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PHILOSOPHY UNIVERSITY OF FLORIDA 2007 2007 ChungJui Wang To my family ACKNOWLEDGMENTS I thank my supervisory committee chair (Dr. Stan Uryasev) and supervisory committee members (Dr. Farid AitSahlia, Dr. Liqing Yan, and Dr. Jason Karceski) for their support, guidance, and encouragement. I thank Dr. Victor E. Cabrera and Dr. Clyde W. Fraisse for their guidance. I am grateful to Mr. Philip Laren for sharing his knowledge of the mortgage secondary market and providing us with the dataset used in the case study. TABLE OF CONTENTS page A CK N O W LED G M EN T S ................................................................. ........... ............. ..... L IST O F TA B LE S ......... .... ........................................................................... 7 LIST OF FIGURES .................................. .. ..... ..... ................. .8 A B S T R A C T ......... ....................... .................. .......................... ................ .. 9 CHAPTER 1 IN T R O D U C T IO N ....................................................................................... .......... .. .. .. 11 2 OPTIMAL CROP PLANTING SCHEDULE AND HEDGING STRATEGY UNDER ENSOBASED CLIMATE FORECAST ........................................ ......................... 16 2 .1 Introduction ............................................................................................ 16 2 .2 M odel ................................................................................................. 19 2.2.1 Random Yield and Price Simulation ................................................... ............... 19 2 .2 .2 M eanC V aR M odel ........................................................................ .................. 22 2.2.3 M odel Im plem entation ...................................................... .......... ............... 23 2.2.4 Problem Solving and Decomposition....................... ... ........................ 29 2 .3 C a se S tu dy ................................................................2 9 2.4 Results and Discussion ................... ......... ...... .... ....... .............. ........ .. 32 2.4.1 Optimal Production with Crop Insurance Coverage ...........................................33 2.4.2 Hedging with Crop Insurance and Unbiased Futures.......................................34 2 .4 .3 B iased F utures M arket......................................... .............................................35 2 .5 C o n clu sio n ...................................... ................................................... 4 3 3 EFFICIENT EXECUTION IN THE SECONDARY MORTGAGE MARKET .....................45 3 .1 In tro d u ctio n ................................................................................................................. 4 5 3.2 M mortgage Securitization .......................................................................... ....................47 3 .3 M o d e l ................... ................... ...................1........ 3 .3 .1 R isk M measure ........................................................................... 52 3.3.2 M odel D evelopm ent .............................................................................. .... ..... 53 3.4 C ase Study ...................................................... ................... ......... ...... 59 3 .4 .1 In p u t D a ta .............................................................................................5 9 3 .4 .2 R e su lt ......................................................................................................................6 1 3.4.4 Sensitivity A analysis ........................................................................63 3 .5 C o n c lu sio n ........................................................................................6 4 4 MORTGAGE PIPELINE RISK MANAGEMENT .................................... ...............69 4 .1 In tro du ctio n ...................................... .................................. ................ 6 9 4 .2 M odel ......................................................................................7 1 4.2.1 Locked Loan Am ount Evaluation ........................................ ....... ............... 71 4.2.2 Pipeline Risk H edge A genda........................................................ ............... 72 4.2.3 M odel D evelopm ent .......................................................................... ............... 73 4.3 C ase Study .............................................................77 4.3.1 D ataset and Experim ent D esign ........................................ ......... ............... 77 4.3.2 A analyses and R results ......... ................. ......................................... ............... 78 4.4 Conclusion .......... .... ..... .........................................................................81 5 CON CLU SION .......... ................................................................. ............. ... 82 APPENDIX A EFFICIENT EXECUTION MODEL FORMULATION ..................................................84 L IS T O F R E F E R E N C E S .................................................................................... .....................87 B IO G R A PH IC A L SK E T C H .............................................................................. .....................90 6 LIST OF TABLES Table page Table 21. Historical years associated with ENSO phases from 1960 to 2003 .............................30 Table 22. Marginal distributions and rank correlation coefficient matrix of yields of four planting dates and futures price for the three ENSO phases........................................31 Table 23. Parameters of crop insurance (2004) used in the farm model analysis......................32 Table 24. Optimal insurance and production strategies for each climate scenario under the 90% CVaR tolerance ranged from $20,000 to $2,000 with increment of $2000............34 Table 25. Optimal solutions of planting schedule, crop insurance coverage, and futures hedge ratio with various 90% CVaR upper bounds ranged from $24,000 to $0 with increment of $4,000 for the three ENSO phases..................................... ............... 35 Table 26. Optimal insurance policy and futures hedge ratio under biased futures prices............36 Table 27. Optimal planting schedule for different biases of futures price in ENSO phases........39 Table 31. Summary of data on mortgages .......................................................................61 Table 32. Summary of data on MBS prices of MBS pools ......................................................61 Table 33. Guarantee fee buyup and buydown and expected retained servicing multipliers......61 Table 34. Summary of efficient execution solution under different risk preferences ..................65 Table 35 Sensitivity analysis in servicing fee multiplier................................... ............... 67 Table 36. Sensitivity analysis in mortgage price................................ ...................67 Table 37. Sensitivity analysis in M B S price.......................................... ........................... 68 Table 41 Mean value, standard deviation, and maximum loss of the 50 outofsample losses of hedged position based on rolling window approach .................................................81 Table 42 Mean value, standard deviation, and maximum loss of the 50 outofsample losses of hedged position based on growing window approach ......... ................................... 81 LIST OF FIGURES Figure page Figure 21. Definition of VaR and CVaR associated with a loss distribution.............................23 Figure 22. Bias of futures price versus the optimal hedge ratio curves associated with different 90% CVaR upper bounds in the La Nifia phase................. .........................38 Figure 23. The efficient frontiers under various biased futures price. (A) El Niho year. (B) N neutral year. (C) La N iia year. ............................................... .............................. 41 Figure 31. The relationship between participants in the passthrough MBS market.. .................48 Figure 32. G guarantee fee buydow n. ................................................................. .....................49 Figure 33. Guarantee fee buyup. ...... ........................... ............................................ 49 F figure 34 : E efficient F rontiers.. ......................................................................... ...................... 62 F igure 41. N negative convexity ........................................................................... .....................70 Figure 42 Value of naked pipeline position and hedged pipeline positions associated with different risk measures ......................... ..... .. .. .. ..................... 79 Figure 43 Outofsample hedge errors associated with eight risk measures using rolling window approach ............... .................. .............. .............. ............ 80 Figure 44. Outofsample hedge errors associated with eight risk measures using growing window approach ............... .................. .............. ................. ......... 80 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 OPTIMIZATION APPROACHES IN RISK MANAGEMENT: APPLICATIONS IN FINANCE AND AGRICULTURE By ChungJui Wang December 2007 Chair: Stanislav Uryasev Major: Industrial and Systems Engineering Along with the fast development of the financial industry in recent decades, novel financial products, such as swaps, derivatives, and structure financial instruments, have been invented and traded in financial markets. Practitioners have faced much more complicated problems in making profit and hedging risks. Financial engineering and risk management has become a new discipline applying optimization approaches to deal with the challenging financial problems. This dissertation proposes a novel optimization approach using the downside risk measure, conditional valueatrisk (CVaR), in the reward versus risk framework for modeling stochastic optimization problems. The approach is applied to the optimal crop production and risk management problem and two critical problems in the secondary mortgage market: the efficient execution and pipeline risk management problems. In the optimal crop planting schedule and hedging strategy problem, crop insurance products and commodity futures contracts were considered for hedging against yield and price risks. The impact of the ENSObased climate forecast on the optimal production and hedge decision was also examined. The Gaussian copula function was applied in simulating the scenarios of correlated nonnormal random yields and prices. Efficient execution is a significant task faced by mortgage bankers attempting to profit from the secondary market. The challenge of efficient execution is to sell or securitize a large number of heterogeneous mortgages in the secondary market in order to maximize expected revenue under a certain risk tolerance. We developed a stochastic optimization model to perform efficient execution that considers secondary marketing functionality including loanlevel efficient execution, guarantee fee buyup or buydown, servicing retain or release, and excess servicing fee. The efficient execution model balances between the reward and downside risk by maximizing expected return under a CVaR constraint. The mortgage pipeline risk management problem investigated the optimal mortgage pipeline risk hedging strategy using 10year Treasury futures and put options on 10year Treasury futures as hedge instruments. The outofsample hedge performances were tested for five deviation measures, Standard Deviation, Mean Absolute Deviation, CVaR Deviation, VaR Deviation, and twotailed VaR Deviation, as well as two downside risk measures, VaR and CVaR. CHAPTER 1 INTRODUCTION Operations Research, which originated from World War II for optimizing the military supply chain, applies mathematic programming techniques in solving and improving system optimization problems in many areas, including engineering, management, transportation, and health care. Along with the fast development of the financial industry in recent decades, novel financial products, such as swaps, derivatives, and structural finance instruments, have been invented and traded in the financial markets. Practitioners have faced much more complicated problems in making profit and hedging risks. Financial engineering and risk management has become a new discipline applying operations research in dealing with the sophisticated financial problems. This dissertation proposes a novel approach using conditional valueatrisk in the reward versus risk framework for modeling stochastic optimization problems. We apply the approach in the optimal crop planting schedule and risk hedging strategy problem as well as two critical problems in the secondary mortgage market: the efficient execution problem, and the mortgage pipeline risk management problem. Since Markowitz (1952) proposed the meanvariance framework in portfolio optimization, variance/covariance has become the predominant risk measure in finance. However, this risk measure is suited only to elliptic distributions, such as normal or tdistributions, with finite variances (Szego 2002). The other drawback of variance risk measure is that it measures both upside and downside risks. In practice, however, finance risk management is concerned mostly with the downside risk. A popular downside risk measure in economics and finance is Valueat Risk (VaR) (Jorion 2000), which measures a percentile of loss distribution. However, as was shown by Artzner et al. (1999), VaR is illbehaved and nonconvex for general distribution. The disadvantage of VaR is that it only considers risk at a percentile of loss distribution and does not consider the magnitude of the losses in the atail (i.e., the worst 1a percentage of scenarios). To address this issue, Rockafellar and Uryasev (2000, 2002) proposed Conditional Value atRisk, which measures the mean value of the atail of loss distribution. It has been shown that CVaR satisfies the axioms of coherent risk measures proposed by Artzner et al. (1999) and has desirable properties. Most importantly, Rockafellar and Uryasev (2000) showed that CVaR constraints in optimization problems can be formulated as a set of linear constraints and incorporated into problems of optimization. This linear property is crucial in formulating the model as a linear programming problem that can be efficiently solved. This dissertation proposes a meanCVaR model which, like a meanvariance model, provides an efficient frontier consisting of points that maximize expected return under various risk budgets measured by CVaR. Since CVaR is defined in monetary units, decision makers are able to decide their risk tolerance much more intuitively than with abstract utility functions. It is worth noting that CVaR is defined on a loss distribution, so a negative CVaR value represents a profit. Although the variance/covariance risk measure has its drawbacks, it has been the proxy of risk measure for modeling the stochastic optimization problem in financial industry. The main reason is that the portfolio variance can be easily calculated given individual variances and a covariance matrix. However, the linear correlation is a simplified model and is limited in capturing the association between random variables. Therefore, a more general approach is needed to model the more complicated relationship between multivariate random variables, e.g., tail dependent. More importantly, the approach should provide a convenient way to create the portfolio loss distribution from marginal ones, which is the most important input data in the meanCVaR optimization model. This dissertation applies the copula function to model the correlation between random variables. In Chapter 2, the use of copulas to generate scenarios of dependent multivariate random variables is discussed. Furthermore, the simulated scenarios are incorporated into the meanCVaR model. Chapter 2 investigates the optimal crop planting schedule and hedging strategy. Crop insurance products and futures contracts are available for hedging against yield and price risks. The impact of the ENSObased climate forecast on the optimal production and hedging decision is examined. Gaussian copula function is applied in simulating the scenarios of correlated non normal random yields and prices. Using data of a representative cotton producer in the Southeastern United States, the best production and hedging strategy is evaluated under various risk tolerances for each of three predicted ENSObased climate phases. Chapters 3 and 4 are devoted to two optimization problems in the secondary mortgage market. In Chapter 3, the efficient execution problem was investigated. Efficient execution is a significant task faced by mortgage bankers attempting to profit from the secondary market. The challenge of efficient execution is to sell or securitize a large number of heterogeneous mortgages in the secondary market in order to maximize expected revenue under a certain risk tolerance. A stochastic optimization model was developed to perform efficient execution that considers secondary marketing functionality including loanlevel efficient execution, guarantee fee buyup or buydown, servicing retain or release, and excess servicing fee. Since efficient execution involves random cash flows, lenders must balance between expected revenue and risk. We employ a CVaR risk measure in this efficient execution model that maximizes expected revenue under a CVaR constraint. By solving the efficient execution problem under different risk tolerances specified by a CVaR constraint, an efficient frontier was found, which provides secondary market managers the best execution strategy associated with different risk budgets. The model was formulated as a mixed 01 linear programming problem. A case study was conducted and the optimization problem was efficiently solved by the CPLEX optimizer. Chapter 4 examines the optimal mortgage pipeline risk hedging strategy. Mortgage lenders commit to a mortgage rate while the borrowers enter the loan transaction process. The process is typically for a period of 3060 days. While the mortgage rate rises before the loans go to closing, the value of the loans declines. Therefore, the lender will sell the loans at a lower price when the loans go to closing. The risk of a fall in value of mortgages still being processed prior to their sale is known as mortgage pipeline risk. Lenders often hedge this exposure by selling forward their expected closing volume or by shorting U.S. Treasury notes or futures contracts. Mortgage pipeline risk is affected by fallout. Fallout refers to the percentage of loan commitments that do not go to closing. As interest rates fall, fallout rises since borrowers who have locked in a mortgage rate are more likely to find better rates with another lender. Conversely, as rates rise, the percentage of loans that go to closing increases. Fallout affects the required size of the hedging instrument because it changes the size of risky pipeline positions. At lower rates, fewer loans will close and a smaller position in the hedging instrument is needed. Lenders often use options on U.S. Treasury futures to hedge against the risk of fallout (Cusatis and Thomas, 2005). A model was proposed for the optimal mortgage pipeline hedging strategy that minimizes the pipeline risks. A case study considered two hedging instruments for hedging the mortgage pipeline risks: the 10 year Treasury futures and put options on 10year Treasury futures. To investigate the impact of different risk measurement practices on the optimal hedging strategies, we tested five deviation measures, standard deviation, mean absolute deviation, CVaR deviation, VaR deviation, and twotailed VaR deviation, as well as two downside risk measures, VaR and CVaR, in the minimum mortgage pipeline risk model. The outofsample performances of the five deviation measures and two downside risk measures were examined. CHAPTER 2 OPTIMAL CROP PLANTING SCHEDULE AND HEDGING STRATEGY UNDER ENSO BASED CLIMATE FORECAST 2.1 Introduction A riskaverse farmer preferring higher profit from growing crops faces uncertainty in the crop yields and harvest price. To manage uncertainty, a farmer may purchase a crop insurance policy and/or trade futures contracts against the yield and price risk. Crop yields depend on planting dates and weather conditions during the growing period. The predictability of seasonal climate variability (i.e., the El Nifio Southern Oscillation, ENSO), gives the opportunity to forecast crop yields in different planting dates. With the flexibility in planting timing, the profit can be maximized by selecting the best planting schedule according to climate forecast. Risk  averse is another critical factor when farmers make the decision. Farmers may hedge the yield and price risks by purchasing crop insurance products or financial instruments. Two major financial instruments for farmers to hedge against crop risks are crop insurances and futures contracts. The Risk Management Agency (RMA) of the United States Department of Agricultural (USDA) offers crop insurance policies for various crops, which could be categorized into three types: the yieldbased insurance, revenuebased insurance, and policy endorsement. The most popular yieldbased insurance policy, Actual Production History (APH), or Multiple Peril Crop Insurance (MPCI), is available for most crops. The policies insure producers against yield losses due to natural causes. An insured farmer selects to cover a percentage of the average yield together with an election price (a percentage of the crop price established annually by RMA). If the harvest yield is less than the insured yield, an indemnity is paid based on the shortfall at the election price. The most popular revenuebased insurance policy, Crop Revenue Coverage (CRC), provides revenue protection. An insured farmer selects a coverage level of the guarantee revenue. If the realized revenue is below the guarantee revenue, the insured farmer is paid an indemnity to cover the difference between the actual and guaranteed revenue. Catastrophic Coverage (CAT), a policy endorsement, pays 55% of the established price of the commodity on crop yield shortfall in excess of 50%. The cost of crop insurances includes a premium and an administration fee1. The premiums on APH and CRC both depend on the crop type, county, practice (i.e., irrigated or nonirrigated), acres, and average yield. In addition, the APH premium depends on price election and yield coverage, and the CRC depends on revenue coverage. The premium on CAT coverage is paid by the Federal Government; however, producers pay the administrative fee for each crop insured in each county regardless of the area planted2 In addition to crop insurance coverage, farmers may manage commodity price risk by a traditional hedge instrument such as futures contract. A futures contract is an agreement between two parties to buy or sell a commodity at a certain time in the future, for a specific amount, at a certain price. Futures contracts are highly standardized and are traded by exchange. The cost of futures contract includes commissions and interest foregone on margin deposit. A riskaverse producer may consider using insurance products in conjunction with futures contacts for best possible outcomes. El Nifio Southern Oscillation refers to interrelated atmospheric and oceanic phenomena. The barometric pressure difference between the eastern and western equatorial Pacific is frequently changed. The phenomenon is known as the Southern Oscillation. When the pressure over the western Pacific is above normal and eastern Pacific pressure is below normal, it creates abnormally warm sea surface temperature (SST) known as El Niho. On the other hand, when the ' The administration fee is $30 for each APH and CRC contract and $100 for each CAT contract. 2 Source: http://www2.rma.usda.gov/policies . eastwest barometric pressure gradient is reversed, it creates abnormally cold SST known as La Nifia. The term "neutral" is used to indicate SSTs within a normal temperature range. These equatorial Pacific conditions known as ENSO phases refer to different seasonal climatic conditions. Since the Pacific SSTs are predictable, ENSO becomes an index for forecasting climate and consequently crop yields. A great deal of research has been done on the connection between the ENSObased climate prediction and crop yields since early 1990s. Cane et al. (1994) found the longterm forecasts of the SSTs could be used to anticipate Zimbabwean maize yield. Hansen et al. (1998) showed that El Nifio Southern Oscillation is a strong driver of seasonal climate variability that impact crop yields in the southeastern U.S. Hansen (2002) and Jones et al. (2000) concluded that ENSO based climate forecasts might help reduce crop risks. Many studies have focused on the crop risk hedging with crop insurance and other derivative securities. Poitras (1993) studied farmers' optimal hedging problem when both futures and crop insurance are available to hedge the uncertainty of price and production. Chambers et al. (2002) examined optimal producer behavior in the presence of areayield insurance. Mahul (2003) investigated the demand of futures and options for hedging against price risk when the crop yield and revenue insurance contracts are available. Coble (2004) investigated the effect of crop insurance and loan programs on demand for futures contract. Some researchers have studied the impacts of the ENSObased climate information on the selection of optimal crop insurance policies. Cabrera et al. (2006) examined the impact of ENSObased climate forecast on reducing farm risk with optimal crop insurance strategy. Lui et al. (2006), following Cabrera et al. (2006), studied the application of Conditional ValueatRisk (CVaR) in the crop insurance industry under climate variability. Cabrera et al. (2007) included the interference of farm government programs on crop insurance hedge under ENSO climate forecast. The purpose of this research is twofold. First, a meanCVaR optimization model was proposed for investigating the optimal crop planting schedule and hedging strategy. The model maximizes the expected profit with a CVaR constraint for specifying producer's downside risk tolerance. Second, the impact of the ENSObased climate forecast on the optimal decisions of crop planting schedule and hedging strategy was examined. To this end, we generate the scenarios of correlated random yields and prices by Monte Carlo simulation with the Gaussian copula for each of the three ENSO phases. Using the scenarios associated with a specific ENSO phase as the input data, the meanCVaR model is solved for the optimal production and hedging strategy for the specified ENSO phase. The remainder of this article is organized as follows. The proposed model for optimal planting schedule and hedging strategy is introduced in section 2.2. Next, section 2.3 describes a case study using the data of a representative cotton producer in the Southeastern United States. Then, section 2.4 reports the results of the optimal planting schedule, crop insurance policy selection, and hedging position of futures contract. Finally, section 2.5 presents the conclusions. 