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CONTINGENT CLAIMS ANALYSIS OF OPTIMAL INVESTMENT DECISION MAKING IN THE MANAGEMENT OF TIMBER STANDS By SHIV NATH MEHROTRA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Shiv Nath Mehrotra ACKNOWLEDGMENTS I am grateful to my supervisory committee chair, Dr. Douglas R. Carter, cochair, Dr. Janaki R. Alavalapati, and Drs. Donald L. Rockwood, Alan J. Long and Charles B. Moss for their academic guidance and support. I particularly wish to thank Dr. Charles Moss for always finding time to help with the finance theory as well as for aiding my research in many ways. I thank my family for their support and encouragement. TABLE OF CONTENTS page ACKNOWLEDGMENT S ................. ................ iii....... .... LIST OF TABLES ........._.._... ..............vi..._.._........ LIST OF FIGURES .............. ....................vii AB S TRAC T ......_ ................. ..........._..._ viii.. CHAPTER 1. INTRODUCTION ................. ...............1.......... ...... Economic Conditions in Timber Markets ................. ...............1............ ... The Forest Industry in Florida ................ ...............2............ ... Outline of the Investment Problem ................. ...............3............ ... Research Objectives............... ............... 2. PROBLEM BACKGROUND .............. ...............9..... Introduction to Slash Pine....................... .... ............ Slash Pine as a Commercial Plantation Crop .............. ...............9..... Slash Pine Stand Density ................. ...............11................ Thinning of Slash Pine Stands............... ...............12. Financial Background ............... ... ........... ............ .............1 The Nature of the Harvesting Decision Problem ................. .......................14 Arbitrage Free Pricing .............. ... .... ... ... ... .. .......... ...........1 Review of Literature on Uncertainty and Timber Stand Management. .................. ....20 3. THE CONTINGENT CLAIMS MODEL AND ESTIMATION METHODOLOGY .............. ...............26.... The OnePeriod Model .............. ...............26.... The Deterministic Case .............. ...............26.... The Stochastic Case....................... ........................2 Form of the Solution for the Stochastic Value Problem. ........._..._.._ ........_.......3 1 The Contingent Claims Model ................. ...............31................ The Lattice Estimation Models............... ...............38. T he B inomi al Latti ce M od el .............. ...............3 8.... The Trinomial Lattice Model for a Mean Reverting Process..............................42 The Multinomial Lattice Model for Two Underlying Correlated Stochastic Assets ........._.___..... .__ ...............43.... 4. APPLICATION OF THE CONTINGENT CLAIMS MODEL ................ ...............45 Who is the Pulpwood Farmer? ............ .... ...............45 The Return to Land in Timber Stand Investments................ ..............5 On the Convenience Yield and the Timber Stand Investment .............. ..................57 Dynamics of the Price Process ................. ......... ...............60..... Modeling the Price Process .............. ...............63.... The Geometric Brownian Motion Process .............. ........... ...............6 Statistical Tests of the Geometric Brownian Motion Model ............... .... ...........67 The Mean Reverting Process............... .. .... ................7 Statistical Tests of the Mean Reverting Process Model .............. ................74 Instantaneous Correlation ............ .....___ ......__ ............7 The D ata..................... ... .... ..........7 Growth and Yield Equations .............. ...............76.... Plantation Establishment Expenses .............. ...............78.... RiskFree Rate of Return ............ .....___ ...............79.. The Model Summarized .............. ...............79.... 5. RE SULT S AND DIS CU SSION............... ..............8 A Single Product Stand and the Geometric Brownian Motion Price Process ............81 Sensitivity Analysis............... .... ... .. ...........8 Comparison with the Dynamic Programming Approach .................. ...............89 A Single Product Stand and the Mean Reverting Price Process ............... .............90 The Multiple Product Stand and Geometric Brownian Motion Price Processes........93 Thinning the Single Product Stand and the Geometric Brownian Motion Price Process .............. ...............96.... D discussion .................. .. ....... .... ........... .............9 Recommendations for Further Research .............. ...............104.... APPENDIX CORRELATION OF FIRST DIFFERENCES OF AVERAGES OF TWO RANDOM CHAINS................ ...............105 LI ST OF REFERENCE S ................. ...............107................ BIOGRAPHICAL SKETCH ................. ...............115......... ...... LIST OF TABLES Table pg 11. Comparison of applied Dynamic Programming and Contingent Claims approaches ................. ...............6................. 21. Area of timberland classified as a slash pine forest type, by ownership class, 1980 and 2000 (Thousand Acres) .............. ...............10.... 31. Parameter values for a three dimensional lattice ......____ ... ......_ ...............44 41. Florida statewide nominal pine stumpage average product price difference and average relative prices (19802005) ...._.. ................ ............... 46 .... 42. The effect of timber product price differentiation on optimal Faustmann rotation...47 43. The effect of timber product relative prices on optimal Faustmann rotation ............47 44. Estimated GBM process parameter values for Florida statewide nominal quarterly average pulpwood prices .............. ...............66.... 45. Results of JarqueBera test applied to GBM model for Florida statewide nominal quarterly average pulp wood stumpage prices .............. ...............70.... 46. Inflation adjusted regression and MR model parameter estimates............._._... .........73 47. Results of JarqueBera test applied to MR model residuals for Florida statewide nominal quarterly average pulpwood stumpage prices ................ ........._ ......75 48. Average per acre plantation establishment expenses for with a 800 seedlings/acre planting density .............. ...............78.... 51. Parameter values used in analysis of harvest decision for single product stand with GBM price process............... ...............82 52. Parameter values used in analysis of harvest decision for single product stand with MR price process............... ...............91 53. Parameter values used in analysis of harvest decision for multiproduct stand with GBM price processes .............. ...............93.... LIST OF FIGURES Figure pg 11. Florida statewide nominal quarterly average pine stumpage prices (19762005 II qtr) .............. ...............1..... 31. Typical evolution of evenaged stand and stumpage values for the Faustmann analy si s............... ............... 2 41. Sample autocorrelation function plot for nominal Florida statewide pulpwood stumpage instantaneous rate of price changes.. .........._..._ ................. ....._._.69 42. Sample autocorrelation function plot for nominal Florida statewide pulpwood stumpage price MR model regression residuals ................. .....___ .........._.....75 51. Total per acre merchantable yield curve for slash pine stand............... ..................8 52. Crossover price line for single product stand with GBM price process...................83 53. Crossover price lines for different levels of intermediate expenses ........................85 54. Crossover price line for different levels of standard deviation .............. .................86 55. Crossover price lines for varying levels of positive constant convenience yield......87 56. Crossover price lines for different levels of current stumpage price.........................88 57. Crossover price line for single product stand with MR price process.......................92 58. Merchantable yield curves for pulpwood and CNS ................. ........................93 59. Crossover price lines for multiproduct stand ................. ...............95......_._. . 510. Single product stand merchantable yield curves with single thinning at different ages ........._._. ._......_.. ...............97..... 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 CONTINGENT CLAIMS ANALYSIS OF OPTIMAL INVESTMENT DECISION MAKING IN THE MANAGEMENT OF TIMBER STANDS By Shiv Nath Mehrotra August 2006 Chair: Douglas R. Carter Cochair: Janaki R. Alavalapati Maj or Department: Forest Resources and Conservation The treatment of timber stand investment problems involving stochastic market prices for timber and multiple options can be considerably improved by the application of real options analysis. The analysis is applied to the dilemma of mature slash pine pulpwood crop holders in Florida facing depressed markets for their product. Using a contingent claims approach an arbitrage free market enforced value is put on the option of waiting with or without commercial thinning, which when compared with the present market value of stumpage allows an optimal decision to be taken. Results for two competing models of timber price process support the decision to wait for a representative unthinned 20yearold cutover slash pine pulpwood stand with site index 60 (age 25) and initial planting density 800 trees per acre. The present (III Qtr 2005) value of stumpage is $567/acre as compared to the calculated option value for the Geometric Brownian motion price process of $966/acre and $1,290/acre for the Mean Reverting price process. When the analysis differentiates the merchantable timber yield between products pulpwood and chipnsaw with correlated Geometric Brownian motion price processes the option value rises to $1,325/acre for a stumpage market value of $585/acre. On the other hand the commercial thinning option holds no value to the single product stand investment when the poor response of the slash pine species to late rotation thinning is accounted for. The analysis shows that the measurement of option values embedded in the timber stand asset is hampered by the lack of availability of market information. The absence of a market for the significant catastrophic risk associated with the asset as well other non marketed risks also hampers the measurement of option values. The analysis highlights the importance of access to market information for optimal investment decision making for timber stand management. It concludes that stand owners can realize the full value of the significant managerial flexibility in their stands only when access to market information improves and markets for trading in risks develop for the timber stand investment. CHAPTER 1 INTTRODUCTION Economic Conditions in Timber Markets Pine pulpwood prices in Florida have been declining since the peaks of the early 1990's (Figure 11). After reaching levels last seen in the early 1980's, in 2005 the prices have shown signs of a weak recovery. The trend in pulpwood markets reflects the impact of downturn in pulp and paper manufacturing resulting from several factors (Ince 2002) like: 50.00 45.00 O *Saw Timber Price S25.00 aChipnsaw Price P Pulpwood Prices 5.00 Year Source: Timber MartSouth Figure 11. Florida statewide nominal quarterly average pine stumpage prices (19762005 II qtr) 1. A strong US dollar, rising imports and weakness in export markets since 1997. 2. Mill ownership consolidation and closures. 3. Increased paper recycling along with continued expansion in pulpwood supply from managed pine plantations, particularly in the US South. In a discussion of the findings and proj sections of the Resource Planning Act (RPA), 2000 Timber Assessment (Haynes 2003), Ince (2002) has noted that the pulp and paper industry sector has witnessed a fall in capacity growth since 1998 with capacity actually declining in 2001. The report proj ects that US wide pulpwood stumpage prices would stabilize in the near term with a gradual recovery, but would not increase appreciably for several decades into the future. With anticipated expansion in southern pine pulpwood supply from maturing plantations, pine stumpage prices are proj ected to further subside after 2015. Pine pulpwood stumpage prices are not proj ected to return to the peak levels of the early 1990's in the foreseeable future (Adams 2002). Nevertheless, the US South is proj ected to remain the dominant region in production of fiber products and pulpwood demand and supply. The Forest Industry in Florida Florida has over 16 million acres of forests, representing 47% of the state' s land area. Nonindustrial private forest (NIPF) owners hold approximately 53% of the over 14 million acres of timberland in the state (Carter and Jokela 2002). The forest based industry in Florida has a large presence with close to 700 manufacturing facilities. The industry produces over 900,000 tons of paper and over 1,700,000 tons of paperboard annually apart from hardwood and softwood lumber and structural panels (AF&PA 2003). Pulpwood and sawlogs are the principal roundwood products in Florida accounting for up to 80% of the output by volume. Pulpwood alone accounted for more than 50% of the roundwood output in 1999. NIPF land contributed 45% of the total roundwood output while an equal percent came from industry held timberlands. Slash and longleaf pine provided 78% of the softwood roundwood output (Bentley et al. 2002). Forest lands produce many benefits for their owners who express diverse reasons for owning them. A survey of private forest land owners in the US South by Birch (1997) found that nearly 3 8% of the private forestland owners hold forestland primarily because it is simply a part of the farm or residence. Recreation and esthetic enj oyment was the primary motive for 17% while 9% of the owners stated farm or domestic use as the most important reason for owning forest land. Amongst commercial motives, land investment was the primary motive for 12% of the owners. At the same time expected increase in land value in the following 10 years was listed as the most important benefit from owning timberland by 27% of landowners accounting for 21% of private forests listed. Significantly, timber production was the primary motive for only 4% of the private forestland owners, but these owners control 35% of the private forestland. Similarly, only 7% of the owners have listed income from the sale of timber as the most important benefit in the following 10 years, but they control 40% of the private forest. Outline of the Investment Problem Timberland is defined as land that either bears or has the potential to bear merchantable quality timber in economic quantities. The US has nearly 740 million acres of forestland, of which 480 million acres is classified as timberland and the rest are either preserves or lands too poor to produce adequate quality or quantity of merchantable timber (Wilson 2000). Small private woodlot ownership (<100 acres) accounts for more than 90% of NIPF timberland holdings in the US and remains a significant part of the investment pattern (Birch 1996). The prolonged depression in pulpwood prices poses a dilemma for NIPF small woodlot timber cultivators in Florida who are holding a mature pulpwood crop. These pulpwood farmers must decide about harvesting or extending the rotation. The option to extend the rotation and wait out the depressed markets brings further options like partial realization of revenues immediately through commercial thinnings. These decisions must be made in the face of uncertainty over the future market prices) for their timber productss. Slash pine pulpwood stand owners must also contend with the fact that the species does not respond well to late rotation thinnings, limiting the options for investing in late rotation products (Johnson 1961). The timber stand investment is subj ect to several risks, marketed as well as non marketed (e.g., risk of damage to the physical assets in the absence of insurance). Understanding and incorporating these risks into management decisions is crucial to increasing the efficiency of the investment. The asset value/price risk is the most common form of risk encountered by all investors. For most forms of investments markets have developed several Einancial instruments for trading in risk. Insurance products are the most common while others such as forwards, futures and options are now widely used. Unfortunately, timberland investments lag behind in this respect. Institutional timberland investors, with their larger resources, deal with specific risk by diversification (geographic, product). Small woodlot owners must contend with the greatest exposure to risk. Investment risk in timber markets has been long recognized and extensively treated in literature. As a result, on the one hand, there is a better appreciation of the nature and importance of correctly modeling the stochastic variables, and on the other hand, there is improved insight into the nature of the investment problem faced by the decision maker. Despite the considerable progress, no single universally acceptable approach or model has yet been developed for analyzing and solving these problems. Due to the Einancial nature of the problem, developments in Einancial literature have mostly preceded progress in forest economics research. In the last decades, the most important and influential development in Einancial theory has been that of the option pricing theory. Several timber investment problems are in the nature of contingent claims and best treated by the application of option pricing theory or what is described as real options analysis (since the investments are real as opposed to Einancial instruments). It is known that for investment decisions characterized by uncertainty, irreversibility, and the ability to postpone, investors set a higher hurdle rate. Stand management decisions like commercial thinning and final harvest share these characteristics. Options analysis provides a means for valuing the flexibility in these investments. There are two approaches to options analysis, namely, the dynamic programming (DP) approach and the contingent claims (CC) approach. Almost all treatment of investment problems in forestry literature uses the DP approach to options analysis. Despite its popularity in research, the applied DP approach has some drawbacks which limit its utility for research or empirical applications. The CC valuation is free from these limitations. Some important features of the application of the two approaches are compared in Table 1.1. The most critical problem is that application of the DP approach requires the determination of an appropriate discount rate. In the absence of theoretical guidance on the subj ect studies are forced to use arbitrary discount rates with little relation to the risk of the asset. For example, Insley (2002) uses a discount rate of 5%, Insley and Rollins (2005) use 3% and 5% real discount rates alternately, while Plantinga (1998) uses a 5% "real riskfree" discount rate even though the analysis uses subj ective probabilities. No justification is offered for the choice of the discount rate (Plantinga (1998) cites Morck et al. (1989) for providing a rate "typical" to timber investment). Hull (2003) illustrates the difference between the discount rate applicable to the underlying instrument and the option on it. For a 16% discount rate applicable to the underlying, the illustration shows that the discount rate on the option is 42.6%. Explaining the higher discount rate required for the option, Hull (2003) mentions that a position on the option is riskier than the position on the underlying. Another problem with the use of arbitrary discount rates is that the results of different studies are not comparable. Table 11. Comparison of applied Dynamic Programming and Contingent Claims approaches Dynamic Programming Approach 1. Requires the use of an externally determined discount rate. This discount rate is unobservable (unless the option itself is traded). The discount rates used in published forestry literature bear no relation to the risk of the asset. 