volatility has positive impact on both project value and option value. This finding is consistent with ma ny observations in real options research (Titman, 1985; Williams, 1991; Quigg, 1993) that greater volat ility increases option value, which is also the reason why the real options methodology should be applied to projects with high uncertainty. PAGE 113 113 2 8 14 20 26 32 1% 3% 5% 7% 9% 100 200 300 400 500 600 700 800 900 Option Value Gross Rent Cap Rate Figure 618. Option value in relation w ith rental rate and Cap rate in 3D. 50 100 150 3%6%9%12%15%18%21%24%27%30%33%36%VolatilityPV with Flexibili t 6% Cap 8% Cap 10% Cap Figure 619. Present value in relation with volatility and Cap rate. PAGE 114 114 20 40 60 80 100 3%6%9%12%15%18%21%24%27%30%33%36%VolatilityOption Valu e 6% Cap 8% Cap 10% Cap Figure 620. Option value in relation with volatility and Cap rate. Summary This chapter applies the separa ted approach to value the in fill land of the 211 Perimeter property in Atlanta. Rental ra te and development cost are id entified as the two major risk drivers. Rental rate is assumed to have jump di ffusion effect due to the uncertainty of the larger neighborhood redevelopment project. Development co st is assumed to be a private risk with no corresponding traded twin asse t and it is estimated subjectively based on managements experience. DTA methodology is app lied and an event tree is constr ucted, in which three options are incorporated: the option to develop immediately, the option to delay development, and the option to sell the land. The RERO approach indicates that the land is worth $22,140,000 and the value of managerial flexibility is worth $45,570,000. In Chapter 5, the building is estimated to be worth $25 million; in this chapter, the land is estimated to be worth $22 million, totaling $47 million. This is very close to reality, because the property was actually sold for $43.5 million in 2005. PAGE 115 115 CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS Conclusions Three main conclusions are drawn from this research: (1) acquisition and development has different characteristics and deserv e different kinds of attention; (2) consideration of managerial flexibility can change investment decisions; a nd (3) many unconventional real option valuation problems can be realized by using binomi al lattice and Monte Carlo simulations. Acquisition and development have different characteristics and thus deserve different kinds of valuation. The option value of acquisition is usually on a mu ch lower scale than that of development, but by no means is it less significant. In the case studies, th e option in the existing building is replacement of the chiller system. Its value is $496,000, or 52% of the replacement cost of $950,000. On the other hand, the opti on on the infill land is development timing and abandonment. The option value is as high as $45.65 million, but only 26% of the development cost of $177.5 million. Due to the scale of the valu ations, it is better to have the option in the building and the options in the la nd valued separately. But the im pact of management flexibility on acquisition and operation is as significant as, if not more than, that on development. The consideration of operating flexibility in ac quisition is important. It adds competitive value to the bid for a property. In the case st udies, the building is worth $25 million, and the land is worth $22 million, totaling $47 million. In other words, the infill land is worth almost as much as the building. This is very close to reality, because the prope rty was actually sold for $43.5 million. Note that the present value of th e development project wit hout any flexibility is negative $23 million. With negative NPV, the pr oject will not break ground. This means if management does not incorporate the flexibilities into the land valuation, the development is deemed worthless, and so is the land. PAGE 116 116 The RERO framework explores a few unconventi onal real option cases, including (1) jump diffusion process that does not go back to normal diffusion, (2) risk drivers that do not follow the multiplicative stochastic movement, (3) private ri sk that has no market equivalent and hence violating the noarbitrage option pricing assumption. All of thes e can be implemented through a binomial lattice with Monte Carlo simulations or the DTA approach The RERO framework is a simple yet powerful tool, intuitiv e to the practitioners, yet math ematically correct and precise. Recommendations for Future Research There are at least three direct ions that future research can go in: model perfection, game theory and phase investment. Model perfection is to improve the precisen ess of outcome from the RERO models. Lattice is a discretetime method for option pr icing. The smaller the time step, the closer the result will be to that calculated by continuoustime methods. At the same time, the development cost is assumed to have three values in our case study: the optimistic value, the most likely value, and the pessimistic value. More branches can be added to produce a more precise result. By dividi ng the lattice into more time step s, and breaking the development cost into more branches, a more pr ecise result will be generated. A significant factor not considered in this study is competition. Without the consideration of competition, in most cases it is optimal to defer exercising an option until the end of the holding period. However, competition erodes the valu e of waiting, affects the value of option as competitors enter or exit the market place and changes the market dynamics (Williams 1993; Myerson, 1991). Should game theory be incorporat ed into the RERO fram ework, we predict the option value would be slightly lower, and hence even closer to the closing price. The other direction is stage i nvestment and phased investment. Real estate development is a lengthy process, and it usually takes 2 to 3 years, if not longer. During this period, a lot of uncertainties can change the managerial strategies Stage investment refers to dividing a real PAGE 117 117 estate development project into di fferent stages: planning, design, c onstruction, sales, etc. This process can be valued similar to pharmaceutical re search and development. Phased investment refers to dividing a large real estate development project into different phases, for example, Phase I retail corridor, Phase II residential condominium, Phase II I office and hotel towers, etc. Decisions at later phases are contingent upon the out come of earlier phases. However, there can be timing overlaps between two phases. 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Equilibrium and Options on Real Assets, The Review of Financial Studies 6 (4), 825850. Williams, Joseph T. (1997). Redevelopment of Real Assets, Real Estate Economics 25 (3), 387407. Yao, Junkui, and Ali Jaafari. (2003). C ombining Real Options and Decision Tree, The Journal of Structured and Project Finance 9 (Fall), 5370. PAGE 122 122 BIOGRAPHICAL SKETCH NgaNa Leung earned her PhD degree in build ing construction from the University of Florida, Gainesville, FL. While earning this de gree, she worked as an acquisition analyst for Parmenter Realty Partners in Miami, FL later fo r Acadia Realty Trust in White Plains, NY, and now for Antares Investment Partners in Greenwich, CT. She also holds a master of science degree in re al estate from the University of Florida, a master's degree in building from the National Univ ersity of Singapore, Singapore, and a bachelor's degree in architecture fr om Tongji University, Shanghai, China. NgaNa worked as an assistant project ma nager in the Environetics Design Group in Shanghai, China prior to coming to the US. At UF, she was supported by the Alumni Fellowship, the highest meritbased award for gr aduate students. After graduation NgaNa will continue her career in commercial real estate investment, including acquisitions, development, and management. 