2.2 Model 2.2.1 Random Yield and Price Simulation To investigate the impact of ENSObased climate forecast on the optimal production and risk management decisions, we calibrate the yield and price distributions for an ENSO phase based on the historical yields and prices of the years classified to the ENSO phase based on the Japan Meteorological Agency (JMA) definition (Japan Meteorological Agency, 1991). Then, random yield and price scenarios associated with the ENSO phase are generated by Monte Carlo simulation. We assume a farmer may plant crops in a number of planting dates across the planting season. The yields of planting dates are positive correlated according to the historical data. In addition, the correlation between the random production and random price is crucial in risk management since negative correlated production and price provide a natural hedge that will affect the optimal hedging strategy (McKinnon, 1967). As a consequence, we consider the correlation between yields of different planting dates and the crop price. Since the distributions of the crop yield and price are not typically normal distributed, a method to simulate correlated multivariate nonnormal random yield and price was needed. Copulas are functions that describe dependencies among variables, and provide a way to create distributions to model correlated multivariate data. Copula function was first proposed by Sklar (1959). The Sklar theorem states that given a joint distribution function F on R" with marginal distributionF, there is a copula function C such that for all x,,..., x, in R, F(x,..., x, = C(F1 (x),...,F(x)). (21) Furthermore, if F, are continuous then C is unique. Conversely, if C is a copula andF, are distribution functions, then F, as defined by the previous expression, is ajoint distribution function with margins F,. We apply the Gaussian copula function to generate the correlated non normal multivariate distribution. The Gaussian copula is given by: C, (F, (x,),..., Fn (x,))= D,, (D 1(F~(xF)),...,_ (Fn(x ))), (22) which transfers the observed variable x,, i.e. yield or price, into a new variable y, using the transformation Y, = ) '[(x()] (23) where (D, is the joint distribution function of a multivariate Gaussian vector with mean zero and correlation matrix p. D is the distribution function of a standard Gaussian random variable. In moving from x, to y, we are mapping observation from the assumed distribution F into a standard normal distribution 0 on a percentile to percentile basis. We use the rank correlation coefficient Spearman's rho p, to calibrate the Gaussian copula to the historical data. For n pairs of bivariate random samples (X,,, X), define R, = rank(X,) and R, = rank(X, ). Spearman's sample rho (Cherubini, 2004) is given by p, = 1 6 k = (24) n(n2 1) Spearman's rho measures the association only in terms of ranks. The rank correlation is preserved under the monotonic transformation in equation 3. Furthermore, there is a onetoone mapping between rank correlation coefficient, Spearman's rho p,, and linear correlation coefficient p for the bivariate normal random variables (y,, y,) (Kruksal, 1958) 6 p(yl, y,) P, (,, ,) = arcsinp (25) Z7 2 To generate correlated multivariate nonnormal random variables with margins F, and Spearman's rank correlation p,, we generate the random variables y, 's from the multivariate normal distribution 0,p with linear correlation p = 2sin P (26) by Monte Carlo simulation. The actual outcomes x, 's can be mapped from y, 's using the transformation x =F [(y,)]( 2.2.2 MeanCVaR Model Since Markowitz (1952) proposed the meanvariance framework in portfolio optimization, variance/covariance has become the predominant risk measure in finance. However, the risk measure is suited only to the case of elliptic distributions, like normal or tdistributions with finite variances (Szego, 2002). The other drawback of variance risk measure is that it measures both upside and downside risks. In practice, finance risk management is concerned only with the downside risk in most cases. A popular downside risk measure in economics and finance is ValueatRisk (VaR) (Jorion, 2000), which measures a percentile of loss distribution. However, as was shown by Artzner et al. (1999), VaR is illbehaved and nonconvex for general distribution. The other disadvantage of VaR is that it only considers risk at a percentile of loss distribution and does not consider the magnitude of the losses in the atail (the worst 1a percentage of scenarios). To address this issue, Rockafellar and Uryasev (2000, 2002) proposed Conditional ValueatRisk, which measures the mean value of atail of loss distribution. Figure 21 shows the definition of CVaR and the relation between CVaR and VaR. It has been shown that CVaR satisfies the axioms of coherent risk measures proposed by Artzner et al. (1999) and has desirable properties. Most importantly, Rockafellar and Uryasev (2000) showed that CVaR constraints in optimization problems can be formulated as a set of linear constraints and incorporated into the problems of optimization. The linear property is crucial to formulate the model as a mixed 01 linear programming problem that could be solved efficiently by the CPLEX solver. This research proposes a meanCVaR model that inherits advantages of the return versus risk framework from the meanvariance model proposed by Markowitz (1952). More importantly, the model utilizes the CVaR risk measure instead of variance to take the (27) advantages of CVaR. Like meanvariance model, the meanCVaR model provides an efficient frontier consisting of points that maximize expected return under various tolerances of CVaR losses. Since CVaR is defined in monetary units, farmers are able to decide their risk tolerance much more intuitively compared to abstract utility functions. It is worth noting that CVaR is defined on a loss distribution. Therefore, a negative CVaR value represents a profit. For example, a $20,000 90% CVaR means the average of the worst 10% scenarios should provide a profit equal to $20,000. Frequency VaR Maximal loss catail (with probability 1 a) CVaR .. ..........,,l ll l ll i lll.,..... .......I .. ................... .. .. Losses Figure 21. Definition of VaR and CVaR associated with a loss distribution 2.2.3 Model Implementation Assume a farmer who plans to grow crops in a farmland of Q acres. There are K possible types of crops and more than one crop can be planted. For each crop k, there are Tk potential planting dates that give different yield distributions based on the predicted ENSO phase, as well as Ik available insurance policies for the crop. The decision variables xk, and 77k represent the acreages of crop k planted in date t with insurance policy i and the hedge position (in pounds) of crop k in futures contract, respectively. The randomness of crop yield and harvest price in a specific ENSO phase is managed by the joint distribution corresponding to the ENSO phase. We sample J scenarios from the joint distribution by Monte Carlo simulation with Gaussian copula, and each scenario has equal probability. Let Ykt denote thejth realized yield (pound per acre) of crop k planted on date t, and Pk denote thejth realized cash price (dollar per pound) for crop k at the time the crop will be sold. The objective function of the model, shown in (28), is to maximize the expectation of random profit f(xkt, 7k ) that consists of the random profit from production f (xk ), from crop insurance f' (x, ), and from futures contract fF (r7 ). maxEf(xk,, 7k )= max[Ef (xk,)+ Ef (xk )+EfF (r'k ) (28) The profit from production of crop k in scenario is equal to the income from selling the Tk k Tk 'k crop, YktjPk xkh, minus the production cost, Ck xk and plus the subsidy, t=1 \ =1 t= 1 =1 Tk 'k Sk xk ., where Ck and Sk are unit production cost and subsidy, respectively. Consequently t=l 1=1 Equation (29) expresses the expected profit from production. 1 J K Tk 'k Eff P kI )= ( Ck +Sk) k, (29) J ]=1 k=1 t=1 =11 Three types of crop insurance policies are considered in the model, including Actual Production History (APH), Crop Revenue Coverage (CRC), and Catastrophic Coverage (CAT). For APH farmers select the insured yield, a percentage a, from 50 to 75 percent with five percent increments of average yield Y, as well as the election price, a percentage ,, ,between 55 and 100 percent, of the of the established price Pk established annually by RMA. If the harvest is less than the yield insured, the farmer is paid an indemnity based on the difference S, cY Yk~ )xk at price 7Pk The indemnity of APH insurance policy i G IAPH for crop k in t=l thejth scenario is given by Dkj =max a, Yk 0 k,0 x A Pk Vi IPH (210) _t=1 For CRC, producers elect a percentage of coverage level 7, between 50 and 75 percent. Tk The guaranteed revenue is equal to the coverage level 7, times the product of Y kxk and the t=l higher of the base price (earlyseason price) Pk and the realized harvest price in thejth scenario of crop k, Ph The base price and harvest price of crop k are generally defined based on the crop's futures price in planting season and harvest season, respectively. If the calculated revenue Tk SYkV XkP, is less than the guaranteed one, the insured will be paid the difference. Equation (2 t=l 11) shows the indemnity of CRC insurance policy i e ICRC for crop k in thejth scenario. Dk = max YXkt x max [Pk h k ,0 Vi ICRC, (211) t=1 t=1 The CAT insurance pays 55 % of the established price of the commodity on crop losses in excess of 50 %. The indemnity of CAT insurance policy i e ICAr for crop k in thejth scenario is given by Dk = max (0.5Yk YJ)xk,,0 x0.55Pk (212) St=1 The cost of insurance policy i for crop k is denoted by R,, which includes a premium and an administration fee. For the case of CAT, the premium is paid by the Federal Government. Therefore, the cost of CAT only contains a $ 100 administrative fee for each crop insured in each county. The expected total profit from insurance is equal to the indemnity from the insurance coverage minus the cost of the insurance is given by I J K Ik Tk Efi(xk)= DkjJ Rkxk (213) Sj=1 k=1 zi=1 t=1 The payoff of a futures contract for crop k in scenario for a seller is given by S= (Fk fk )1k (214) where Fk is the futures price of crop k in the planting time, fk, is thejth realized futures price of crop k in the harvest time, and rk is the hedge position (in pounds) of crop k in futures contract. It is worth noting that the futures price fkA is not exactly the same as the local cash pricePk, at harvest time. Basis, defined in (215), refers to the difference that induces the uncertainty of futures hedging known as the basis risk. The random basis can be estimated from comparing the historical cash prices and futures prices. Basis = Cash Price Futures Price. (215) The cost of a futures contract, C', includes commissions and interest foregone on margin deposit. Equation (216) expresses the expected profit from futures contract. Ef F k)F = L C) (216) J ,=1k=1 We introduce binary variables zk in constraint (217) and (218) to ensure only one insurance policy can be selected for each crop k. Tk x Ik Izk =1 Vk (218) where S= if crop k is insured by policy i, S O otherwise. Constraint (219) restricts the total planting area to a given planting acreage Q. The equality in this constraint can be replaced by an inequality (<) to represent farmers choosing not to grow the crops when the production is not profitable. K Tk Ik IZZ Xkt, =Q (219) k=l t=l z1 To model producer's risk tolerance, we impose the CVaR constraint aCVaR(L(xkt, k )) < U. (220) where L(xk,,, k) is a random loss equal to the negative random profit f(xk,,' k) defined in (8). The definition of aCVaR(L(xk,, rk)) is given by aCVaR(L(xt,,rk)) = E[L(xk,rik) L(xk,rik )> (L(xk,, 7k))], (221) where ', (L(xk,, 17k))is the aquintile of the distribution of L(xk,,r7 k). Therefore, constraint (2 20) enforces the conditional expectation of the random loss L(xk,, rk ) given that the random loss exceeds aquintile to be less than or equal to U. In other words, the expected loss of atail, i.e. (1 a)100% worst scenarios, is upper limited by an acceptable CVaR upper bound U. Rockafellar & Uryasev (2000) showed that CVaR constraint (220) in optimization problems can be expressed by linear constraints (222), (223), and (224) J(Ca ( ) + Z J(1 a) Ji K Tk Ik z, 2 Z L(xk, rk) a(L(xk, r k)) Vj, (223) k=l t=l 1z= z> 0 Vj, (224) where z are artificial variables introduced for the linear formulation of CVaR constraint. Note that the maximum objective function contains indemnities Dk that include a max term shown in equation (210), (211), and (212). To implement the model as a mix 01 linear problem, we transform the equations to an equivalent linear formulation by disjunctive constraints (Nemhauser and Wolsey, 1999). For example, equation (210), Dk, = max (a k. )xk ,0 x iPk, can be represented by a set of mix 01 linear constraints _t=1 Dk > 0, Dk LaY k Yb)xktlx/Pk DTk k kt )k xktPk + MZk , Dk S(aYk )xk X APk I (az,k ktY )k x. ] X > MZAZ. t=1 (225) where Mis a big number and Zk is a 01 variable. Similarly, equation (211) and (212) can be transformed into a set of mix 01 linear constraints in the same way. Consequently, the optimal crop production and hedging problem has been formulated as a mix 01 linear programming problem. 2.2.4 Problem Solving and Decomposition Although the mix 01 linear programming problem can be solved with optimization software, the solving time increases exponentially when the problem becomes large. To improve the solving efficiency, we may decompose the original problem into subproblems that could be solved more efficiently than the original problem. Since only one insurance policy could be selected for each crop, we decomposed the original problem into subproblems in which each crop is insured by a specific insurance policy. The original problem contains K types of crops, and for the kth type of crop there are Ik eligible insurance policies. Therefore, the number of the subproblems is equal to the number of all possible insurance combinations of the K crops, K HIk k=1 The formulation of the subproblem is the same as the original problem except that the index i's are fixed and the equation (217) and (218) are removed. Solving subproblems gives the optimal production strategy and futures hedge amount under a specific combination of insurance policies for K crops. The solution of the subproblem with the highest optimal expected profit among all subproblems gives the optimal solution of the original problem in which the optimal production strategy and futures hedge position are provided from the sub problem solution and the optimal insurance coverage is the specific insurance combination of the subproblem. 2.3 Case Study Following the case study in Cabrera et al. (2006), we consider a representative farmer who grows cotton on a nonirrigated farm of 100 acres in Jackson County, Florida. Dothan Loamy Sand, a dominant soil type in the region, is assumed. The farmer may trade futures contracts from the New York Board of Trade and/or purchase crop insurance to hedge the crop yield and price risk. Three types of crop insurances, including Actual Production History (APH), Crop Revenue Coverage (CRC), and Catastrophic Coverage (CAT), are eligible for cotton and the farmer may select only one eligible insurance policy to hedge against the risk or opt for none. For APH, the eligible coverage levels of yield are from 65% to 75% with 5% increments, and the election price is assumed to be 100% of the established price. In addition, the available coverage levels of revenue for CRC are from 65% to 85% with 5% increments. To investigate the impact of ENSObased climate forecast in the optimal decisions of production and hedging strategy, we select historical climate data from 1960 to 2003 for the numerical implementation. ENSO phases during this period included 11 years of El Niho, 9 years of La Nifia, and the remaining 25 years of Neutral, according to the Japan Meteorological Index (Table 21). Table 21. Historical years associated with ENSO phases from 1960 to 2003 EL Nifio Neutral La Nifia 1964 1987 1960 1975 1984 1994 1965 1989 1966 1988 1961 1978 1985 1995 1968 1999 1970 1992 1962 1979 1986 1996 1971 2000 1973 1998 1963 1980 1990 1997 1972 1977 2003 1967 1981 1991 2001 1974 1983 1969 1982 1993 2002 1976 The cotton yields during the period of 19602003 were simulated using the CROPGRO Cotton model (Messina et al., 2005) in the Decision Support System for Agrotechnology Transfer (DSSAT) v4.0 (Jones et al., 2003) based on the historical climate data collected at Chipley weather station. The input for the simulation model followed the current management practices of variety, fertilization and planting dates in the region. More specifically, a medium to full season Delta & Pine Land variety (DP55), 110 kg/ha Nitrogen fertilization in two applications, and four planting dates, 16 Apr, 23 Apr, 1 May, and 8 May, were included in the yield simulation, which was further stochastically resampled to produce series of synthetically generated yields following the historical distributions (for more details see Cabrera et al., 2006). Assume cotton would be harvested and sold in December. The December cotton futures contact was used to hedge the price risk. In addition, assume the farmer will settle the futures contract on the last trading date, i.e. seventeen days from the end of December. The historical settlement prices of the December futures contract on the last trading date from 1960 to 2003 were collected from the New York Board of Trade. The statistics and the rank correlation coefficient Spearman's rho matrix of yields and futures price are summarized in Table 22, which shows that crop yields for different planting dates are highly correlated and the correlation of yields is decreasing when the corresponding two planting dates are getting farther. In addition, the negative correlation between yields and futures price is found in the El Nifio and Neutral phases, but not in La Nifia. We assumed the random yields and futures price follow the empirical distributions of yields and futures price. Table 22. Marginal distributions and rank correlation coefficient matrix of yields of four planting dates and futures price for the three ENSO phases Statistics of Marginal Rank Correlation Coefficient Matrix Distribution Spearman's rho ENSO Variable Yield Yield SStandard Yield Yield Future Mean on on Deviation on 5/1 on 5/8 s Price 4/16 4/23 Yield on 4/16 (lb) 815.0 71.7 1.00 0.93 0.75 0.74 0.36 Yield on 4/23 (lb) 804.6 79.4 0.93 1.00 0.63 0.57 0.23 El Nifio Yield on 5/1 (lb) 795.4 99.8 0.75 0.63 1.00 0.75 0.22 Yield on 5/8 (lb) 793.7 79.1 0.74 0.57 0.75 1.00 0.42 Futures Price ($/lb) 0.5433 0.1984 0.36 0.23 0.22 0.42 1.00 Yield on 4/16 (lb) 808.9 108.8 1.00 0.84 0.77 0.62 0.16 Yield on 4/23 (lb) 818.4 100.6 0.84 1.00 0.75 0.64 0.28 Neutral Yield on 5/1 (lb) 825.8 86.2 0.77 0.75 1.00 0.75 0.01 Yield on 5/8 (lb) 824.5 68.0 0.62 0.64 0.75 1.00 0.19 Futures Price ($/lb) 0.5699 0.1872 0.16 0.28 0.01 0.19 1.00 Yield on 4/16 (lb) 799.1 99.8 1.00 0.97 0.67 0.60 0.13 Yield on 4/23 (lb) 790.7 85.3 0.97 1.00 0.73 0.68 0.20 La Nifia Yield on 5/1 (lb) 793.9 90.6 0.67 0.73 1.00 0.97 0.13 Yield on 5/8 (lb) 809.3 94.1 0.60 0.68 0.97 1.00 0.08 Futures Price ($/lb) 0.4669 0.1851 0.13 0.20 0.13 0.08 1.00 We further estimated the local basis defined in Equation (215). The monthly historical data on average cotton prices received by Florida farmers from the USDA National Agricultural Statistical Service were collected (1979 to 2003) as the cotton local cash prices. By subtracting the futures price from the local cash price, we estimated the historical local basis. Using the Input Analyzer in the simulation software Arena, the best fitted distribution based on minimum square error method was a beta distribution with probability density function 0.13+0.15x BETA (2.76, 2.38). We calibrated the Gaussian copula based on the sample rank correlation coefficient Spearman's rho matrix for the three ENSO phases. For each ENSO phase, we sampled 2,000 scenarios of correlated random yields and futures price based on the Gaussian copula and the empirical distributions of yields and futures price by Monte Carlo simulation. Furthermore, we simulated the basis and calculated the local cash price from the futures price and basis. We assumed the futures commission and opportunity cost of margin to be $0.003 per pound, the production cost of cotton was $464 per acre, and the subsidy for cotton in Florida was $349 per acre. Finally, the parameters of crop insurance are listed in Table 23. Table 23. Parameters of crop insurance (2004) used in the farm model analysis Crop Insurance Parameters Values APH premium 65%75% $19.5/acre $38/acre CRC premium 65%85% $24.8/acre$116.9/acre Established Price for APH $0.61/lb Average yield 814 lb/acre Source: www.rma.usda.gov 2.4 Results and Discussion This section reports the results of optimal planting schedule and hedging strategy with crop insurance and futures contract for the three predicted ENSO phases. In section 2.4.1 we assumed crop insurances were the only risk management tool for crop yield and price risk together with an unbiased futures market3. In section 2.4.2 we considered both insurance and futures contracts were available and assumed the future market being unbiased. In section 2.4.3 we investigated the optimal decision under biased futures markets. 2.4.1 Optimal Production with Crop Insurance Coverage This section considers crop insurance as the only crop risk management tool. Since the indemnity of CRC depends on the futures price, we assume the futures market is unbiased, i.e., F = Ef where F is the futures price in planting time andfis the random futures price in harvest time. Table 24 shows that the optimal insurance and production strategies for each ENSO phase with various 90%CVaR upper bounds ranged from $20,000 to $2,000 with increments of $2000. Remarks in Table 24 are summarized as follows. First, the ENSO phases affected the expected profit and the feasible region of the downside risk. The Neutral year has highest expected profit and lowest downside loss. In contrast, the La Nifia year has lowest expected profit and highest downside loss. Second, the 65%CRC and 70%CRC crop insurance policies are desirable to the optimal hedging strategy in all ENSO phases when 90%CVaR constraint is lower than a specific value depending on the ENSO phase. In contrast, the APH insurance policies are not desirable for all ENSO phases and 90%CVaR upper bounds. Third, risk management can be conducted through changing the planting schedule. The last two rows associated with the Neutral phase shows that planting 100 acres in date 3 provides a 90% CVaR of$6,000 that can be reduced to $8,000 by changing the planting schedule to 85 acres in date 3 and 15 acres in date 4. Last, changing the insurance coverage together with the planting schedule may reduce the downside risk. In the La Nifia phase, planting 100 acres in date 4 provides a 90%CVaR of  3 Although only crop insurance contracts were considered, the unbiased futures market assumption is need since the indemnity of CRC depends on the futures price. $4,000 that can be reduced to $10,000 by purchasing a 65%CRC insurance policy and shifting the planting date from date 4 to date 1. Table 24. Optimal insurance and production strategies for each climate scenario under the 90% CVaR tolerance ranged from $20,000 to $2,000 with increment of $2000 ENSO 90%CVaR Upper Optimal Expected Optimal Optimal Planting Schedule Insurance Phases Bound Profit Strategy Datel Date2 Date3 Date4 El Nifo <18000 infeasible 16000 28364 CRC70% 100 0 0 0 14000 to 4000 28577 CRC65% 100 0 0 0 >2000 28691 No 100 0 0 0 Neutral <20000 infeasible 18000 31149 CRC70% 0 0 100 0 16000 to 10000 31240 CRC65% 0 0 100 0 8000 31779 No 0 0 85 15 >6000 31793 No 0 0 100 0 La Nifia <12000 infeasible 10000 to 6000 20813 CRC65% 100 0 0 0 >4000 21572 No 0 0 0 100 No = no insurance. Planting dates: Datel = April 16, Date2 = April 23, Date3 = May 1, Date4 = May 8. Negative CVaR upper bounds represent profits. 2.4.2 Hedging with Crop Insurance and Unbiased Futures In this section, we consider managing the yield and price risk with crop insurance policies and futures contracts when the futures market is unbiased. Since the crop yield is random, we define the hedge ratio of the futures contract as the hedge position in the futures contract divided by the expected production. The optimal solutions of the planting schedule, crop insurance coverage, and futures hedge ratio with various 90%CVaR upper bounds ranged from $24,000 to $0 with increment of $4,000 for the three ENSO phases (Table 25). From Table 25, when the futures market is unbiased, the futures contract dominating all crop insurance policies is the only desirable risk management tool. The optimal hedge ratio increases when the upper bound of 90%CVaR decreases. This means that to achieve lower downside risk, higher hedge ratio is needed. Next, we compare the hedge ratio in different ENSO phases with the same CVaR upper bound, the La Nifia phase has the highest optimal hedge ratio Table 25. Optimal solutions of planting schedule, crop insurance coverage, and futures hedge ratio with various 90% CVaR upper bounds ranged from $24,000 to $0 with increment of $4,000 for the three ENSO phases 90%CVaR Optimal Optimal Optimal Optimal Optimal Planting ENSO Upper Expected Insurance Expected Hedge Hedge Schedule Phases Bound Profit Strategy Production Amount Ratio Datel Date2 Date3 Date4 24000 infeasible 20000 28520 No 81422 57092 0.70 100 0 0 0 16000 28563 No 81422 4482 0.52 100 0 0 0 ElNifio 12000 28605 No 81422 28895 0.35 100 0 0 0 8000 28646 No 81422 15238 0.19 100 0 0 0 4000 28686 No 81422 1726 0.02 100 0 0 0 <0 28691 No 81422 0 0 100 0 0 0 24000 infeasible 20000 31645 No 82482 49598 0.60 0 0 100 0 Neutral 16000 31695 No 82482 32721 0.40 0 0 100 0 12000 31744 No 82482 16547 0.20 0 0 100 0 8000 31792 No 82482 529 0.01 0 0 100 0 4000 31793 No 82482 0 0 0 0 100 0 16000 infeasible LaNifia 12000 21438 No 80911 44812 0.55 0 0 0 100 8000 21516 No 80911 18705 0.23 0 0 0 100 4000 21572 No 80911 0 0 0 0 0 100 No = no insurance. Planting dates: Datel= April 16, Date2= April 23, Date3= May 1, Date4= May 8. Negative CVaR upper bounds represent profits. and the Neutral phase has the lowest one. Similar to the result in Table 24, the Neutral phase has the highest expected profit and the lowest feasible downside loss. In contrast, the La Nifia phase has the lowest expected profit and the highest feasible downside loss. Furthermore, we compare two risk management tools: insurance (in Table 24) and futures (in Table 25). The futures contract provides higher expected profit under the same CVaR upper bound, as well as a larger feasible region associated with the CVaR constraint. Finally, the optimal production strategy with futures hedge is to plant 100 acres in date 1 for the El Nifio phase, in date 3 for the Neutral phase, and in date 4 for the La Nifia phase. 2.4.3 Biased Futures Market In the section 2.4.2 we assumed the futures market is unbiased. However, the futures prices observed from futures market in the planting time may be higher or lower than the expected futures price in the harvest time. This section examines the impact of biased futures prices on the optimal insurance and futures hedging decisions in the three ENSO phases. We first illustrate the optimal hedge strategies and the optimal planting schedule. Then, the performance of the optimal hedge and planting strategies is introduced by the efficient frontiers on the expected profit versus CVaR risk diagram. Table 26. Optimal insurance policy and futures hedge ratio under biased futures prices Bias 10% 5% 0%0 5% 10% ENSO 90 Hedge Hedge Hedge Insurance Hedge Hedge Phases CVaR Upper Insurance Insurance Rat Insurance Ratio Insurance Rat Insurance Ratio Phases BondRatio Ratio Ratio Ratio Ratio Bound 28000 x 26000 x No 1.03 24000 x No 1.00 No 1.24 22000 x No 0.83 No 1.18 No 1.41 20000 x 65%CRC 0.48 No 0.70 No 1.33 No 1.57 18000 65%CRC 0.45 65%CRC 0.29 No 0.61 No 1.47 No 1.72 ElNino 16000 70%CRC 0.20 65%CRC 0.15 No 0.52 No 1.60 No 1.87 14000 70%CRC 0.04 65%CRC 0.03 No 0.44 No 1.73 No 2.02 12000 65%CRC 0.01 65%CRC 0.07 No 0.35 No 1.86 No 2.16 10000 65%CRC 0.10 65%CRC 0.15 No 0.27 No 1.98 No 2.30 8000 65%CRC 0.20 65%CRC 0.24 No 0.19 No 2.11 No 2.44 6000 65%CRC 0.29 65%CRC 0.33 No 0.10 No 2.23 No 2.58 4000 65%CRC 0.39 65%CRC 0.41 No 0.02 No 2.35 No 2.73 2000 65%CRC 0.48 65%CRC 0.50 No 0.00 No 2.48 No 2.87 0 65%CRC 0.58 65%CRC 0.58 No 0.00 No 2.60 No 3.01 30000 x 28000 x No 1.06 26000 No 0.99 No 1.27 24000 x No 1.19 No 1.42 22000 x No 0.73 No 1.33 No 1.56 20000 x No 0.66 No 0.60 No 1.46 No 1.68 Neutral 18000 70%CRC 0.28 70%CRC 0.12 No 0.50 No 1.57 No 1.80 16000 70%CRC 0.10 70%CRC 0.02 No 0.40 No 1.68 No 1.92 14000 70%CRC 0.04 70%CRC 0.12 No 0.30 No 1.79 No 2.04 12000 70%CRC 0.16 No 0.20 No 0.20 No 1.89 No 2.15 10000 70%CRC 0.28 No 0.09 No 0.10 No 2.00 No 2.27 8000 70%CRC 0.39 No 0.00 No 0.01 No 2.10 No 2.38 6000 70%CRC 0.51 No 0.11 No 0.00 No 2.20 No 2.49 4000 70%CRC 0.62 No 0.21 No 0.00 No 2.30 No 2.59 2000 70%CRC 0.74 No 0.31 No 0.00 No 2.39 No 2.70 0 70%CRC 0.86 No 0.42 No 0.00 No 2.49 No 2.81 20000 x 18000 x No 1.06 16000 x No 1.04 No 1.23 14000 x No 0.75 No 1.20 No 1.37 12000 x No 0.67 No 0.55 No 1.33 No 1.51 La Nifia 10000 No 0.56 No 0.46 No 0.39 No 1.45 No 1.64 8000 No 0.33 No 0.27 No 0.23 No 1.57 No 1.77 6000 No 0.11 No 0.09 No 0.08 No 1.68 No 1.90 4000 No 0.10 No 0.08 No 0.00 No 1.80 No 2.02 2000 No 0.30 No 0.25 No 0.00 No 1.91 No 2.15 0 No 0.49 No 0.41 No 0.00 No 2.02 No 2.27 x = infeasible. No = No insurance. Negative CVaR upper bounds represent profits. Table 26 shows the optimal insurance policy and futures hedge ratio associated with different 90% CVaR upper bounds under biased futures prices for the three ENSO phases. When the futures price is unbiased or positive biased, the futures contract is the only desirable instrument for crop risk management and all insurance policies are not needed in the optimal hedging strategy. On the other hand, when the futures price is negative biased, the optimal hedging strategy includes 65%CRC (or 70% CRC in some cases) insurance policies and futures contract in the El Nifio phase for all feasible 90%CVaR upper bounds. In the Neutral phase, the optimal hedging strategy consists of the 70%CRC insurance policy and futures contract for all CVaR upper bounds with the deep negative biased (10%) futures price and for the CVaR upper bounds between 18000 and 14000 with the negative biased (5%) futures price. In addition, no insurance policy is desirable under the La Nifia phase. Mahul (2003) showed that the hedge ratio contains two parts: a pure hedge component and a speculative component. The pure hedge component refers to the hedge ratio associated with unbiased futures price. A positive biased futures price induces the farmer to select a long speculative position and a negative biased futures price implies a short speculative position. Therefore, the optimal futures hedge ratio under positive (negative) biased futures prices should be higher (lower) than that under the unbiased futures price. However, the optimal hedge ratios in Table 26 do not agree with the conclusion when the futures price is negative biased. We use the optimal hedge ratios in the La Nifia phase to illustrate how optimal futures hedge ratios change corresponding to the bias of the futures price4. Figure 22 shows the bias of futures price versus the optimal hedge ratio curves associated with different 90% CVaR upper bounds in the La Nifia phase. 4 Since the optimal hedging strategy in the La Nifia phase contains only futures contracts, the optimal hedge ratios are compatible. When the CVaR constraint is not strict (i.e. the upper bound of 90%CVaR equals zero) the optimal hedge ratio curve follows the pattern claimed in Mahul (2003). The hedge ratio increases (decreases) with the positive (negative) bias of futures price in a decreasing rate. However, when the CVaR constraint becomes stricter (i.e., the CVaR upper bound equals $8,000), the optimal hedge ratio increases not only with the positive bias but also with the negative one. It is because the higher negative bias of futures price implies a heavier cost (loss) is involved in futures hedge. It makes the CVaR constraint become stricter such that a higher pure hedge component is required to satisfy the constraint. The net change of the optimal hedge ratio, including an increment in the pure hedge component and a decrement in the speculative component, depends on the loss distribution, the CVaR upper bound, and the bias of futures price. Optimal Hedge Ratio 2.50 2.00  0 15 2000 1.50 . 