2. Published forestry literature does not specify whether the marketed, the nonmarketed or both components of the asset's risk are being treated. 3. Risk preferences are treated inconsistently in published forestry literature . 4. Requires use of historical estimates of mean return or drift which is susceptible to large statistical errors. Contingent Claims Approach Uses a riskfree discount rate that is reliably estimated from existing market instruments. Distinguishes between marketed and nonmarketed components of the assets risk. Applies only to marketed risk. Extensions have been proposed to account for nonmarketed risk It is a risk neutral analysis. Replaces the drift with the riskfree rate of return. Estimates of historical variance are relatively stable. Similarly, none of the published research on options analysis in forestry specifies whether the marketed, the nonmarketed or both risks are being treated. Since the only stochasticity allowed is in the timber price, it may be possible to infer that the marketed risk is the obj ect of the analysis. But such inference would challenge the validity of some of their conclusions. For example, Plantinga (1998) concludes that reservation price policies, on an average, increase rotation lengths in comparison to the Faustmann rotation, while management costs decrease rotation lengths. By including a notional cost of hedging against nonmarketed risks (insurance purchase) in the analysis as a management cost any conclusion regarding the rotation extension effect of reservation prices policies would be cast in doubt without better market data on the size of these hedging costs. Failure to highlight the treatment of risk preferences in the analysis is another source of confusion. Some studies like Brazee and Mendelsohn (1988) specify that the decision maker is risk neutral. Knowing this helps individuals to interpret the results according to their risk preferences. But when risk preferences are not specified, as in Insley (2002) for example, and there is confusion over the discount rate applied, the results produced by the analysis lose interpretative value. Real options analysis as it is applied through contingent claims valuation is itself a nascent branch of the option pricing theory which has developed principally by extending option pricing concepts to the valuation of real assets. There is increasing recognition of the shortcomings of the techniques developed for pricing financial asset options when applied to real assets and several modified approaches have been proposed. Nevertheless, application of real options analysis to timber investment decisions offers an opportunity to take advantage of a unified financial theory to treat the subject and thus obtain a richer interpretation of the results. Research Objectives The general obj ective of this study is to apply contingent claims analysis to examine typical flexible investment decisions in timber stand management, made under uncertainty. The analysis is applied to the options facing the NIPF small woodlot owner in Florida holding a mature even aged slash pine pulpwood crop. The specific obj ectives are 1. To analyze and compare the optimal clearcut harvesting decision for a single product, i.e., pulpwood, producing stand with Geometric Brownian Motion (GBM) and Mean Reverting (MR) price process alternately. 2. To analyze the optimal clearcut harvesting decision for a multiple product, i.e. pulpwood and chipnsaw, producing stand with their prices following correlated GBM processes. 3. To analyze the optimal clearcut harvesting decision with an option for a commercial thinning for a single product, i.e., pulpwood, producing stand with a GBM price process. CHAPTER 2 PROBLEM BACKGROUND Introduction to Slash Pine Slash pine (Pinus elliottii var. elliottii) is one of the hard yellow pines indigenous to the southeastern United States. Other occasional names for the specie are southern pine, yellow slash pine, swamp pine, pitch pine, and Cuban pine. Along with the most frequently encountered variety P. elliottii var. elliottii the other recognized variety is P. elliottii var. densa, which grows naturally only in the southern half of peninsula Florida and in the Keys (Lohrey and Kossuth 1990). The distribution of slash pine within its natural range (80 latitude and 100 longitude) was initially determined by its susceptibility to fire injury during the seedling stage. Slash pine grew throughout the flatwoods of north Florida and south Georgia as well as along streams and the edges of swamps and bays. Within these areas either ample soil moisture or standing water protected young seedlings from frequent wildfires in young forests (Lohrey and Kossuth 1990). Slash pine is a frequent and abundant seed producer and is characterized by rapid early growth. After the sapling stage it can withstand wildfires and rooting by wild hogs which has helped it to spread to drier sites (Lohrey and Kossuth 1990). Slash Pine as a Commercial Plantation Crop Florida has the largest area of timberland (Barnett and Sheffield 2004) classified as slash pine forest type (49%) while nonindustrial private landowners hold the largest portion of slash pine timberland (Table 21) Table 21. Area of timberland classified as a slash pine forest type, by ownership class, 1980 and 2000 (Thousand Acres) Ownership Class 1980 2000 National Forest 522 493 Other Public 569 684 Forest Industry 4,649 3,719 Nonindustrial Private 7,039 5,479 Total 12,779 10,375 Source: Barnett and Sheffield, 2004 Slash pine makes rapid volume growth at early ages and is adaptable to short rotations under intensive management. Almost threefourths of the 50year yield is produced by age 30, regardless of stand basal area. Below age 30, maximum cubic volume yields are usually produced in unthinned plantations, so landowners seeking maximum yields on a short rotation will seldom find commercial thinning beneficial. Where sawtimber is the obj ective, commercial thinnings provide early revenues while improving the growth and quality of the sawtimber and maintaining the stands in a vigorous and healthy condition (Lohrey and Kossuth 1990). A study by Barnett and Sheffield (2004) found that a maj ority (59%) of the slash pine inventory volume in plantations and natural stands was in the <10" dbh class while about 25% of the stands were less than 8 years old. The study concluded that this confirmed the notion that slash pine rotations are typically less than 30 years and that the stands are intensively managed. Plantation yields are influenced by previous land use and interspecies competition. Early yields are usually highest on recently abandoned fields where the young trees apparently benefit from the residual effects of tillage or fertilizer and the nearly complete lack of vegetative competition. Plantations established after the harvest of natural stands and without any site treatment other than burning generally have lower survival and, consequently, lower basal area and volume than stands on old fields. Yields in plantations established after timber harvest and intensive site preparation such as disking or bedding are usually intermediate. Comparing slash pine to loblolly pine (Pinus taeda L.), Shiver (2004) notes that slash pine may be preferred over loblolly pine for reasons other than wood yields. For instance, slash pine would be the favored species for landowners who want to sell pine straw. Slash pine also prunes itself much better than loblolly, and for solid wood products the lumber grade will probably be higher for slash pine. Slash pine is more resistant to southern pine beetle (Dendroctonus frontalis Zimmermann) attack than loblolly and it is rarely bothered with pine tip moth (Rhyacionia frustr anar~rtrt~t~r (Comstock)), which can decimate young loblolly stands. Slash Pine Stand Density Dickens and Will (2004) discuss the effects of stand density choices on the management of slash pine stands. The choice of initial planting density and its management during the rotation depends on landowner obj ectives like maximizing revenues from pine straw, obtaining intermediate cash flows from thinnings or growing high value large diameter class timber products. High planting density in slash pine stands decreases tree diameter growth as well as suppresses the tree height growth to a lesser extent, but total volume production per unit of land is increased. However, the volume increment observed for early rotation ages soon peaks and converges to that of lower density stands as the growth rate of high density stands reach a maximum earlier. Citing a study at the Plantation Management Research Cooperative, Georgia, Dickens and Will (2004) remark that management intensity does not change the effects of stand density. Dickens and Will (2004) mention that higher density plantings achieve canopy closure, site utilization, and pine straw production earlier than lower density plantings under the same level of management. Higher planting densities also may be beneficial on cutover sites with low site preparation and management inputs. The higher planting densities help crop trees occupy the site, whereas the lower planting densities may permit high interspecific competition until much later during stand development, reducing early stand volume production. Thinning of Slash Pine Stands Mann and Enghardt (1972) describe the results of subj ecting slash pine stands to three levels of thinnings at ages 10, 13 & 16. Early thinnings removed the diseased trees while later thinnings concentrated on release of better stems. Their study concluded that early and heavy thinnings increased diameter growth but reduced volume growth. The longer thinnings were deferred, the slower was the response in diameter growth. They concluded that age 10 was too early for a thinning as most of the timber harvested was not merchantable and volume growth was lost, even though the diameter increment results were the best. The decision between thinning at ages 13 and 16 depended on the end product, the ability to realize merchantable volumes in thinnings and the loss of volume growth. They recommend that short rotation pulpwood crops were best left unthinned as the unthinned stands had good volume growth. Quoting Mann and Enghardt (1972) "volume growth is good, no costs are incurred for marking, there are fewer small trees to harvest and stand disturbances that may attract bark beetle are avoided" (Mann and Enghardt 1972, p.10). Johnson (1961) has discussed the results of a study of thinning conducted on heavily stocked industrial slash pine stands of merchantable size. The study found that slash pine does not respond well to late release i.e., if it has been grown in moderately dense stands for the first 20 to 25 years of its life. It does not stagnate, except perhaps on the poorest sites, but it cannot be expected to respond to cultural treatments such as thinnings as promptly or to the degree desired. Johnson (1961) observes that the typical thinning operation that removes four to six cords of wood from wellstocked stands is nothing more than an interim recovery of capital from the forestry enterprise. These thinnings do not stimulate growth of the residual stand or total production. The study found no real increase in total volume production or in average size of trees fr~om commercial thrinning\ in slash pine stands being managed on short rotations for small products. Johnson (1961) concludes that silvicultural considerations for commercial thinning in small product slash pine forest management are secondary to commercial considerations because of its response to intermediate cuttings. Financial Background The timber farming investment exposes the investor to the risks that the asset carries. These risks come in the form of marketed risks like the volatile market price for the timber products or nonmarketed risks that also effect the value of the investment such as hazards that threaten the investment in the form of fire, pests, adverse weather etc. Usually, investors separate the spectrum of risks taken on by them from an investment into core and noncore risks. The core risk could be the market price of the investments output or product. This is the risk the investor expects to profit out of and likes to retain. The noncore risk like the nonmarketed risks listed above are undesirable and the investor would ideally like to transfer such risks. A common market instrument for risk transfer is the insurance product. By paying a price one can transfer the undesirable risk to the market. If the nonmarketed risks associated with the timber investment were marketed, the market data available can be incorporated into investment analysis. In the absence of markets for a part or all of an assets risk, the common asset pricing theories are not applicable and alternate methods have to be applied. The analysis in this study is restricted to the marketed risk in the form of timber price risk only. The Nature of the Harvesting Decision Problem Following a price responsive harvesting regime, the slash pine pulpwood farming investor holding a mature crop and facing a stochastically evolving pulpwood market price would like to know the best time for selling his crop. From his knowledge of past movements of market price for pulpwood the investor knows that the present price is lower than the average of prices in the recent past. He may sell the crop at the present price but significantly he has the option to hold the crop. The crop is still growing, both in size and possibly in value, and that provides incentive to hold the harvest. But the market price is volatile. The future market price for pulpwood cannot be predicted with certainty. How does the investor decide his immediate action; sell or hold? While equilibrium asset values are determined by their productive capacities their instantaneous market values are determined by the ever changing market forces. Asset holders would like to eamn a fair compensation on their investment i.e., the principal plus a return for the risk undertaken by holding the investment over time. But there is no guarantee to earning a 'fair' return in the market place. Usually investors have a finite time frame for holding an asset and must realize the best value for their asset in this period. The decision to hold the asset for a future sale date is a gamble, an act of speculation. It carries the risk of loss as well as the lure of profit. But all investments in risky assets are speculative activities. One investment may be more risky than another but one market equilibrium theory in the form of the Capital Asset Pricing Model (CAPM) assures us that their expected returns are proportional to their risk, specifically to the systematic or nondiversifiable portion of their risk. The CAPM theory, development of which is simultaneously attributed to Sharpe (1963, 1964) and Lintner (1965a, 1965b) amongst others, has it that at any point in time each marketed asset has an associated equilibrium rate of return which is a function of its covariance with the market portfolio and proportional to the market price of risk. The expression 'rate of return' refers to the capital appreciation plus cash payout, if any, over a period of time, expressed as a ratio to the asset value at the commencement of the period. If all risky investments are gambles, how does one choose amongst the enormous variety of gambles that are available in the market place? Once again, financial theory informs us that the choice amongst risky assets depends on the risk attitudes of individuals. Individuals would apportion their wealth amongst a portfolio of assets (which serves to eliminate the nonsystematic risk of the assets). The portfolio is constructed to match the riskreturn tradeoff sought by the individual. Once chosen, how does one decide how long to hold an asset? The risk associated with every asset as well as its expected return changes over time. Over a period of time the riskreturn characteristic of a particular asset may lose its appeal to the individual's portfolio which itself keeps changing with maturing of risk attitudes over time. Returning to the pulpwood farmer' s decision problem, the question boils down to this: How does the pulpwood farmer decide whether his investment is worth holding anymore? It follows from the arguments above that the crop would be worth holding as long it can be expected to earn a return commensurate with its risk. But, how is the comparison between the expected rate of return and the required rate of return achieved? The usual financial technique is to subj ectively estimate the expected cash flows from the asset, discount them to the present using a riskadjusted discount rate, and compare the resulting value to the present market value of the asset. If the expected discounted value is higher, then the expected rate of return over the future relevant period under consideration is higher than the required rate of return. And how does this work? It works because the required rate of return and the riskadjusted discount rate are different names for the same value. The expected equilibrium rate of return generated by the CAPM represents the average return for all assets sharing the same risk characteristics or in other words, the opportunity cost. When we use the riskadjusted discount rate to calculate the present value of the future cash flows, we are in effect accounting for the required rate of return. The discounting apportions the future cash flows between the required rate of return and residual value, if any. Can discounted cash flow (DCF) analysis be used to solve the pulpwood farmer' s harvesting problem? The pulpwood farmer' s valuation problem is compounded by the ability to actively manage the investment (flexibility) or more specifically, the ability to postpone the harvest decision should the