4000 1.00 6000 0.50 8000 0.00 10000 S+12000 050 14000 1.00 16000 10% 5% 0% 5% 10% 18000 Bias of Futures Price Figure 22. Bias of futures price versus the optimal hedge ratio curves associated with different 90% CVaR upper bounds in the La Nifia phase Table 27 reported the optimal planting schedules for different biases of futures price in the three ENSO phases. For the El Nifio phase, the optimal planting schedule (i.e., planting 100 acres in date 1) was not affected by the biases of futures price and 90%CVaR upper bounds. Table 27. Optimal planting schedule for different biases of futures price in ENSO phases 90%CVaR El Nifio Neutral La Nifia upper Date Date Date Date Date Date Date Date Date Date Date Date Bias bound 1 2 3 4 1 2 3 4 1 2 3 4 20000 x 18000 100 0 0 0 16000 100 0 0 0 14000 100 0 0 0 12000 100 0 0 0 10% 10000 100 0 0 0 8000 100 0 0 0 6000 100 0 0 0 4000 100 0 0 0 2000 100 0 0 0 0 100 0 0 0 22000 x 20000 100 0 0 0 18000 100 0 0 0 16000 100 0 0 0 14000 100 0 0 0 12000 100 0 0 0 5% 10000 100 0 0 0 8000 100 0 0 0 6000 100 0 0 0 4000 100 0 0 0 2000 100 0 0 0 0 100 0 0 0 24000 x 22000 100 0 0 0 20000 100 0 0 0 18000 100 0 0 0 16000 100 0 0 0 14000 100 0 0 0 0% 12000 100 0 0 0 10000 100 0 0 0 8000 100 0 0 0 6000 100 0 0 0 4000 100 0 0 0 2000 100 0 0 0 0 100 0 0 0 x 0 97 3 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 x 0 39 61 0 100 0 0 100 0 0 100 0 0 44 56 0 46 54 0 85 15 0 85 15 0 92 8 0 100 0 0 93 7 x 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 0 100 x = infeasible. Date 1 = 16 Apr, Date 2 = 23 Apr, Date 3 = 1 May, Date 4 = 8 May. Negative CVaR upper bounds represent profits. Table 27. Optimal planting schedule for different biases of futures price in ENSO phases (cont'd) 90%CVaR El Nifio Neutral La Nifia upper Date Date Date Date Date Date Date Date Date Date Date Date Bias bound 1 2 3 4 1 2 3 4 1 2 3 4 28000 26000 x 24000 100 0 0 0 22000 100 0 0 0 20000 100 0 0 0 18000 100 0 0 0 5% 16000 100 0 0 0 14000 100 0 0 0 12000 100 0 0 0 10000 100 0 0 0 8000 100 0 0 0 6000 100 0 0 0 4000 100 0 0 0 2000 100 0 0 0 0 100 0 0 0 30000 28000 x 26000 100 0 0 0 24000 100 0 0 0 22000 100 0 0 0 20000 100 0 0 0 18000 100 0 0 0 10% 16000 100 0 0 0 14000 100 0 0 0 12000 100 0 0 0 10000 100 0 0 0 8000 100 0 0 0 6000 100 0 0 0 4000 100 0 0 0 2000 100 0 0 0 0 100 0 0 0 x 0 30 70 0 47 53 0 59 41 0 68 32 0 82 18 0 92 8 0 99 1 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 x 0 31 69 0 45 55 0 54 46 0 67 33 0 75 25 0 87 13 0 96 4 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 0 100 0 x 38 0 0 62 33 0 0 67 35 0 0 65 35 0 0 65 33 0 0 67 34 0 0 66 38 0 0 62 39 0 0 61 37 0 0 63 x 33 17 0 50 45 0 0 55 47 0 0 53 47 0 0 53 50 0 0 50 54 0 0 46 56 0 0 44 59 0 0 41 61 0 0 39 63 0 0 37 x = infeasible. Date 1 = 16 Apr, Date 2 = 23 Apr, Date 3 = 1 May, Date 4 = 8 May. Negative CVaR upper bounds represent profits. Efficient frontier in El Nino phase under various biased futures price 5 10%  5%  0%  5%  10% ,.4Y 26000 22000 18000 14000 10000 6000 2000 90ACVaR upper bound Efficient frontier in Neutral phase under various biased futures price , 10% * 5%  0% , 5% * 10% 28000 24000 20000 16000 12000 8000 4000 0 90C/VaR upper bound Efficient frontier in La Nina phase under various biased futures price  10%  5%  0%  5%  10% 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 90%CVaR upper bound Figure 23. The efficient frontiers under various biased futures price. (A) El Nifio year. (B) Neutral year. (C) La Nifia year. 45000 41000 S37000  33000 29000 29000 45000 41000  37000 S33000 x 29000 qcrnnn ~~~~~ __ 30000 28000  26000 C. S 24000 x 22000 w 20000 18000 z5UUU For the Neutral phase, however, the optimal planting strategy was to plant on date 3 and date 4 depending on the 90%CVaR upper bounds. More exactly, the date 3 is the optimal planting date for all risk tolerances under unbiased futures market. When futures prices are positive biased, the lower the 90%CVaR upper bounds (i.e., the stricter the CVaR constraint) was, the more planting acreages moved to date 4 from date 3. This result was based on the fact that there was no insurance coverage involved in the optimal hedging strategy. When the futures prices are negative biased, the optimal planting schedule had the same pattern as positive biased futures markets but was affected by the existence of insurance coverage in the optimal hedging strategy. For example, when the 90%CVaR upper bounds were within the range of $8,000 and  $14,000 under a 5% biased futures price, the optimal planting acreage in date 4 went down to zero due to a 75%CRC in the optimal hedging strategy. For the La Nifia phase, the optimal planting schedule was to plant 100 acres in date 4 when future prices were unbiased or negative biased. When future price was negative biased, the stricter CVaR constraint was, the more planting acreage shifted from date 4 to date 1. For deep negative biased futures price together with strict CVaR constraint (i.e., 10% biased futures price and $18,000 90%CVaR upper bound), the optimal planting schedule included date 1, date 2, and date 4. Figure 23 shows meanCVaR efficient frontiers associated with various biased futures prices for three ENSO phases. With the efficient frontiers, the farmer may make the optimal decision based on his/her downside risk tolerance and trade off between expected profit and downside risk. The three graphs show that the Neutral phase has highest expected profit and lowest feasible CVaR upper bound. In contrast, the La Nifia phase has the lowest expected profit and highest feasible CVaR upper bound. The pattern of the efficient frontiers in the three graphs is the same. The higher positive bias of futures price is, the higher expected profit would be provided. However, the higher negative bias of futures price provides a higher expected profit under a looser CVaR constraint and a lower expected profit under a stricter CVaR constraint. 2.5 Conclusion This research proposed a meanCVaR model for investigating the optimal crop planting schedule and hedging strategy when the crop insurance and futures contracts are available for hedging the yield and price risk. Due to the linear property of CVaR, the optimal planting and hedging problem could be formulated as a mixed 01 linear programming problem that could be efficiently solved by many commercial solvers such as CPLEX. The meanCVaR model is powerful in the sense that the model inherits the advantage of the return versus risk framework (Markowitz, 1952) and further utilizes CVaR as a (downside) risk measure that can cope with general loss distributions. Compared to using utility functions for modeling risk aversion, the meanCVaR model provides an intuitive way to define risk. In addition, a problem without nonlinear side constraints could be formulated linearly under the meanCVaR framework, which could be solved more efficiently comparing to the nonlinear formulation from the utility function framework. A case study was conducted using the data of a representative cotton producer in Jackson County in Florida to examine the optimal crop planting schedule and risk hedging strategy under the three ENSO phases. The eligible hedging instruments for cotton include futures contracts and three types of crop insurance policies: APH, CRC, and CAT. We first analyzed the best production and risk hedging problem with three types of insurance policies. The result showed that 65%CRC or 70%CRC would be the optimal insurance coverage when the CVaR constraint reaches a strict level. Furthermore, we examined the optimal hedging strategy when crop insurance policies and futures contracts are available. When futures price are unbiased or positive biased, the optimal hedging strategy only contains futures contracts and all crop insurance policies are not desirable. However, when the futures price is negative biased, the optimal hedging strategy depends on the ENSO phases. In the El Niho phase, the optimal hedging strategy consists of the 65%CRC (or 70%CRC for some CVaR upper bounds) and futures contracts for all CVaR upper bound values. In the Neutral phase, when futures price is deep negative biased (10%), the optimal hedging strategy consists of the 70%CRC and futures contracts for all CVaR upper bound values. Under a 5% negative biased futures price, optimal hedging strategy contains the 70%CRC and futures contracts when the CVaR upper bound is within the range of$18,000 and $14,000. Otherwise, the optimal hedging strategy contains only futures contract. In the La Nifia phase, the optimal hedging strategy contains only futures contract for all CVaR upper bound values and all biases of futures prices between 10% and 10%. The optimal futures hedge ratio increases with the increasing CVaR upper bound when the insurance strategy is unchanged. For a fixed CVaR upper bound, the optimal hedge ratio increases when the positive bias of futures price increases. However, when the futures price is negative biased, the optimal hedge ratio depends on the value of CVaR upper bound. The case study provides some insight into how planting schedule incorporated with insurance and futures hedging may manipulate the downside risk of a loss distribution. In our model, we used a static futures hedging strategy that trades the hedge position in the planting time and keeps the position until the harvest time. A dynamic futures hedging strategy may be considered in the future research. The small sample size for the El Niho and La Nifia phases may limit the case study results. In addition, we assumed the cost of futures contract as the commission plus an average interest foregone for margin deposit and the risk of daily settlement that may require a large amount of cash for margin account was not considered. It may reduce the value of futures hedging for riskaverse farmers. CHAPTER 3 EFFICIENT EXECUTION IN THE SECONDARY MORTGAGE MARKET 3.1 Introduction Mortgage banks (or lenders) originate mortgages in the primary market. Besides keeping the mortgages as a part of the portfolio, a lender may sell the mortgages to mortgage buyers (or conduits) or securitize the mortgages as mortgagebacked securities (MBSs) through MBS swap programs in the secondary market. In the United States, three governmentsponsored enterprises (GSEs) (Fannie Mae, Freddie Mac, and Ginnie Mae) provide MBS swap programs in which mortgage bankers can deliver their mortgages into appropriate MBS pools in exchange for MBSs. In practice, most mortgage bankers prefer to participate in the secondary market based on the following reasons. First, mortgage banks would get funds from secondary marketing and then use the funds to originate more mortgages in the primary market and earn more origination fees. Second, the value of a mortgage is risky and depends on several sources of uncertainties, i.e., default risk, interest rate risk, and prepayment risk. Mortgage bankers could reduce risks by selling or securitizing mortgages in the secondary market. More exactly, when mortgages are sold as a whole loan, all risks would be transferred to mortgage buyers. On the other hand, when mortgages are securitized as MBSs, the risky cash flows of mortgages are split into guarantee fees, servicing fees, and MBS coupon payments, which belong to MBS issuers, mortgage services, and MBS investors, respectively. In this case, mortgage bankers are exposed only to risk from retaining the servicing fee and other risky cash flows are transferred to different parties. A significant task faced by mortgage bankers attempting to profit from the secondary market is efficient execution. The challenge of efficient execution is to sell or securitize a large number of heterogeneous mortgages in the secondary market in order to maximize expected revenue through complex secondary marketing functionality. In addition, to deal with the uncertain cash flows from the retained servicing fee, the balance between mean revenue and risk is also an important concern for mortgage bankers. In this chapter, we develop a stochastic optimization model to perform an efficient execution that considers secondary marketing functionality, including loanlevel efficient execution, guarantee fee buyup or buydown, servicing retain or release, and excess servicing fee. Further, we employ Conditional ValueatRisk (CVaR), proposed by Rockafellar and Uryasev (2000), as a risk measure in the efficient execution model that maximizes expected revenue under a CVaR constraint. By solving the efficient execution problem under different risk tolerances specified by a CVaR constraint, an efficient frontier could be found. A great deal of research has focused on mortgage valuation (Kau, Keenan, Muller, and Epperson (1992); Kau (1995); Hilliard, Kau and Slawson (1998); and Downing, Stanton, and Wallace (2005)), MBS valuation (Schwartz and Torous (1989); Stanton (1995); Sugimura (2004)), and mortgage servicing right valuation (Aldrich, Greenberg, and Payner (2001); Lin, Chu, and Prather (2006)). However, academic literature addressing topics of mortgage secondary marketing is scant. Hakim, Rashidian, and Rosenblatt (1999) addressed the issue of fallout risk, which is an upstream secondary marketing problem. To the best of our knowledge, we have not seen any literature focusing on efficient execution. The organization of this chapter is as follows: Section 2 discusses mortgage securitization. We describe the relationship between MBS market participants and introduce the Fannie Mae MBS swap program. Section 3 presents our model development. Section 4 reports our results, and the final section presents our conclusions. 3.2 Mortgage Securitization Mortgage bankers may sell mortgages to conduits at a price higher than the par value to earn revenue from the whole loan sales. However, for lenders who possess efficient execution knowledge, mortgage securitization through MBS swap programs of GSEs may bring them higher revenue than the whole loan sale strategy. In this research, we consider passthrough MBS swap programs provided by Fannie Mae (FNMA). To impose considerations of MBS swap programs of other GSEs is straightforward. In this section, we describe the relationship between participants in the passthrough MBS market and detail the procedure of mortgage securitization through a MBS swap program. Participants in the MBS market can be categorized into five groups: borrowers, mortgage bankers, mortgage services, MBS issuers, and mortgage investors. The relationship between these five participants in the passthrough MBS market is shown in Figure 31. In Figure 31, solid lines show cash flows between participants and dashed lines represent mortgage contracts and MBS instruments between them. Mortgage bankers originate mortgage loans by signing mortgage contracts with borrowers who commit to making monthly payments in a fixed interest rate known as the mortgage note rate. To securitize those mortgages, mortgage bankers deliver the mortgages into an MBS swap in exchange for MBSs. Further, mortgage bankers sell the MBSs to MBS investors and receive MBS prices in return. MBS issuers provide MBS insurance to protect the MBS investors against default losses and charge a base guarantee fee. The base guarantee fee is a fixed percentage, known as the 5 Mortgage bankers underwrite mortgages at a certain mortgage note rate. The par value is the value of the mortgage when the discount interest rate equals the mortgage note rate. In other words, the par value of a mortgage is its initial loan balance. Loan Borrowers 4 Mortgage banker Mortgage A I (MBS swap) Mortgage payment (pay mortgage note rate) MBS Moage MBS ce I MBS MBS price II (charges servicing fee) Guarantee fee (charges guarantee fee) Mortgage service MBS issuer (FNMA) MBS coupon payment MBS investors MBS investors (MBS coupon rate = mortgage note rate servicing fee rate guarantee fee rate) Figure 31. The relationship between participants in the passthrough MBS market. Mortgage bankers originate mortgage loans by signing mortgage contracts with borrowers who commit to making monthly payments with a fixed interest rate known as the mortgage note rate. To securitize those mortgages, mortgage bankers deliver the mortgages into an MBS swap in exchange for MBSs. Further, mortgage bankers sell the MBSs to MBS investors and receive MBS prices in return. The MBS issuer provides MBS insurance and charges a base guarantee fee. Mortgage services provide mortgage servicing and a base servicing fee is disbursed for the servicing. Both fees are a fixed percentage (servicing fee rate or guarantee fee rate) of the outstanding mortgage balance and decline over time as the mortgage balance amortizes. Deducting guarantee fees and servicing fees from mortgage payments, the remaining cash flows that passthrough to the MBS investors are known as MBS coupon payments with a rate of return equal to the mortgage note rate minus the servicing fee rate minus the guarantee fee rate. guarantee fee rate, of the outstanding mortgage balance, and which declines over time as the mortgage balance amortizes. Mortgage bankers negotiate the base guarantee fee rate with Fannie Mae and have the opportunity to "buydown" or "buyup" the guarantee fee. When lenders buydown the guarantee fee, the customized guarantee fee rate is equal to the base guarantee fee rate minus the guarantee fee buydown spread. Further, lenders have to make an upfront payment to Fannie Mae. On the other hand, the buyup guarantee fee allows lenders to increase the guarantee fee rate from the base guarantee fee rate and receive an upfront payment from Fannie Mae. For example, if a lender wants to include a 7.875% mortgage with a 0.25% base guarantee fee and a 0.25% base servicing fee in a 7.5% passthrough MBS (Figure 32), the lender can buydown the guarantee fee rate to 0.125% from 0.25% by paying Fannie Mae an upfront amount equal to the present value of the cash flows of the 0.125% difference and maintaining the 0.25% base servicing fee. Servicing Fee I Guarantee Fee MBS Coupon Rate 0.25% 0.125% 7.5%  Mortgage Note Rate 7.875% Figure 32. Guarantee fee buydown. A lender may include a 7.875% mortgage in a 7.5% pass through MBS by buyingdown the guarantee fee to 0.125% from 0.25% and maintaining the 0.25% base servicing fee. If a lender chooses to include an 8.125% mortgage in the 7.5% passthrough MBS (Figure 33), the lender can buyup the guarantee fee by 0.125% in return for a present value of the cash flows of the 0.125% difference. The buydown and buyup guarantee fee features allow lenders to maximize the present worth of revenue. 1 Servicing Fee Guarantee Fee MBS Coupon Rate 0.25% 0.375% 7.5% Mortgage Note Rate 8.125% Figure 33. Guarantee fee buyup. A lender may include an 8.125% mortgage in a 7.5% pass through MBS by buyingup the guarantee fee to 0.375% from 0.25% and maintaining the 0.25% base servicing fee. Mortgage services provide mortgage services, including collecting monthly payments from borrowers, sending payments and overdue notices, and maintaining the principal balance report, etc. A base servicing fee is disbursed for the servicing, which is a fixed percentage, known as the base servicing fee rate, of the outstanding mortgage balance, and which declines over time as the mortgage balance amortizes. Mortgage bankers have the servicing option to sell the mortgage servicing (bundled with the base servicing fee) to a mortgage service and receive an upfront payment from the service or retain the base servicing fee and provide the mortgage servicing. Deducting guarantee fees and servicing fees from mortgage payments, the remaining cash flows that passthrough to the MBS investors are known as MBS coupon payments, which contain a rate of return known as the MBS coupon rate (or passthrough rate), equal to the mortgage note rate minus the servicing fee rate minus the guarantee fee rate. Fannie Mae purchases and swaps more than 50 types of mortgages on the basis of standard terms. This research focuses on passthrough MBS swaps of 10, 15, 20, and 30year fixedrate mortgages. Mortgages must be pooled separately by the time to maturity. For instance, 30year fixedrate mortgages are separated from 15year fixedrate ones. For each maturity, Fannie Mae provides different MBS pools characterized by MBS coupon rates that generally trade on the half percent (4.5%, 5.0%, 5.5%, etc.). Mortgage lenders have the option to deliver individual mortgages into one of these eligible MBS pools, which allows lenders to maximize revenue. Further, the mortgage note rate must support the MBS coupon rate plus the servicing fee rate plus the guarantee fee rate. Therefore, when securitizing a mortgage as an MBS, mortgage bankers have to manipulate the servicing fee rate and guarantee fee rate so that Equation (31) is satisfied. Mortgage Note Rate Servicing Fee Rate + Guarantee Fee Rate (31) + MBS Coupon Rate. Mortgage bankers could retain an excess servicing fee from the mortgage payment, which, like the base servicing fee, is a fixed percentage, known as the excess servicing fee rate, of the outstanding mortgage balance and which declines over time as the mortgage balance amortizes. In Equation (32), the excess servicing fee rate is equal to the excess of the mortgage note rate over the sum of the MBS coupon rate, customized guarantee fee rate, and base servicing fee rate. In other words, the servicing fee rate in Equation (31) consists of the base servicing fee rate and the excess servicing fee rate. Excess Servicing Fee Rate = Mortgage Note Rate MBS Coupon Rate (32) Guarantee Fee Rate Base Servicing Fee Rate. In the example shown in Figure 33, the excess servicing fee may be sold to Fannie Mae by buyingup the guarantee fee. Another option for mortgage bankers is to retain the excess servicing fee in their portfolio and to receive cash flows of the excess servicing fee during the life of the mortgage. The value of the excess servicing fee is equal to the present value of its cash flows. This value is stochastic since borrowers have the option to terminate mortgages before maturity and the interest rate used to discount the future cash flows is volatile. Therefore, efficient execution becomes a stochastic optimization problem. Similar to the guarantee fee buy up and buydown features, the excess servicing fee allows lenders to maximize the expected revenue. 3.3 Model Efficient execution is a central problem of mortgage secondary marketing. Mortgage bankers originate mortgages in the primary market and execute the mortgages in the secondary market to maximize their revenue through different secondary marketing strategies. In the secondary market, each mortgage can be executed in two ways, either sold as a whole loan, or pooled into an MBS with a specific coupon rate. When mortgages are allocated into an MBS pool, we consider further the guarantee fee buyup/buydown option, the mortgage servicing retain/release option, and excess servicing fee to maximize the total revenue. Based on secondary marketing strategies, mortgage bankers may retain the base servicing fee and the excess servicing fee when mortgages are securitized. The value of the retained servicing fee is random and affected by the uncertainty of interest rate term structure and prepayment. Therefore, a stochastic optimization model is developed to maximize expected revenue under a risk tolerance and an efficient frontier can be found by optimizing expected revenue under different risk tolerances specified by a risk measure. 3.3.1 Risk Measure Since Markowitz (1952), variance (and covariance) has become the predominant risk measure in finance. However, the risk measure is suited only to the case of elliptic distributions, like normal or tdistributions with finite variances (Szego (2002)). The other drawback of variance risk measure is that it measures both upside and downside risks. In practice, finance risk managers are concerned only with the downside risk in most cases. A popular downside risk measure in economics and finance is ValueatRisk (VaR) (Duffie and Pan (1997)), which measures a percentile of loss distribution. However, as was shown by Artzner et al. (1999), VaR is illbehaved and nonconvex for general distribution. The other disadvantage of VaR is that it only considers risk at a percentile of loss distribution and does not consider how much worse the atail (the worst 1a percentage of scenarios) could be. To address this issue, Rockafellar and Uryasev ((2000) and (2002)) proposed Conditional ValueatRisk (CVaR), which is the mean value of atail of loss distribution. It has been shown that CVaR satisfies the axioms of coherent risk measures proposed by Artzner et al. (1999) and has desirable properties. Most importantly, Rockafellar and Uryasev (2000) showed that CVaR constraints in optimization problems can be expressed by a set of linear constraints and incorporated into problems of optimization. This research uses CVaR as the measure of risk in developing the efficient execution model that maximizes expected revenue under a CVaR constraint. Thanks to CVaR, an efficient execution model could be formulated as a mixed 01 linear programming problem. It is worth mentioning that the prices of MBSs, prices of whole loan sale mortgages, upfront payment of released servicing, and upfront payment of guarantee fee buyup or buy down are deterministic numbers that can be observed from the secondary market. However, the revenue from retained base servicing fees and excess servicing fees are equal to the present value of cash flows of the fees. Because of the randomness of interest rate term structure and prepayment, the revenue from the fees is varied with different scenarios. Lenders could simulate the scenarios based on their own interest rate model and prepayment model. This research assumes the scenarios are given input date. 3.3.2 Model Development In this subsection, we present the stochastic optimization model. The objective of the model is to maximize the revenue from secondary marketing. Four sources of revenue are included in the model: revenue from MBSs or whole loan sale, revenue from the base servicing fee, revenue from the excess servicing fee, and revenue from the guarantee fee buyup/buy down. (1) Revenue from MBSs or whole loan sale: fi (z = Lm x P X z )+P7 xz ,m (33) m=l c=l where M = total number of mortgages, m = index of mortgages (m = 1, 2, ..., M), L" = loan amount of mortgage m, C" = number of possible MBS coupon rates of mortgage m, c = index of MBS coupon rate, P," = price ofMBS with coupon rate index c and maturity of t", t" = maturity of mortgage m, P' = whole loan sale price of mortgage m, 1 =, if mortgage m is pooled into MBS with coupon rate index c, Zc 0, otherwise, { 1, if mortgage m is sold as a whole loan, Z 0, otherwise. Equation (34) enforces that each mortgage could be either sold as a whole loan or delivered into a specific MBS pool. Cm z +z = 1. (34) c=1 If mortgage m is securitized as an MBS with coupon rate index c, then zm = 1, zc = 0 for all c c, and z = 0, and the revenue from mortgage m equals L" x Pm x zm. On the other hand, if mortgage m is sold as a whole loan, then zj = 0 for all c, and zm = 1, and the revenue from mortgage m equals Pm x z The total revenue from the M mortgages is shown in Equation (3 3). (2) Revenue from the base servicing fee of securitized mortgages: f, (zm', z ) (L' xB"x z'"b + (" x R" x K (35) sbo sb r I sr / s (sbr') m=l m=1 \k= where S1, if the servicing of mortgage m is retained, zsbr = 0, otherwise, m 1, if the servicing of mortgage m is sold, zsbo 0, otherwise, B" = base servicing value of mortgage m, Pk = the probability of scenario k, K = number of scenarios, c' = servicing cost of mortgage m, K'k = retained servicing fee multiplier of mortgage m under scenario k. The retained servicing multipliers Kk for scenario k could be generated by simulation. Mortgage bankers first simulate the random discounted cash flows of the retained servicing fee by using their own interest rate model and prepayment model. Then, a retained servicing multiplier Kk could be found associated with each scenario. In this research, we treat the retained servicing multiplier Kk as input data. Details of how to get the retained servicing multiplier Kk is beyond the scope of this research. Equation (36) enforces that when a mortgage is securitized, the servicing of the mortgage can be either released or retained, and the revenue from mortgage servicing exists only if the mortgage is securitized as an MBS instead of being sold as a whole loan. m + zo + z' =l. (36) More exactly, if mortgage m is sold as a whole loan, then zm = 1, zs = Zs7r = 0, and the revenue from the base servicing fee equals zero. On the other hand, if mortgage m is securitized as an MBS and the servicing of mortgage m is sold, then zso = 1, zs' = 0, zm = 0, and the upfront payment from the mortgage service equals L" x B' x zs,; otherwise, z~r = 1 zm = 0, z" = 0, and the expected revenue from the released servicing equals the expected revenue from K base servicing fee p L'" x Rm x Kk X Zs ) minus servicing cost c x zb. The total revenue k=l from the M mortgages is expressed in Equation (35). (3) Revenue from the excess servicing fee of securitized mortgages: M K f3 (() = ZZp (LxxKk xS), (37) m=l k=l where r" = retained excess servicing fee of mortgage m. If mortgage m is securitized as an MBS, the mortgage generates expected revenue K Pk (L" x Kk s rie) from retaining the excess servicing fee. The total revenue from the M k=l mortgages is shown in Equation (37). (4) Revenue from the guarantee fee buyup/buydown of securitized mortgages: MI M f4(r = "x "(K x xr (KxL xr'), (38) m=l m=l where r2 = guarantee fee buyup spread of mortgage m, rg = guarantee fee buydown spread of mortgage m, K, = guarantee fee buyup multiplier of mortgage m, K, = guarantee fee buydown multiplier of mortgage m. Guarantee fee buyup and buydown multipliers (or ratios) K" and K', announced by Fannie Mae, are used to calculate the upfront payment of a guarantee fee buyup and buydown. Lenders buyup the guarantee fee of mortgage m to receive an upfront K x L" x r; from Fannie Mae. On the other hand, they can buydown the guarantee fee of mortgage m and make an upfront payment K" x L" x r" to Fannie Mae. The total revenue from the Mmortgages is shown in Equation (38). Guarantee fee buyup/buydown and retaining the excess servicing fee are considered only when mortgage m is securitized. Equation (39) enforces ri r, and r" to be zero when mortgage m is sold as a whole loan. zw" + r' + rg + r' _1 (39) From equation (31), when mortgages are securitized as MBSs, the mortgage note rate has to support the MBS coupon rate, servicing fee rate, and guarantee fee rate. Equation (310) places a mathematic expression in the restriction. Cm R7z' + r' r"' + r" < Rm' R'" R 's g (310) c 1 (310) where R, = MBS coupon rate related to index c, Rm = note rate of mortgage m, Rm = base servicing fee of mortgage m, Rm = base guarantee fee of mortgage m, Next, we introduce the CVaR constraint CVaR, (L)< U (311) where L is the loss function, a is the percentile of CVaR, and U is the upper bound of CVaR losses. Equation (311) restricts the average of atail of loss distribution to be less than or equal to U. In other words, the average losses of the worst 1a percentage of scenarios should not exceed U. It is worth mentioning that CVaR is defined on a loss distribution. Therefore, we should treat revenue as negative losses when we use CVaR constraint in maximum revenue problem. Rockafellar & Uryasev (2000) proposed that CVaR constraints in optimization problems can be expressed by a set of linear constraints. K +(1 a) pkZk k=l zk>Lk; Vk =,...,K (313) zk >0 Vk ,...,K (314) M Cm M M Lk L x P cxz1 + woe x z.m _) + Bxzo) L x K xzsbr m=1 c=1 m=1 m=l M M M (LE x K xr) (K: x Lrx rm)+ (Km x L xirm) (315) m=l m=l m=l whereLk is loss value in scenario k, and 4 and Zk are real variables. Users of the model could specify their risk preference by selecting the value of a and U. More constraints could be considered in the model based on the mortgage banker's preference. For instance, mortgage bankers may want to control the retained excess servicing fee based on the future capital demand or risk consideration. The excess servicing fee could be limited by an upper bound in three levels, including aggregate level, group level, and loan level. Constraint (316) limits average excess servicing fee across all mortgages by an upper bound 1xm xr j m=l .m=l We further categorized mortgages into groups according to the year to maturity. Constraint (317) limits the grouplevel average excess servicing fee by an upper bound U, for each group j. L xr U (U L" (1z:) (317) In addition, constraint (318) restrict the loan level excess servicing fee to an upper bound U:. 0 < r'" < U" (318) Furthermore, constraints (319) and (320) impose an upper bound Utm and Utm in the guarantee fee buyup and buydown spreads, respectively. 0 rgM UMm (319) 0 < Ugd (320) Those upper bounds are determined by the restrictions of an MBS swap program or sometimes by the decision of mortgage bankers. For example, the maximum guarantee fee buy down spread accepted by Fannie Mae is the base guarantee fee rate. Finally, we impose nonnegativity constraints r1m r"' r' >0 (321) and binary constraints zc, e z b z b {0,1} Ve, m = 1,2,...., C" (322) The notations and model formulation are summarized in the Appendix. 3.4 Case Study In this section, a case study is conducted. First, we present the data set of mortgages, MBSs, base servicing fee and guarantee fee multipliers, and scenarios of retained servicing fee multiplier. Next, we introduce the solver that is used to solve the mixed 01 linear programming problem. Then, we show the results including efficient frontier and sensitivity analysis. 3.4.1 Input Data In the case study, we consider executing 1,000 fixedrate mortgages in the secondary market. For each mortgage, the data includes years to maturity (YTM), loan amount, note rate, and guarantee fee. Table 31 summarizes the data on mortgages. Mortgages are categorized into four groups according to YTM. For each group, the table shows number of mortgages, and the minimum, mean, and maximum value of loan amount, note rate, and guarantee fee. MBS pools are characterized by the MBS coupon rate and YTM. In this case study, we consider 13 possible MBS coupon rates from 3.5% to 9.5%, increasing in increments of 0.5%, and four different YTM: 10, 15, 20, and 30 years. There are a total of 50 MBS pools since the MBS pool with a 3.5% coupon rate is not available for 20year and 30year mortgages. Table 32 shows the prices of MBSs for different MBS pools. This case study assumes the base servicing rate is 25 bp (Ibp = 0.01%) for all mortgages. The base servicing value is 1.09 for mortgages with maturity of 10, 15, and 20 years and 1.29 for mortgages with maturity of 30 years. In addition, the servicing cost are assumed to be zero. The guarantee fee buyup and buydown multipliers are summarized in Table 33, which shows that guarantee fee buyup and buydown multipliers depend on mortgage note rate and maturity. A mortgage with a higher note rate and longer maturity has larger multipliers. In addition, buy down multipliers are larger than the buyup multipliers under the same note rate and maturity. In the case study, we consider 20 scenarios that are uniformly distributed across the range between 0 and 2K,, where Km is the expected retained servicing fee multiplier for mortgage m summarized in Table 33. Equation (323) defines the probability mass function for the variable of retained servicing fee multiplier. f0.05,xe{ S:S =(.05+.lz)K;, O From Table 33, the expected retained servicing fee multiplier for mortgage m depends on note rate and maturity. A mortgage with a larger note rate and longer maturity has a higher expected retained servicing fee multiplier. The largescale mixed 01 linear programming problem was solved by CPLEX90 on an Intel Pentium 4, 2.8GHz PC. The running times for solving the instances of the efficient execution problem are approximately one minute with a solution gap6 of less than 0.01%. 6 Solution gap defines a relative tolerance on the gap between the best integer objective and the object of the best node remaining. When the value best nodebest integer /(le10 + best integer) falls below this value, the mixed integer programming (MIP) optimization is stopped. 3.4.2 Result The efficient execution model solved the optimal execution solution under different upper bounds of CVaR losses across the range from $193,550,000 to $193,200,000, increasing in increments of $50,000, under a fixed a value to get an efficient frontier in the expected revenue versus CVaR risk diagram. The procedure was repeated for a values of 0.5, 0.75, 0.9, and 0.95 to get efficient frontiers under different risk preferences associated with a values. The efficient frontiers are shown in Figure 34, and the solutions of efficient execution under different a and Uvalues are listed in Table 34. Table 31. Summary of data on mortgages Loan Amount ($) Note Rate (%) Guarantee Fee (%) YTM # of Mortgages Min Mean Max Sum Min Mean Max Min Mean Max 10 13 $539,323 $164,693 $330,090 $2,141,009 4.38 5.00 5.75 0.125 0.146 0.40 15 148 $34,130 $172,501 $459,000 $25,530,131 4.00 5.29 8.00 0.125 0.187 0.80 20 11 $38,920 $137,024 $291,320 $1,507,260 5.13 5.97 7.25 0.125 0.212 0.80 30 828 $24,000 $194,131 $499,300 $160,740,083 4.75 5.92 7.875 0.125 0.212 1.05 Table 32. Summary of data on MBS prices of MBS pools MBS coupon rate (%) YTM 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 10 107.69 107.69 107.69 107.66 107.31 106.94 105.84 104.94 103.69 102.45 101.13 99.703 93.381 15 107.69 107.69 107.69 107.66 107.31 106.97 105.84 104.63 103.19 101.5 99.688 97.531 93.381 20 107.73 107.73 107.72 107.63 106.69 105.78 104.69 103.64 101.78 99.719 97.047 90.263 N/A 30 107.73 107.73 107.72 107.63 106.69 105.78 104.5 103.13 100.91 98.469 95.313 90.263 N/A Table 33. Guarantee fee buyup and buydown and expected retained servicing multipliers Note 30YR 20YR 15YR 10YR 30YR 10, 15, 20 YR Expected Retained Expected Retained Rate Buyup Buydown Buyup Buydown Buyup Buydown Buyup Buydown Servicing Servicing 4 5.65 7.6 4.80 6.52 3.60 5.55 3.42 5.27 5.75 4.00 5 4.95 6.9 4.20 5.93 2.95 4.90 2.80 4.65 5.74 3.60 6 3.15 5.0 2.67 4.41 1.50 3.29 1.38 3.18 4.19 2.76 7 1.65 3.3 1.40 3.00 0.95 2.75 0.90 2.70 2.22 2.00 8 0.95 2.5 0.75 2.35 0.55 2.35 0.50 2.30 1.60 0.62 Figure 34 shows the tradeoff between CVaR (atail risk) and expected revenue. For a fixed a value, when the upper bound of CVaR increases, the optimal expected revenue increases in a decreasing rate. On the other hand, for a fixed upper bound of CVaR, a high a value implies high risk aversion. Therefore, the associated optimal expected revenue becomes lower. Efficient Frontiers Expected Revenue ?=0.5 ?=0.75 ~?=0.9 x?=0.95 193680000 193660000 193640000 . XK 193620000 193600000 193580000 193560000 193540000 193550000 193500000 193450000 193400000 193350000 193300000 193250000 193200000 Upper bound of CVaR Loss (Negative Revenue) Figure 34: Efficient Frontiers. Plot of maximum expected revenue associated with different upper bound of CVaR losses across the range from $193,550,000 to $193,200,000 in increments of $50,000 under a fixed a value in the expected revenue versus CVaR risk diagram to get an efficient frontier. Repeat the procedure for different a values of 0.5, 0.75, 0.9, and 0.95 to get efficient frontiers under different risk preferences associated with a values. Table 34 summaries the solution of efficient execution under different risk preferences specified by a and U, which includes the number of mortgages sold as a whole loan, number of mortgages securitized as MBSs with a specific coupon rate ranging from 3.5% to 9.5% that increases in 0.5% increments, the number of retained mortgage servicing and released mortgage servicing, the sum of guarantee fee buyup and buydown amount, and the sum of excess servicing fee amount. 3.4.4 Sensitivity Analysis A sensitivity analysis was conducted in servicing fee multipliers, mortgage prices, and MBS prices. The sensitivity analysis is performed under a CVaR constraint with a =75% and U =$193,400,000. Table 35 shows that when all servicing fee multipliers increase by a fixed percentage, the number of mortgages sold as a whole loan decreases since lenders could get higher revenue from securitization due to the increasing servicing fee. In addition, the number of retained servicing and the amount of excess servicing fee increase due to the increasing retained servicing fee multipliers, and the number of released servicing decreases because the base servicing values, i.e. the upfront payments of released servicing, do not increase associated with retained servicing multipliers. An interesting result is that the increasing servicing fee multipliers increase (decrease) the guarantee fee buydown (buyup) amount, and the number of mortgages pooled into low coupon rate MBSs (4.5% and 4%) increases and the number of mortgages pooled into high coupon rate MBSs (7%, 6.5%, and 6%) decreases. Table 36 shows that when all mortgage prices increase by a fixed percentage, the number of mortgages sold as a whole loan increases so that lenders can take advantage of high mortgage price. Since the number of whole loan sale mortgages increases which implies that the number of securitized mortgages decreases, the sum of buyup and buydown guarantee fee and excess servicing fee slightly decrease. Furthermore, the number of released servicing decreases since the number of securitized mortgage decreases. Table 37 shows that when MBS price increases, the number of whole loan sale decreases, the number of securitized mortgages increases in the lower MBS coupon rate pools (4%, 4.5%, and 5%), and the number of retained servicing slightly increases. 3.5 Conclusion This research proposed a stochastic optimization model to perform the efficient execution analysis. The model considers secondary marketing functionalities, including the loanlevel execution for an MBS/whole loan, guarantee fee buyup/buydown, servicing retain/release, and excess servicing fee. Since secondary marketing involves random cash flows, lenders must balance between expected revenue and risk. We presented the advantages of CVaR risk measure and employed it in our model that maximizes expected revenue under a CVaR constraint. By solving the efficient execution problem under different risk tolerances, efficient frontiers could be found. We conducted a sensitivity analysis in parameters of expected retained servicing fee multipliers, mortgage prices, and MBS prices. The model is formulated as a mixed 01 linear programming problem. The case study shows that realistic instances of the efficient execution problem can be solved in an acceptable time (approximately one minute) by using CPLEX90 solver on a PC. Table 34. Summary of efficient execution solution under different risk preferences Upper Bound of CVaR Losses U # of Total Whole Revenue Loan Sale # of mortgages pooled into MBS (with coupon rate %) 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 S Sum of Sum of # of # of Sum of Excess buy Released Retained buyup don servicing down svi Servicin Servicin amount fee amount g g (%) ) amount () (%) 193550000 193500000 193450000 193400000 0.5 193350000 193300000 193250000 193200000 193550000 193500000 193450000 193400000 0.75 193350000 193300000 193250000 193200000 195380000 193626000 193655000 193656000 193656000 193656000 193656000 193656000 193570000 193600000 193621000 193639000 193655000 193656000 193656000 193656000 5 30 183 429 134 59 5 0 5 30 183 428 135 56 8 0 5 21 192 428 134 54 11 0 5 14 199 428 116 70 13 0 5 14 199 428 116 70 13 0 5 14 199 428 116 70 13 0 5 14 199 428 116 70 13 0 5 14 199 428 116 70 13 0 5 30 183 429 134 59 5 0 5 30 183 428 135 59 5 0 5 30 183 428 135 56 8 0 5 30 183 428 135 56 8 0 5 21 192 428 134 55 10 0 5 14 199 428 116 70 13 0 5 14 199 428 116 70 13 0 5 14 199 428 116 70 13 0 40.41 16.72 0.077 0 0 0 0 0 45.10 28.94 20.04 7.67 0.69 0 0 0 63.76 63.64 60.87 56.80 56.80 56.80 56.80 56.80 63.76 63.64 63.64 63.64 60.87 56.80 56.80 56.80 11.29 36.85 57.22 66.73 66.73 66.73 66.73 66.73 6.60 23.13 33.53 45.89 56.11 66.73 66.73 66.73 Table 34. Summary of efficient execution solution under different risk preferences (cont'd.) Upper Bound of CVaR Losses U # of Total Whole Revenue Loan Sale # of mortgages pooled into MBS (with coupon rate %) 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 S Sum of Sum of Sum of Excess # of # of buy Released Retained uu down service amount fee SServicing Servicing amount am 3.5 (%) amount 193550000 193500000 193450000 193400000 0.9 193350000 193300000 193250000 193200000 193550000 193500000 193450000 193400000 0.95 193350000 193300000 193250000 193200000 193568000 193592000 193611000 193626000 193639000 193652000 193656000 193656000 193567000 193590000 193608000 193622000 193635000 193648000 193656000 193656000 5 30 183 429 134 59 5 5 30 183 428 135 59 5 5 30 183 428 135 59 5 5 30 183 428 135 56 8 5 30 183 428 135 56 8 5 30 183 428 135 56 8 5 20 193 428 133 54 12 5 14 199 428 116 70 13 5 30 183 429 134 59 5 5 30 183 428 135 59 5 5 30 183 428 135 59 5 5 30 183 428 135 56 8 5 30 183 428 135 56 8 5 30 183 428 135 56 8 5 20 193 428 133 54 12 5 14 199 428 116 70 13 46.12 33.06 24.66 16.99 9.40 1.04 0 0 46.46 34.57 25.96 20.22 12.18 5.16 0 0 63.76 63.64 63.64 63.64 63.64 63.64 60.57 56.80 63.76 63.64 63.64 63.64 63.64 63.64 60.57 56.80 5.57 19.01 27.41 36.58 44.17 52.53 58.50 66.73 5.23 17.50 26.11 33.35 41.39 48.41 58.50 66.73 Table 35 Sensitivity analysis in servicing fee multiplier Servicing # of # of mortgages pooled into MBS (with coupon rate %) Fee Whole Multipliers Loan 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 Increment Sale 0% 155 0 0 0 0 0 5 30 183 428 135 56 8 0 10% 155 0 0 0 0 0 4 15 197 321 221 71 16 0 20% 114 0 0 0 0 0 4 8 157 64 496 114 43 0 30% 78 0 0 0 0 0 3 3 126 48 505 149 88 0 40% 74 0 0 0 0 0 3 2 127 7 495 174 118 0 50% 73 0 0 0 0 0 3 0 84 45 235 430 130 0 # of # of Released Retained Servicing Servicing Sum of buyup amount (%) 7.6741 0 0 0 0 0 Sum of Sum of buydown Excess amount servicing fee (%) amount (%) 63.6374 45.8949 38.4944 107.926 74.0746 371.006 137.755 537.436 165.085 626.538 175.951 804.253 Table 36. Sensitivity analysis in mortgage price Mortgage # of # of mortgages pooled into MBS # of Price Whole (with coupon rate %) Released Increment Loan Sale 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 Servicing 0% 155 0 0 0 0 0 5 30 183 428 135 56 8 0 845 0.5% 196 0 0 0 0 0 5 14 199 426 82 65 13 0 804 1% 277 0 0 0 0 0 5 14 198 406 36 52 12 0 723 1.5% 313 0 0 0 0 0 5 14 197 388 36 35 12 0 687 2% 429 0 0 0 0 0 5 14 192 301 32 17 10 0 571 # of Sum of Sum of buy Sum of Excess Retained buyup down servicing fee Servicing amount (%) amount (%) amount (%) 0 7.6741 63.6374 45.8948 0 0 55.144 62.6578 0 0 48.773 51.4078 0 0 45.1942 49.2382 0 0 30.587 44.2382 Table 37. Sensitivity analysis in MBS price MBS P of # of mortgages pooled into MBS increment Whole (with coupon rate %) Increment Loan Sale 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 0% 155 0 0 0 0 0 5 30 183 428 135 56 8 0 1% 70 0 0 0 0 0 5 17 196 428 206 54 24 0 2% 43 0 0 0 0 0 5 20 193 428 210 74 27 0 3% 29 0 0 0 0 0 5 20 193 427 212 86 28 0 4% 10 0 0 0 0 0 5 20 193 428 212 104 28 0 5% 1 0 0 0 0 0 5 21 192 428 212 113 27 1 # of Released Servicing 845 858 864 866 866 867 # of Retained Servicing 0 72 93 105 124 132 Sum of buy Sum of buy up amount down amount (%) (%) 7.6746 63.6374 0.25 63.7476 0.25 65.3748 0.25 65.523 0.25 66.3766 0.25 67.6766 Sum of Excess servicing fee amount (%) 45.8949 62.0756 66.4206 68.1438 67.9206 67.4706 CHAPTER 4 MORTGAGE PIPELINE RISK MANAGEMENT 4.1 Introduction Mortgage lenders commit to a mortgage interest rate while the loan is in process, typically for a period of 3060 days. If the interest rate rises before the loan goes to closing, the value of the loans will decline and the lender will sell the loans at a lower price. The risk that mortgages in process will fall in value prior to their sale is known as interest rate risk. In addition, the profit of mortgage origination is affected by fallout. The loan borrowers have the right to walk away at any time before closing. This right is actually a put option for the borrowers of the loan commitments. As interest rates fall, fallout rises because borrowers who have locked in a mortgage rate are more likely to find better rates with another lender. Conversely, as rates rise the percentage of loans that close increases. Fallout affects the required size of the hedging instrument because it affects the size of the pipeline position to be hedged. At lower rates, fewer rate loans will close and a smaller position in the hedging instrument is needed. Those risks in the mortgage underwriting process are known as the mortgage pipeline risks. Lenders often hedge this exposure by selling forward their expected closing volume or by shorting U.S. Treasury notes or futures contracts. In addition, options on U.S. Treasury note futures are used to hedge against the risk of fallout (Cusatis and Thomas, 2005). Mortgage lenders are concerned with two types of risk: volatility and extreme loss. The volatility measures the uncertainty of the value of loan commitments. High volatility of the exposure means the mortgage institution has high uncertainty in the value of locked loans. The extreme loss measures the worse case scenario. The downside risk measure is used to decide the reservation of economic capital prepared for the extreme loss. To reduce the volatility and downside risk of locked loan values, mortgage pipeline managers hedge the exposure by purchasing financial instruments whose values are sensitive to mortgage rates. A direct way is to sell the MBS forward contracts. However, MBS forward contracts are traded over the counter. In other words, the contract is not standardized and the lack of liquidity may induce high transaction costs. An alternative approach is to sell an equivalent amount of Treasury futures. As rates go up, the value of the short position will rise, offsetting the losses on the loan commitments in the pipeline. The fallout is also a critical source of risk in pipeline management. Mortgage loans provide a put option to borrowers allowing them to not close the loan commitment. This embedded put option reduces the number of loans that go to closing as rates go down and vise versa. In other words, the fallout effect varies the loan amount that need to be hedged. This phenomenon is called "negative convexity" in the fixed income market. Figure 4.1 shows the negative convexity in the pricerate diagram. Curve A shows the convexity of fixed income security value with respect to interest rate. For locked mortgages in pipeline, when rates drop below the locked rate, the value of loans in pipeline is increasing in a decreasing rate because of fallout. Therefore, the value of locked mortgages with respect to rates has a negative convexity property shown in Curve B. Value of loans in pipeline A B Locked Rate Rates Figure 41. Negative convexity. When rates drop below the locked rate, the value of loans in pipeline is increasing in a decreasing rate because of fallout. Put options on Treasury futures is a preferred instrument for hedging the fallout risk. It is well known that the 10year Treasury notes are the preferred choice for hedging pipeline risk, mainly because the duration more closely matches the loan portfolio (Taflia, 2003). This research considers using the put options on 10year Treasury futures and the 10year Treasury futures for hedging the mortgage pipeline risk. The other advantage in selecting these two hedge instruments is that those instruments are traded in exchange with high liquidity. So the hedge portfolio could be rebalanced without unexpected transaction costs. It is worth mentioning that the spread between mortgage rates and Treasury rates creates basis risk. So an optimal cross hedge model for the pipeline risk management is developed in next section. This research investigates the best risk measures for mortgage pipeline risk hedging strategy. Five deviation risk measures and two downside risk measures are selected for the optimal pipeline risk hedge model. We tested the hedging performance by optimizing the hedging strategy associated with each risk measure based on an (insample) historical dataset. Then, the outofsample performance of the optimal hedging strategy is examined. 4.2 Model 4.2.1 Locked Loan Amount Evaluation To hedge the pipeline risk, we first model the value of locked loan amount with respected to the change in mortgage rate. The fallout of mortgage loans varies for different regions. A general way to estimate the fallout is using a regression model based on the regional historical data. This research defined the fallout as a function of the price change in the first generic 10 year Treasury futures given by Fallout = aexp(b ATY1) (41) where a and b are coefficients, and ATY1 is the price change on the mortgage backed security. Although the fallout effect increases when rates drop, there are always some loans that do not go to closing even the rates go up. To model this effect, the fallout is assumed to be greater than a minimum fallout level Fallout > Fallout (42) where Fallout is the lower bound of the fallout, which is also regional dependent. The value change of the locked loan amount can be formulated as the value change of the fallout adjusted loan amount. We assume that the value change of mortgage loans can be approximately equivalent to the value change of 10year Treasury futures. ALV = (1 Fallout) ATY1 (43) where ALV is the value change of locked loan amount. 4.2.2 Pipeline Risk Hedge Agenda Assume a pipeline risk manager has 100 million dollars loan amount flowing into pipeline every two weeks. The lock period of the mortgage loans is four weeks. The risk manager hedges the loan exposure twice during the lock period, one at the time point when the loans are originated, i.e., in the start date of the first week of the lock period, and the other in the middle of the lock period, i.e., on the beginning of the third week. In this hedge agenda, the risk manager hedges the pipeline risk every two weeks with two different locked loan amounts. The first part of the loans was locked in the current mortgage rate with a loan amount of 100 million dollars. The second part of the loans was locked in the rate two weeks earlier. The loan amount is fallout adjusted. Therefore, the total value change is given by ALV, = AL V + AL V2, (44) where ALVl, is the value change of the loans originated at time t1 during [tl,t], and ALV2, is the value change of the loans originated at time t2 during [tl,t]. ALV1I is given by ALV1 =LV1, LVL1,_ (1 Fallout,).(TY1, TYI)/100.LVl t (45) = ( max[a exp(b(TY1, TYl,_,)),Fallout. (TY1, TY ,_,) /100 LVI,_,, where Fallout, is the fallout during [tl,t], and LVl_1 is the loan amount originated at time t1. ALV2, is given by LV2t = LV2t LV2t,_ = (1 Fallout,,,1). (TY1, TY1,_)/100. LV2,_ (46) The value change of the loans originated at time t2 during [tl,t] considered fallouts in two consecutive periods. The second fallout, Fallout,,, is calculated based on the value change of 10year Treasury futures during [t2,t] and adjusted by the first one, Fallout,,, Fallout ,, = max[a exp(b(TY1, TYI~ 2)) Fallout, Fallout] (47) and the time t1 value of the locked loans originated at time t2, LV2,,, is given by LV2, = (1 Fallout,,) LV2, (48) 4.2.3 Model Development An optimization model for the mortgage pipeline risk hedge was developed. The model considers shorting the first generic 10year Treasury futures (TY1) and buying the put options on 10year Treasury futures, such as TYZ7P 118, for pipeline risk hedge. As mortgage rates rise, the locked loan values as well as the value of 10year Treasury futures drop. The increasing value in the short position of 10year Treasury futures offsets the losses in the locked loan value. On the other hand, the decreasing value of the 10year Treasury futures increases the value of long position of the put option on 10year Treasury futures. To investigate the performance of different risk measures on the pipeline hedging strategies, five deviation measures: standard deviation, mean absolute deviation, CVaR deviation, VaR deviation, and twotailed VaR deviation, and two downside risk measures: CVaR and VaR, were tested in the pipeline hedge model The notations used in the model formulation are listed as follows. x = vector [x,, x,] of decision variables (hedge positions), x, = hedge position of 10year Treasury futures, x2 = hedge position of put options on 10year Treasury futures, 0 = random vector [00,01,02] of parameters (prices), 0 = thejth scenario of random vector [00, 0,02 ], 0 = thejth scenario of value change in the locked loan amount in one analysis period, 0 = thejth scenario of price change in 10year Treasury futures in one analysis period, 2 = thejth scenario of price change in put options on 10year Treasury futures in one analysis period, 00 = value change in the locked loan amount in one analysis period, 01 = price change in 10year Treasury futures in one analysis period, 02 = price change in put options on 10year Treasury futures in one analysis period, J = number of scenarios, L(x, OJ)= thejth scenario of loss based on hedge position x andjth scenario of prices J,, The model formulation associated with each risk measure is described as follows. The objective function (49) of the minimum standard deviation model is equivalent to the standard deviation since the value of the sum of the square is larger than one. A drawback of the standard deviation measure is that the squared term amplifies the effect of outliers. The formulation of minimum standard deviation model is given by {Y L (x,0W J MinZ L(x, Oj) J (49) s.t. L(x,0, )= 0 (01x, +0 x2) Vj. To overcome the drawback of standard deviation mentioned above, the mean absolute deviation takes the absolute value instead of the squared value in calculating dispersion. The formulation of minimum mean absolute standard deviation model is given by J ZL(x,O,) jl J s.t. L(x, O )= 00j (jx + 0 X2) Vj. The nonlinear programming problem was formulated as an equivalent linear programming problem Min z s.t. z,> L(x,Oj) J= L(x,O,) zi L(xO)+  L(x, O ) = 00 (o,1x1 + 0x2) Conditional ValueatRisk (CVaR,) deviation measures the distance between the mean value and CVaR, of a distribution. In other words, the minimum CVaR, deviation model minimizes the difference between 50% percentile and mean of a tail of a distribution. The minimum CVaR, deviation model can be formulated as a linear programming problem J JL(x,O,) Min + I z Jz SJ(1 a)J1 J s.t. z L(x, O )C Vj L(x, 0 0)= (, ( x, + 02x) where is a real number variable representing the a percentile of a distribution. where C4, is a real number variable representing the a percentile of a distribution. ValueatRisk (VaR,) deviation measures the distance between the mean value and VaR, of a distribution. The minimum CVaR, deviation model minimizes the difference between 50% percentile and a percentile of a distribution. The minimum VaRa deviation model can be formulated as a 01 mixed linear programming problem J L(x,O,) Min a L J s.t.L(x, O), j i y,(1 a)J J=1 L(x,)O) = , (o1,x, +O ) Vj y, { (0,1} Vj where Mis a big number used to enforce y, = 1 when L(x, O,) C > 0. The twotailed VaR, deviation measures the distance between two VaRy values in two tails of a distribution. This measure ignores the outliers beyond a and 1 a tails of a distribution. The property can be used in forecast and robust regression. Min + ; s.t.L(x,, ) y, ( a)J ,y<(i a)J J=1 L(x, O) l < Mz Vj .=1 L(x,,)= 0 (1x, +ox2) Vj y,, z {0,1} Vj In addition to the deviation measures, two downside risk measures are considered in the mortgage pipeline risk models. Conditional ValueatRisk (CVaR,) measures the mean of the a tail of a distribution. The minimum CVaR, model minimizes the mean of the worst (1 a) *100 % loss scenarios. The minimum CVaR, model can be formulated as a linear programming problem. 1 J Min + 1 J(1 a) s.t. z> L(x, ) Vj L(x,0 )= 0, (ox, + Ox) Vj z,>0 Vj ValueatRisk (VaR,) measures the a percentile of a distribution. The minimum VaR, model minimizes the loss value at a percentile of a distribution. The minimum VaR, model can be formulated as a 01 mixed linear programming problem Min ; s.t.L(x,o,), < My Vj Sy, <(I a) j=1 L(x,0) = 0 ( (ox +ox,) Vj yJ G {0,1} Vj. 4.3 Case Study 4.3.1 Dataset and Experiment Design The dataset for the case study includes biweekly data of the price of the generic first 10 year Treasury furthers (TY1)7, the price of put options on the 10year Treasury futures8, and the 7 The TY1 is the near contract generic TY future. A genetic is constructed by pasting together successive "Nth" contract prices from the primary months of March, June, September, and December. 8 The strike prices of the put options are selected based on the price of TY 1. mortgage rate index MTGEFNCL9. All prices and index are the close price on Wednesday from 8/13/2003 to 10/31/2007. The dataset includes 110 scenarios. To test the performance of six deviation measures and two downside risk measures in the pipeline risk management, we used the first 60 time series data as known historical data and solve the optimal hedge positions associated with each risk measure. Next, the optimal hedge positions associated with different risk measures are applied in hedging the next (the 61st) scenario, and a hedge error can be calculated for each model. Then, the window is rolled one step forward. The dataset from the second to the 61st scenarios are used to compute the optimal hedging positions, and the one step forward outofsample hedge error can be get from the 62nd scenario. This procedure was repeated for the remaining 50 scenarios, and 50 outofsample hedge errors were used to evaluate the performances of eight risk measures in pipeline risk hedge. To investigate whether more historical data provides better solution in outofsample performance, an alternative approach is to fix the window start date. The growing window approach keeps all the known historical data for solving the optimal hedge positions. The results from two approaches are compared. 