need arise. Not only do decision makers have to deal with an uncertain future market value for the pulpwood crop but they must also factor in the response to the possible values. The termination date or harvest date of the timber stand investment and thus its payoff is not fixed or predetermined. Traditional DCF analysis can deal with the price uncertainty by the technique of subj ective expectations but has no answer for flexibility of cash flow timings. This shortcoming has been overcome by decision analysis tools like decision trees or simulation to account for the state responsive future cash flows. So, are tools like decision trees or simulation techniques the answer to the pulpwood farmer's dilemma? Almost, except that the appropriate discount rate still needs to be determined. Arbitrage Free Pricing Despite widespread recognition of its shortcomings, the CAPM generated expected rate of return is most commonly used as the riskadjusted discount rate appropriate to an investment. It turns out that while the meanvariance analysis led school of equilibrium asset pricing does a credible job of explaining expected returns on assets with linear risk they fail to deal with nonlinear risk of the type associated with assets whose payoffs are contingent. Hull (2003) provides an illustration to show that the risk (and hence discount rates) of contingent claims is much higher than that of the underlying asset. The pulpwood farmer holds an asset with a contingent claim because the payoff from his asset over any period is contingent on a favorable price being offered by the market for his crop. There are two alternate though equivalent techniques for valuing a risky asset by discounting its expected future cash flows. One, as already described involves an adjustment to the discount rate to account for risk. The other method adjusts the expected cash flows (or equivalently, the probability distribution of future cash flows) and uses the riskfree rate to discount the resulting certainty equivalent of the future cash flows. The CC valuation procedure follows this certainty equivalent approach. The argument is based on the Law of One Price (LOP). The LOP argues that in a perfect market, in equilibrium, only one price for each asset, irrespective of individual risk preferences, can exist as all competing prices would be wiped out by arbitrageurs. Baxter and Rennie (1996) illustrate the difference between expectation pricing and arbitrage pricing using the example of a forward trade. Suppose one is asked by a buyer to quote today a unit price for selling a commodity at a future date T A fair quote would be one that yields no sure profit to either party or in other words provides no arbitrage opportunities. Using expectation pricing, the seller may believe that the fair price to quote would be the statistical average or expected price of the commodity, E [S, ], where S, is the unit price of the commodity at time T and E is the expectation operator. But a statistical average would turn out to be the true price only by coincidence and thus could be the source of significant loss to the seller. The market enforces an arbitrage free price for such trades using a different mechanism. If the borrowing/lending rate is r then the market enforced price for the forward trade is Sne'T This price follows the logic that it is the cost that either party would incur by borrowing funds at the rate r to purchase the commodity today and store it for the necessary duration (assuming no storage costs). This price would be different from the expected price, yet offer no arbitrage opportunities. The arbitrage free approach to the problem of valuing financial options was first solved by Black and Scholes (1973) using a replicating portfolio technique. The replicating portfolio technique involves finding an asset or combination of assets with known values, with payoffs that exactly match the payoffs of the contingent claim. Then, using the LOP it can be argued that the contingent claim must have the same value as the replicating portfolio. Financial options are contingent claims whose payoffs depend on some underlying basic Einancial asset. These instruments are very popular with hedgers or risk managers. The underlying argument to the equilibrium asset pricing methods is the no arbitrage condition. The no arbitrage condition requires that the equilibrium prices of assets should be consistent in a way that there is no possibility of riskless profit. A complete market offers no arbitrage opportunities as there exists a unique probability distribution under which the prices of all marketed assets are proportional to their expected values. This unique distribution is called a risk neutral probability distribution of the market. The expected rate of return on every risky asset is equal to the riskfree rate of return when expectations are calculated with respect to the market risk neutral di stributi on. Copeland et al. (2004) define a complete market as one in which for every future state there is a combination of traded assets that is equivalent to a pure state contingent claim. A pure state contingent claim is a security with a payoff of one unit if a particular state occurs, and nothing otherwise. In other words, when the number of unique linearly independent securities equals the total number of alternative future states of nature, the market is said to be complete. Equilibrium asset pricing theories have been developed with a set of simplifying assumptions regarding the market. In addition to completeness and pure competition, CC analysis theory assumes that the market is perfect i.e., it is characterized by 1.An absence of transaction costs & taxes 2.Infinite divisibility of assets. 3.A common borrowing and lending rate. 4.No restrictions on short sales or the use of its proceeds. 5.Continuous trading. 6.Costless access to full information. Review of Literature on Uncertainty and Timber Stand Management The published literature on treatment of uncertainty in timber stand management is reviewed here from an evolutionary perspective. A selected few papers are reviewed as examples of a category of research. The literature dealing with static analysis of financial maturity of timber stands is vast and diverse. Including the seminal analysis of Faustmann (1849) several approaches to the problem have been developed. The early work on static analysis has been summarized by Gaffney (1960) and Bentley and Teeguarden (1965). These approaches range from the zero interest rate models to present net worth models and internal rate of return models. The Soil Rent/Land Expectation Value (LEV) model, also known as the FaustmannOhlinPressler model, is now accepted as the correct static financial maturity approach. However, the static models are built on a number of critical assumptions which erode the practical value of the analysis. Failure to deal with the random nature of stand values is a prominent shortcoming. Uncertain future values mean that the date of optimal harvest cannot be determined in advance but must be price responsive i.e., it must depend on the movement of prices and stand yield amongst other things. The harvest decision is local to the time of decision and it is now recognized that a dynamic approach to address the stochastic nature of timber values is appropriate. Amongst the first to treat stochasticity in stand management, Norstom (1975) uses DP to determine the optimal harvest with a stochastic timber market price. The stochastic variable was modeled using transition matrices as in Gassmann (1988), who dealt with harvesting in the presence of Gire risk. The use of transition matrices has persisted with Teeter et al. (1993) in the determination of the economic strategies for stand density management with stochastic prices. However, much advance followed in modeling stochasticity with the introduction of the use of diffusion processes in investment theory. Brock et al. (1982) illustrated the optimal stopping problem in stochastic Einance using the example of a harvesting problem over a single rotation of a tree with a value that grows according to a diffusion process. Miller and Voltaire (1980, 1983) followed up, extending the analysis to the multiple rotation problems. Clarke and Reed (1989) obtained an analytical solution using the Myopic Look Ahead (MLA) approach, allowing for simultaneous stochasticity in timber price and yield. These papers illustrate the use of stochastic dynamic programming for stylized problems which are removed from the practical problems in forestry e.g., they ignore the costs in forestry. Modeling the empirical forestry problem, Yin and Newman (1995) modified Clarke and Reed (1989) to incorporate annual administrative and land rental costs as exogenous parameters. However, while acknowledging option costs, they chose to ignore them for simplicity. Also, as noted by Gaffney (1960) the solution to the optimal harvest problem is elusive because the land use has no predetermined cost and the solution calls for simultaneous determination of site rent and financial maturity. Since land in forestry investment is typically owned, not leased or rented, accounting for the unknown market land rental has been one obj ective of Einancial maturity analysis since Faustmann (1849). In the meanwhile, the use of search models to develop a reservation price approach gained popularity with papers by Brazee and Mendelsohn (1988) and others. The technique of the search models is not unlike the DP approach to contingent claims. The approach differs from the CC approach in solution methodology and in the interpretation of the results. Fina et al. (2001) presents an extension of the reservation price approach using search models to consider debt repayment amongst other things. Following the landmark Black and Scholes (1973) paper the development of methodology for the valuation of contingent claims has progressed rapidly. A useful simplification in the form of the discrete time binomial lattice to approximate the stochastic process was presented by Cox et al. (1979). Other techniques for obtaining numerical approximations have been developed including the trinomial approximation, the finite difference methods, Monte Carlo simulations and numerical integration. Geske and Shastri (1985) provide a review of the approximation techniques developed for valuation of options. An important simultaneous line of research has been the study of the nature of stochasticity in timber prices. Washburn and Binkley (1990a) tested for weak form efficiency in southern pine stumpage markets and reported that annual and quarterly average prices display efficiency, but also point out that monthly averages display serial correlation. Yin and Newman (1996) found evidence of stationarity in monthly and quarterly southern pine time series price data. Since reported prices for timber are in the form of period averages, researchers have to contend with unraveling the effect of averaging on the statistical properties of the price series. Working (1960) demonstrated the introduction of serial correlation in averaged price series, not present in the original series. However, Haight and Holmes (1991) demonstrated that serially correlated averaged price series tends to behave as a random walk. The lack of conclusive data on the presence or absence of stationarity in timber price data is because of the imperfections of the data available for analysis. Despite the lack of unanimity on the empirical evidence there is some theoretical support for the mean reversion (negative autoregression) arising from the knowledge that commodity prices could not exhibit arbitrarily large deviations from long term marginal cost of production without feeling the effects of the forces of demand and supply (Schwartz 1997). The use of contingent claim analysis is a relatively recent development in stand management literature. Morck et al. (1989) use real options analysis to solve for the problem of operating a fixed term lease on a standing forest with the option to control the cut rate. Zinkhan (1991, 1992) and Thomson (1992b) used option analysis to study the optimal switching to alternate land use (agriculture). Thomson (1992a) used the binomial approximation method to price the option value of a timber stand with multiple rotations for a GBM price process. The paper demonstrates a comprehensive treatment of the harvest problem, incorporating the option value of abandonment and switching to an alternate land use. Plantinga (1997) illustrated the valuation of a contingent claim on a timber stand for the meanreverting and driftless random walk price processes, using a DP approach attributed to Fisher and Hanemann (1986). Yoshimoto and Shoji (1998) use the binomial tree approach to model a GBM process for timber prices in Japan and solve for the optimal rotation ages. Insley (2002) advocated the meanreverting process for price stochasticity. The paper incorporates amenity values and uses harvesting costs as an exercise price to model the harvesting problem over a single rotation as an American call option. In order to obtain a numerical solution, the paper uses a discretization of the linear complementarity formulation with an implicit finite difference method. All these studies use a stochastic DP approach with an arbitrary discount rate. Hughes (2000) used the BlackScholes call option valuation equation to value the forest assets sold by the New Zealand Forestry Corporation in 1996. The option value estimated by him was closer to the actual sale value than the alternate discounted cash flow analysis. It is a unique case of a study applying real options analysis to value a real forestry transaction. Insley and Rollins (2005) solve for the land value of a public forest with mean reverting stochastic timber prices and managerial flexibility. They use a DP approach to show that by including managerial flexibility, the option value of land exceeds the Faustmann value (at mean prices) by a factor of 6.5 for a 3% discount rate. The land value is solved endogenously for an infinite rotation framework. In a break from analysis devoted to the problems of a single product timber stand Forboseh et al. (1996) study the optimal clear cut harvest problem for a multiproduct pulpwoodd and sawtimber) stand with joint normally distributed correlated timber prices. The study extends the reservation price approach of Brazee and Mendelsohn (1988) to multiple products and looks at the effect of various levels of prices and correlation on the expected land value and the probability of harvest at different rotation ages. A discrete time DP algorithm is used to obtain the solutions. In a similar study, Gong and Yin (2004) study the effect of incorporating multiple autocorrelated timber products into the optimal harvest problem. The paper models the timber prices pulpwoodd and sawtimber) as discrete first order autoregressive processes. Dynamic programming is used to solve for reservation prices. Teeter and Caulfied (1991) use dynamic programming to demonstrate the determination of optimal density management with stochastic prices using a first order autoregressive price process modeled using a transition probability matrix. The study uses a Eixed rotation age and allows multiple thinnings. Brazee and Bulte (2000) analyze an optimal evenaged stand management strategy with the option to thin (fixed intensity) with stochastic timber prices. Using a random draw mechanism for the price process and a backward recursive DP algorithm for locating the reservation prices, the study finds the existence of an optimal reservation price policy for the thinning option. Lu and Gong (2003) use an optimal stocking level function to determine the optimal thinning as well as a reservation price function to determine the optimal harvest strategy for a multiproduct stand with stochastic product prices without autocorrelations. CHAPTER 3 THE CONTINTGENT CLAIMS MODEL AND ESTIMATION METHODOLOGY The OnePeriod Model In order to develop the application of options analysis to investment problems it is helpful to first examine the nature of oneperiod optimization models. Oneperiod models for investment decision making operate by comparing the value of the investment in the beginning of period with its value at the end of the period. The model is first explained in the context of the deterministic Faustmann problem. This is followed by an extension of the logic to the stochastic problem. The Deterministic Case The problem of finding the optimal financial stand rotation age is an optimal stopping problem. In the deterministic Faustmann framework, the optimal rotation age is achieved by holding the stand as long as the (optimal) investment in the stand is compensated by the market at the required rate of return. The value of the immature stand is the value of all net investments in the stand up to the present including the land rental costs and the cost of capital. This means that the value of all investments in the stand (adjusted for positive intermediate cash flows like revenue from thinnings) up to the present compounded at the required rate of return represents the stand value. This value represents fair compensation to the stand owner for his investment and fair cost to the purchaser who would incur an identical amount in a deterministic world. Therefore, this value represents the fair market value of the premature stand. The market value of the merchantable timber in the stand, if any, is less than the stand market value in this period. The stand owner continues to earn the required return on his (optimal) investments only till the rotation age is reached when the value of the merchantable timber in the stand exactly equals the compounded value of investments. Beyond this rotation age the market will only pay for the value of the merchantable timber in the stand. If the stand is held longer than this rotation age, even if no fresh investments other than land rent are made, the market compensation falls short of the compounded value of investments as the value of merchantable timber grows at a lower rate. The optimal rotation age represents the unique point of financial maturity of the stand. Before this age the stand is financially immature and after this age the stand is financially over mature. A typical evolution of the two values is depicted in Figure 31. 