4.3.2 Analyses and Results The experiment was conducted in a UNIX workstation with 2 Pentium4 3.2 GHz processor and 6GB of memory. The optimization models were solved using CPLEX optimizer. Figure 42 shows the values of naked unhedgedd) and hedged pipeline positions. It shows that no matter which risk measure is selected, the pipeline hedge reduces the volatility of the value of locked loans dramatically. Figure 43 and Figure 44 show the outofsample hedge 9 The MTGEFNCL index represents the 30 Year FNMA current coupon, which has been used as an index for the 30 year mortgage rate. performances associated with risk measures using rolling window approach and growing window approach, respectively. The risk measures selected for the optimal hedging model includes: standard deviation (STD), mean absolute deviation (MAD), CVaR deviation with 90% confident interval (CVaR90 Dev), VaR deviation with 90% confident interval (VaR9o Dev), two tailed VaR deviation with 90% confident interval (2TVaR9o), twotailed VaR deviation with 99% confident interval (2TVaR9o), CVaR90, and VaR90. The Figures show that the standard deviation has the worst outofsample hedge performance among all risk measures. Table 41 and Table 42 show the mean value, standard deviation, and maximum loss of the 50 outofsample losses of hedged position based on rolling window and growing window approach, respectively. Comparing these two tables we can see that the two approaches are not significantly different. For the rolling window approach, the two tailed 99% VaR has the best hedge performance among all risk measures. In addition, the mean standard deviation has the Value of loans in pipeline 4000000 3000000 2000000 1000000 A 0 1 04 711 10 1 6 1\ 2 28 1 4 37 40 43 4 1000000 2000000 3000000 4000000 STD  MAD CVAR90Dev VAR90Dev  2TVAR90 CVaR90 VAR90 2TVAR99 UnHedge Figure 42 Value of naked pipeline position and hedged pipeline positions associated with different risk measures. Value of hedged position in pipeline 500000 400000 300000 200000 100000 0 100000 200000 300000 STD  MAD CVAR90Dev VAR90Dev  2TVAR90 CVaR90  VAR90 2TVAR99 Figure 43 Outofsample hedge errors associated with eight risk measures using rolling window approach Value of hedged position in pipeline 500000 400000 300000 200000  100000 MAD 100000 CVAR90Dev VAR90Dev 1 4 7 10 13 19 22 5 28 3 3M 4 46 49 43 46 49 2TVAR90 100000 CVAR90 VAR90 200000 2TVAR99 300000 Figure 44. Outofsample hedge errors associated with eight risk measures using growing window approach best performance in reducing the volatility. For the growing window approach, CVaR deviation has best performance in mean value and standard deviation, and the CVaR has the best performance in maximum loss. Table 41 Mean value, standard deviation, and maximum loss of the 50 outofsample losses of hedged position based on rolling window approach CVaR9o VaR9o STD MAD Dev Dev 2TVaR9o 2TVaR99 CVaR9o VaR9c Mean 45260 36648 31863 42667 37571 30178 31317 384 StDev 94182 54407 71278 66014 62836 55409 77217 622 Max Loss 394484 273282 270711 295226 290365 261490 278274 2952 Table 42 Mean value, standard deviation, and maximum loss of the 50 outofsample losses of hedged position based on growing window approach CVaR9o VaR9o STD MAD Dev Dev 2TVaR90 2TVAR99 CVaRo9 VaR9c Mean 45260 42028 35546 47363 37571 38296 35944 445 StDev 94182 59247 54723 76232 62836 59040 60523 658 Max Loss 394484 276070 278462 314817 290365 296030 252958 3148 91 09 26 66 64 17 4.4 Conclusion This chapter studies the optimal mortgage pipeline risk management strategy. We developed an optimization model to minimize the pipeline risk under five different deviation risk measures and two downside risk measures. A case study using 10year Treasury futures and options on the 10year Treasury futures as the hedging instruments shows that under the growing window approach, the CVaR deviation has a better performance for hedging the volatility and CVaR is better for hedging the downside risk. On the other hand, when the rolling window approach is used, the two tailed 99% VaR has a better performance in general and the mean absolute deviation performs well in hedging volatility. It is worth mentioning that the result of the case study shows the standard deviation is the worst risk measure for the outofsample pipeline risk hedge. CHAPTER 5 CONCLUSION This dissertation has shown a novel stochastic optimization model using conditional value atrisk as the risk measure in the reward versus risk framework. This optimization approach was applied in modeling the optimal crop production and risk hedging strategy. To get a better performance in capturing the correlation between marginal distributions of multivariate random variables, the copula function was used to generate the portfolio loss distribution. Incorporating the copula based loss distribution within the CVaR optimization model produces a powerful model for solving the optimal planting schedule and hedging strategy problem. In addition, the ENSObased climate forecast information was used to get a better prediction of the crop yield in the coming season. A case study showed that 65%CRC or 70%CRC would be the optimal insurance coverage when the CVaR constraint reaches a strict level. Furthermore, the optimal hedging strategy with crop insurance products and futures contracts was examined. When futures price are unbiased or positive biased, the optimal hedging strategy only contains futures contracts and no crop insurance policies are desirable. However, when the futures price is negative biased, the optimal hedging strategy depends on the ENSO phases. The optimal futures hedge ratio increases with the increasing CVaR upper bound when the insurance strategy is unchanged. For a fixed CVaR upper bound, the optimal hedge ratio increases when the positive bias of futures price increases. However, when the futures price is negative biased, the optimal hedge ratio depends on the value of CVaR upper bound. For the secondary mortgage market efficient execution problem, a stochastic optimization model is proposed to perform the efficient execution analysis. The model considers secondary marketing functionalities, including the loanlevel execution for an MB S/whole loan, guarantee fee buyup/buydown, servicing retain/release, and excess servicing fee. Since secondary marketing involves random cash flows, lenders must balance between expected revenue and risk. The advantage of the CVaR risk measure was introduced and employed in the model that maximizes expected revenue under a CVaR constraint. By solving the efficient execution problem under different risk tolerances, efficient frontiers could be found, which provides optimal execution strategies associated with different risk budgets. A sensitivity analysis was conducted in parameters of expected retained servicing fee multipliers, mortgage prices, and IMBS prices. In the optimal mortgage pipeline risk management strategy problem, we developed an optimization model to minimize the mortgage pipeline risks under five different deviation risk measures and two downside risk measures. A case study using 10year Treasury futures and out options on the 10year Treasury futures as the hedging instruments was conducted. The result shows that the CVaR Deviation and CVaR have better outofsample performances in controlling the volatility and maximum loss, respectively, under the growing window approach. In addition, the twotailed 99% VaR deviation performs better on mean loss and maximum loss for the rolling window approach. In contrast, standard deviation performed the worst. APPENDIX A EFFICIENT EXECUTION MODEL FORMULATION Notations: Indices: m = index of mortgages (1,2,...,M), M = total number of mortgages, j = index of mortgage groups (1,2,...,J), J = total number of groups, k = index of scenarios (1,2,...,K), K = total number of scenarios, c = index of MBS coupon rate (1,2,...,C"), C" = number of possible MBS coupon rates of mortgage m. Decision Variables: 1, if mortgage m is pooled into MBS with coupon rate index c, c 0, otherwise, 1, if mortgage m is sold as a whole loan, S 0, otherwise, j_ 1, if the servicing of mortgage m is retained, [ 0, otherwise, 1_ if the servicing of mortgage m is sold, [ 0, otherwise, r = guarantee fee buyup spread of mortgage m, r, = guarantee fee buydown spread of mortgage m, rr" = retained excess servicing fee spread of mortgage m, zk = real variables used in CVaR constraint formulation, = real variables used in CVaR constraint formulation. Input Data: L = loan amount of mortgage m, p'" = price of MBS with coupon rate index c and maturity t", t" = maturity of mortgage m, Pw = whole loan sale price of mortgage m, Rc = MBS coupon rate related to index c, K, = guarantee fee buyup multiplier of mortgage m, K7 = guarantee fee buydown multiplier of mortgage m, R" = note rate of mortgage m, R, = base servicing fee of mortgage m, R, = base guarantee fee of mortgage m, B" = base servicing value of mortgage m, c" = servicing cost of mortgage m, pk = the probability of scenario k, Knk = retained servicing fee multiplier of mortgage m under scenario k, U' = upper bound of guarantee fee buyup spread of mortgage m, U" = upper bound of guarantee fee buydown spread of mortgage m, U = upper bound of retained excess servicing fee of mortgage m, U = upper bound of average retained excess servicing fee of all mortgages, U = upper bound of average retained excess servicing fee of mortgages in group, U = upper bound of CVaR losses, a = percentile of CVaR. Model Formulation: [L x f (p'tm xz;)+P; x zm m=1 c=1 Y(L xB xz )+m p L xR xK ) c M K +'Z Pk (L xK" Kx) m=l k=l + (Kl x rxr;) (Km x rx) m=l m=l zc + z =1 m = 1,2,...,3 c z+z0' +zl, =1 Vm = 1,2,...,M z" +r"+r" +r < 1 Vm = 1,2,...,M C. Rcz + +r : R< rR' R Vm = 1,2,...,M c=l +(1 a)1pkzk g k=1 zk Lk V k=L ...,K zk 0, Vk = ,...,K [revenue from MBSs or whole loan sale revenue from the base servicing fee of securitized mortgages Revenue from the excess servicing fee of securitized mortgages revenue from the guarantee fee buyup/buydown of securitized mortgages mortgage m must be either securitized as an MBS or sold as a whole loan if mortgage m is sold, there is no servicing; otherwise, servicing either be sold or be retained if mortgage m is sold, there is no guarantee fee and excess servicing fee if mortgage m is securitized, note rate = MBS coupon rate + servicing fee rate + guarantee fee rate CVaRa (loss) < U : the expected loss (negative revenue) of the worst 1a percent of senarios should be less than or equal to U Max L, L' L P X Z,+)+ oT, z LX B, Yo) m=1L c=l m=l x Lm x Bk xmr Z(LK xRX KxXr Z i) K"z(Lx xtrx) m=l m=l M M Z(L x x )r+U x b(1 xz X m=l m=l se se YLx r" U, rL (1 z;) mj \mej OIr r U: Vm = 1,2,...,M O0 r< U; Vm = 1, 2,...,M 0g rU; Vmm = 1,2,...,M z, zL,, z, z e {0,1} Vm: r",,r r >0 Vj = 1, 2,..., J : 1,2,...... C" the average excess servicing fee of all securitized mortgages is restricted to an upper bound. Sthe average excess servicing fee of securitized mortgages of groups is restricted to an upper bound for each securitized mortgage m, the excess servicing fee is restricted to upper bound for each securitized mortgage m, the guarantee fee buyup spread is restricted to an upper bound for each securitized mortgage m, the guarantee fee buydown spread is restricted to an upper bound [binary constraints [nonnegative constraints LIST OF REFERENCES Aldrich S., W. Greenberg, and B. Payner. 2001. "A Capital Markets View of Mortgage Servicing Rights." The Journal ofFixed Income 11: 3754. Artzner, P., F. Delbaen, J.M. Eber, and D. Heath. 1999. "Coherent measures of risk." Mathematical Finance 9: 203228. Cabrera, V.E., C. Fraisse, D. Letson, G. Podesta, and J. Novak. 2006. "Impact of climate information on reducing farm risk by optimizing crop insurance strategy." Transactions of the American Society ofAgricultural and Biological Engineers (ASABE) 49: 12231233. Cabrera, V.E., D. Letson, and G. Podesta. 2007. "The Value of the Climate Information when Farm Programs Matter." Agricultural Systems 93: 2542. Cane, M.A., G. Eshel, and R. Buckland. 1994. "Forecasting Zimbabwean maize yield using eastern equatorial Pacific sea surface temperature." Nature 370: 204205. Cherubini U., E. Luciano, and W. Vecchiato. 2004. Copula Methods in Finance, Wiley. Coble K.H., J.C. Miller, and M. Zuniga. 2004. "The joint effect of government crop insurance and loan programmes on the demand for futures hedging." European Review of Agricultural Economics. 31: 309330 Cusatis, P.J. and M.R. Thomas. 2005. Hedging Instruments and Risk Management. McGrawHill Downing, C., R. Stanton, and N. Wallace. 2005. "An Empirical Test of a TwoFactor Mortgage Valuation Model: How Much Do House Prices Matter?" Real Estate Economics 33: 681 710. Duffie, D. and J. Pan. 1997. "An Overview of ValueatRisk." Journal ofDerivatives 4: 749. Fabozzi, F. 2001. The Handbook of MortgageBacked Securities, 5th edition, McGrawHill. Fabozzi, F. 2002. The Handbook ofFixedIncome Securities, 6th edition, McGrawHill. Hakim, S., M. Rashidian, and E. Rosenblatt. 1999. "Measuring the Fallout Risk in the Mortgage Pipeline." The Journal ofFixed Income 9: 6275. Hansen, J.W. 2002. "Realizing the potential benefits of climate prediction to agriculture: issues, approaches, challenges." Agricultural Systems 74: 309330. Hansen, J.W., A.W. Hodges, and J.W. Jones. 1998. "ENSO influences on agriculture in the Southeastern US." Journal of Climate 11: 404411. Hilliard, J.E., J.B. Kau, and V.C. Slawson. 1998. "Valuing Prepayment and Default in a Fixed Rate Mortgage: A Bivariate Binomial Option Pricing Technique." Real Estate Economics, 26,431468. Jones, J.W., J.W. Hansen, F.S. Royce, and C.D. Messina. 2000. "Potential benefits of climate forecast to agriculture." Agriculture, ecosystems & environment. 82: 169184. Jones, J.W., G. Hoogenboom, C.H. Porter, K.J. Boote, W.D. Batchelor, L.A. Hunt, P.W. Wilkens, U. Singh, A.J. Gijsman, J.T. Ritchie. 2003. "The DSSAT cropping system model." European Journal ofAgronomy 18: 23565. nd Jorion, P. 2000. Value at Risk: The New Benchmark for Managing Financial Risk, 2 edition, McGrawHill. Kau, J.B., D.C. Keenan, W.J. Muller, III, and J.F. Epperson. 1992. "A Generalized Valuation Model for Fixed Rate Residential Mortgages." Journal of Money, Credit, andBanking 24: 279299. Kau, J.B., D.C. Keenan, W.J. Muller, III, and J.F. Epperson. 1994. "The Value at Origination of FixedRate Mortgages with Default and Prepayment." Journal of Real Estate Finance and Economics 11: 536. Kau, J.B. 1995. An Overview of the OptionTheoretic Pricing of Mortgages. Journal of Housing Research 6: 217244. Kruksal, W.H. 1958. "Ordinal measures of association." Journal of the American Statistical Association 53: 814861. Lederman, J. 1997. Handbook of Secondary Marketing. Mortgage Bankers Association of America. Lin, C.C., T.H. Chu, L.J. Prather. 2006. "Valuation of Mortgage Servicing Rights with Foreclosure Delay and Forbearance Allowed." Review of Quantitative Finance and Accounting 26: 4154. Liu J., C. Men, V.E. Cabrera, S. Uryasev, and C.W. Fraisse. 2006. "CVaR Model for Optimizing Crop Insurance under Climate Variability." working paper. Mahul O. 2003. "Hedging Price Risk in the Presence of Crop Yield and Revenue Insurance." European Review ofAgricultural Economics 30: 217239. Markowitz, H.M. 1952. "Portfolio Selection." Journal ofFinance 7: 7791. McKinnon, R. I. 1967. "Futures Market, buffer stocks, income stability for primary producers." Journal of Political Economy 75: 844861. Messina, C.R., J.W. Jones, C.W. Fraisse. 2005. "Development of cotton CROPGRO crop model." Sw,,iew, /%t Climate Consortium StaffPaper Series 0505: Gainesville, FL. Nemhauser, G.L. and L.A. Wolsey. 1999. Integer and Combinatorial Optimization, Wiley. Poitras, G. 1993. "Hedging and Crop Insurance." The Journal of Futures Markets 13: 373389. Robert G.C. and J. Quiggin. 2002. "Optimal producer behavior in the presence of areayield crop insurance." American Journal ofAgricultural Economics 84: 320334. Rockafellar, R.T. and S.P. Uryasev. 2000. "Optimization of Conditional ValueatRisk." Journal ofRisk 2: 2141. Rockafellar, R.T. and S.P. Uryasev. 2002. "Conditional ValueatRisk for General Loss Distribution." Journal ofBanking and Finance 26: 14431471. Schwartz, E.S. and W.N. Torous. 1989. "Prepayment and the Valuation of MortgageBacked Securities." Journal ofFinance 44: 375392. Sklar A. 1959. "Fonctions de repartition a n dimensions et leures marges." Publications de 'l'nstitut de Statistique de L'Universite de Paris 8: 229231. Stanton, R. 1995. "Rational Prepayment and the Value of MortgageBacked Securities." Review of Financial Studies 8: 677708. Sugimura, T. 2004. "Valuation of Residential MortgageBacked Securities with Default Risk Using an IntensityBased Approach." AsiaPacific Financial Markets 11: 185214. Szego G. 2002. "Measures of Risk." Journal of Banking andFinance 26: 12531272. Taglia, P. 2003. "Risk Management Case Study: How Mortgage Lenders Use Futures to Hedge Pipeline Risk." Futures Industry Magazine, September/October. BIOGRAPHICAL SKETCH ChungJui Wang was born in Taipei, Taiwan. He earned his bachelor's (1994) and master's (1996) degrees in civil engineering from National Central University. After 2 years military service, he worked for Evergreen Construction Co. for two years. In 2002, he began his master's and doctoral studies in the Department of Industrial and Systems Engineering at the University of Florida. He finished his Ph.D. in operation research with concentration in quantitative finance in December 2007. PAGE 1 1 OPTIMIZATION APPROACHES IN RISK MANAGEMENT: APPLICATIONS IN FINANCE AND AGRICULTURE By CHUNGJUI WANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 PAGE 2 2 2007 ChungJui Wang PAGE 3 3 To my family PAGE 4 4 ACKNOWLEDGMENTS I thank my supervisory committee chair (Dr. Stan Uryasev) and supervisory committee members (Dr. Farid AitSahlia, Dr. Liqing Yan, and Dr. Jason Karceski) for their support, guidance, and encouragement. I thank Dr. Victor E. Cabrera and Dr. Clyde W. Fraisse for their guidance. I am grateful to Mr. Philip Laren for sharing his knowledge of the mortgage secondary market and providing us with the dataset used in the case study. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ..............9 CHAPTER 1 INTRODUCTION..................................................................................................................11 2 OPTIMAL CROP PLANTING SCHEDU LE AND HEDGING STRATEGY UNDER ENSOBASED CLIMATE FORECAST...............................................................................16 2.1 Introduction............................................................................................................... ........16 2.2 Model...................................................................................................................... ..........19 2.2.1 Random Yield and Price Simulation......................................................................19 2.2.2 MeanCVaR Model................................................................................................22 2.2.3 Model Implementation...........................................................................................23 2.2.4 Problem Solving and Decomposition.....................................................................29 2.3 Case Study................................................................................................................. .......29 2.4 Results and Discussion.....................................................................................................32 2.4.1 Optimal Production with Crop Insurance Coverage..............................................33 2.4.2 Hedging with Crop Insurance and Unbiased Futures.............................................34 2.4.3 Biased Futures Market............................................................................................35 2.5 Conclusion................................................................................................................. .......43 3 EFFCIENT EXECUTION IN THE SECONDARY MORTGAGE MARKET.....................45 3.1 Introduction............................................................................................................... ........45 3.2 Mortgage Securitization....................................................................................................47 3.3 Model...................................................................................................................... ..........51 3.3.1 Risk Measure..........................................................................................................52 3.3.2 Model Development...............................................................................................53 3.4 Case Study.................................................................................................................59 3.4.1 Input Data...............................................................................................................59 3.4.2 Result................................................................................................................... ...61 3.4.4 Sensitivity Analysis................................................................................................63 3.5 Conclusion................................................................................................................. .......64 4 MORTGAGE PIPELINE RISK MANAGEMENT...............................................................69 4.1 Introduction............................................................................................................... ........69 PAGE 6 6 4.2 Model...................................................................................................................... ..........71 4.2.1 Locked Loan Amount Evaluation..........................................................................71 4.2.2 Pipeline Risk Hedge Agenda..................................................................................72 4.2.3 Model Development...............................................................................................73 4.3 Case Study................................................................................................................. .......77 4.3.1 Dataset and Experiment Design.............................................................................77 4.3.2 Analyses and Results..............................................................................................78 4.4 Conclusion................................................................................................................. .......81 5 CONCLUSION..................................................................................................................... ..82 APPENDIX A EFFICIENT EXECUTION MODEL FORMULATION.......................................................84 LIST OF REFERENCES............................................................................................................. ..87 BIOGRAPHICAL SKETCH.........................................................................................................90 PAGE 7 7 LIST OF TABLES Table page Table 21. Historical years associated with ENSO phases from 1960 to 2003.............................30 Table 22. Marginal distributions and rank correlation coefficien t matrix of yields of four planting dates and futures price for the three ENSO phases..............................................31 Table 23. Parameters of crop insurance ( 2004) used in the farm model analysis........................32 Table 24. Optimal insurance and production stra tegies for each climate scenario under the 90% CVaR tolerance ranged from $20,000 to $2,000 with increment of $2000............34 Table 25. Optimal solutions of planting sche dule, crop insurance coverage, and futures hedge ratio with various 90% CVaR upper bounds ranged from $24,000 to $0 with increment of $4,000 for the three ENSO phases................................................................35 Table 26. Optimal insurance policy and future s hedge ratio under bias ed futures prices............36 Table 27. Optimal planting schedule for different biases of futures price in ENSO phases........39 Table 31. Summary of data on mortgages....................................................................................61 Table 32. Summary of data on MBS prices of MBS pools..........................................................61 Table 33. Guarantee fee buyup and buydown and expected retained se rvicing multipliers......61 Table 34. Summary of efficient execution solution under different risk preferences..................65 Table 35 Sensitivity analysis in servicing fee multiplier..............................................................67 Table 36. Sensitivity analys is in mortgage price..........................................................................67 Table 37. Sensitivity an alysis in MBS price.................................................................................68 Table 41 Mean value, standard deviation, a nd maximum loss of the 50 outofsample losses of hedged position based on rolling window approach.....................................................81 Table 42 Mean value, standard deviation, a nd maximum loss of the 50 outofsample losses of hedged position based on growing window approach...................................................81 PAGE 8 8 LIST OF FIGURES Figure page Figure 21. Definition of VaR and CVaR associated with a loss distribution...............................23 Figure 22. Bias of futures price versus the optimal hedge ratio curves associated with different 90% CVaR upper bounds in the La Nia phase..................................................38 Figure 23. The efficient frontiers under various biased futures price. (A) El Nio year. (B) Neutral year. (C) La Nia year..........................................................................................41 Figure 31. The relationship between partic ipants in the passthrough MBS market...................48 Figure 32. Guarantee fee buydown.............................................................................................49 Figure 33. Guarantee fee buyup............................................................................................... ...49 Figure 34: Efficient Frontiers................................................................................................ .......62 Figure 41. Negative convexity................................................................................................. .....70 Figure 42 Value of naked pipe line position and hedged pipeline positions associated with different risk measures.......................................................................................................79 Figure 43 Outofsample hedge errors associat ed with eight risk measures using rolling window approach...............................................................................................................80 Figure 44. Outofsample hedge errors associat ed with eight risk measures using growing window approach...............................................................................................................80 PAGE 9 9 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMIZATION APPROACHES IN RISK MANAGEMENT: APPLICATIONS IN FINANCE AND AGRICULTURE By ChungJui Wang December 2007 Chair: Stanislav Uryasev Major: Industrial and Systems Engineering Along with the fast development of the financ ial industry in recent decades, novel financial products, such as swaps, derivatives, and structure financial instruments, have been invented and traded in financial markets. Practitioners have faced much more complicat ed problems in making profit and hedging risks. Financ ial engineering and risk management has become a new discipline applying optimization a pproaches to deal with the ch allenging financial problems. This dissertation proposes a nove l optimization approach using the downside risk measure, conditional valueatrisk (CVaR), in the reward versus risk framework for modeling stochastic optimization problems. The approach is appl ied to the optimal crop production and risk management problem and two critical problems in the secondary mortgage market: the efficient execution and pipeline risk management problems. In the optimal crop planting schedule a nd hedging strategy problem, crop insurance products and commodity futures contracts were considered for hedging against yield and price risks. The impact of the ENSObased climat e forecast on the optimal production and hedge decision was also examined. The Gaussian c opula function was applied in simulating the scenarios of correlated nonnor mal random yields and prices. PAGE 10 10 Efficient execution is a significant task faced by mortgage bankers attempting to profit from the secondary market. The challenge of effi cient execution is to sell or securitize a large number of heterogeneous mortgages in the seco ndary market in order to maximize expected revenue under a certain risk tolerance. We deve loped a stochastic optimization model to perform efficient execution that considers secondary marketing functionality including loanlevel efficient execution, guarantee fee buyup or buydow n, servicing retain or release, and excess servicing fee. The efficient execution model bala nces between the reward and downside risk by maximizing expected return under a CVaR constraint. The mortgage pipeline risk management pr oblem investigated the optimal mortgage pipeline risk hedging strategy using 10year Treasury future s and put options on 10year Treasury futures as hedge instruments. The outo fsample hedge performances were tested for five deviation measures, Standa rd Deviation, Mean Absolute Deviation, CVaR Deviation, VaR Deviation, and twotailed VaR De viation, as well as two downside risk measures, VaR and CVaR. PAGE 11 11 CHAPTER 1 INTRODUCTION Operations Research, which originated from World War II for optimizing the military supply chain, applies mathematic programming techniques in solving and improving system optimization problems in many areas, including engineering, management, transportation, and health care. Along with the fast development of the financial industry in recent decades, novel financial products, such as swap s, derivatives, and structural fi nance instruments, have been invented and traded in the financial markets. Practitioners have faced much more complicated problems in making profit and hedging risks. Fi nancial engineering and risk management has become a new discipline applying operations research in dealing with the sophisticated financial problems. This dissertation proposes a nove l approach using conditional va lueatrisk in the reward versus risk framework for modeling stochastic op timization problems. We apply the approach in the optimal crop planting schedule and risk hedging strategy probl em as well as two critical problems in the secondary mortgage market: the efficient execution problem, and the mortgage pipeline risk management problem. Since Markowitz (1952) proposed the meanvar iance framework in portfolio optimization, variance/covariance has become the predominant risk measure in finance. However, this risk measure is suited only to elliptic distributions, such as normal or tdistributions, with finite variances (Szeg 2002). The other dr awback of variance risk measure is that it measures both upside and downside risks. In practice, however, finance risk management is concerned mostly with the downside risk. A popular downside risk measure in economics and finance is ValueatRisk (VaR) (Jorion 2000), which measures percentile of loss dist ribution. However, as was shown by Artzner et al. (1999), VaR is illbeha ved and nonconvex for general distribution. The PAGE 12 12 disadvantage of VaR is that it only considers risk at percentile of loss di stribution and does not consider the magnitude of the losses in the tail (i.e., the worst 1percentage of scenarios). To address this issue, Rockafellar and Uryasev (2000, 2002) proposed Conditional ValueatRisk, which measures the mean value of the tail of loss distribution. It has been shown that CVaR satisfies the axioms of coherent risk m easures proposed by Artzner et al. (1999) and has desirable properties. Most importantly, Rockaf ellar and Uryasev (2000) showed that CVaR constraints in optimization problems can be fo rmulated as a set of linear constraints and incorporated into proble ms of optimization. This linear prope rty is crucial in formulating the model as a linear programming problem that can be efficiently solved. This dissertation proposes a meanCVaR m odel which, like a meanvariance model, provides an efficient frontier c onsisting of points that maximize expected return under various risk budgets measured by CVaR. Since CVaR is de fined in monetary