4500 4000 3500 3000 S Value of Merchantable limber t 2500 * Present Value of net S2000 investments gj 1500 1000 500 Rotation age (Years) Figure 31. Typical evolution of evenaged stand and stumpage values for the Faustmann analysis Equivalently, a more familiar way of framing this optimization problem is to let the stand owner compare the value of harvesting the stand in the present period to the net (of cost of waiting) discounted values of harvests at all possible future rotation ages. The cost of waiting includes land rent and all other intermediate cash flows. More specifically, the comparison is between the value of a harvest decision today and the net discounted value of the stand in the next period assuming that similar optimal decisions are taken in the future. In this case the stand value represents the discounted value of a future optimal harvest which exactly equals the earlier defined stand market value consisting of net investment value. Thus, the problem is cast as a oneperiod problem. The oneperiod deterministic Faustmann optimization problem in discrete time can be summarized mathematically by Equation 31. F(t)= m;lnr, xlax ~ O+tr8) (31) Here, F()= Stand Value function t = Rotation age 0Z = Stand termination value or the market value of the merchantable timber in the stand ri = Rate of cash flow (land rental expenses, thinnings etc) B = Constant discount rate A = A discrete interval of time For the period that the decision to hold the stand dominates, the second expression in the bracket is relevant and we have for the holding period Equation 32. F(t + At) 32 It may be noted that the only decision required of the decision maker is whether to hold the stand or to harvest it. In the standard deterministic case, any intervention requiring new investments like thinnings is assumed optimally predetermined and the resulting cash flows are only a function of rotation age. This holding expression can be simplified to yield Equation 33 for the continuous time OF~t) rt+Ftl) (33) Here, the limit of A 4 0 has been taken. Equation 33 clearly expresses the holding condition in perfect competition as one in which the yield (Right Hand Side (RHS)) in the form of the dividend and the capital appreciation/depreciation or change in market value over the next infinitesimal period equals the required rate of return on the current market value of the asset (Left Hand Side (LHS)). The optimal stopping conditions are F(T) = OZ(T) (3 4) F,(T)= O,(T) (35) In Equation 35 the subscript t denotes the derivative of the respective function with respect to the time variable. The first condition is simply that at the optimal rotation age Tthe market value should equal the termination value and the second condition is the tangency or the smooth pasting condition (Dixit and Pindyck 1994) requiring that the slopes of the two functions should be equal. The Stochastic Case In the stochastic value framework, the problem of optimal rotation is equivalent to holding the asset as long as it is expected to eamn the required return. With stochastic parameter values, not only are future asset values dependent on the realizations of the parameters but the ability to actively manage the asset by responding to revealed parameter values induces an option value. Dixit and Pindyck (1994) derive the holding condition for the stochastic framework using the Bellman equation, which expresses the value as F(x,t) =max z~(x,u~,t)A(+(1+A) lEF(x~t+ A)l x~u) (36) Here, F()= Stand value function x = The (vector of) stochastic variable(s). For this analysis it represents the timber prices) t = Rotation age u = The control or decision variable (option to invest) xi = The rate of cash flow B = The discount rate E = Expectation operator A = A discrete interval of time This relation means that the present value F(x, t) from holding the asset is formed as a result of the optimal decision taken at the present, which determines the cash flow ai in the next period A and the expected discounted value resulting from taking optimal decisions thereon. Distinct from the deterministic case, in this case, the value (and possibly cash flows) depends on the stochastic timber price. Also, the decision can be expanded to include the decisions to make new investments in the stand (like thinning) which effects the immediate cash flows as well as expectations of future market values. Similar to the deterministic case, the holding condition can be reexpressed as OF(x, t) = max zcx,u~~t)+ E FxL ) (3 7) To quote Dixit and Pindyck (1994): The equality becomes a no arbitrage or equilibrium condition, expressing the investor' s willingness to hold the asset. The maximization with respect tou means the current operation of the asset is being managed optimally, bearing in mind not only the immediate payout but also the consequences for future values. (Dixit and Pindyck 1994, p.105) Form of the Solution for the Stochastic Value Problem In general the solution to the problem has the form of ranges of values of the stochastic variable(s) x Continuation is optimal for a range(s) of values and termination for otherss. But as elaborated by Dixit and Pindyck (1994), economic problems in general have a structured solution where there is a single cutoff x* with termination optimal on one side and continuation on the other. The threshold itself is a continuous function of time, referred to as the crossover line. The continuation optimal side is referred to as the continuation region and the termination optimal side as the termination region. As pointed out by Plantinga (1998) the values of the crossover stumpage price line for timber harvesting problems are equivalent to the concept of reservation prices popular in forestry literature. Consequently, the optimal stopping conditions for the stochastic case for all t are (Dixit and Pindyck 1994) F(x* (t), t) = O(x*(t), t) (38) Fx (x* (t), t) = 2x (x*(t), t) (39) In Equation 39 the subscript x denotes the derivative of the respective function with respect to the variable x . The Contingent Claims Model In this section the general theory of CC valuation is developed in the context of the harvest problem. The CC valuation approach is also built on a oneperiod optimization approach and the discussion of the last section should help to put the following discussion into perspective. The simplest harvest problem facing the decision maker is as follows: Should the stand be harvested immediately, accepting the present market value of the timber or should the harvest decision be postponed in expectation of a better outcome? That is, the possibility for all optimal interventions other than harvest is ignored. In a dynamic programming formulation of the problem, using the Bellman equation, the problem can be expressed mathematically as follows (Dixit and Pindyck 1994) I 1 F~(x,ti)=max O(x,t), r(x,ti)+18E[F(x,ti+1) x] (310) Here F(x, t) is the expected net present value of all current and future cash flows associated with the investment at time t, when the decision maker makes all decisions optimally from this point onwards. The stochastic state variable, timber price in the present problem, is represented by x The immediate cash flow from a decision to hold the investment is denoted by zi(x, t) The result of optimal decisions taken in the next period and thereafter will yield value F(x, t + 1) which is a random variable today. The expected value of F(x, t + 1) is discounted to the present at the discount rate B Finally, OZ(x, t) represents the present value of termination or the value realized when the investment is fully disposed off today. While we know the present termination value, we are interested in learning the value of waiting or the continuation value. If the decision to wait is optimally taken then the continuation value is given by _I 1 F(x, t) =z(x,t)++E [F(x,t +1)I x] (311) If the increments of time are represented by A and A 4 0, the continuation value expressed in continuous time after algebraic manipulation will be OF(x,ta)= z(x, t)+ E[dF] (3 12) If it is assumed that the state variable x (timber price) follows a general diffusion process of the form dx~ = pU(x, t)dt + a(x, t)dz (313) then, using Ito's Lemma, after algebraic manipulation and simplification we obtain the partial differential equation (PDE) a2FXY+ pUFx + Ft BF + z= 0 (314) Here, pu = pu(x, t), a = o(x, t) and ai = zi(x, t) In typical economic problems the continuation equation will hold for the value of the asset for all x > x*, where x* is a critical value of the state variable x with the property that continuation is optimal when the state variable value is on one side of it and stopping or termination is optimal when the state variable value is on its other side. This yields the boundary conditions for all t, given by Equations 38 and 39, which the value of the asset must meet at the critical value of the state variable The DP formulation of the problem assumes that the appropriate discount rate B is known or can be determined by some means. An equivalent formulation of the problem can be found using CC valuation. In this form the PDE for the continuation region value is given by G2FXY + (r 3)FX + Ft rF + = 0 (315) Here r represents the riskfree rate of return and 3 represents the rate of return shortfall which could be a dividend and/or convenience yield. Dixit and Pindyck (1994) illustrate the derivation of the contingent claim PDE by using the replicating portfolio method. In an alternate general derivation the procedure is to first show that under certain assumption all traded derivative assets must satisfy the noarbitrage equilibrium relation ar = A re, Here a is the expected return on the derivative security, a, represents the component of its volatility attributable to an underlying stochastic variable i and ii, represents the market price of risk for the underlying stochastic variable. Where there is only one underlying stochastic variable the relation simplifies to a r = Aso. Constantinides (1978) derived the condition for changing the asset valuation problem in the presence of market risk to one where the market price of risk was zero. The derivation, presented below, proceeds from Merton' s (1973) proof of equilibrium security returns satisfying the CAPM relationship a r =i 2pm (316) where ii= (s)is the market price of risk, the subscript p refers to the proj ect (asset, option etc) and subscript m refers to the market portfolio which forms the single underlying stochastic variable. Merton (1973) assumed that 1. The markets are perfect with no transaction costs, no taxes, infinitely divisible securities and continuous trading of securities. Investors can borrow and lend at the same interest rate and short sale of securities with full use of proceeds is allowed. 2. The prices of securities are lognormally distributed. For each security, the expected rate of return per unit time as and variance of return over unit time 0z2 OXiSt and are finite with 0,2 > 0 The opportunity set is nonstochastic in the sense that az >"22 and the covariance of returns per unit time o, and the riskless borrowinglending rate r are all nonstochastic functions of time. 3. Each investor maximizes his strictly concave and timeadditive utility function of consumption over his lifespan. Investors have homogenous expectations regarding the opportunity set. Let F(x, t) denote the market value of a proj ect. The market value is completely specified by the state variable x and time t, and represents the time and riskadjusted value of the stream of cash flows generated from the proj ect. Let the change in the state variable x be given by dre = udt + odz (317) The drift u and variance 02 may have the general form u = u(x, t) and 02 __ 2(x, t) .Let aidt denote the cash flow generated by the proj ect in time interval (t, t + dt) with ai = zi(x, t) Then, the return on the proj ect in the time interval (t, t + dt) is the sum of the capital appreciation dF(x, t) and the cash retuma~idt Assuming that the function F(x, t) is twice differentiable w.r.t. x and atleast once differentiable w.r.t. t, Ito's Lemma can be used to expand dF(x, t) as dFi(xt)=( F+uF +F,; dt + Fxdz (318) The rate of return on the proj ect is dF(x, t) +zidt__ 1 2 F z+,+~ +F dt+ dz (319) F (x, t) F 2 : Fto with expected value per unit time a,, and covariance with the market per unit time a, given by 1 a2 a = F +F uF +F ,; (320) F 2 J, = pa, smnce p = FF where p = p(x, t) is the instantaneous correlation coefficient between &z and the return on the market portfolio. By substitution in the Equation 316 we obtain the PDE  Fxx + (u_ Alp)Fx + Ft rF + 7r = 0 (3 21) First, it may be noted that p is the correlation coefficient between &z and the return on the market portfolio. Since &z is the only source of stochasticity in the proj ect and the underlying, p is also equal to the correlation coefficient between return on the underlying and the return on the market portfolio. Second, when compared with the DP formulation using the discount rate B it can be seen that the CC analysis modifies the total expected rate of return pu by a factor of Alpo which allows the use of the riskfree rate of return r In this manner the CC analysis converts the problem of valuing a risky asset to one of valuing its certainty equivalent. It does away with the need to determine the discount rate B but does require an additional assumption regarding the completeness of the market or in other words only the marketed risk of the asset can be valued. Further, as shown in Hull and White (1988), if the state variable is a traded security and pays a continuous proportional dividend at rate 3, then in equilibrium, the total return provided by the security in excess of the riskfree rate must still be Alpo, so that; pu + 3 r = Alpo (3 22) pu Alpe = r 3 (323) Substituting in Equation 321 we obtain the PDE derived by Dixit and Pindyck (1994) using the replicating portfolio i.e., cr2F, + (r 3)Fx + F rF + 7r = 0 (3 24) Hull (2003) differentiates between the investment and the consumption asset. An investment asset is one that is bought or sold purely for the purpose of investment by a significant number of investors. Conversely, a consumption asset is held primarily for consumption. Commodities like timber are consumption assets and can earn a below equilibrium rate of return. Lund and Oksendal (1991) discuss that generally investors will not like to hold an asset that earns a belowequilibrium rate of return. But empirically commodities that earn a below equilibrium rate of return are stored in some quantities. To quote Lund and Oksendal (1991): In order to explain storage of commodities whose prices are belowequilibrium, it is assumed that the stores have an advantage from the storage itself. This is known as gross convenience yield of the commodity. The net convenience yield (or simply the convenience yield) is defined as the difference between the marginal gross convenience yield and the marginal cost of storage. (Lund and Oksendal 1991, p.8) If we assume a continuous proportional convenience yield then the assumption is completely analogous to an assumption of a continuous proportional dividend yield. Therefore, the 3 can represent the continuous proportional convenience yield from holding the timber and the PDE will hold. The Lattice Estimation Models The Binomial Lattice Model In order to determine the holding value of the asset i.e., the value in the continuation region, it is necessary to solve the PDE. As it is not always possible to obtain an analytical solution, several numerical procedures have been devised. Amongst the popular methods for obtaining a numerical solution are the lattice or tree approximations (that work by approximating the stochastic process) and the finite difference methods, explicit and implicit (that work by discretizing the partial differential equation). Monte Carlo simulations and numerical integration are other popular techniques. This study uses the lattice approximation approach for its simplicity and intuitive appeal. Depending on the nature of the problem the binomial or higher dimension lattice models were used. The binomial approximation approach is suitable for valuation of options on a single underlying stochastic state variable and was first presented by Cox et al. (1979). For an underlying asset that follows a GBM process of the form = pudt + odz (325) where the drift pu and the variance 02 are assumed constant, the binomial approach works by translating the continuous time GBM process to a discrete time binomial process. The price of a nondividend paying underlying asset denoted by P is modeled to follow a multiplicative binomial generating process. The current asset price is allowed to either move up over the next period of length A by a multiplicative factor u to uP with subjective probability p or fall by the multiplicative factor dto dP with probability (1 p) To prevent arbitrage the relation u > 1+ r > d must hold where r represents the riskfree interest rate. The asset price follows the same process in every period thereafter. Following Ross (2002) it can be shown that, the binomial model approximates the lognormal GBM process as A becomes smaller. Let Y equal 1 if the price goes up at time iA and 0 otherwise. Then, in the first n increments the number of times the price goes up is [Y and the asset price would be P = (d" u I Letting n = gives ',P = d^ Taking logarithms we obtain In( =,, In d + Y I n (326) The Y are independent, identically distributed (iid) Bernoulli random variables with mean p and variance p(1 p) Then, by the central limit theorem, the summation I Y, which has a Binomial distribution, approximates a normal distribution t t t with mean p and variance p(1 p) as Abecomes smaller (and grows larger). Therefore, the distribution of In converges to the normal distribution as ~hgrows. Following the moment matching procedure Luenberger (1998) shows that the derived expressions for the parameters ,d and p are p =2 +1 Io 2 :" (327) u =e" (328) d =e "J (329) The DP procedure for analysis of an option on an underlying asset that follows GBM process would proceed by using a binomial lattice parameterized by these expressions. The DP procedure would obtain the option value by recursively discounting the next period values using the subj ective probability value p and an externally determined discount rate. In contrast the contingent analysis procedure is illustrated using the replicating portfolio argument as follows. In addition to the usual assumptions of frictionless and competitive markets without arbitrage opportunities, as noted earlier, it is assumed that the price of a nondividend paying asset denoted by P follows a multiplicative binomial generating process. The asset price is allowed to either move up in the next period by a multiplicative factor u or fall by the multiplicative factor d .If there exists an option on the asset with an exercise price of X, then the present value of the option denoted by c would depend on the contingent payoffs in the next period denoted by cl = M4lX [0,ugq X] and cd = M4X [0,d~o X] where 4 denotes the current price of the asset. In order to price the option a portfolio consisting of one unit of the asset and na units of the option written against the asset is constructed such that the end of the period payoff on the portfolio are equal i.e., upo nac,, = dP, nacd (330) Solving for na we get nz= 4()(331) c,, cd If the end of the period payoff is equal the portfolio will be risk free and if we multiply the present value of the portfolio by l +r we should obtain the end of period payoff (1 + r)(P, mc) = uP, nac, (332) P, [(1 +r) uj +nc c = (333) nz(1+r) Substituting Equation 331 for na in Equation 333 yields C='((1+ r)d 1 edu(1+r): (r 34 ud ud Letting (1+ r) d q = (335) ud where q is known as the riskneutralprobability/, we can express the present value of the option as c = [qc, + (1 q)cd j+t(1 +r) (3 3 6) From Equations 328 and 329 we have u = e"i and d = e" To find the value of the riskneutral probability q these values of u and d can be substituted in Equation 335 to obtain (1+ r) en 1 1 2 q~ = + 0 (337) Compared with the expression for the subjective probability p in Equation 327, it can be seen that under the riskneutral valuation the drift of the GBM process pu is replaced by the riskfree rater . In general, if the asset pays out a continuous proportional dividend 6 then, under CC analysis the drift is modified tor 3 (Equation 324). The corresponding risk neutral probability is q=1 2 (33 8) 2 20 For the treatment of previsible nonstochastic intermediate cash flows (costs) with a fixed value (si) the Equation 336 is modified to c = [qcu + (1q4)cd jt(1+ r)+ zi (339) It is implicit that i represents the discounted net present value of all such cash flows in the period. The CC procedure for a single period outlined above is easily extended to multiple periods and the option value is derived by recursively solving through the lattice. The Trinomial Lattice Model for a Mean Reverting Process For an asset that follows a MR process of the form dx= 9 x x dt + odz (3 40) the contingent claims PDE is given by aF~U + [r 3(x)]F~ + , rF + z = 0 (341) where 3(x) = pu (x x) i s a function of the underlying asset x . Hull (2003) describes a general twostage procedure for building a trinomial lattice to represent a MR process for valuation of an option on a single underlying state variable. For trinomial lattice the state variable can move up by a multiple u, down by a multiple d or remain unchanged represented by na Since MR processes tend to move back to a mean when disturbed, the trinomial lattice has three kinds of branching. Depending on the current value of the state variable the next period movements can follow one of the three branching patterns ((na, d, d), {u, na, d), {u, u, na) with associated probabilities. The parameters of the lattice are determined by matching the moments of the trinomial and MR processes. The procedure can be adapted for most forms of the MR process. Details of the procedure can be found in Hull (2003). The Multinomial Lattice Model for Two Underlying Correlated Stochastic Assets When the problem is to find the value of an option on two underlying assets with values that follow the GBM processes dxC = pU,xldt +a~xldZ i = 1, 2 (3 42) which are correlated with instantaneous correlation coefficient given by p ( i.e., they have a joint lognormal distribution) the contingent claim PDE has the form 12 1F' 2ea FI2 F22 3 (343) +[r32 2Fx2+F, rF+7t=0 A multinomial lattice approach is used to value such options, also called rainbow options. The development of the multinomial process for the correlated assets is similar to that described for the binomial lattice with a single underlying asset. The parameters of the multinomial lattice are derived by matching the moments of the underlying asset value processes. Hull (2003) discusses alternate lattice parameterization methods developed for the multinomial lattice valuation approach. This study uses the method discussed in Hull and White (1988). At each node on the lattice the assets can move jointly to four states in the next period. The resulting parameter values for a three dimensional lattice are summarized in Table 31. Table 31. Parameter values for a three dimensional lattice Period 1 state Risk neutral probability ugu2 0.25(1+ p) uz,d2 0.25(1 p) dz,u2 0.25(1 p) dz, d2 0.25(1+ p) Here uI =e 2 (344) d. = e 2~ (345) represent, respectively, the constant up and down movement multiplicands for asseti . Parameter o, represents the volatility and 3, represents the dividend/convenience yield of asseti while r represents the riskfree rate and A the size of the discrete time step CHAPTER 4 APPLICATION OF THE CONTINTGENT CLAIMS MODEL Who is the Pulpwood Farmer? Before applying the CC model to the pulpwood farmer' s dilemma, it is necessary to establish a mathematical description of a pulpwood farmer. For a commercial timber production enterprise, the choice of timber products) to be produced (or rotation length chosen) is guided by the prevailing and expected future timber market prices amongst other things. The following discussion describes the role of relative timber product prices in this decision. A slash pine stand will produce multiple timber products over its life. For products that are principally differentiated by log diameter, the early part of the rotation produces the lowest diameter products like pulpwood. As the rotation progresses the trees gain in diameter resulting in production of higher diameter products like sawtimber. Since individual tree growth rates vary there is no en bloc transition of the stand from the lower to a higher diameter product, but rather, for most part of the merchantable timber yielding rotation ages the stand would contain a mix of products with the mix changing in favor of the higher diameter products with increasing rotation age. The average pine stumpage price data series reported by Timber Mart South (TMS) for different timber products reveal that on an average the large diameter products garner prices that are significantly higher than lower diameter product prices (Table 41). This implies that the value of merchantable timber in the stand increases sharply with rotation age from the combined effects of larger merchantable yields and increasing proportion of higher diameter timber. More important, it also implies that short rotation farming may be difficult to justify using the TMS reported prices. Table 41. Florida statewide nominal pine stumpage average product price difference and average relative prices (19802005) Timber Products Average absolute Average relative price difference prices ($/Ton) Sawtimber vs. CNS 7.09 1.35 CNS vs. Pulpwood 9.79 1.99 Source: Timber MartSouth In general, the cultivation of early rotation products is differentiable from that of the late rotation products by the silvicultural choices. High density planting and absence of precommercial thinnings are some choices that could characterize the cultivation of pulpwood. For slash pine, the decision to plant dense and not resort to precommercial thinnings limits the stand owner' s choices with respect to switching to higher diameter product farming by prolonging the rotation. For a general slash pine stand with two products pulpwoodd and sawtimber) the results of price differentiation on the optimal Faustmann rotation age are shown in Table 42. The illustration uses a year 0 establishment cost of $120/acre, no intermediate cash flows, a 5% constant annual discount rate, a cutover site index of 60 and 600 surviving trees per acre (tpa) at age 2 with the Pienaar and Rheney (1995) slash pine growth and yield equations. Pulpwood was defined as merchantable timber from trees with minimum diameter at breast height (dbh) of 4 inches up to a diameter 2 inches outside bark and sawtimber as trees with minimum dbh 8 inches to 6 inches outside bark. The undifferentiated single timber product price was assumed $10/ton. Table 42. The effect of timber product price differentiation on optimal Faustmann rotation Price difference Optimal rotation age Absolute Relative $/Ton Years 0 1.0 21 5 1.5 23 10 2.0 25 20 3.0 27 30 4.0 28 Similarly, Table 43 shows that it is the relative product prices (for the purpose of this study, relative product price was defined as the price of the late rotation product expressed as a proportion of the price of the early rotation product) that are important to the determination of the optimal rotation changes. Table 43 maintains the absolute increments while changing the size of the relative increments. For this illustration the initial common timber product market price was assumed to be $20/ton. Table 43. The effect of timber product relative prices on optimal Faustmann rotation Price difference Optimal rotation age Absolute Relative $/Ton Years 0 1.0 20 5 1.25 21 10 1.50 23 20 2.00 24 30 2.50 26 Economic theory has it that relative pricing of goods is an important market signal which allows the efficient allocation of resources. In the context of the timber stumpage markets, relative product pricing serves as a signal to the timber producers to produce (more/less of) one or the other timber product. In order to induce producers to increase the production of late rotation products (which involve greater investment and/or risk) the market must offer a higher relative price. Since relative prices are not constant, an investment decision based on these prices must consider an average or mean of relative prices over an appropriate period of time. A timber land owner who bases his investment decision on the average relative prices deduced from the prices reported by TMS would never choose the lower rotation ages associated with pulpwood farming (1925 years for slash pine from Yin et al. 1998). Yet pulpwood farming is chosen by substantial numbers of timber land owners. Certainly, the average relative price of products could not be the only reason for choosing the pulpwood rotation. The practice of pulpwood farming with slash pine could be the result of several considerations. Short rotations are attractive in themselves for the early realization of timber sale revenues (capital constraint consideration). Other considerations like earning regular income from the sale of pine straw in denser stands with no thinnings also influence the choice. However, the alternate considerations do not diminish the importance of relative product prices. The stumpage prices applicable to a particular stand can be vastly different from the average prices reported by TMS. Some stands can experience greater relative prices at the same point in time than others. This varying relative prices experienced by stands can be easily explained by the nature of timber markets. Once the maximum FOB price that a timber purchaser can offer is determined for a period of time, the stumpage price applicable to prospective suppliers is determined by the cost of harvesting and transporting the timber to the location of the purchaser' s consumption/storage facility. From the average difference between TMS reported FOB and stumpage prices, it can be seen that these costs form a very high (as much as 2/3 for pulpwood) portion of the FOB price. At a point in time harvesting costs may vary little from one stand location to another in a region but the transportation costs can vary significantly. For a pulpwood stand located close to a purchaser (pulp mill) the relative product prices would always be lower than those for other distant stands (say with respect to a sawtimber purchaser in the region), justifying the pulpwood farming decision. This argument implies that a significant number of the stands located close to pulp mills would be choosing pulpwood farming and this should be empirically borne out. It also implies that once multiple products are considered there can be no single Faustmann rotation age that suits all evenaged single same specie stands even if their site quality was the same; rather, there would be a continuum of optimal rotation ages depending on the average relative stumpage prices applicable for the stand. For the present analysis it was assumed that the slash pine pulpwood plantation was located close to a pulpwood purchaser (who was not expected to stop operations) resulting in experiencing low average (long run) relative prices. For sufficiently small average relative prices it may also justify treating the entire merchantable timber output, irrespective of diameter size, as pulpwood. Despite the location advantage, in the shorter run, a stand would still experience wide differences in relative product prices from fluctuating market conditions. In the present conditions where pulpwood prices have been depressed for several years while other products have fared relatively better, there is a market signal in favor of higher diameter products to all stands irrespective of location. Therefore, the pulpwood stand under analysis would be experiencing higher than normal relative prices. This situation was analyzed as a multiproduct option problem though with modest relative prices as compared to TMS reported prices. The Return to Land in Timber Stand Investments This section deals with the calculation of return to timberland. Land serves as a store of value as well as a factor of production. As a factor of production used for the timber stand investment, land must earn a return appropriate to the investment. For the pulpwood farmer' s harvest problem land rent is a cost that will be incurred if the option to wait is chosen. In the contingent claim analysis the land rent is modeled as a parameter observed by the decision maker and hence having a known present value. If land is not owned but rented/leased the explicit portion of this return is in the form of rent charged by the renter/leaser. However, whether explicit or implicit, there is very little useful data available on either timber land values or lease/rent values. Most decision makers do not have access to a reliable estimate of even the present value of their timber land. This makes it necessary to determine the appropriate return to be charged to land for the purpose of the analysis. The following discussion uses the term 'timberland value' to refer to the value of the bare land, unless it is specifically stated otherwise. In its report on large timberland transactions in the US, the TMS newsletter (2005) reports a weighted average transaction price in 2005 for the southern US of $1160/acre. Smaller timberland transactions at $2000/acre or higher in Florida are routinely reported. A part of these valuations must arise from the value of the land itself while some of the balance could be for the standing trees (if any). Recent literature discusses other important sources of the valuation like high nontimber values in the form of leisure and recreation values etc. and expectations of future demand for alternative higher uses. Aronsson and Carlen (2000) studies empirical forest land price formation and notes that nontimber services, amongst other reasons, may explain the divergence of the valuation from present value of future timber sale incomes. Wear and Newman (2004) discuss the high timber land values in the context of using empirical timberland prices to predict migration of forest land to alternate uses. Zhang et al. (2005) look at the phenomena of timber land fragmentation or 'retailization' through sales to purchasers looking for aesthetic/recreation values and its implications for forestry. Since this analysis assumes that timber sales is the only maj or source of value the appropriate timberland value is the value of bare land the present best use of which is timber farming and for which nontimber values are insignificant. Assuming that the market price of such timberland could be observed, the question is: Can this value be used for the purpose of analysis? Is information on the traded price of bare timber land appropriate for analysis? Chang (1998) has proposed a modified version of the Faustmann model suggesting that empirical land values could be used with the Faustmann optimality condition to determine optimal rotations. Chang (1998) discusses a generalized Faustmann formula that allows for changing parameters (stumpage price, stand growth function, regeneration costs and interest rate) from rotation to rotation. The form of the optimal condition derived by him is 8Rk (Tk > S= BR(Rk k)+ BLEYk+1 (41) Here, R(T) is the net revenue from a clearcut sale of an even aged timber stand at the optimal rotation age T and 6 is the required rate of return. The subscript k refers to the rotation. The condition is interpreted to mean that instead of the constant LEV of the standard Faustmann condition, the discounted value of succeeding harvest net revenues (LEYk+1) must be substituted. Chang (1998) interprets this to mean that the market value of bare land existing at the time of taking the harvest decision can replace the standard constant Faustmann LEV. But, if the observed land value is very high as compared to the LEV the RHS of the equation increases significantly, resulting in a drastic lowering of the optimal rotation age T In the state of Florida, which is experiencing high rates of urbanization, it is not unusual to find timber land valued at several multiples of the LEV. Failure to account for nonmarket values like aesthetic or recreational values in the model alone may not explain the failure to observe the rotation shortening effect. Klemperer and Farkas (2001) discuss this effect of using empirical land values while using Chang' s (1998) version of the Faustmann model. By definition, the value of any asset is the discounted value of net surpluses that it is expected to provide over its economic lifetime in its best use. This suggests that the market value of land may be differentiable into two parts. One part of the market value is derived from the current best use and the other is the speculative or expected future best use (Castle and Hoch 1982). This means that the present market value of timber land, if known, does not provide information on the value in current use without the separation of the speculative value component. The critical fact is that the land rent chargeable to current best use cannot exceed the expected net surplus in current best use. No investor would pay a land rent higher than the net surplus he expects to earn by putting it to use. Using empirical land values could result in overcharging rents as the land values may be inflated by the speculative value component. This gives rise to the question: What about the opportunity cost to the speculative component of land value? Does the landowner lose on that account? The answer is that if a parcel of land is being held by the landowner despite its current market value being higher than its valuation in current best use, an investment or speculation motive can be ascribed to the landowner. The landowner treats the land not only as a productive factor in the timber stand investment but also as a speculative asset. The landowner could earn a capital gain over and above the value of future rents in the current best use by selling the land in such a market. If the investor chooses to hold the land, it is because he expects to profit from doing so. And this profit is in the form of expected capital appreciation. It is this expected capital appreciation that compensates the landowner for the opportunity cost on the speculative component of land value A formal derivation of this argument follows from economic theory. According to the economic theory of capital, in a competitive equilibrium, an asset holder will require compensation for the opportunity cost on the current market value of a capital asset plus the depreciation cost for allowing the use of his asset (Nicholson 2002). Representing the present market price of the capital asset by P, the