units, decision makers are able to decide their risk tolerance much more intu itively than with abstract utility functions. It is worth noting that CVaR is defined on a loss dist ribution, so a negative CVaR value represents a profit. Although the variance/cova riance risk measure has its drawbacks, it has been the proxy of risk measure for modeling the stochastic optimi zation problem in financial industry. The main reason is that the portfolio variance can be eas ily calculated given indi vidual variances and a covariance matrix. However, the linear correlation is a simplified model and is limited in capturing the association between random variable s. Therefore, a more general approach is needed to model the more complicated relations hip between multivariate random variables, e.g., tail dependent. More importantl y, the approach should provide a convenient way to create the portfolio loss distribution from marginal ones, which is the most important input data in the PAGE 13 13 meanCVaR optimization model. This dissertati on applies the copula function to model the correlation between random variable s. In Chapter 2, the use of copul as to generate scenarios of dependent multivariate random variables is discu ssed. Furthermore, the simulated scenarios are incorporated into the meanCVaR model. Chapter 2 investigates the optimal crop planting schedul e and hedging strategy. Crop insurance products and futures contracts are availa ble for hedging against yield and price risks. The impact of the ENSObased climate for ecast on the optimal production and hedging decision is examined. Gaussian copula function is applie d in simulating the scen arios of correlated nonnormal random yields and prices. Using data of a representative cotton producer in the Southeastern United States, th e best production and hedging stra tegy is evaluated under various risk tolerances for each of three pr edicted ENSObased climate phases. Chapters 3 and 4 are devoted to two optimi zation problems in the secondary mortgage market. In Chapter 3, the efficient execution prob lem was investigated. Efficient execution is a significant task faced by mortgage bankers atte mpting to profit from the secondary market. The challenge of efficient execution is to sell or securitize a large number of heterogeneous mortgages in the secondary market in order to maximize expected revenue under a certain risk tolerance. A stochastic optimization model was developed to perform efficient execution that considers secondary marketing functionality incl uding loanlevel efficient execution, guarantee fee buyup or buydown, servicing retain or releas e, and excess servicing fee. Since efficient execution involves random cash flows, lenders must balance between expect ed revenue and risk. We employ a CVaR risk measure in this effi cient execution model that maximizes expected revenue under a CVaR constraint. By solving th e efficient execution problem under different risk tolerances specified by a CVaR constraint, an efficient frontier was found, which provides PAGE 14 14 secondary market managers the best execution st rategy associated with different risk budgets. The model was formulated as a mixed 01 linear programming problem. A case study was conducted and the optimization pr oblem was efficiently solved by the CPLEX optimizer. Chapter 4 examines the optimal mortgage pipe line risk hedging stra tegy. Mortgage lenders commit to a mortgage rate while the borrowers ente r the loan transaction process. The process is typically for a period of 3060 days. While the mort gage rate rises before the loans go to closing, the value of the loans declines. Th erefore, the lender will sell the loans at a lower price when the loans go to closing. The risk of a fall in value of mortgages still being pr ocessed prior to their sale is known as mortgage pipeline risk. Lende rs often hedge this e xposure by selling forward their expected closing volume or by shorting U.S. Treasury notes or futures contracts. Mortgage pipeline risk is affected by fall out. Fallout refers to the percentage of loan commitments that do not go to closing. As interest ra tes fall, fallout rises since bo rrowers who have locked in a mortgage rate are more likely to find better rates with another lender. Conversely, as rates rise, the percentage of loans that go to closing in creases. Fallout affects the required size of the hedging instrument because it changes the size of ri sky pipeline positions. At lower rates, fewer loans will close and a smaller position in the hedging instrument is n eeded. Lenders often use options on U.S. Treasury futures to hedge against the risk of fallout (Cusatis and Thomas, 2005). A model was proposed for the optimal mortgage pipeline hedging strategy that minimizes the pipeline risks. A case study considered tw o hedging instruments fo r hedging the mortgage pipeline risks: the 10 year Treasury futures a nd put options on 10year Treasury futures. To investigate the impact of different risk measur ement practices on the optimal hedging strategies, we tested five deviation measur es, standard deviation, mean abso lute deviation, CVaR deviation, VaR deviation, and twotailed VaR deviation, as well as two downside risk measures, VaR and PAGE 15 15 CVaR, in the minimum mortgage pipeline risk m odel. The outofsample performances of the five deviation measures and two downs ide risk measures were examined. PAGE 16 16 CHAPTER 2 OPTIMAL CROP PLANTING SCHEDULE A ND HEDGING STRATEGY UNDER ENSOBASED CLIMATE FORECAST 2.1 Introduction A riskaverse farmer preferring higher profit from growing crops faces uncertainty in the crop yields and harvest price. To manage uncertainty, a farmer may purchase a crop insurance policy and/or trade futures contracts against th e yield and price risk. Crop yields depend on planting dates and weather conditions during the growing period. The predictability of seasonal climate variability (i.e., the El Nio Southern Oscillation, ENSO), gives the opportunity to forecast crop yields in different planting dates. With the flexibility in planting timing, the profit can be maximized by selecting the best planting schedule according to climate forecast. Risk averse is another critical fact or when farmers make the decision. Farmers may hedge the yield and price risks by purchasing crop insurance products or financial instruments. Two major financial instruments for farm ers to hedge against crop risks are crop insurances and futures contracts. The Risk Ma nagement Agency (RMA) of the United States Department of Agricultural (USDA) offers cr op insurance policies for various crops, which could be categorized into three types: the yiel dbased insurance, revenu ebased insurance, and policy endorsement. The most popular yieldba sed insurance policy, Actual Production History (APH), or Multiple Peril Crop Insurance (MPCI), is available for most crops. The policies insure producers against yield losses due to natural causes. An insured farmer selects to cover a percentage of the average yield together with an election price (a percen tage of the crop price established annually by RMA). If th e harvest yield is less than the insured yield, an indemnity is paid based on the shortfall at the election pr ice. The most popular revenuebased insurance policy, Crop Revenue Coverage ( CRC), provides revenue protection. An insured farmer selects a coverage level of the guarantee revenue. If th e realized revenue is be low the guarantee revenue, PAGE 17 17 the insured farmer is paid an indemnity to cover the difference between the actual and guaranteed revenue. Catastrophic Coverage (C AT), a policy endorsement, pays 55% of the established price of the commod ity on crop yield shortfall in ex cess of 50%. The cost of crop insurances includes a premiu m and an administration fee1. The premiums on APH and CRC both depend on the crop type, county, practice (i.e., irri gated or nonirrigated) acres, and average yield. In addition, the APH premium depends on price election and yiel d coverage, and the CRC depends on revenue coverage. The premium on CAT coverage is paid by the Federal Government; however, producers pay the administra tive fee for each crop insured in each county regardless of the area planted2. In addition to crop insurance coverage, farm ers may manage commodity price risk by a traditional hedge instrument such as futures contra ct. A futures contract is an agreement between two parties to buy or sell a commodity at a certai n time in the future, for a specific amount, at a certain price. Futures contracts are highly standa rdized and are traded by exchange. The cost of futures contract includes commissions and intere st foregone on margin deposit. A riskaverse producer may consider using insurance products in conjunction with futures contacts for best possible outcomes. El Nio Southern Oscillation refers to in terrelated atmospheric and oceanic phenomena. The barometric pressure difference between th e eastern and western e quatorial Pacific is frequently changed. The phenomenon is known as the Southern Oscillation. When the pressure over the western Pacific is above normal and eastern Pacific pressure is below normal, it creates abnormally warm sea surface temperature (SST) know n as El Nio. On the other hand, when the 1 The administration fee is $30 for each APH and CRC contract and $100 for each CAT contract. 2 Source: http://www2.rma.usda.gov/policies PAGE 18 18 eastwest barometric pressure gradient is reve rsed, it creates abnormally cold SST known as La Nia. The term neutral is used to indicate SSTs within a normal temperature range. These equatorial Pacific conditions known as ENSO phases refer to different seasonal climatic conditions. Since the Pacific SST s are predictable, ENSO becomes an index for forecasting climate and consequently crop yields. A great deal of research has been done on the connection between the ENSObased climate prediction and crop yields since early 1990s. Cane et al. (1994) found the longterm forecasts of the SSTs could be used to anticipate Zimbabwean maize yield. Hansen et al. (1998) showed that El Nio Southern Oscillation is a strong driver of seasonal climate variability that impact crop yields in the southeastern U.S. Hansen (2002) and Jones et al. (2000) concluded that ENSObased climate forecasts might help reduce crop risks. Many studies have focused on the crop risk hedging with crop insurance and other derivative securities. Po itras (1993) studied farmers optimal hedging problem when both futures and crop insurance are available to hedge the unc ertainty of price and production. Chambers et al. (2002) examined optimal producer behavior in the presence of areay ield insurance. Mahul (2003) investigated the demand of futures and opt ions for hedging against price risk when the crop yield and revenue insurance co ntracts are available. Coble (2004) investigated the effect of crop insurance and loan programs on demand for futures contract. Some researchers have studied the impacts of the ENSObased climate information on the selection of optimal crop insurance policies. Ca brera et al. (2006) examined the impact of ENSObased climate forecast on reducing farm risk with optimal crop insurance strategy. Lui et al. (2006), following Cabrera et al (2006), studied the applicati on of Conditional ValueatRisk (CVaR) in the crop insurance industry under clim ate variability. Cabrera et al. (2007) included PAGE 19 19 the interference of farm government programs on crop insurance hedge under ENSO climate forecast. The purpose of this research is twofold. First, a meanCVaR optimization model was proposed for investigating the optimal crop pl anting schedule and hedgi ng strategy. The model maximizes the expected profit with a CVaR cons traint for specifying producers downside risk tolerance. Second, the impact of the ENSObased climate forecast on the optimal decisions of crop planting schedule and hedgi ng strategy was examined. To this end, we generate the scenarios of correlated random yi elds and prices by Monte Carlo simulation with the Gaussian copula for each of the three ENSO phases. Using the scenarios associated with a specific ENSO phase as the input data, the meanCVaR model is solved for the optimal production and hedging strategy for the specified ENSO phase. The remainder of this article is organized as follows. The proposed model for optimal planting schedule and hedging strategy is introduced in section 2.2. Next, section 2.3 describes a case study using the data of a representative cott on producer in the Southeastern United States. Then, section 2.4 reports the resu lts of the optimal planting schedule, crop insurance policy selection, and hedging position of futures contract. Finally, section 2.5 presents the conclusions. 2.2 Model 2.2.1 Random Yield and Price Simulation To investigate the impact of ENSObased climate forecast on the optimal production and risk management decisions, we calibrate the yi eld and price distributions for an ENSO phase based on the historical yields and prices of the years classified to the ENSO phase based on the Japan Meteorological Agency (JMA) definiti on (Japan Meteorological Agency, 1991). Then, random yield and price scenarios associated with the ENSO phase are generated by Monte Carlo simulation. PAGE 20 20 We assume a farmer may plant crops in a number of planting dates across the planting season. The yields of planting date s are positive correlated accordi ng to the historical data. In addition, the correlation between the random pro duction and random price is crucial in risk management since negative correlated production and price provide a na tural hedge that will affect the optimal hedging strategy (McKinnon, 1967). As a consequence, we consider the correlation between yields of diffe rent planting dates and the crop price. Since the distributions of the crop yield and price are not typically norma l distributed, a method to simulate correlated multivariate nonnormal random yield and price was needed. Copulas are functions that describe depende ncies among variables, and provide a way to create distributions to model correlated multiv ariate data. Copula function was first proposed by Sklar (1959). The Sklar theorem states that given a joint distribution function Fon n R with marginal distributioniF there is a copula function C such that for all nx x ,...,1 in R )). ( ),..., ( ( ) ,..., (1 1 1 n n nx F x F C x x F (21) Furthermore, if iF are continuous then C is unique. Conversely, if C is a copula andiF are distribution functions, then F as defined by the previous e xpression, is a joint distribution function with marginsiF. We apply the Gaussian copula func tion to generate the correlated nonnormal multivariate distribution. The Gaussian copula is given by: ))), ( ( )),..., ( ( ( )) ( ),..., ( (1 1 1 1 1 1 n n n n nx F x F x F x F C (22) which transfers the observed variable ix i.e. yield or price, into a new variable iy using the transformation ) (1 i i ix F y (23) PAGE 21 21 where n is the joint distribution function of a mu ltivariate Gaussian vector with mean zero and correlation matrix is the distribution function of a st andard Gaussian random variable. In moving from ix to iy we are mapping observation fr om the assumed distribution iF into a standard normal distribution on a percentile to percentile basis. We use the rank correlation coefficient Spearmans rho s to calibrate the Gaussian copula to the historical data. For n pairs of bivariate random samples j iX X define ) (i iX rank Rand ) (j jX rank R. Spearmans sample rho (C herubini, 2004) is given by ) 1 ( 6 12 1 n n R Rn k jk ik s. (24) Spearmans rho measures the association only in terms of ranks. The rank correlation is preserved under the monotonic transformation in e quation 3. Furthermore, there is a onetoone mapping between rank correlation coefficient, Spearmans rho s and linear correlation coefficient for the bivariate normal random variables) (2 1y y (Kruksal, 1958) 2 ) ( arcsin 6 ) (2 1 2 1y y y ys (25) To generate correlated multivariate nonnormal random variables with margins iF and Spearmans rank correlation s we generate the random variables iy s from the multivariate normal distribution n with linear correlation 6 sin 2s (26) by Monte Carlo simulation. The actual outcomes ix s can be mapped from iy s using the transformation PAGE 22 22 ) (1 i i iy F x (27) 2.2.2 MeanCVaR Model Since Markowitz (1952) proposed the meanvariance framework in portfolio optimization, variance/covariance has become the predominant risk measure in finance. However, the risk measure is suited only to the case of elliptic di stributions, like normal or tdistributions with finite variances (Szeg, 2002). The other drawback of variance risk measure is that it measures both upside and downside risks. In practice, finance risk management is concerned only with the downside risk in most cases. A popular downsid e risk measure in economics and finance is ValueatRisk (VaR) (Jori on, 2000), which measures percentile of loss di stribution. However, as was shown by Artzner et al. (1999), VaR is illbehaved and nonconvex for general distribution. The other disa dvantage of VaR is that it only considers risk at percentile of loss distribution and does not consider the magnitude of the losses in the tail (the worst 1percentage of scenarios). To address this i ssue, Rockafellar and Uryasev (2000, 2002) proposed Conditional ValueatRisk, which measures the mean value of tail of loss distribution. Figure 21 shows the definition of CVaR and the relation between CVaR and VaR. It has been shown that CVaR satisfies the axioms of coherent ri sk measures proposed by Ar tzner et al. (1999) and has desirable properties. Most importantly, Rockafellar and Urya sev (2000) showed that CVaR constraints in optimization problems can be fo rmulated as a set of linear constraints and incorporated into the problems of optimization. The linear property is crucial to formulate the model as a mixed 01 linear programming problem that could be solved efficiently by the CPLEX solver. This research proposes a meanC VaR model that inherits advantages of the return versus risk framework from the mean variance model proposed by Markowitz (1952). More importantly, the model utili zes the CVaR risk measure inst ead of variance to take the PAGE 23 23 advantages of CVaR. Like meanvariance mode l, the meanCVaR model provides an efficient frontier consisting of points that maximize expect ed return under various tolerances of CVaR losses. Since CVaR is defined in monetary units, farmers are able to decide their risk tolerance much more intuitively compared to abstract utility functions. It is worth noting that CVaR is defined on a loss distribution. Th erefore, a negative CVaR va lue represents a profit. For example, a $20,000 90% CVaR mean s the average of the worst 10% scenarios should provide a profit equal to $20,000. Figure 21. Definition of VaR and CVaR associated with a loss distribution 2.2.3 Model Implementation Assume a farmer who plans to grow crops in a farmland of Q acres. There are K possible types of crops and more than one crop can be planted. For each crop k there are Tk potential planting dates that give different yield distributions based on the predicted ENSO phase, as well as Ik available insurance policies for the crop. The decision variables ktix and k represent the acreages of crop k planted in date t with insurance policy i and the hedge position (in pounds) of crop k in futures contract, respectively. Losses Maximal loss tailVaR CVaR(with pr obabili ty 1 )Frequency PAGE 24 24 The randomness of crop yield and harvest pric e in a specific ENSO phase is managed by the joint distribution corresponding to the ENSO phase. We sample J scenarios from the joint distribution by Monte Carlo simu lation with Gaussian copula, and each scenario has equal probability. Let ktjYdenote the jth realized yield (pou nd per acre) of crop k planted on date t and kjP denote the jth realized cash price ( dollar per pound) for crop k at the time the crop will be sold. The objective function of the model, shown in (28), is to maximize the expectation of random profit k ktix f that consists of the random profit from productionkti Px f, from crop insurance kti Ix f, and from futures contract k Ff. k F kti I kti P k ktiEf x Ef x Ef x Ef max max, (28) The profit from production of crop k in scenario j is equal to the income from selling the crop, kkT t I i kti kj ktjx P Y11, minus the production cost, kkT t I i kti kx C11, and plus the subsidy, kkT t I i kti kx S11., where kC and kS are unit production cost and subsi dy, respectively. Consequently Equation (29) expresses the expect ed profit from production. J j K k T t I i kti k k kj ktj kti Pkkx S C P Y J x Ef11111 (29) Three types of crop insurance policies are considered in the model, including Actual Production History (APH), Crop Re venue Coverage (CRC), and Ca tastrophic Coverage (CAT). For APH farmers select the insured yield, a percentage i from 50 to 75 percent with five percent increments of average yield kY as well as the election price, a percentage i between PAGE 25 25 55 and 100 percent, of the of the established price kPestablished annually by RMA. If the harvest is less than the yield in sured, the farmer is paid an indemnity based on the difference kT t kti ktj k ix Y Y1 at price k iP The indemnity of APH insurance policy APHI i for crop k in the jth scenario is given by For CRC, producers elect a percentage of coverage level i between 50 and 75 percent. The guaranteed revenue is e qual to the coverage level i times the product of kT t kti kx Y1and the higher of the base price (earlyseason price) b kP and the realized harvest price in the jth scenario of crop k, h kjP. The base price and harvest price of crop k are generally defined based on the crops futures price in planting season and harvest season, respectively. If the calculated revenue kT t kj kti ktjP x Y1 is less than the guaranteed one, the insure d will be paid the di fference. Equation (211) shows the indemnity of CRC insurance policy CRCI i for crop k in the jth scenario. CRC T t kj kti ktj T t h kj b k kti k i kijI i P x Y P P x Y Dk k 0 max max1 1, (211) The CAT insurance pays 55 % of the establis hed price of the comm odity on crop losses in excess of 50 %. The indemnity of CAT insurance policy CATI i for crop k in the jth scenario is given by 55 0 0 5 0 max1 k T t kti ktj k kijP x Y Y Dk (212) APH k i T t kti ktj k i kijI i P x Y Y Dk 0 max1 (210) PAGE 26 26 The cost of insurance policy i for crop k is denoted bykiR, which includes a premium and an administration fee. For the case of CAT, the premium is paid by the Federal Government. Therefore, the cost of CAT only contains a $ 100 administrative fee for each crop insured in each county. The expected total profit from insurance is equal to the indemnity from the insurance coverage minus the cost of the insurance is given by k kT t kti J j K k I i ki kij kti Ix R D J x Ef1 1111. (213) The payoff of a futures contract for crop k in scenario j for a seller is given by k kj k F kjf F ) ( (214) where kF is the futures price of crop k in the planting time, kjf is the jth realized futures price of crop k in the harvest time, and k is the hedge position (in pounds) of crop k in futures contract. It is worth noting that the futures price kjf is not exactly the same as the local cash pricekjP at harvest time. Basis, defined in (215), refers to the difference that induces the uncertainty of futures hedging known as the basis risk. The rand om basis can be estimated from comparing the historical cash prices and futures prices. Basis = Cash Price Futures Price. (215) The cost of a futures contract, F kC, includes commissions and interest foregone on margin deposit. Equation (216) expresses the e xpected profit from futures contract. J j K k F k F kj k FC J Ef111 (216) PAGE 27 27 We introduce binary variables kiz in constraint (217) and (218) to ensure only one insurance policy can be selected for each crop k. k i z Q xki T t ktik,1 (217) k zkI i ki 11 (218) where kiz otherwise. 0 policy by insured is crop if 1i k Constraint (219) restricts the total pl anting area to a given planting acreage Q. The equality in this constraint can be replaced by an inequality ) ( to represent farmers choosing not to grow the crops when the production is not profitable. Q xK k T t I i ktikk 111 (219) To model producers risk tolerance, we impose the CVaR constraint where k ktix L is a random loss equal to the negative random profit k ktix f defined in (8). The definition of )) ( (k ktix L CVaR is given by where ) (k ktix L is the quintile of the distribution of k ktix L ,. Therefore, constraint (220) enforces the conditional e xpectation of the random loss k ktix L given that the random loss exceeds quintile to be less than or equal to U. In other words, the expected loss of tail, i.e. (1)100% worst scenarios, is upper limited by an acceptable CVaR upper bound U. Rockafellar U x L CVaRk kti )) ( ( (220) ) ( ) ( ) ( )) ( (k kti k kti k kti k ktix L x L x L E x L CVaR (221) PAGE 28 28 & Uryasev (2000) showed that CVaR constrai nt (220) in optimization problems can be expressed by linear constraints (222), (223), and (224) U z J x LJ j j k kti 1) 1 ( 1 (222) j x L x L zk kti K k T t I i k kti jkk ,111, (223) j zj 0 (224) where jzare artificial variables introduced for the linear formulation of CVaR constraint. Note that the maximum objective function contains indemnitieskijD that include a max term shown in equation (210), (211), and (21 2). To implement the model as a mix 01 linear problem, we transform the equations to an equivalent linear formulation by disjunctive constraints (Nemhauser and Wolsey, 1999). For example, equation (210), k i T t kti ktj k i kijP x Y Y Dk 10 max, can be represented by a set of mix 01 linear constraints where M is a big number and kijZ is a 01 variable. Similarly, eq uation (211) and (212) can be transformed into a set of mix 01 linear constrai nts in the same way. Consequently, the optimal crop production and hedging problem has been formulated as a mix 01 linear programming problem. 0 kijD, k i T t kti ktj k i kijP x Y Y Dk 1, MZ P x Y Y Dkij k i T t kti ktj k i kijk 1, ) Z M( Dkij kij1 ) Z M( P x Y Ykij k i T t kti ktj k ik11 kij k i T t kti ktj k iMZ P x Y Yk 1. (225) PAGE 29 29 2.2.4 Problem Solving and Decomposition Although the mix 01 linear programming probl em can be solved with optimization software, the solving time increas es exponentially when the problem becomes large. To improve the solving efficiency, we may decompose the orig inal problem into subproblems that could be solved more efficiently than the original pr oblem. Since only one insurance policy could be selected for each crop, we decomposed the original problem into subproblems in which each crop is insured by a specific insurance policy. The original problem contains K types of crops, and for the kth type of crop there are Ik eligible insurance policies. Therefore, the number of the subproblems is equal to the number of a ll possible insurance combinations of the K crops, K k kI1. The formulation of the subproblem is the same as the original problem except that the index is are fixed and the equation (217) and (218) are removed. Solving subproblems gives the optimal production strategy and futures hedg e amount under a specific combination of insurance policies for K crops. The solution of the subpr oblem with the highest optimal expected profit among all subproblems gives the optimal solution of the original problem in which the optimal production strategy and futures hedge position are provided from the subproblem solution and the optimal in surance coverage is the specifi c insurance combination of the subproblem. 