required compensation v will be v = P(6 + d) (42) where 6 is the percent opportunity cost and d is the assumed proportional depreciation on the asset value. When the asset market value is not constant over time the required compensation will be a function of time v(t) The present value of the asset would equal the discounted value of future compensation incomes. At the present time t the present discounted value (PDV) of the compensation received at time s (t < s) would be v(s)es'"t'" and the present discounted value of all future compensation incomes would PD'LV =P(t) =~ Iy(s)e "(" 'ds e"' my(s)e ""ds (43) Differentiating P(t) with respect to t and ignoring depreciation we have dP(t) = Be' v(s)e"d s e" v(t)e '= OP( t) v(t) (44) Therefore, dP(t) v(t) = OP(t) (45) So, the required compensation income at any time is equal to the opportunity cost on the current market value of the asset less the expected change in the market value of the asset. The interpretation is that the 'fair' or competitive compensation for leasing an asset consist of both the interest cost as well as the expected change in the value of the asset. If the expected change in the value is positive the rental charges are decreased to that extent since the value appreciation compensates the asset owner for a part of the interest cost. The net compensation v(t) is the opportunity cost for the asset value in its current best use. It cannot be more than this cost as prospective renters cannot afford to pay more as already argued. It cannot be less because a lower charge would transfer a surplus to the renter attracting competition amongst renters. Therefore, to find the amount to be appropriated as return to timberland it is required that its value in current (best) use be determined and then the return would be given by the opportunity cost of holding the land in its current (best) use. The static Faustmann framework determines the timber land value or Land Expectation Value (LEV) as the present value of net harvest revenues arising from infinite identical rotations in timber farming use. In contrast, in the stochastic framework, the ability to actively manage the investment adds an option value which must be incorporated in the valuation method. As argued and shown by Plantinga (1998), Insley (2002, 2005), and Hughes (2000) a price responsive harvest strategy adds a significant option value to the investment. In this study the land value was determined within the CC analysis assuming that timber farming was the current best use. The parameters for the valuation are the current values of timber price and plantation establishment cost as well as annual maintenance costs. In the risk neutral analysis the current riskfree rate serves as the discount rate. The infinite identical rotations methodology was used to capture the tradeoff with future incomes meaning that the net expected surplus value over the first rotation was used as the expected average value for future rotations. This may not effect the land value significantly since, as observed by Bright and Price (2000), the present value of net surplus in the first rotation forms most of (>80%) the estimated timberland value when calculated in this way, for a sufficiently long rotation and high discount rate. Therefore, the land value in current use can be estimated with information available to the decision maker. And the land rent is the opportunity cost of this value. The mathematical formulation of the land value estimation problem is given by Equation 46. L V = max (46) Ly =maxI E~[PIP,])(ert C Here, L V = Present value of land Eq = Risk neutral expectation operator P, = Stochastic timber price at t period from present time 0 Q~t) = Deterministic merchantable timber yield function (of rotation age t) at = Annual recurring plantation administration expenses treated as riskfree asset Co = Value of plantation establishment expenses to be incurred today (at year 0) r = Present riskfree interest rate assumed constant in future Then, the present value of estimated land rent is rL V There are some points to note about the above arguments and methodology outlined for determining the land value in current best use. First, it is implied that the rent value is calculated afresh by the decision maker every period. This is empirically true for shorter duration uses like agriculture farming and there is no reason why it should not be so for timber farming if decision makers are efficient information processors as normally assumed and information is easily and freely accessible. If market information on comparable land rents was available, it would be stochastic and the decision maker would utilize the new information available every period for decision making. It is also important that a stochastic rent value calculated as argued above captures and transfers fresh information about the expected future to the decision making process. That is, if the estimated land/rent value is high, it will increase the cost of rotation extension and vice versa. For example, if the stand owner learns of a demonstrable technological advance improving the financial returns to stand investments, in the midst of the rotation, the stand owner will seek to apply the technology to the present rotation, thereby adopting the 'best use'. However, if the improved technology cannot be applied to the current rotation then there should be pressure on the decision maker to shorten the present rotation so that the improved technology could be applied to the next rotation. There is no empirical evidence known to support this result regarding timberland owner behavior but it can be argued that timberland owners never have free and easy access to the necessary information. It is also important to note that market prices of timberland themselves provide no valuable input to the stand decision making process but rather it is the value in current best use that is relevant. Thus, in periods of speculative inflation of land prices one may not expect to observe any appreciable changes in stand decision making behavior. It is the direction of changes in real input costs and output values which result in changes in land rent. Second, it is implied that the rent value is a function of the current timber pricess. However, it need not be perfectly correlated with timber prices(s) since other parameters of the valuation (the costs and discount rates) would be expected to follow (largely) independent stochastic processes. Third, the method outlined for estimation of land rent provides an estimate for a single period i.e., for the present period only. Ideally, the rent value should be modeled as a stochastic variable. But that would require information regarding the stochastic process defining the plantation expenses (or nontimber sources of cash flow) and discount rate. In the absence of data on the stochastic process for the other flows, in the following analysis, land rent was assumed to behave like a riskfree asset. On the Convenience Yield and the Timber Stand Investment To solve the harvesting problem using the lattice approach the constant volatility of the underlying variable is estimated from historical price data. The riskfree rate is estimated from yields on treasury bills of matching maturity. However, the estimation of the convenience yield poses a problem. The concept of convenience yield, as it is popularly interpreted, was first proposed by Working (1948, 1949) in a study of commodity futures markets. The phenomena of "prices of deferred futures....below that of the near futures" (Working 1948, p. 1) was labeled an inverse carrying charge. The carrying charge or storage cost is the cost of physically holding an asset over a period. The concept can be illustrated as follows: ignoring physical storage costs, the arbitrage free forward price F for future delivery of a commodity is determined by the relation Poe't where P is the current unit price of the commodity, r is the borrowing/lending rate while t is the period of the contract. Therefore, the forward price should be proportional to the length of the contract. The inverse carrying charge or convenience yield discussed by Working (1948, 1949) is said to accrue to the contract writer when the no arbitrage relation does not hold for some contract lengths and F an opportunity for arbitrage exists, arbitrageurs are unable to take advantage as nobody that is holding the commodity in inventory is willing to lend the commodity for shorting. Inventory holder may be unwilling to lend the commodity when markets are tight (Luenberger, 1998) i.e., supply shortage is anticipated. Brennan (1991) defines convenience yield of a commodity as: ...the flow of services which accrue to the owner of a physical inventory but not to the owner of a contract for future delivery. ....the owner of the physical commodity is able to determine where it will be stored and when to liquidate the inventory. Recognizing the time lost and the costs incurred in ordering and transporting a commodity from one location to another, the marginal convenience yield includes both the reduction in costs of acquiring inventory and the value of being able to profit from temporary local shortages of the commodity through ownership of a larger inventory. The profit may arise from either local price variations or from the ability to maintain a production process despite local shortages of a raw material. (Brennan 1991, p.3334) The convenience yield is not constant but would vary with the gross inventory of the commodity in question, amongst other things. If there exists a futures market for the commodity then the futures prices represent the risk neutral expected values of the commodity. The risk neutral drift pu(t) which will be a function of time since the convenience yield 3(t) and forward risk free rate r(t) are empirically stochastic, can be calculated from the futures prices as (Hull 2003) 8[1n~t) F=+1 p~lt) = ]d =I In 1Ft (47) Here, F(t) is the futures price at time t. In the absence of a futures market, theoretically it should be possible to estimate the convenience yield by comparing with equilibrium returns on an investment asset that spans the commodity's risk (replicating portfolio).As discussed by McDonald and Siegel (1985), the difference between the equilibrium rate of return on a financial asset that shares the same covariance as the asset and expected rate of return on the asset will yield 3 But, empirically, such an asset is difficult to locate or construct from existing traded assets. Similarly, we could estimate r 3 from the equivalent pu Alpo Usually, the Capital Asset Pricing model (CAPM) for timber stands is estimated by regressing excess returns on the historical timber price against the excess returns on the market portfolio. Thus, this methodology suffers from the failure to incorporate the convenience yield in total returns on timber stands. Using the estimated ilp by this method will only yield pu Alpo = r i.e., a 3 value of zero. To the best of this authors' knowledge no method for estimating convenience yield for timber is available in published literature. Therefore, this study proceeds by assuming that the convenience yield 3 = 0 and pu Alpo = r The results are tested for sensitivity to different levels of constant 6 . Dynamics of the Price Process Modeling the empirical price process is the key to the development and results of the real options analysis. Beginning with Washburn and Binkley (1990a) there has been debate over whether the empirical stumpage price returns process is stationary (mean reverting) or nonstationary (random walk). The debate has remained inconclusive due to the conflicting evidence on the distorting effect of period averaging on prices. Working (1960) was the first to show that the first differences of a period averaged random chain would exhibit first order serial correlation of the magnitude of 0.25 (approximated as the number of regularly spaced observations in the averaged period increased). Washburn and Binkley (1990a) found consistent negative correlation at the first lag for several quarterly and annual averaged stumpage price series though most were less than 0.25 and statistically significant only for prices in one case. On the other hand, Haight and Holmes (1991) have provided heuristic proof to the effect that a stationary first order autoregressive process, when averaged over a period, would behave like a random walk as the size of the averaging period was increased. They used this proof to explain away the observed nonstationarity in the quarterly averaged stumpage prices. The stationarity of the price process has implications for the efficiency of stumpage markets. "A market in which prices always "fully reflect" available information is called "efficient"" (Fama 1970, p.3 83). Utilizing the expected rate of return format, market efficiency is described as E Pr ; .=,,i = +Ery : O $ (48) Here, E is the expectation operator, P~, is the price of security j at time t, P,,r I its random price at time t +1 with intermediate cash flows reinvested, r ,~t~l is the random P, ri P oneperiod percent rate of return .'' O, represents the information set assumed P~, to be fully reflected in the price at t . The information set O, is further characterized according to the form of efficiency implied i.e., weak form efficiency which is limited to the historical data set, semistrong form efficiency which includes other publicly available information and strong form efficiency that also includes the privately available information. As Fama (1970) discusses, the hypothesis that asset prices at any point fully reflect all available information is extreme. It is more common to use historical data to test prices for weak form efficiency in support of the random walk model of prices. Washburn and Binkley (1990a) tested for the weak form efficiency using the equilibrium model of expected returns with alternate forms of Sharpe' s (1963) single index market models. Expost returns to stumpage were regressed on a stock market index and an inflation index. The residuals from the regressions were then tested for serial correlation the presence of which would lead to rej section of the weak form efficiency hypothesis. Since these tests required the assumption of a normal distribution for the residuals, this was tested using the higher moments skewnesss and kurtosis). The nonparametric turning points test was also conducted as an alternate test for serial dependence. They found evidence of stationarity in returns generated from monthly averaged data but returns generated from quarterly and annually averaged data displayed nonstationarity. Significantly, they did not find evidence to support the normal distribution assumption of the residuals. Haight and Holmes (1991) used an Augmented DickeyFuller test and found stationarity in instantaneous returns on monthly and quarterly spot stumpage prices and nonstationarity in instantaneous returns on quarterly averaged stumpage prices. Hultkrantz (1993) contended that the stationarity found in returns generated from monthly averaged price series by Washburn and Binkley (1990a) could be consistent with market efficiency when producers were risk averse. He used a panel data approach to DickeyFuller tests and found that southern stumpage prices were stationary. Washburn and Binkley (1993) in reply argued that the results of Hultkrantz' s analysis were by and large similar to their analysis and point out that if Haight and Holmes (1991) proof of the behavior of averaged prices is considered, then both (Hultkrantz 1993 and Washburn & Binkley 1990) their analyses could be biased away from rej section of the weak form market efficiency. Yin and Newman (1996) used the Augmented DickeyFuller and arrived at conclusions similar to Hultkrantz' s (1993). Gj olberg and Guttormsen (2002) applied the variance ratio test to timber prices to check the null hypothesis of a random walk for the instantaneous returns. Their tests could not rej ect the random walk hypothesis in the shorter periods (1 month and 1 year) but over longer horizons, they found evidence of mean reversion. Prestemon (2003) found that most southern pine stumpage price series returns were nonstationary. He noted that tests of time series using alternate procedures may not agree regarding stationarity or market informational efficiency as time series of commodity asset prices may not be martingales. McGough et al. (2004) argue that a first order autoregressive process for timber prices is consistent with efficiency in the timber markets. They advocate the use of complex models (VARMA) that include dynamics of the timber inventory while noting that such models would be difficult to estimate and apply to harvesting problems. In summary, in the absence of better data and models or stronger tests, it is difficult to conclusively establish the efficiency or otherwise of stumpage markets and or choose between the random walk or autoregressive models. This study considered both, the stationary and nonstationary models, for the price process alternately. Modeling the Price Process Two alternate models for the stochastic price process were applied to the real options model. The first model is the Geometric Brownian motion which a form of the random walk process that incorporates a drift and conforms with the efficient market hypothesis. Expressed mathematically it is dP = puPdt + oPd: (49) Here, P =Price of the asset at time t pu = Constant drift a = Constant volatility dz = Increment of a Weiner process Geometric Brownian motion processes tend to wander far away from their starting points. This may be realistic for some economic variables like investment asset prices. It is argued that commodity prices (Schwartz 1997) must be related to their long run marginal cost of production. Such asset prices are modeled by Mean Reverting processes, which is the second model used for the stochastic process in this analysis. While in the short run the price of a commodity may fluctuate randomly, in the long run they are drawn back to their marginal cost of production. The OrnsteinUhlenbeck process is a simple form of the MR process expressed as dP= 9 P Pdt+ rd (410) Here, r = Coefficient of reversion P = Mean or normal level ofP The r is interpreted as the speed of reversion. Higher values of r correspond to faster mean reversion. P is the level to which P tends to revert. P may be the long run marginal cost of production.. The expected change in P depends on the difference between P and P. If P is greater (less) than P, it is more likely to fall (rise) over the next short interval of time. Hence, although satisfying the Markov property, the process does not have independent increments. The Weiner process in discrete time is expressed as z, q,= Z,+ e,~ (411) Here, E,= Realization of a Normal Random variable with mean 0 variance 1 and Cov(e,,e,,)= 0 for jf0 In continuous time, the process is diz, = e,Jd A Weiner process z, is a random walk in continuous time with the properties (Luenberger 1998) i. For any s < t the quantity z(t) z(s) is a normal random variable with mean zero and vaniancet s . ii. For any 0 < t, < t, < t, < t4 the random variables z(t,) z(t) land z(t4) 3~, r uncorrelated. iii. z(t,) = with probability 1. The Geometric Brownian Motion Process Applying Ito's lemma, the Geometric Brownian motion (GBM) process can be expressed in logarithmic form as dn =pi o1\dt +odz (412) In discrete logarithmic form the equation becomes IIn'~:.l g n t + odz (413) Thus, the logdifference or the instantaneous rate of price change is normally di stributed. In order to model the GBM process an estimate of the volatility was required. Following Tsay (2002), let r, = In P, In P Then, r, is normally distributed with mean pY a At and vrianac a At, wher e Atii is inite timeinterval. If s denotes the sample standard deviation i.e., i; r, r s t= g1 (414) then a = (415) Here a denotes the estimated values of a from the data. For the nominal F.O.B. and stumpage statewide pulpwood quarterly price data for Florida, using the above methodology we obtain the estimates listed in Table 34. However, as the TMS data is available in period average form while the GBM process models the behavior of spot prices, it is necessary to account for any distortion to the statistical properties of the data from averaging. Working (1960) has demonstrated that to an approximation, the variance of rates of change calculated from arithmetic averages of n consecutive regular spaced values of a random chain will be of the variance of first difference of correspondingly positioned terms in the unaveraged chain, as n increases. The prices reported by TMS are calculated as an arithmetic average of all reported prices in a quarter. As discussed by Washburn and Binkley (1990b), the price averages will be unbiased estimates of the arithmetic mean of prices at any n regular intervals within the period so long as the likelihood of a timber sale occurring and the expected transaction size are constant throughout each period. Making the necessary correction to the estimated variance we obtain the revised estimate of the variance listed in Table 44. Table 44. Estimated GBM process parameter values for Florida statewide nominal quarterly average pulpwood prices Estimated Parameter FOB Stumpage Uncorrected Values Standard Deviation 0.10 0.24 Corrected Values Standard Deviation 0.12 0.29 It may be noted that the calculated standard deviation for the F.O.B. price was significantly lower than the standard deviation of the stumpage price. One possible explanation is that pulp mills revise their mill delivered prices relatively infrequently, whether they are gate purchase prices or supplier contracted prices. It is also possible that while gate purchase prices are public knowledge, mill delivered price of pulpwood purchased from other sources may be incompletely reported due to mill concerns with strategic competitive disadvantage from revealing prices. On the other hand harvesting and transportation costs change drastically from one stand to another, resulting in higher volatility of reported stumpage prices. The stumpage prices reported are not the prices experienced by a particular stand or a common price experienced by all stands but prices experienced by different stands that reported selling timber in the period. The harvesting and transportation costs are themselves volatile and likely imperfectly correlated with FOB prices but it is possible that they do not account for the entire difference in the reported standard deviations. Since stand owners experience the stumpage price and not the FOB price, in the absence of data on volatility of harvesting and transportation costs, this study uses the estimated standard deviation of reported stumpage prices to replace the estimated volatility of F.O.B. prices while treating the harvesting and transportation costs as nonstochastic. To account for the possibility of overestimation of timber price variance the analysis was subj ected to tests of sensitivity to price volatility. Statistical Tests of the Geometric Brownian Motion Model The GBM process in discrete logarithmic form is a discrete random walk with drift i.e., it has the general form y, = y,/ + a + E, where E, I N(0, a At) and Clov(e,E, ) = 0 for j f 0 (416) The process is clearly nonstationary with a unit root. But if we take the first difference we obtain a stationary process Ay, = a + E, (4 17) The first difference process has mean a and variance o At Further, the covariance Cov(Ay,, Ay,,) = 0 for s+ 0. To see how well the empirical data fits the GBM model, the sample autocorrelation function (ACF) at several lags was calculated and plotted for the difference logarithmic form of the price data series. The sample ACF [p(h)] at lag h was calculated using the formula p(h)= n n tZ =1 For large n, the sample autocorrelations of an independent identically distributed (iid) sequence with finite variance are approximately iid with distribution N(0, 1 (Brockwell and Davis 2003). Hence, for the iid sequence, about 95% of the sample 1.96 autocorrelations should fall between the bounds+ . For the GBM process the instantaneous rate of price change r, = In P, In P, is zero for all he 0 If the empirical price data is modeled by the GBM process, then the sequence tr, J should be white noise i.e., it should be a sequence of uncorrelated random variables. A plot of the sample autocorrelations for the instantaneous rate of price changes of the reported nominal pulpwood statewide stumpage prices along with the 95% confidence intervals are presented in Figure 41, plotted using the ITSM 2000 statistical software (Brockwell and Davis 2003). 1 OO Sample ACF 80 40 2 5 O364 Figure 41. Sample autocorrelation function plot for nominal Florida statewide pulpwood stumpage instantaneous rate of price changes The dashed lines on either side of the centre plot the 95% confidence interval. If the sequence is stationary, for the 40 lags plotted, 2 or less ACF's should fall beyond the 95% confidence bounds. For the stumpage price series, no more than 1 ACF beyond lag 0 fell outside the 95% confidence bounds. Significantly, as proved by Working (1960), the ACF at lag 1 for stumpage price instantaneous rate of change sequence was approx. 0.25 ( p(1) = 0.27 ). This could be the effect of the averaging process. To check if the (r \ sequennncwa Gaussia ie.,;~ if al of~1 its+E joint distributions are normal the JarqueBera test was used. The JarqueBera test statistic is given by (Brockwell and Davis 2003) n +m distributed asymptotically as X (2) (419) if (Yt>, } I ID N,,i~ cr)whr m The results of the JarqueBera test applied to the stumpage prices, given in Table 4 5, indicate that the normality hypothesis was weakly supported by the empirical data at the 5% level of significance. Table 45. Results of JarqueBera test applied to GBM model for Florida statewide nominal quarterly average pulpwood stumpage prices Test Value Stumpage JarqueBera Test Statistic 6.0984 Pvalue 0.0474 The ACF test indicates that the GBM process can be used to model the empirical data. However, this does not exclude the possibility of the true price process having a nonconstant drift and/or variance. Lutz (1999) tested stumpage price series for constant variance. He found that the variance for stumpage price series examined was not constant for the early parts of the series i.e., up to 1920. From 1920 onwards, the examined series were found to display constant variance. Even if stumpage price data is heteroskedastic, it was implicitly assumed that the logarithmic transformation rendered the data homescedastic. The test results also did not exclude the possibility of an alternate model providing a better fit. On the basis of the ACF test it was only possible to conclude that there was insufficient evidence to rej ect the GBM model. The Mean Reverting Process The simple MR process is given by Hence, Pis normally distributed with E(P,  P, )= P+ P et and 2r Using the expected value and variance we can express P, as 1e 2')9 41 ,= GP1P + e, when (1e") = 1 (420) where E, IN 0,a2 The last equation provides a discrete time first order autoregressive equivalent of the continuous time OrnsteinUhlenbeck process. In order to estimate the parameters an OLS regression of the form p, 4F = a + bP + e, (421) with a = yP and b = r was run. Then, the estimated parameters are given by P= a = b and a = o, One problem with using this simple form of the MR process is that it allows negative values for the stochastic variable. Plantinga (1998) justifies the choice of this model for timber stumpage price by referring to the possibility of harvesting and transporting costs exceeding the FOB timber price. In such a case the effective stumpage price would be negative. However, it can be argued that the negative stumpage price would still be bound by the harvesting and transportation costs i.e., if the FOB price were zero, the negative stumpage price cannot be larger than the cost for harvesting and removal. But the MR process described above is unbounded in the negative direction. Hull (2003) describes an alternate log normal form of the MR process dlni = 9 InPIng dtrcrd (422) This model restricts the price process to positive values. Thus, In gF given InP, is normally distributed with mean In P+~ In In Pu et and vanianceor 1 e2,) .The deviations from the longrun mean are expected to decay following an exponential decline. This analysis uses this form of the MR process to model the FOB prices. When the harvesting and transportation costs are deducted from the stochastic values of the FOB price the magnitude of resulting negative stumpage price is restricted to these costs. However, adopting Equation 421 for estimation of P implies that the mean to which the process reverts is constant over any period of time. Considering that P is interpreted as the long run cost of production, over a short interval of a few days or weeks, it may be feasible to assume that the value is constant. But when the analysis must cover several years, this assumption is questionable. One common correction method employed (Smith and McCardle 1999) is to regress the inflated values (present) of the historical asset price, which yields an inflation adjusted estimate ofP It also implies that for an analysis conducted in nominal terms, the future values of P must be inflated at an estimated average inflation rate. The average rate of annual inflation computed from the PPI (19212005) was approx.3.0%. The possibility of a constant real or inflation adjusted nominal P for pulpwood prices was corroborated by the historical performance of pulpwood prices over the 30 years or so of TMS reporting period as well as the RPA (2003) proj sections of future performance. This phenomenon can be partly attributed to technological advances and in some measure to adverse demand and supply movements. Considering that the other parameter estimates are only marginally effected this analysis uses the inflation adjusted parameter values. To check for the effect of period averaging on the estimated parameter values, simulation was carried out. The simulation revealed that regression of period averaged data generated consistent estimates of P while cr was consistently underestimated by a factor of approx. 0.67 or the Working' s correction. The result of regressing inflated past values of the pulpwood prices using the Producers Price IndexAll Commodities (PPI) on estimated P and other parameters are listed in Table 46. Table 46. Inflation adjusted regression and MR model parameter estimates a b P r a* Stumpage Price 0.2979 0.1245 2.3922 0.1245 0.2230 Standard Error 0.3285 0.1278 FOB Price 0.5694 0.1781 3.1963 0.1781 0.0945 Standrad Error 0.4338 0.1350 * Estimates corrected for period averaging effect. Of particular importance are the reversion coefficient values. For both price series the reversion coefficient values are low indicating that the annual price series exhibit low or insignificant reversion behavior. The 'half life' of the MR process or the time it takes 10 0.5 to revert half way back to the long run mean, given by , was approximately 2.6 years for stumpage price, illustrating the extremely slow reversion process. Regarding the low values of mean reversion coefficients Dixit and Pindyck (1994) observe that this seems to be the case for many economic variables and that it is usually difficult to rej ect the random walk hypothesis using just 30 or 40 years data. Secondly, the estimated variance for the FOB price process was sharply lower than that for the stumpage price. Once again, this difference can be attributed to the stochastic harvesting and transportation costs but may also partly be the result of the unsuitable data. As in the GBM process case, the analysis was conducted by attributing the stumpage price process variance to the FOB price process and using a nonstochastic harvesting and transportation cost. Finally, it must be noted that the lattice model for MR process used in this analysis was based on the existence of futures markets for the commodity and hence knowledge of futures prices, which represent the risk neutral expected future values. In the absence of futures markets for timber, the value of the inflation adjusted estimated P was used. This was justified for a long interval since mean reverting prices (and hence futures prices) are expected to converge to P in the long run. However, in the short run this only serves as an approximation. Statistical Tests of the Mean Reverting Process Model Examination of the stumpage price regression residuals shows first order serial correlation (Figure 42) as shown by Working (1964). Sample ACF 40 2 5 0364 OO mpg Irc IR Ioe rersso I  1a Figre 2.Sampe atcrrlto fnto po o nominal Floridal statewie pulpwood tma pie Tst ttsi tumpage priceMRmdlrgesorsius JarqueBera Test Statistic 9.6299 Pvalue 0.0081 Instantaneous Correlation In order to model the simultaneous stochastic evolution of two correlated stochastic processes following the GBM, an estimate of the instantaneous correlation between the two time series was required. The estimation of the instantaneous correlation of two period averaged GBM processes is not effected by period averaging (Appendix). The estimated instantaneous correlation for the TMS reported Florida statewide average stumpage quarterly prices of pulpwood and chipnsaw assuming GBM processes was 0.43. The Data Time series data on prices of the timber products was acquired from Timber Mart South (TMS). Price data for Florida extending from the last quarter of 1976 for pulpwood and the first quarter of 1980 for chipnsaw to the second quarter of 2005 are used for the analysis. The stumpage price data are reported by Timber MartSouth as quarterly average of final sale prices recorded in auctions for timber products in the reporting region. The data was used to represent spot timber prices in the analysis. However, due to the nature of data generation, collection and reporting processes, the validity of the data for this purpose is suspect. For example, the process starting from bidding for the timber to removal of the timber from the stand is usually a few months long. This means that the auction bid prices are a reflection of the bidder' s expectation regarding future prices when the timber will actually reach the market, not the immediate price. Errors in recording, approximations etc also undermine the data. Other shortcomings have been discussed in various contexts above. Harvesting and transportation cost was calculated using the difference between reported F.O.B. and stumpage prices of timber. The appropriateness of this method is questionable because of the time difference between auctions and actual movement of timber from the stand to the market. Growth and Yield Equations Slash pine growth and yield equations developed by Pienaar and Rheney (1996) are used. These equations for cutover forest land were developed using data from plantations sites in Georgia. The average site index for the sites was 60 ft (at age 25). The equations used in the analysis are 77 i. Expected Average dominant height (H in ft) H = 1.3679S(1 eC)7345dge 1 8()4 + (0.678Z, + 0.546Z2 + 1.395Z3 0.412Z Z )Age "eC)(691Age where =Site Index =1 if fertilized, zero otherwise =1 if bedded, zero otherwise =1 if herbicided, zero otherwise N,e)()(41(Age Age Age, and Age, ii. Survival after the second growing season (in trees/acre) N, where N, and N, are trees per acre surviving at respectively (Age, > Age, ) . iii. Basal area (B in ft2 /ce B =e3 39435 668 4e 1 336+6 2)5.4g () 366+3 155.4g (0.557Z, + 0.436Z, + 2. 134Z3 0.3 54Z,Z )ag eOe 09.4ge where Z, = 1 if burned, zero otherwise iv. Stem Volume outside bark( V in ft3/ce ~~~r( ) O82 17 1 (1eg )16+"""~;S v. Merchantable volume prediction (T in ft3ice T38 15" where V 9, =per acre volume of trees with dbh>d inches to a merchantable diameter t inches outside bark D = quadratic mean dbh in inches = 0.005454N\/ tr = pl The merchantable yield output from the growth and yield models is in units of ft3 outside bark/acre. To convert the yield to tons/acre conversion factors of 90 ft3/COrd and 2.68 tons/cord (TimberMart South) were used. These equations were developed from experimental plantations reaching an age of 16 growing seasons. For this reason, their use for extrapolating growth and yield to higher rotation ages is questionable (Yin et al.1998) and may not represent the true stand growth. Nevertheless, for the purpose of this analysis, these equations are the best source for modeling the growth and yield of slash pine. Plantation Establishment Expenses Average plantation establishment expenses for cutover land in the US South reported by Dubois et al. (2003) were used. The relevant reported costs are listed in Table 48. Table 48. Average per acre plantation establishment expenses for with a 800 seedlings/acre planting density Expense Head 2002 2005 $/acre $/seedling $/acre $/seedling Mechanical Site preparation* 166.50 195.82 Burning** 15.02 17.66 Planting cost* 49.99*** 58.79 Seedling cost 0.04 0.05 Total Cost for 800 seedlings/acre 280.00 329.21 * All Types ** Others *** Planting cost for average 602 seedlings/acre The Producers Price IndexAll Commodities was obtained from the Federal Reserve Economic data and used to extrapolate the nominal plantation expense data reported for 2002 to 2005. The index stood at 132.9 in December 2002 and rose to 156.3 by July 2005. For an acre planted with 800 seedlings the total planting cost under the above listed expense heads in July 2005 was estimated at $329.21 or approx. $330/acre. RiskFree Rate of Return The yield on Treasury bills with 1 year maturity (Federal Reserve Statistical Release) was used as the estimated riskfree rate of return. The reported riskfree rate for July 2005 was 3.64%. The Model Summarized The value of options available to the decision maker were analyzed using a CC valuation procedure. The analysis also highlighted the form of the optimal strategy. The following are the important points of the model 1. The model considers an evenaged mature (20 year age) slash pine pulpwood plantation in 2005. Only revenues from sale of timber are considered significant for the analysis. Since the analysis focuses on the pulpwood crop, the plantation was assumed to have been planted dense (800 trees/acre initial planting density) with no thinnings up to the present age. The plantation was assumed to be cutover with site index 60 ft (rotation age 25). Site preparation activities assumed are mechanical site preparation (shear/rake/pile) and burning only. A clearcut harvest was considered for the final harvest. When the thinning option was the subj ect of analysis only a single thinning in the form of a row thinning that removes every third row of trees was considered. 