2.3 Case Study Following the case study in Cabrera et al. (2006) we consider a representative farmer who grows cotton on a nonirrigated fa rm of 100 acres in Jackson County, Florida. Dothan Loamy Sand, a dominant soil type in the region, is assumed. The farmer may trade futures contracts from the New York Board of Trade and/or purch ase crop insurance to hedge the crop yield and PAGE 30 30 price risk. Three types of crop insurances, including Actual Production History (APH), Crop Revenue Coverage (CRC), and Cata strophic Coverage (CAT), are eligible for cotton and the farmer may select only one eligible insurance po licy to hedge against th e risk or opt for none. For APH, the eligible coverage le vels of yield are from 65% to 75% with 5% increments, and the election price is assumed to be 100% of the establ ished price. In addition, the available coverage levels of revenue for CRC are from 65 % to 85% with 5% increments. To investigate the impact of ENSObased cl imate forecast in the optimal decisions of production and hedging strategy, we select historical climate data from 1960 to 2003 for the numerical implementation. ENSO phases during this period included 11 years of El Nio, 9 years of La Nia, and the remaining 25 years of Neutral, according to the Japan Meteorological Index (Table 21). Table 21. Historical years associat ed with ENSO phases from 1960 to 2003 EL Nio Neutral La Nia 1964 1987 1960 1975 1984 1994 1965 1989 1966 1988 1961 1978 1985 1995 1968 1999 1970 1992 1962 1979 1986 1996 1971 2000 1973 1998 1963 1980 1990 1997 1972 1977 2003 1967 1981 1991 2001 1974 1983 1969 1982 1993 2002 1976 The cotton yields during the period of 19602003 were simulated using the CROPGROCotton model (Messina et al., 2005) in th e Decision Support System for Agrotechnology Transfer (DSSAT) v4.0 (Jones et al., 2003) based on the historical climate data collected at Chipley weather station. The input for the si mulation model followed the current management practices of variety, fertilizati on and planting dates in the region. More specifically, a medium to full season Delta & Pine Land variety (DP 55), 110 kg/ha Nitrogen fe rtilization in two applications, and four planting dates, 16 Apr, 23 Apr, 1 May, and 8 May, were included in the PAGE 31 31 yield simulation, which was furthe r stochastically resampled to produce series of synthetically generated yields following the historical distribu tions (for more details see Cabrera et al., 2006). Assume cotton would be harvested and sold in December. The December cotton futures contact was used to hedge the pr ice risk. In addition, assume th e farmer will settle the futures contract on the last trading date, i.e. seventeen days from the end of December. The historical settlement prices of the December futures cont ract on the last trading date from 1960 to 2003 were collected from the New York Board of Trade. The statistics and the rank correlation coe fficient Spearmans rho matrix of yields and futures price are summarized in Table 22, whic h shows that crop yields for different planting dates are highly correlated and the correlation of yields is decreasi ng when the corresponding two planting dates are getting farther. In addi tion, the negative correlation between yields and futures price is found in the El Nio and Neutra l phases, but not in La Nia. We assumed the random yields and futures price follow the empirica l distributions of yields and futures price. Table 22. Marginal distributions and rank correlation coefficien t matrix of yields of four planting dates and futures price for the three ENSO phases Statistics of Marginal Distribution Rank Correlation Coefficient Matrix Spearmans rho ENSO Variable Mean Standard Deviation Yield on 4/16 Yield on 4/23 Yield on 5/1 Yield on 5/8 Future s Price Yield on 4/16 (lb) 815.0 71.7 1.00 0.93 0.75 0.74 0.36 Yield on 4/23 (lb) 804.6 79.4 0.93 1.00 0.63 0.57 0.23 El Nio Yield on 5/1 (lb) 795.4 99.8 0.75 0.63 1.00 0.75 0.22 Yield on 5/8 (lb) 793.7 79.1 0.74 0.57 0.75 1.00 0.42 Futures Price ($/lb) 0.5433 0.1984 0.36 0.23 0.22 0.42 1.00 Yield on 4/16 (lb) 808.9 108.8 1.00 0.84 0.77 0.62 0.16 Yield on 4/23 (lb) 818.4 100.6 0.84 1.00 0.75 0.64 0.28 Neutral Yield on 5/1 (lb) 825.8 86.2 0.77 0.75 1.00 0.75 0.01 Yield on 5/8 (lb) 824.5 68.0 0.62 0.64 0.75 1.00 0.19 Futures Price ($/lb) 0.5699 0.1872 0.16 0.28 0.01 0.19 1.00 Yield on 4/16 (lb) 799.1 99.8 1.00 0.97 0.67 0.60 0.13 Yield on 4/23 (lb) 790.7 85.3 0.97 1.00 0.73 0.68 0.20 La Nia Yield on 5/1 (lb) 793.9 90.6 0.67 0.73 1.00 0.97 0.13 Yield on 5/8 (lb) 809.3 94.1 0.60 0.68 0.97 1.00 0.08 Futures Price ($/lb) 0.4669 0.1851 0.13 0.20 0.13 0.08 1.00 PAGE 32 32 We further estimated the local basis defined in Equation (215). Th e monthly historical data on average cotton prices received by Florid a farmers from the USDA National Agricultural Statistical Service were collected (1979 to 2003) as the cotton local cash prices. By subtracting the futures price from the local cash price, we estimated the historical local basis. Using the Input Analyzer in the simulation software Arena, the best fitted distribution based on minimum square error method was a beta distribution with probability density function 0.13+0.15 BETA (2.76, 2.38). We calibrated the Gaussian copula based on the sample rank correlation coefficient Spearmans rho matrix for the three ENSO pha ses. For each ENSO phase, we sampled 2,000 scenarios of correlated random yi elds and futures price based on the Gaussian copula and the empirical distributions of yields and futures pr ice by Monte Carlo simulation. Furthermore, we simulated the basis and calculated the local cas h price from the futures price and basis. We assumed the futures commission and opportu nity cost of margin to be $0.003 per pound, the production cost of cotton was $464 per acre, and the subsidy for cotton in Florida was $349 per acre. Finally, the parameters of crop insurance are listed in Table 23. Table 23. Parameters of crop insurance ( 2004) used in the farm model analysis Crop Insurance Parameters Values APH premium 65%~75% $19.5/acre ~$38/acre CRC premium 65%~85% $24.8/acre~$116.9/acre Established Price for APH $0.61/lb Average yield 814 lb/acre Source: www.rma.usda.gov 2.4 Results and Discussion This section reports the results of optimal planting schedule and he dging strategy with crop insurance and futures contract for the three pred icted ENSO phases. In section 2.4.1 we assumed crop insurances were the only risk management tool for crop yield a nd price risk together with an PAGE 33 33 unbiased futures market3. In section 2.4.2 we considered bo th insurance and futures contracts were available and assumed the future market be ing unbiased. In section 2.4.3 we investigated the optimal decision under biased futures markets. 2.4.1 Optimal Production with Crop Insurance Coverage This section considers crop insurance as the only crop risk management tool. Since the indemnity of CRC depends on the futures price, we assume the futures market is unbiased, i.e., Ef F where F is the futures price in planting time and f is the random futures price in harvest time. Table 24 shows that the optimal insuran ce and production strategies for each ENSO phase with various 90%CVaR upper bounds ranged from $20,000 to $2,000 with increments of $2000. Remarks in Table 24 are summarized as fo llows. First, the ENSO phases affected the expected profit and the feasible region of the downside risk. The Neutral year has highest expected profit and lowest downsid e loss. In contrast, the La Nia year has lowest expected profit and highest downside loss. Second, th e 65%CRC and 70%CRC crop insurance policies are desirable to the optimal hedging strategy in all ENSO phases when 90%CVaR constraint is lower than a specific value depending on the ENSO phase In contrast, the APH insurance policies are not desirable for all ENSO phases and 90%CVaR upper bounds. Third, risk management can be conducted through changing the planting schedule. The last two rows associated with the Neutral phase shows that planting 100 acr es in date 3 provides a 90% CVaR of $6,000 that can be reduced to $8,000 by changing the planting schedule to 85 acres in date 3 a nd 15 acres in date 4. Last, changing the insurance c overage together with the pl anting schedule may reduce the downside risk. In the La Nia phase, planting 100 acres in date 4 provides a 90%CVaR of 3 Although only crop insurance contracts were considered, the unbiased futures market assumption is need since the indemnity of CRC depends on the futures price. PAGE 34 34 $4,000 that can be reduced to $10,000 by pur chasing a 65%CRC insurance policy and shifting the planting date from date 4 to date 1. Table 24. Optimal insurance and production strate gies for each climate scenario under the 90% CVaR tolerance ranged from $20,000 to $2,000 with increment of $2000 Optimal Planting Schedule ENSO Phases 90%CVaR Upper Bound Optimal Expected Profit Optimal Insurance Strategy Date1 Date2 Date3 Date4 El Nio <18000 infeasible 16000 28364 CRC70% 100 0 0 0 14000 to 4000 28577 CRC65% 100 0 0 0 >2000 28691 No 100 0 0 0 Neutral <20000 infeasible 18000 31149 CRC70% 0 0 100 0 16000 to 10000 31240 CRC65% 0 0 100 0 8000 31779 No 0 0 85 15 >6000 31793 No 0 0 100 0 La Nia <12000 infeasible 10000 to 6000 20813 CRC65% 100 0 0 0 >4000 21572 No 0 0 0 100 No = no insurance. Planting dates: Date1 = April 16, Date2 = April 23, Date3 = May 1, Date4 = May 8. Negative CVaR upper bounds represent profits. 2.4.2 Hedging with Crop Insurance and Unbiased Futures In this section, we consider managing the yiel d and price risk with crop insurance policies and futures contracts when the futures market is unbiased. Since the cr op yield is random, we define the hedge ratio of the futures contract as the hedge position in the futures contract divided by the expected production. The optimal solutio ns of the planting schedule, crop insurance coverage, and futures hedge ratio with vari ous 90%CVaR upper bounds ranged from $24,000 to $0 with increment of $4,000 for the three ENSO phases (Table 25). From Table 25, when the futures market is unbiased, the futures c ontract dominating all crop insurance policies is the onl y desirable risk management tool. The optimal hedge ratio increases when the upper bound of 90%CVaR decreases. This means that to achieve lower downside risk, higher hedge ratio is needed. Next, we compare the hedge ratio in different ENSO phases with the same CVaR upper bound, the La Nia phase has the highest optimal hedge ratio PAGE 35 35 Table 25. Optimal solutions of planting schedul e, crop insurance covera ge, and futures hedge ratio with various 90% CVaR upper bounds ranged from $24,000 to $0 with increment of $4,000 for the three ENSO phases Optimal Planting Schedule ENSO Phases 90%CVaR Upper Bound Optimal Expected Profit Optimal Insurance Strategy Expected Production Optimal Hedge Amount Optimal Hedge Ratio Date1 Date2 Date3 Date4 24000 infeasible 20000 28520 No 81422 57092 0.70 100 0 0 0 16000 28563 No 81422 4482 0.52 100 0 0 0 El Nio 12000 28605 No 81422 28895 0.35 100 0 0 0 8000 28646 No 81422 15238 0.19 100 0 0 0 4000 28686 No 81422 1726 0.02 100 0 0 0 <0 28691 No 81422 0 0 100 0 0 0 24000 infeasible 20000 31645 No 82482 49598 0.60 0 0 100 0 Neutral 16000 31695 No 82482 32721 0.40 0 0 100 0 12000 31744 No 82482 16547 0.20 0 0 100 0 8000 31792 No 82482 529 0.01 0 0 100 0 4000 31793 No 82482 0 0 0 0 100 0 16000 infeasible La Nia 12000 21438 No 80911 44812 0.55 0 0 0 100 8000 21516 No 80911 18705 0.23 0 0 0 100 4000 21572 No 80911 0 0 0 0 0 100 No = no insurance. Planting dates: Date1= April 16, Date2= April 23, Date3= May 1, Date4= May 8. Negative CVaR upper bounds represent profits. and the Neutral phase has the lowest one. Similar to the result in Table 24, the Neutral phase has the highest expected profit and th e lowest feasible downside loss. In contrast, the La Nia phase has the lowest expected profit a nd the highest feasible downside loss. Furthermore, we compare two risk management tools: insurance (in Tabl e 24) and futures (in Table 25). The futures contract provides higher exp ected profit under the same CVaR upper bound, as well as a larger feasible region associated with the CVaR c onstraint. Finally, the opt imal production strategy with futures hedge is to plant 100 acres in date 1 for the El Nio phase, in date 3 for the Neutral phase, and in date 4 for the La Nia phase. 2.4.3 Biased Futures Market In the section 2.4.2 we assumed the futures mark et is unbiased. However, the futures prices observed from futures market in the planting tim e may be higher or lower than the expected PAGE 36 36 futures price in the harvest time. This section examines the impact of biased futures prices on the optimal insurance and futures hedging decisions in the three ENSO phases. We first illustrate the optimal hedge strategies and the optimal planting schedule. Then, the performance of the optimal hedge and planting strategies is introduced by the efficient frontiers on the expected profit versus CVaR risk diagram. Table 26. Optimal insurance policy and futu res hedge ratio under bi ased futures prices Bias 10% 5% 0% 5% 10% ENSO Phases 90% CVaR Upper Bound Insurance Hedge Ratio Insurance Hedge Ratio Insurance Hedge Ratio Insurance Hedge Ratio Insurance Hedge Ratio 28000 x 26000 x No 1.03 24000 x No 1.00 No 1.24 22000 x No 0.83 No 1.18 No 1.41 20000 x 65%CRC 0.48 No 0.70 No 1.33 No 1.57 18000 65%CRC 0.45 65%CRC 0.29 No 0.61 No 1.47 No 1.72 El Nio 16000 70%CRC 0.20 65%CRC 0.15 No 0.52 No 1.60 No 1.87 14000 70%CRC 0.04 65%CRC 0.03 No 0.44 No 1.73 No 2.02 12000 65%CRC 0.01 65%CRC 0.07 No 0.35 No 1.86 No 2.16 10000 65%CRC 0.10 65%CRC 0.15 No 0.27 No 1.98 No 2.30 8000 65%CRC 0.20 65%CRC 0.24 No 0.19 No 2.11 No 2.44 6000 65%CRC 0.29 65%CRC 0.33 No 0.10 No 2.23 No 2.58 4000 65%CRC 0.39 65%CRC 0.41 No 0.02 No 2.35 No 2.73 2000 65%CRC 0.48 65%CRC 0.50 No 0.00 No 2.48 No 2.87 0 65%CRC 0.58 65%CRC 0.58 No 0.00 No 2.60 No 3.01 30000 x 28000 x No 1.06 26000 No 0.99 No 1.27 24000 x No 1.19 No 1.42 22000 x No 0.73 No 1.33 No 1.56 20000 x No 0.66 No 0.60 No 1.46 No 1.68 Neutral 18000 70%CRC 0.28 70%CRC 0.12 No 0.50 No 1.57 No 1.80 16000 70%CRC 0.10 70%CRC 0.02 No 0.40 No 1.68 No 1.92 14000 70%CRC 0.04 70%CRC 0.12 No 0.30 No 1.79 No 2.04 12000 70%CRC 0.16 No 0.20 No 0.20 No 1.89 No 2.15 10000 70%CRC 0.28 No 0.09 No 0.10 No 2.00 No 2.27 8000 70%CRC 0.39 No 0.00 No 0.01 No 2.10 No 2.38 6000 70%CRC 0.51 No 0.11 No 0.00 No 2.20 No 2.49 4000 70%CRC 0.62 No 0.21 No 0.00 No 2.30 No 2.59 2000 70%CRC 0.74 No 0.31 No 0.00 No 2.39 No 2.70 0 70%CRC 0.86 No 0.42 No 0.00 No 2.49 No 2.81 20000 x 18000 x No 1.06 16000 x No 1.04 No 1.23 14000 x No 0.75 No 1.20 No 1.37 12000 x No 0.67 No 0.55 No 1.33 No 1.51 La Nia 10000 No 0.56 No 0.46 No 0.39 No 1.45 No 1.64 8000 No 0.33 No 0.27 No 0.23 No 1.57 No 1.77 6000 No 0.11 No 0.09 No 0.08 No 1.68 No 1.90 4000 No 0.10 No 0.08 No 0.00 No 1.80 No 2.02 2000 No 0.30 No 0.25 No 0.00 No 1.91 No 2.15 0 No 0.49 No 0.41 No 0.00 No 2.02 No 2.27 x = infeasible. No = No insurance. Ne gative CVaR upper bounds represent profits. PAGE 37 37 Table 26 shows the optimal insurance policy and futures hedge ratio associated with different 90% CVaR upper bounds unde r biased futures prices for the three ENSO phases. When the futures price is unbiased or positive biased the futures contract is the only desirable instrument for crop risk management and all in surance policies are not needed in the optimal hedging strategy. On the other hand, when the futures price is negative biased, the optimal hedging strategy includes 65%CRC (or 70% CRC in so me cases) insurance policies and futures contract in the El Nio phase for all feasible 90%CVaR upper bounds In the Neutral phase, the optimal hedging strategy consists of the 70%C RC insurance policy and futures contract for all CVaR upper bounds with the deep negative biased (10%) futures price and for the CVaR upper bounds between 18000 and 14000 with the negative bi ased (5%) futures price. In addition, no insurance policy is desirable under the La Nia phase. Mahul (2003) showed that the hedge ratio c ontains two parts: a pure hedge component and a speculative component. The pure hedge component refers to the hedge ratio associated with unbiased futures price. A positive biased future s price induces the farmer to select a long speculative position and a negative biased future s price implies a short speculative position. Therefore, the optimal futures hedge ratio under positive (negative) biased futures prices should be higher (lower) than that under the unbiased futures price. However, the optimal hedge ratios in Table 26 do not agree with the conclusion when the futures price is negative biased. We use the optimal hedge ratios in the La Nia phase to illustrate how optimal futures hedge ratios change corresponding to the bias of the futures price4. Figure 22 shows the bias of futures price versus the optimal hedge ratio curves associat ed with different 90% CVaR upper bounds in the La Nia phase. 4 Since the optimal hedging strategy in the La Nia phase contains only futures contracts, the optimal hedge ratios are compatible. PAGE 38 38 When the CVaR constraint is not strict (i .e. the upper bound of 90%CVaR equals zero) the optimal hedge ratio curve follows the pattern cl aimed in Mahul (2003). The hedge ratio increases (decreases) with the positive (negative) bias of fu tures price in a decreasing rate. However, when the CVaR constraint becomes stricter (i.e., the CVaR upper bound equals $8,000), the optimal hedge ratio increases not only with the positive bias but also with the negative one. It is because the higher negative bias of futures price implies a h eavier cost (loss) is involved in futures hedge. It makes the CVaR constraint become stricter such that a higher pure hedge component is required to satisfy the constraint. The net ch ange of the optimal hedge ratio, including an increment in the pure hedge component and a decrement in the speculative component, depends on the loss distribution, the CVaR upper bound, and the bias of futures price. Figure 22. Bias of futures price versus the optim al hedge ratio curves associated with different 90% CVaR upper bounds in the La Nia phase Table 27 reported the optimal pl anting schedules for different biases of futures price in the three ENSO phases. For the El Nio phase, the optimal planting schedule (i.e., planting 100 acres in date 1) was not affected by the biases of futures price a nd 90%CVaR upper bounds. 18000 Optimal Hedge Ratio 1.00 0.50 0.00 0.50 1.00 1.50 2.00 2.50 10% 5% 0% 5% 10%Bias of Futures Price 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 PAGE 39 39 Table 27. Optimal planting schedule for different biases of futures pr ice in ENSO phases 90%CVaR El Nio Neutral La Nia Bias upper bound Date 1 Date 2 Date 3 Date 4 Date 1 Date 2 Date 3 Date 4 Date 1 Date 2 Date 3 Date 4 20000 x x 18000 100 0 0 0 0 0 97 3 16000 100 0 0 0 0 0 100 0 14000 100 0 0 0 0 0 100 0 12000 100 0 0 0 0 0 100 0 x 10% 10000 100 0 0 0 0 0 100 0 0 0 0 100 8000 100 0 0 0 0 0 100 0 0 0 0 100 6000 100 0 0 0 0 0 100 0 0 0 0 100 4000 100 0 0 0 0 0 100 0 0 0 0 100 2000 100 0 0 0 0 0 100 0 0 0 0 100 0 100 0 0 0 0 0 100 0 0 0 0 100 22000 x x 20000 100 0 0 0 0 0 39 61 18000 100 0 0 0 0 0 100 0 16000 100 0 0 0 0 0 100 0 14000 100 0 0 0 0 0 100 0 x 12000 100 0 0 0 0 0 44 56 0 0 0 100 5% 10000 100 0 0 0 0 0 46 54 0 0 0 100 8000 100 0 0 0 0 0 85 15 0 0 0 100 6000 100 0 0 0 0 0 85 15 0 0 0 100 4000 100 0 0 0 0 0 92 8 0 0 0 100 2000 100 0 0 0 0 0 100 0 0 0 0 100 0 100 0 0 0 0 0 93 7 0 0 0 100 24000 x x 22000 100 0 0 0 0 0 100 0 20000 100 0 0 0 0 0 100 0 18000 100 0 0 0 0 0 100 0 16000 100 0 0 0 0 0 100 0 x 14000 100 0 0 0 0 0 100 0 0 0 0 100 0% 12000 100 0 0 0 0 0 100 0 0 0 0 100 10000 100 0 0 0 0 0 100 0 0 0 0 100 8000 100 0 0 0 0 0 100 0 0 0 0 100 6000 100 0 0 0 0 0 100 0 0 0 0 100 4000 100 0 0 0 0 0 100 0 0 0 0 100 2000 100 0 0 0 0 0 100 0 0 0 0 100 0 100 0 0 0 0 0 100 0 0 0 0 100 x = infeasible. Date 1 = 16 Apr, Date 2 = 23 Apr, Da te 3 = 1 May, Date 4 = 8 May. Negative CVaR upper bounds represent profits. PAGE 40 40 Table 27. Optimal planting schedule for differe nt biases of futures price in ENSO phases (contd) 90%CVaR El Nio Neutral La Nia Bias upper bound Date 1 Date 2 Date 3 Date 4 Date 1 Date 2 Date 3 Date 4 Date 1 Date 2 Date 3 Date 4 28000 x 26000 x 0 0 30 70 24000 100 0 0 0 0 0 47 53 22000 100 0 0 0 0 0 59 41 20000 100 0 0 0 0 0 68 32 18000 100 0 0 0 0 0 82 18 x 5% 16000 100 0 0 0 0 0 92 8 38 0 0 62 14000 100 0 0 0 0 0 99 1 33 0 0 67 12000 100 0 0 0 0 0 100 0 35 0 0 65 10000 100 0 0 0 0 0 100 0 35 0 0 65 8000 100 0 0 0 0 0 100 0 33 0 0 67 6000 100 0 0 0 0 0 100 0 34 0 0 66 4000 100 0 0 0 0 0 100 0 38 0 0 62 2000 100 0 0 0 0 0 100 0 39 0 0 61 0 100 0 0 0 0 0 100 0 37 0 0 63 30000 x 28000 x 0 0 31 69 26000 100 0 0 0 0 0 45 55 24000 100 0 0 0 0 0 54 46 22000 100 0 0 0 0 0 67 33 20000 100 0 0 0 0 0 75 25 x 18000 100 0 0 0 0 0 87 13 33 17 0 50 10% 16000 100 0 0 0 0 0 96 4 45 0 0 55 14000 100 0 0 0 0 0 100 0 47 0 0 53 12000 100 0 0 0 0 0 100 0 47 0 0 53 10000 100 0 0 0 0 0 100 0 50 0 0 50 8000 100 0 0 0 0 0 100 0 54 0 0 46 6000 100 0 0 0 0 0 100 0 56 0 0 44 4000 100 0 0 0 0 0 100 0 59 0 0 41 2000 100 0 0 0 0 0 100 0 61 0 0 39 0 100 0 0 0 0 0 100 0 63 0 0 37 x = infeasible. Date 1 = 16 Apr, Date 2 = 23 Apr, Da te 3 = 1 May, Date 4 = 8 May. Negative CVaR upper bounds represent profits. PAGE 41 41 Efficient frontier in El Nino phase under various biased futures price25000 29000 33000 37000 41000 45000 260002200018000140001000060002000 90%CVaR upper boundExpected profit 10% 5% 0% 5% 10% Efficient frontier in Neutral phase under various biased futures price25000 29000 33000 37000 41000 45000 2800024000200001600012000800040000 90%CVaR upper boundExpected profit 10% 5% 0% 5% 10% Efficient frontier in La Nina phase under various biased futures price18000 20000 22000 24000 26000 28000 30000 180001600014000120001000080006000400020000 90%CVaR upper boundExpected profit 10% 5% 0% 5% 10% Figure 23. The efficient frontie rs under various biased futures price. (A) El Nio year. (B) Neutral year. (C) La Nia year. (A) (B) (C) PAGE 42 42 For the Neutral phase, however, the optimal planting strategy was to plant on date 3 and date 4 depending on the 90%CVaR upper bounds. Mo re exactly, the date 3 is the optimal planting date for all risk tolerances under unbias ed futures market. When futures prices are positive biased, the lower the 90%CVaR upper bounds (i.e., the stricter the CVaR constraint) was, the more planting acreages moved to date 4 from date 3. This result was based on the fact that there was no insurance cove rage involved in the optimal he dging strategy. When the futures prices are negative biased, the optimal planting sc hedule had the same pattern as positive biased futures markets but was affected by the existenc e of insurance coverage in the optimal hedging strategy. For example, when the 90%CVaR upper bounds were within the range of $8,000 and $14,000 under a 5% biased futures price, the optim al planting acreage in date 4 went down to zero due to a 75%CRC in the optimal hedging stra tegy. For the La Nia phase, the optimal planting schedule was to plant 100 acres in date 4 when future prices were unbiased or negative biased. When future price was negative biased, the stricter CVaR constraint was, the more planting acreage shifted from date 4 to date 1. Fo r deep negative biased futures price together with strict CVaR constraint (i.e., 10% bi ased futures price and $18,000 90%CVaR upper bound), the optimal planting schedule included date 1, date 2, and date 4. Figure 23 shows meanCVaR efficient frontiers associated with various biased futures prices for three ENSO phases. With the effici ent frontiers, the farmer may make the optimal decision based on his/her downside risk toleranc e and trade off between expected profit and downside risk. The three graphs show that the Neutral phase has highest expected profit and lowest feasible CVaR upper bound. In contrast, the La Nia phase has the lowest expected profit and highest feasible CVaR upper bound. The pattern of the efficient frontiers in the three graphs is the same. The higher positive bias of futures price is, the higher expected profit would be PAGE 43 43 provided. However, the higher negative bias of futures price provides a higher expected profit under a looser CVaR constraint and a lower expect ed profit under a stri cter CVaR constraint. 2.5 Conclusion This research proposed a meanCVaR model for investigating the optimal crop planting schedule and hedging strategy when the crop insurance and futures contracts are available for hedging the yield and price risk. Due to the lin ear property of CVaR, th e optimal planting and hedging problem could be formulated as a mixed 01 linear programming problem that could be efficiently solved by many commercial solver s such as CPLEX. The meanCVaR model is powerful in the sense that the model inherits the advantage of the return versus risk framework (Markowitz, 1952) and further utilizes CVaR as a (downside) risk measure that can cope with general loss distributions. Compar ed to using utility functions for modeling risk aversion, the meanCVaR model provides an intuitive way to define risk. In additi on, a problem without nonlinear side constraints could be formulated linearly under the meanCVaR framework, which could be solved more efficiently comparing to the nonlinear formulation from the utility function framework. A case study was conducted using the data of a representative cott on producer in Jackson County in Florida to examine the optimal crop planting schedule and risk hedging strategy under the three ENSO phases. The eligible hedging inst ruments for cotton include futures contracts and three types of crop insurance policies: APH, CRC, and CAT. We first analyzed the best production and risk hedging problem with three types of insurance policies. The result showed that 65%CRC or 70%CRC would be the optimal insu rance coverage when the CVaR constraint reaches a strict level. Furthermore, we ex amined the optimal hedging strategy when crop insurance policies and futures co ntracts are available. When futures price are unbiased or positive biased, the optimal hedging strategy onl y contains futures contracts and all crop PAGE 44 44 insurance policies are not desirable. However, when the futures price is negative biased, the optimal hedging strategy depends on the ENSO phases. In the El Ni o phase, the optimal hedging strategy consists of the 65%CRC (o r 70%CRC for some CVaR upper bounds) and futures contracts for all CVaR upper bound values. In the Neutral phase, when futures price is deep negative biased (10%), the optimal he dging strategy consists of the 70%CRC and futures contracts for all CVaR upper bound values. Under a 5% negative biased futures price, optimal hedging strategy contains the 70%CRC and futu res contracts when the CVaR upper bound is within the range of $18,000 and $14,000. Othe rwise, the optimal hedging strategy contains only futures contract. In the La Nia phase, the optimal hedging strategy contains only futures contract for all CVaR upper bound values and all biases of futures prices between 10% and 10%. The optimal futures hedge ratio increase s with the increasing CVaR upper bound when the insurance strategy is unchanged. For a fixe d CVaR upper bound, the optimal hedge ratio increases when the positive bias of futures price increases. However, when the futures price is negative biased, the optimal hedge ratio de pends on the value of CVaR upper bound. The case study provides some insight into how planting schedule incorporated with insurance and futures hedging may manipulate the dow nside risk of a loss distribution. In our model, we used a static futures hedging strate gy that trades the hedge position in the planting time and keeps the position until the harvest time. A dynamic futures hedging strategy may be considered in the future research. The small sample size for the El Nio and La Nia phases may limit the case study results. In addition, we a ssumed the cost of futures contract as the commission plus an average intere st foregone for margin deposit a nd the risk of daily settlement that may require a large amount of cash for ma rgin account was not considered. It may reduce the value of futures hedging for riskaverse farmers. PAGE 45 45 CHAPTER 3 EFFCIENT EXECUTION IN THE SECONDARY MORTGAGE MARKET 3.1 Introduction Mortgage banks (or lenders) originate mortga ges in the primary market. Besides keeping the mortgages as a part of the portfolio, a lender may sell the mo rtgages to mortgage buyers (or conduits) or securitize th e mortgages as mortgagebacked se curities (MBSs) through MBS swap programs in the secondary market. In the United States, three governmentsponsored enterprises (GSEs) (Fannie Mae, Freddie Mac, and Ginnie Mae) provide MBS swap programs in which mortgage bankers can deliver their mortgages into appropriate MBS pools in exchange for MBSs. In practice, most mortgage ba nkers prefer to participate in the secondary market based on the following reasons. First, mortgage banks woul d get funds from secondary marketing and then use the funds to originate more mortgages in the primary market and earn more origination fees. Second, the value of a mortgage is risky and de pends on several sources of uncertainties, i.e., default risk, interest rate risk, and prepayme nt risk. Mortgage bankers could reduce risks by selling or securitizing mortgages in the seconda ry market. More exactly, when mortgages are sold as a whole loan, all risks would be transferred to mortgage buyers. On the other hand, when mortgages are securitized as MBSs, the risky ca sh flows of mortgages are split into guarantee fees, servicing fees, and MBS coupon payments, which belong to MBS issuers, mortgage servicers, and MBS investors, re spectively. In this case, mortgage bankers are exposed only to risk from retaining the servic ing fee and other risky cash flow s are transferred to different parties. A significant task faced by mortgage bankers attempting to profit from the secondary market is efficient execution. The challenge of effi cient execution is to sell or securitize a large PAGE 46 46 number of heterogeneous mortgages in the seco ndary market in order to maximize expected revenue through complex secondary marketing func tionality. In addition, to deal with the uncertain cash flows from the retained servicing fee, the balance between mean revenue and risk is also an important concern for mortgage bankers. In this chapter, we develop a stochastic optimization model to perform an efficient execution that considers secondary marketing functionality, including loanlevel efficient execution, guarantee fee buyup or buydown, servic ing retain or release, and excess servicing fee. Further, we employ Conditional ValueatRisk (CVaR), proposed by Rockafellar and Uryasev (2000), as a risk measure in the effi cient execution model that maximizes expected revenue under a CVaR constraint. By solving th e efficient execution problem under different risk tolerances specified by a CVaR constraint an efficient frontier could be found. A great deal of research has focused on mo rtgage valuation (Kau, Keenan, Muller, and Epperson (1992); Kau (1995); Hilliard, Kau and Sl awson (1998); and Downing, Stanton, and Wallace (2005)), MBS valuation (Schwartz a nd Torous (1989); Stanton (1995); Sugimura (2004)), and mortgage servicing right valuation (Aldrich, Gree nberg, and Payner (2001); Lin, Chu, and Prather (2006)). However, academic liter ature addressing topics of mortgage secondary marketing is scant. Hakim, Rashidian, and Rose nblatt (1999) addressed the issue of fallout risk, which is an upstream secondary marketing problem To the best of our knowledge, we have not seen any literature focusing on efficient execution. The organization of this chapter is as follows : Section 2 discusses mortgage securitization. We describe the relationship between MBS market participants and introduce the Fannie Mae MBS swap program. Section 3 presents our model development. Section 4 reports our results, and the final section presents our conclusions. PAGE 47 47 3.2 Mortgage Securitization Mortgage bankers may sell mortgages to condui ts at a price higher than the par value5 to earn revenue from the whole loan sales. Howeve r, for lenders who possess efficient execution knowledge, mortgage securitization through MBS swap programs of GSEs may bring them higher revenue than the whole loan sale strategy. In this research, we consider passthrough MBS swap programs provided by Fannie Mae (FNMA). To impose considerations of MBS swap programs of other GSEs is straightforward. In this section, we describe the relationship between partic ipants in the passthrough MBS market and detail the procedure of mortga ge securitization through a MBS swap program. Participants in the MBS market can be categ orized into five groups: borrowers, mortgage bankers, mortgage servicers, MBS issuers, and mortgage investors. The relationship between these five participants in the passthr ough MBS market is shown in Figure 31. In Figure 31, solid lines show cash flows betw een participants and dashed lines represent mortgage contracts and MBS inst ruments between them. Mortgage bankers originate mortgage loans by signing mortgage contracts with borro wers who commit to making monthly payments in a fixed interest rate known as the mortgage no te rate. To securitize those mortgages, mortgage bankers deliver the mortgages in to an MBS swap in exchange for MBSs. Further, mortgage bankers sell the MBSs to MBS investors and receive MBS prices in return. MBS issuers provide MBS insurance to protect the MBS investors against default losses and charge a base guarantee fee. The base guarantee fee is a fi xed percentage, known as the 5 Mortgage bankers underwrite mortgages at a certain mortgage note rate. The par value is the value of the mortgage when the discount interest rate equals the mortgage note rate. In other words, the par value of a mortgage is its initial loan balance. PAGE 48 48 Figure 31. The relationship between participan ts in the passthrough MBS market. Mortgage bankers originate mortgage loans by signing mortgage contracts with borrowers who commit to making monthly payments with a fixed interest rate known as the mortgage note rate. To securitize