2. For a stand with the chosen initial planting density and site index on a cutover site, the growth and yield equations produce a single product yield curve that peaks approximately at age 43. Rotation age 43 was selected as the terminal age for the options on the stand in this study. This terminal age was applied uniformly to all models for comparability of results. Even though later stand products would have later yield peaks, current empirical practices and unreliability of the yield curve for higher rotation ages were arguments in favor of the lower terminal age. 3. Only the timber price (prices for multiple product analysis) was modeled as a stochastic variable. It was assumed that the stand growth and yield models provide a reliable forecast of the future merchantable timber yields. 4. The present values of other parameters of the valuation model like the riskfree interest rate, the land rent and the intermediate expenditure/cash flows on plantation were assumed known to the decision maker. Some intermediate cash flows could be positive in the form of regular realizations of amenity values or sale of some minor/nontimber products while others could be negative in the form of annual taxes and overhead expenditures associated with maintenance activities. The basic analysis assumes that the net result from a combination of both positive and negative cash flows was a negative cash flow of $10/acre/year. For the purpose of consistency, all intermediate cash flows are treated as occurring at the beginning of a period. This arrangement does not effect the analysis since the intermediate cash flows are assumed nonstochastic. Nonstochastic variables are unaffected by the expectation operator but are effected by discounting. So, regardless of where they occur in the period their value at the beginning of the period can be considered as the appropriately discounted value. These known values were extrapolated like riskfree variables i.e., with close to zero variance (and no correlation with the stochastic variables). Harvesting and transportation costs per unit merchantable timber were assumed constant i.e., the effect of economies of scale observed for older or larger stands was ignored for want of data. 5. The unit FOB price of the timber product was modeled as the stochastic variable and the unit harvesting and transportation costs deduced from empirical data served as the strike price for the option on the stand. The estimated empirical values of variance for the timber stumpage price were used to model the variance of the FOB price. 6. The GBM and MR models applied were assumed to have constant parameters i.e., the drift and variance for the GBM model and the reversion coefficient and variance parameters for the MR models were assumed constant. 7. Land rent was estimated for the stochastic price process using the CC valuation as detailed earlier. 8. Taxes are not specifically treated in the analysis. 9. Ideally the term structure of interest rates should be used to model the riskfree rate. For simplicity, a single constant riskfree interest rate was used instead. 10. An assumption was made that the pulpwood stand was located so as to experience low/moderate relative timber product prices. 11. For the basic model the convenience yield was assumed to be zero. Sensitivity analysis to consider the effect of positive values of the convenience yield was conducted. CHAPTER 5 RESULTS AND DISCUSSION A Single Product Stand and the Geometric Brownian Motion Price Process In the following section the entire merchantable output of the stand at any rotation age was treated as a single undifferentiated product, in this case pulpwood. As argued earlier, this would be the case for a stand experiencing low relative timber product prices. Figure 51 plots the per acre merchantable timber yield curve for a cutover slash pine stand in Florida with the following site description and management history 140.00 120.00 2 100.00 t 80.00 *Total Merchantable Wood 60.00 T 40.00 20.00 0.00 Rotation age (Years) Figure 51. Total per acre merchantable yield curve for slash pine stand * Site Index 60 (age 25) * Site Preparation Burning only * Initial planting density 800 Trees per acre * Thinnings None Parameter values used for the analysis are listed in Table 51. Table 51. Parameter values used in analysis of harvest decision for single product stand with GBM price process Parameter Effective Date/ Unit Value Period FOB price II Qtr 2005 $/Ton 21.96 Stumpage price II Qtr 2005 $/Ton 7.42 Harvesting and transportation cost II Qtr 2005 $/Ton 14.54 Initial plantation expenses II Qtr 2005 $/acre 330.00 Estimated land rent II Qtr 2005 $/acre/year 34.00 Other annual expenses II Qtr 2005 $/acre/year 10.00 Estimated standard deviation 07012005 Annual 0.29 of GBM price process Risk free rate 07012005 %//annum 3.64 Constant convenience yield Annual 0.00 Present age of stand 07012005 Years 20.00 In order to value the option to postpone the clearcut harvest the FOB price for pulpwood was modeled as a stochastic variable following a GBM process with a constant standard deviation of 0.29. A binomial lattice was constructed using Equations 328, 329 and 337 for this stochastic variable with a one year period. The backward recursive option pricing procedure was then implemented to determine the option value. The GAUSS Light version 5.0 (Aptech Systems, Inc.) software was used for finding solutions. The per acre pretax value of an immediate harvest and sale as pulpwood of the entire merchantable yield at rotation age 20 at current stumpage price of $7.42/ton was $567. The maximum or terminal rotation age considered was 43 years. At $966/acre or $12.64/ton the calculated option value was higher than the value of immediate harvest. Figure 52 plots the upper bounds of the stumpage prices for the harvest region or the crossover price line. Since a discrete time approximation with large period values (annual) was used continuity was sacrificed i.e., the reported values of crossover prices display large jumps. The crossover line has also been smoothened to remove the incongruities in the data recovered fr~om the discrete lattice structure. 200.00 180.00 160.00 1 40.00 S120.00 S100.00 mOptimal crosso\Rr prices a. 80.00 S60.00 40.00 20.00 0.00 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Rotation Age (Years) Figure 52. Crossover price line for single product stand with GBM price process The region to the RHS of the line is the harvesting region and to its LHS is the continuity region. The form of the crossover line suggests that the optimal strategy will comprise of harvesting only if the rotation age approaches the terminal age and the stumpage prices decrease to zero. As the rotation age approaches the terminal age harvesting at higher stumpage prices becomes feasible. These results conform to the findings of Thomson (1992a) and the discussion in Plantinga (1998) on the results of a Geometric Brownian motion price process. The form of the optimal harvesting strategy implies that harvesting was only feasible to avoid uneconomic outcomes or when there was a low probability of improving returns by waiting any further in the time left to the terminal date. This result for the GBM price process is also confirmed from the results for plain financial (American) options on nondividend bearing stocks that are always optimally held to the maturity/terminal date. The crossover line for stand harvesting is observed because of the presence of the intermediate expenses. Sensitivity Analysis Sensitivity of the results to changes in values of various parameters was considered next. The land rent was reestimated to reflect the change in value of the parameter under consideration. First, the response of the results to changes in intermediate expenses was considered. The option value corresponding to an increase in intermediate expenses by $10/annum/acre was $859/acre, a decrease of more than $100/acre. On the other hand the option value for an increase in intermediate expenses by $40/annum/acre decreased the option value to $617/acre a drop of about $3 50/acre. If the higher intermediate expenses are considered to arise from payments for purchase of insurance against nonmarketed undesirable risks, it is possible to see the effect that catastrophic risks have on the harvesting strategy and option values. Thus, the observed empirical rotations of less than 30 years could be partly explained by the presence of nonmarketed undesirable risks. On the other hand, lowering intermediate expenses by $10/annum/acre increased the option value to $1,088/acre, an increase of more than $100/acre. Thus, positive and 20 2122 23 2425 2627 2829 30 3132 33 3435 3637 3839 4041 42 Rotation age (Years) previsible cash flows in the form of, say, nontimber incomes or aesthetic values would lead to longer optimal rotations. 400 350 300 S250 S200 E 150 100 50 0 oUnchanged Intermediate Expenses m Less $5 a Less $10 x Less $20 Figure 53. Crossover price lines for different levels of intermediate expenses Figure 53 illustrates the effect of changing the magnitude of intermediate cash flows. As the intermediate cash flows in the form of expenses or negative cash flows increase the crossover line shifts to the left towards lower rotation ages From the option pricing theory it is known that option value is directly related to the magnitude of the variance of the underlying stochastic asset value. The variance for stumpage prices estimated from the TMS data may be higher than the variance experienced by individual pulpwood stand owners for reasons discussed earlier. Higher variances mean the possibility of higher positive payoffs even while the effect of the higher negative values is limited to zero. The results of the sensitivity analysis for different levels of variances confirmed the known behavior of option values. The option value for a standard deviation value of 0.20 was $765/acre as compared to $966/acre for the base standard deviation of 0.29. The option value dropped further to $658/acre for a standard deviation of 0. 10. Figure 54 shows that when the variance level is lower the crossover lines lie to the left of higher variance models so that optimal harvesting at lower rotation ages as well as lower stumpage prices becomes feasible. This implies that in a situation of large expected variances arising, say, from an unpredictable regulatory environment, harvesting should be optimally postponed. 200 180 160 1 40 S120 Unchanged Standard Deviation 0.29 100 A Standard Deviation 0.2 Rotation age (Years) Figure 54. Crossover price line for different levels of standard deviation Instead of using a convenience yield value of zero, the use of a positive constant convenience yield will alter the riskneutral expected drift of the process (Equation 3 3 8). For a constant convenience yield of 0.005 the option value dropped to $874/acre, dropping further to $803/acre for convenience yield value 0.01 and to $755/acre for convenience yield value 0.015. Since higher levels of convenience yield are associated with low levels of inventory and associated higher market prices, it suggests that optimal rotations should be shorter when the markets are tight. Figure 55 plots the effect of different levels of constant convenience yield on the crossover price lines. It shows the leftward shift of the crossover lines in response to higher levels of constant positive convenience yields. 200 180 160 1 40 ~ 120 ~Convenience Yield =0 8 'Convenience Yield = 0.005 S100 A Convenience Yield = 0.01 80t Convenience Yield = 0.015 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Rotation age (Years) Figure 55. Crossover price lines for varying levels of positive constant convenience yield It iz also of interest to know if the optimal decision changes for a different current price i.e., does a higher or lower current stumpage price induce earlier harvesting. The per acre option value corresponding to a present stumpage price of $1/ton ($76/acre) was $7/ton ($548/acre). On the other hand the per acre option value for a present stumpage price of $20/ton ($1,529/acre) was $25/ton ($1,933/acre). The possibility of higher payoffs as a result of higher current prices inflates the land rent reducing the relative option values. Figure 56 plots the results of considering different levels of current stumpage prices. The plots show that the crossover price lines shift to the left for a price increase and vice versa. This results partly from the effect of a direct relation between land rent and current prices. All other things being constant, a higher current timber price increases the present land value which increases the cost of waiting through the land rent. At the same time higher present stumpage price also means lower possibility of unfavorable outcomes but this effect is overwhelmed by the increase in land values. 200 180 160 140 12 o ~Unchanged Stumpage Price = $7.42/Ton 100 xStumpage Price = $1.00/Ton Stumpage Price = $20/Ton 0.. 8 0 S60 40 20 0L L L`rb~~~0 C~O~'b~9 ~ r Rotation Age (Years) Figure 56. Crossover price lines for different levels of current stumpage price Next, by changing the present rotation age of the stand from 20 to 25 and 30 we can observe the drop in option values commonly associated with financial options as the time remaining till the terminal date is reduced. This is due to the lower probability of higher payoffs in the remaining time. For the timber stand, for a present rotation age of 25 the associated option value was $9.97/ton ($999/acre). Similarly, for a present rotation age of 30 the option value was $8.88/ton ($1,027/acre). Finally, the effect of a change in the initial planting density was studied. The current plantation establishment expenses were adjusted to reflect the cost of planting less plants which effects the estimated land rent, though only marginally. The important observation is that lower initial planting densities did not change option values which were $12.62/ton ($909/acre) for 700 tpa and $12.64/ton ($781/acre) for 500 tpa. The option value calculated earlier for a 800 tpa initial planting density was $12.64/ton. Comparison with the Dynamic Programming Approach This section applies the DP approach to the single product slash pine pulpwood stand with timber prices following a GBM process. The binomial lattice was set up using Equations 328 and 329 with the subjective probability given by Equation 327 instead of the riskneutral probability given by Equation 337. The estimated value of the drift pu for pulpwood stumpage prices was 0.05 (with a standard error of =0.053 or >100%). A variety of discount rates have been used in published forestry literature using the DP approach, the most common being a real rate of 5%. Since this analysis was conducted in nominal terms and the average inflation estimated from the PPI series was 3%, a nominal discount rate of 8% was used in this DP analysis. Further, some of the published literature assumes the intermediate costs are constant in real terms. Therefore, for comparability, future values of intermediate expenses including the estimated land rent and harvesting and transportation cost were inflated at the average inflation rate of 3% computed from the PPI series. The land rent was estimated using the DP procedure. All other parameter values used were unchanged from Table 51. The option value derived from the DP approach, parameterized as above, was $2,393/acre. This value was more than twice the option value derived using the CC approach i.e., $966/acre. The use of a discount rate of approximately 12.5% brought down the estimated option value using the DP approach close to the option value estimated using the CC approach. As noted and illustrated in Hull (2003) the appropriate discount rate for options is much higher than the discount rate applicable to the underlying asset. First, it should be noted that at 12.5% the discount rate is much higher than typical rates considered in forestry literature on options analysis. Second, this discount rate is not a constant but would vary according to the parameter values of the problem. This is evident from the sensitivity of the option values to parameters exhibited above. This illustration serves to highlight the problems associated with using the DP approach in the absence of a method for determining the appropriate discount rate. A Single Product Stand and the Mean Reverting Price Process In this section the optimal harvesting strategy for the single product pulpwoodd) stand is analyzed with a mean reverting FOB price process. The stand description and management history were identical to those considered for the GBM price process analysis. The parameters used in the basic analysis are listed in Table52. The problem was modeled by considering only the FOB price for pulpwood as stochastic following a MR process of the form given by Equation 422 with a constant standard deviation of 0.22 and constant reversion coefficient with value 0. 18. Table 52. Parameter values used in analysis of harvest decision for single product stand with MR price process Parameter Effective Date/ Unit Value Period FOB price II Qtr 2005 $/ton 21.96 Stumpage price II Qtr 2005 $/ton 7.42 Harvesting and transportation cost II Qtr 2005 $/ton 14.54 Mean FOB price level II Qtr 2005 $/ton 24.44 Initial plantation expenses II Qtr 2005 $/acre 330.00 Estimated land rent II Qtr 2005 $/acre/year 24.00 Other annual expenses II Qtr 2005 $/acre/year 10.00 Estimated standard deviation 07012005 Annual 0.22 of MR price process Estimated constant reversion 07012005 Annual 0.18 coefficient Risk free rate 07012005 %//annum 3.64 Constant convenience yield Annual 0.00 Present age of stand 07012005 Years 20.00 Estimated average inflation rate %//annum 3.00 The option value at $1,290 was higher than present stumpage value of $567. From the form of the crossover price line (drawn after smoothing) for the mean reverting FOB prices (Figure 57) it is evident that the strategy for the optimal harvest is significantly different than that for the GBM prices. The form of the crossover line suggests that the optimal strategy would be to harvest if a sufficiently high stumpage was received at each rotation age, the crossover price declining with the rotation age. These results are consistent with those reported for the reservation prices obtained using search algorithms and for other studies with first order autoregressive or mean reverting prices. 