those mortgages, mortgage bankers deliver the mortgages into an MBS swap in exchange for MBSs. Further, mortgage bankers sell the MBSs to MBS investors and receive MBS prices in return. The MBS issuer provides MBS insurance and charges a base guarantee fee. Mortgage servicers provide mortgage servicing and a base serv icing fee is disbursed for the servicing. Both fees are a fixed percentage (servici ng fee rate or guarant ee fee rate) of the outstanding mortgage balance and declin e over time as the mortgage balance amortizes. Deducting guarantee fees and serv icing fees from mortgage payments, the remaining cash flows that passthrough to the MBS investors are known as MBS coupon payments with a rate of return e qual to the mortgage note rate minus the servicing fee rate minus the guarantee fee rate. guarantee fee rate, of the outstanding mortgage balance, and which declines over time as the mortgage balance amortizes. Mortgage bankers negotiate the base guarant ee fee rate with Fannie Mae and have the opportunity to buydown or buyup the guara ntee fee. When lenders buydown the guarantee fee, the customized guarantee fee rate is equal to the base guarantee fee rate minus the guarantee fee buydown spread. Further, lenders have to ma ke an upfront payment to Fannie Mae. On the other hand, the buyup guarantee fee allows lenders to increase the guarantee fee rate from the base guarantee fee rate and receive an upfront payment from Fannie Mae. For example, if a lender wants to include a 7.875% mortgage with a 0.25% base guarantee fee and a 0.25% base Mortgage banker Borrowers Loan Mortgage payment (charges servicing fee ) Mort g a g e service r (charges guarantee fee ) MBS issuer ( FNMA ) MBS investors Guarantee fee ( MBS swap)(MBS coupon rate = mortgage note rate servicing fee rate guarantee fee rate) (pay mortgage note rate ) MBS Mortgage MBS Mortgage MBS coupon payment MBS price PAGE 49 49 servicing fee in a 7.5% passthrough MBS (Figure 32), the lender can buydown the guarantee fee rate to 0.125% from 0.25% by paying Fannie Mae an upfront amount equal to the present value of the cash flows of the 0.125% difference a nd maintaining the 0.25% base servicing fee. Figure 32. Guarantee fee buydown. A lender ma y include a 7.875% mortgage in a 7.5% passthrough MBS by buyingdown the guarantee fee to 0.125% from 0.25% and maintaining the 0.25% base servicing fee. If a lender chooses to include an 8.125% mortgage in the 7.5% passthrough MBS (Figure 33), the lender can buyup the guarantee fee by 0.125% in return for a present value of the cash flows of the 0.125% difference. The buydown a nd buyup guarantee fee feat ures allow lenders to maximize the present worth of revenue. Figure 33. Guarantee fee buyup. A lender may include an 8.125% mortgage in a 7.5% passthrough MBS by buyingup the guarantee fee to 0.375% from 0.25% and maintaining the 0.25% base servicing fee. Mortgage servicers provide mortgage serv ices, including collecting monthly payments from borrowers, sending payments and overdue notices, and maintaining the principal balance report, etc. A base servicing f ee is disbursed for the servicing, which is a fixed percentage, known as the base servicing fee rate, of the outstanding mortgage balance, and which declines over time as the mortgage balance amortizes. Mort gage bankers have the se rvicing option to sell Mortgage Note Rate MBS Coupon Rate Servicing Fee Guarantee Fee 0.25% 0.375% 7.5% 8.125% MBS Coupon Rate Servicing Fee Guarantee Fee 0.25% 0.125%7.5% Mortgage Note Rate 7.875% PAGE 50 50 the mortgage servicing (bundled with the base serv icing fee) to a mortgage servicer and receive an upfront payment from the servicer or retain the base servicing fee an d provide the mortgage servicing. Deducting guarantee fees and servicing fees from mortgage payments, the remaining cash flows that passthrough to the MBS invest ors are known as MBS coupon payments, which contain a rate of return know n as the MBS coupon rate (or pa ssthrough rate), equal to the mortgage note rate minus the servicing fee rate minus the guarantee fee rate. Fannie Mae purchases and swaps more than 50 t ypes of mortgages on the basis of standard terms. This research focuses on passthrough MBS swaps of 10, 15, 20, and 30year fixedrate mortgages. Mortgages must be pool ed separately by the time to maturity. For instance, 30year fixedrate mortgages are separa ted from 15year fixedrate ones For each maturity, Fannie Mae provides different MBS pools characterized by MBS coupon rates that generally trade on the half percent (4.5%, 5.0%, 5.5%, etc. ). Mortgage lenders have the option to deliver individual mortgages into one of these el igible MBS pools, which allows lenders to maximize revenue. Further, the mortgage note rate must suppor t the MBS coupon rate plus the servicing fee rate plus the guarantee fee rate. Therefore, when securitizing a mortgage as an MBS, mortgage bankers have to manipulate the se rvicing fee rate and guarantee fee rate so that Equation (31) is satisfied. Mortgage Note Rate = Servicing Fee Rate + Guarantee Fee Rate + MBS Coupon Rate. (31) Mortgage bankers could retain an excess servic ing fee from the mortgage payment, which, like the base servicing fee, is a fixed percentage, known as the ex cess servicing fee rate, of the outstanding mortgage balance and which declines over time as the mortgage balance amortizes. In Equation (32), the excess se rvicing fee rate is equal to the excess of the mortgage note rate PAGE 51 51 over the sum of the MBS coupon rate, customized gua rantee fee rate, and ba se servicing fee rate. In other words, the servicing fee rate in Equation (31) consists of the base servicing fee rate and the excess servicing fee rate. Excess Servicing Fee Rate = Mortgage Note Rate MBS Coupon Rate Guarantee Fee Rate Base Servicing Fee Rate. (32) In the example shown in Figure 33, the exce ss servicing fee may be sold to Fannie Mae by buyingup the guarantee fee. Another option for mortgage bankers is to retain the excess servicing fee in their portfolio and to receive cash flows of the excess servicing fee during the life of the mortgage. The value of the excess servicing fee is equal to the present value of its cash flows. This value is stochastic since borrowers have the option to terminate mortgages before maturity and the interest rate used to discount the future cas h flows is volatile. Therefore, efficient execution becomes a stochastic optim ization problem. Similar to the guarantee fee buyup and buydown features, the excess servicing f ee allows lenders to maximize the expected revenue. 3.3 Model Efficient execution is a central problem of mortgage sec ondary marketing. Mortgage bankers originate mortgages in the primary market and execute the mortgages in the secondary market to maximize their revenue through differe nt secondary marketing strategies. In the secondary market, each mortgage can be executed in two ways, either sold as a whole loan, or pooled into an MBS with a specific coupon rate. When mortgages are allocated into an MBS pool, we consider further the guarantee fee buyup/buydown option, the mortgage servicing retain/release option, and excess servicing fee to maximize the total revenue. Based on secondary marketing strategies, mortga ge bankers may retain the base servicing fee and the excess servicing fee when mortgages are securitized. The value of the retained PAGE 52 52 servicing fee is random and affected by the uncer tainty of interest ra te term structure and prepayment. Therefore, a stochastic optimiza tion model is developed to maximize expected revenue under a risk tolerance and an efficient frontier can be found by optimizing expected revenue under different ri sk tolerances specified by a risk measure. 3.3.1 Risk Measure Since Markowitz (1952), vari ance (and covariance) has become the predominant risk measure in finance. However, the risk measure is suited only to the case of elliptic distributions, like normal or tdistributions with finite va riances (Szeg (2002)). The other drawback of variance risk measure is that it measures both upside and downside risks. In practice, finance risk managers are concerned only with the downside risk in most cases. A popular downside risk measure in econom ics and finance is ValueatRisk (VaR) (Duffie and Pan (1997)), which measures percentile of loss distri bution. However, as was shown by Artzner et al. (1999), VaR is illbeha ved and nonconvex for general distribution. The other disadvantage of VaR is th at it only considers risk at percentile of lo ss distribution and does not consider how much worse the tail (the worst 1percentage of scenarios) could be. To address this issue, Rockafellar and Uryasev ((2000) and (2002)) proposed Conditional ValueatRisk (CVaR), which is the mean value of tail of loss distribution. It has been shown that CVaR satisfies the axioms of coherent ri sk measures proposed by Ar tzner et al. (1999) and has desirable properties. Most importantly, Rockafellar and Urya sev (2000) showed that CVaR constraints in optimization problems can be expressed by a set of linear constraints and incorporated into prob lems of optimization. PAGE 53 53 This research uses CVaR as the measure of risk in developing the efficient execution model that maximizes expected revenue under a CV aR constraint. Thanks to CVaR, an efficient execution model could be formulated as a mixed 01 linear programming problem. It is worth mentioning that the prices of MB Ss, prices of whole loan sale mortgages, upfront payment of released servicing, and upfront payment of guarantee fee buyup or buydown are deterministic numbers that can be obser ved from the secondary market. However, the revenue from retained base servicing fees and ex cess servicing fees are e qual to the present value of cash flows of the fees. Because of the ra ndomness of interest rate term structure and prepayment, the revenue from the fees is varied with different scenarios. Lenders could simulate the scenarios based on their own interest rate model and prepay ment model. This research assumes the scenarios are given input date. 3.3.2 Model Development In this subsection, we present the stochast ic optimization model. The objective of the model is to maximize the revenue from secondary marketing. Four s ources of revenue are included in the model: revenue from MBSs or whol e loan sale, revenue from the base servicing fee, revenue from the excess servicing fee, and revenue fr om the guarantee fee buyup/buydown. (1) Revenue from MBSs or whole loan sale: ) (1m w m cz z f M m m w m w C c m c t c mz P z P Lm m11, (33) where M = total number of mortgages, m = index of mortgages (m = 1, 2, M), mL = loan amount of mortgage m, mC = number of possible MBS coupon rates of mortgage m, c = index of MBS coupon rate, PAGE 54 54 m t cP = price of MBS with coupon rate index c and maturity of mt, mt = maturity of mortgage m, m wP = whole loan sale price of mortgage m, m cz = otherwise, 0, index rate coupon with MBS into pooled is m mortgage if 1cm wz = otherwise. 0, loan, whole a as sold is mortgage if 1m Equation (34) enforces that each mortgage c ould be either sold as a whole loan or delivered into a specific MBS pool. 11mC mm cw czz (34) If mortgage m is securitized as an MBS with coupon rate index c then 1m cz, 0m cz for all c c and 0m wz, and the revenue from mortgage m equals m c t c mz P Lm On the other hand, if mortgage m is sold as a whole loan, then 0m cz for all c, and 1m wz, and the revenue from mortgage m equals m w m wz P The total revenue from the M mortgages is shown in Equation (33). (2) Revenue from the base servici ng fee of securitized mortgages: ) (2 m sbr m sboz z f M m m sbr m s K k mk sr m sb m k M m m sbo m mz c K R L p z B L11 1, (35) where m sbrz otherwise, 0 retained, is mortgage of servicing the if 1m m sboz otherwise, 0 sold, is mortgage of servicing the if 1 m m B = base servicing value of mortgage m, kp = the probability of scenario k, K = number of scenarios, m sc = servicing cost of mortgage m, PAGE 55 55 mk srK = retained servicing fee multiplier of mortgage m under scenario k. The retained servicing multipliers mk srK for scenario k could be generated by simulation. Mortgage bankers first simulate the random disc ounted cash flows of the retained servicing fee by using their own interest rate model and pr epayment model. Then, a retained servicing multiplier mk srK could be found associated with each scen ario. In this res earch, we treat the retained servicing multiplier mk srK as input data. Details of how to get the retained servicing multiplier mk srK is beyond the scope of this research. Equation (36) enforces that when a mortgage is securitized, the serv icing of the mortgage can be either released or retained, and the revenue from mortgage servicing exists only if the mortgage is securitized as an MBS in stead of being sold as a whole loan. 1mmm wsbosbrzzz (36) More exactly, if mortgage m is sold as a whole loan, then 1m wz, 0 m sbr m sboz z, and the revenue from the base servicing fee equals zero. On the other hand, if mortgage m is securitized as an MBS and the servicing of mortgage m is sold, then 1m sboz, 0m sbrz, 0m wz, and the upfront payment from the mortgage servicer equals m sbo m mz B L ; otherwise, 1 m sbrz, 0 m sboz, 0 m wz, and the expected revenue from the released servicing equals the expected revenue from base servicing fee K k m sbr mk sr m sb m kz K R L p1 minus servicing cost m sbr m sz c The total revenue from the M mortgages is expressed in Equation (35). (3) Revenue from the excess servicin g fee of securitized mortgages: ) (3 m serr f11 MK mmkm ksrser mkpLKr, (37) where m serr= retained excess servic ing fee of mortgage m. PAGE 56 56 If mortgage m is securitized as an MBS, the mortgage generates expected revenue K k m ser mk sr m kr K L p1 from retaining the excess servicing fee. The total revenue from the M mortgages is shown in Equation (37). (4) Revenue from the guarantee fee buyup/ buydown of securitized mortgages: ) (4m gd m gur r f11 MM mmmmmm ugudgd mmKLrKLr, (38) where m gur = guarantee fee buyup spread of mortgage m, m gdr = guarantee fee buydown spread of mortgage m, m uK = guarantee fee buyup multiplier of mortgage m, m dK = guarantee fee buydown mu ltiplier of mortgage m. Guarantee fee buyup and buydown multipliers (or ratios) m uK and m dK, announced by Fannie Mae, are used to calculate the upfront payment of a guarantee fee buyup and buydown. Lenders buyup the guarantee fee of mo rtgage m to receive an upfront m gu m m ur L K from Fannie Mae. On the other hand, they can buydown th e guarantee fee of mortgage m and make an upfront payment m gd m m dr L K to Fannie Mae. The total revenue from the M mortgages is shown in Equation (38). Guarantee fee buyup/buydown and retaining th e excess servicing fee are considered only when mortgage m is securitized. Equation (39) enforces m serr, m gur, and m gdr to be zero when mortgage m is sold as a whole loan. 1mmmm wgugdserzrrr (39) From equation (31), when mortgages are secu ritized as MBSs, the mortgage note rate has to support the MBS coupon rate, se rvicing fee rate, and guarant ee fee rate. Equation (310) places a mathematic expression in the restriction. PAGE 57 57 1mC mmmmmmm ccgugdsernsbgb c R zrrrRRR, (310) where cR = MBS coupon rate related to index c, m nR = note rate of mortgage m, m sbR = base servicing fee of mortgage m, m gbR = base guarantee fee of mortgage m, Next, we introduce the CVaR constraint CVaRLU (311) where L is the loss function, is the percentile of CVaR, and U is the upper bound of CVaR losses. Equation (311) restricts the average of tail of loss distribution to be less than or equal to U. In other words, the av erage losses of the worst 1percentage of scenarios should not exceed U. It is worth mentioning that CVaR is defined on a loss distribution. Therefore, we should treat revenue as negative losses when we use CVaR constraint in maximum revenue problem. Rockafellar & Uryasev (2000) proposed that CVaR constraints in optimization problems can be expressed by a set of linear constraints. 1 1(1)K kk k p zU (312) 1,...,kkzLkK (313) 01,...,kzkK (314) 1111 111 .mMCMM mmmmmmmmmmmkm kccwholewsbosbsrsbr mcmm MMM mmkmmmmmmm srserugudgd mmmLLPzPzLBzLRKz LKrKLrKLr (315) PAGE 58 58 wherekLis loss value in scenario k, and and kz are real variables. Users of the model could specify their risk preference by selecting the value of and U. More constraints could be considered in the model based on the mortgage bankers preference. For instance, mortgage bankers may wa nt to control the retained excess servicing fee based on the future capital demand or risk cons ideration. The excess servicing fee could be limited by an upper bound in three leve ls, including aggregat e level, group level, and loan level. Constraint (316) limits average excess se rvicing fee across all mortgages by an upper bound a seU. 11(1)MM mmamm sesew mmLrULz (316) We further categorized mortgages into groups a ccording to the year to maturity. Constraint (317) limits the grouplevel average excess servicing fee by an upper bound j seU for each group j. (1)mmjmm sesew mjmjLrULz (317) In addition, constraint (318) restrict the loan level exce ss servicing fee to an upper bound m seU. 0mm s erserU (318) Furthermore, constraints (319) and (320) impose an upper bound m guU and m gdU in the guarantee fee buyup and buydown spreads, respectively. 0mm g ugurU (319) 0mm g dgdrU (320) PAGE 59 59 Those upper bounds are determined by the rest rictions of an MBS swap program or sometimes by the decision of mortgage bankers For example, the maximum guarantee fee buydown spread accepted by Fannie Mae is the base guarantee fee rate. Finally, we impose nonnegativity constraints ,,0mmm sergugdrrr (321) and binary constraints ,,,{0,1}1,2,....,mmmmm cwholesbrsbozzzzmC (322) The notations and model formulation are summarized in the Appendix. 3.4 Case Study In this section, a case study is conducted. First, we presen t the data set of mortgages, MBSs, base servicing fee and guarantee fee multip liers, and scenarios of retained servicing fee multiplier. Next, we introduce the solver that is used to solve the mixed 01 linear programming problem. Then, we show the results including efficient frontier and sensitivity analysis. 3.4.1 Input Data In the case study, we consider executing 1, 000 fixedrate mortgage s in the secondary market. For each mortgage, the data includes year s to maturity (YTM), loan amount, note rate, and guarantee fee. Table 31 summarizes the data on mortgages. Mortgages are categorized into four groups according to YTM. For each group, th e table shows number of mortgages, and the minimum, mean, and maximum value of loan amount, note rate, and guarantee fee. MBS pools are characterized by the MBS coupon rate and YTM. In this case study, we consider 13 possible MBS coupon rates from 3.5% to 9.5%, increasing in increments of 0.5%, and four different YTM: 10, 15, 20, and 30 year s. There are a total of 50 MBS pools since the MBS pool with a 3.5% coupon rate is not availabl e for 20year and 30year mortgages. Table 32 shows the prices of MBSs for different MBS pools. PAGE 60 60 This case study assumes the base servicing rate is 25 bp (1bp = 0.01%) for all mortgages. The base servicing value is 1.09 for mortgages w ith maturity of 10, 15, and 20 years and 1.29 for mortgages with maturity of 30 year s. In addition, the servicing co st are assumed to be zero. The guarantee fee buyup and buydown multipliers are summ arized in Table 33, which shows that guarantee fee buyup and buydown multipliers depend on mortgage note rate and maturity. A mortgage with a higher note rate and longer ma turity has larger multipliers. In addition, buydown multipliers are larger than the buyup multipliers under th e same note rate and maturity. In the case study, we consider 20 scenarios th at are uniformly distri buted across the range between 0 and m srK2, where m srK is the expected retained servicing fee multiplier for mortgage m summarized in Table 33. Equati on (323) defines the probability mass function for the variable of retained servicing fee multiplier. otherwise. 0 }, 19 0 ) 1 05 (. : { 05 0 ) ( I z z K z S S x x fm sr X (323) From Table 33, the expected retained servicing fee multiplier for mortgage m depends on note rate and maturity. A mortga ge with a larger note rate an d longer maturity has a higher expected retained serv icing fee multiplier. The largescale mixed 01 linear programming problem was solved by CPLEX90 on an Intel Pentium 4, 2.8GHz PC. The running times for solving the instances of the efficient execution problem are approximately one minute with a solution gap6 of less than 0.01%. 6 Solution gap defines a relative tolerance on the gap between the best integer objective and the object of the best node remaining. When the value best nodebest integer /(1e10 +best integer) falls below this value, the mixed integer programming (MIP) optimization is stopped. PAGE 61 61 3.4.2 Result The efficient execution model solved the optim al execution solution under different upper bounds of CVaR losses across th e range from $193,550,000 to $193,200,000, increasing in increments of $50,000, under a fixed value to get an efficient frontier in the expected revenue versus CVaR risk diagram. The procedure was repeated for values of 0.5, 0.75, 0.9, and 0.95 to get efficient frontiers under different risk preferences associated with values. The efficient frontiers are shown in Figure 34, and the solutions of efficient execution under different and U values are listed in Table 34. Table 31. Summary of data on mortgages Loan Amount ($) Note Rate (%) Guarantee Fee (%) YTM # of Mortgages Min Mean Max Sum Min Mean Max Min Mean Max 10 13 $539,323 $164,693 $330,090 $2,141,009 4.38 5.00 5.75 0.125 0.146 0.40 15 148 $34,130 $172,501 $459,000 $25,530,131 4.00 5.29 8.00 0.125 0.187 0.80 20 11 $38,920 $137,024 $291,320 $1,507,260 5.13 5.97 7.25 0.125 0.212 0.80 30 828 $24,000 $194,131 $499,300 $160,740,0834.75 5.92 7.875 0.125 0.212 1.05 Table 32. Summary of data on MBS prices of MBS pools MBS coupon rate (%) YTM 9.5 9 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 10 107.69 107.69 107.69 107.66 107.31 106.94 105.84 104.94 103.69 102.45 101.13 99.703 93.381 15 107.69 107.69 107.69 107.66 107.31 106.97 105.84 104.63 103.19 101.5 99.688 97.531 93.381 20 107.73 107.73 107.72 107.63 106.69 105.78 104.69 103.64 101.78 99.719 97.047 90.263 N/A 30 107.73 107.73 107.72 107.63 106.69 105.78 104.5 103.13 100.91 98.469 95.313 90.263 N/A Table 33. Guarantee fee buyup and buydown and expected retained se rvicing multipliers Note 30YR 20YR 15YR 10YR 30YR 10, 15, 20YR Rate Buyup Buydown Buyup Buydown BuyupBuydownBuyupBuydown Expected Retained Servicing Expected Retained Servicing 4 5.65 7.6 4.80 6.52 3.60 5.55 3.42 5.27 5.75 4.00 5 4.95 6.9 4.20 5.93 2.95 4.90 2.80 4.65 5.74 3.60 6 3.15 5.0 2.67 4.41 1.50 3.29 1.38 3.18 4.19 2.76 7 1.65 3.3 1.40 3.00 0.95 2.75 0.90 2.70 2.22 2.00 8 0.95 2.5 0.75 2.35 0.55 2.35 0.50 2.30 1.60 0.62 PAGE 62 62 Figure 34 shows the tradeoff between CVaR ( tail risk) and expected revenue. For a fixed value, when the upper bound of CVaR increases, the optimal expected revenue increases in a decreasing rate. On the other hand, for a fixed upper bound of CVaR, a high value implies high risk aversion. Therefore, the associated optimal expected revenue becomes lower. Figure 34: Efficient Frontiers. Plot of maximum expected reve nue associated with different upper bound of CVaR losses across the range from $193,550,000 to $193,200,000 in increments of $50,000 under a fixed value in the expected revenue versus CVaR risk diagram to get an efficient frontie r. Repeat the procedure for different values of 0.5, 0.75, 0.9, and 0.95 to get efficient frontiers under different risk preferences associated with values. Table 34 summaries the soluti on of efficient execution under different risk preferences specified by and U, which includes the number of mortgages sold as a whole loan, number of mortgages securitized as MBSs with a specific coupon rate rangi ng from 3.5% to 9.5% that increases in 0.5% increments, th e number of retained mortgage servicing and released mortgage servicing, the sum of guar antee fee buyup and buydown amount, and the sum of excess servicing fee amount. Efficient Frontiers193540000 193560000 193580000 193600000 193620000 193640000 193660000 193680000 193550000193500000 193450000193400000 193350000 193300000193250000 193200000Upper bound o f CVaR Loss (Negative Revenue) Expected Revenue ?=0.5 ?=0.75 ?=0.9 ?=0.95 PAGE 63 63 3.4.4 Sensitivity Analysis A sensitivity analysis was conducted in servic ing fee multipliers, mortgage prices, and MBS prices. The sensitivity analysis is performed under a CVaR constraint with =75% and U = $193,400,000. Table 35 shows that when all servicing fee multipliers increase by a fixed percentage, the number of mortgages sold as a whole loan decr eases since lenders could get higher revenue from securitization due to the increasing servicing fee. In addition, the number of retained servicing and the amount of excess servicing fee increase due to the increasing re tained servicing fee multipliers, and the number of released servici ng decreases because the base servicing values, i.e. the upfront payments of rel eased servicing, do not increase asso ciated with retained servicing multipliers. An interesting result is that th e increasing servicing fee multipliers increase (decrease) the guarantee fee buydown (buyup) amount, and the number of mortgages pooled into low coupon rate MBSs (4.5% and 4%) increa ses and the number of mortgages pooled into high coupon rate MBSs (7%, 6.5%, and 6%) decreases. Table 36 shows that when all mortgage prices increase by a fixed percentage, the number of mortgages sold as a whole loan increases so that lenders can take a dvantage of high mortgage price. Since the number of whole loan sale mortga ges increases which implies that the number of securitized mortgages decrea ses, the sum of buyup and buydown guarantee fee and excess servicing fee slightly decrease. Fu rthermore, the number of releas ed servicing decreases since the number of securitized mortgage decreases. Table 37 shows that when MBS price increases the number of whole loan sale decreases, the number of securitized mortgages increases in the lower MBS coupon ra te pools (4%, 4.5%, and 5%), and the number of retained servicing slightly increases. PAGE 64 64 3.5 Conclusion This research proposed a stochastic optimiza tion model to perform the efficient execution analysis. The model considers secondary market ing functionalities, incl uding the loanlevel execution for an MBS/whole loan, guarantee f ee buyup/buydown, servici ng retain/release, and excess servicing fee. Since s econdary marketing involves rand om cash flows, lenders must balance between expected revenue and risk. We pr esented the advantages of CVaR risk measure and employed it in our model that maximizes expected revenue under a CVaR constraint. By solving the efficient execution pr oblem under different risk tolera nces, efficient frontiers could be found. We conducted a sensitivity analysis in parameters of e xpected retained servicing fee multipliers, mortgage prices, and MBS prices. The model is formulated as a mixed 01 linear programming problem. The case study shows that realistic instances of the efficient execution problem can be solved in an acceptable time (approximately one minute) by using CPLEX90 solver on a PC. PAGE 65 65 Table 34. Summary of efficient execution solution under different risk preferences # of mortgages pooled into MBS (with coupon rate %) Upper Bound of CVaR Losses U Total Revenue # of Whole Loan Sale 9.59 8.58 7.57 6.56 5.55 4.54 3.5 # of Released Servicin g # of Retained Servicin g Sum of buyup amount (%) Sum of buydown amount (%) Sum of Excess servicing fee amount (%) 193550000 195380000 155 0 0 0 0 0 5 30183429134 59 5 0 845 0 40.41 63.76 11.29 193500000 193626000 155 0 0 0 0 0 5 30183428135 56 8 0 845 0 16.72 63.64 36.85 193450000 193655000 155 0 0 0 0 0 5 21192428134 54 110 845 0 0.077 60.87 57.22 193400000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 193350000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 193300000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 193250000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 0.5 193200000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 193550000 193570000 155 0 0 0 0 0 5 30183429134 59 5 0 845 0 45.10 63.76 6.60 193500000 193600000 155 0 0 0 0 0 5 30183428135 59 5 0 845 0 28.94 63.64 23.13 193450000 193621000 155 0 0 0 0 0 5 30183428135 56 8 0 845 0 20.04 63.64 33.53 193400000 193639000 155 0 0 0 0 0 5 30183428135 56 8 0 845 0 7.67 63.64 45.89 193350000 193655000 155 0 0 0 0 0 5 21192428134 55 100 845 0 0.69 60.87 56.11 193300000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 193250000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 0.75 193200000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 PAGE 66 66 Table 34. Summary of efficient execution solu tion under different risk preferences (contd.) # of mortgages pooled into MBS (with coupon rate %) Upper Bound of CVaR Losses U Total Revenue # of Whole Loan Sale 9.59 8.58 7.57 6.56 5.55 4.54 3.5 # of Released Servicing # of Retained Servicing Sum of buyup amount (%) Sum of buydown amount (%) Sum of Excess servicing fee amount (%) 193550000 193568000 155 0 0 0 0 0 5 30183429134 59 5 0 845 0 46.12 63.76 5.57 193500000 193592000 155 0 0 0 0 0 5 30183428135 59 5 0 845 0 33.06 63.64 19.01 193450000 193611000 155 0 0 0 0 0 5 30183428135 59 5 0 845 0 24.66 63.64 27.41 193400000 193626000 155 0 0 0 0 0 5 30183428135 56 8 0 845 0 16.99 63.64 36.58 193350000 193639000 155 0 0 0 0 0 5 30183428135 56 8 0 845 0 9.40 63.64 44.17 193300000 193652000 155 0 0 0 0 0 5 30183428135 56 8 0 845 0 1.04 63.64 52.53 193250000 193656000 155 0 0 0 0 0 5 20193428133 54 120 845 0 0 60.57 58.50 0.9 193200000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 193550000 193567000 155 0 0 0 0 0 5 30183429134 59 5 0 845 0 46.46 63.76 5.23 193500000 193590000 155 0 0 0 0 0 5 30183428135 59 5 0 845 0 34.57 63.64 17.50 193450000 193608000 155 0 0 0 0 0 5 30183428135 59 5 0 845 0 25.96 63.64 26.11 193400000 193622000 155 0 0 0 0 0 5 30183428135 56 8 0 845 0 20.22 63.64 33.35 193350000 193635000 155 0 0 0 0 0 5 30183428135 56 8 0 845 0 12.18 63.64 41.39 193300000 193648000 155 0 0 0 0 0 5 30183428135 56 8 0 845 0 5.16 63.64 48.41 193250000 193656000 155 0 0 0 0 0 5 20193428133 54 120 845 0 0 60.57 58.50 0.95 193200000 193656000 155 0 0 0 0 0 5 14199428116 70 130 845 0 0 56.80 66.73 PAGE 67 67 Table 35 Sensitivity analysis in servicing fee multiplier # of mortgages pooled into MBS (with coupon rate %) Servicing Fee Multipliers Increment # of Whole Loan Sale 9.5 9 8.5 8 7.57 6.56 5.55 4.54 3.5 # of Released Servicing # of Retained Servicing Sum of buyup amount (%) Sum of buydown amount (%) Sum of Excess servicing fee amount (%) 0% 155 0 0 0 0 0 5 30 18342813556 8 0 845 0 7.6741 63.6374 45.8949 10% 155 0 0 0 0 0 4 15 19732122171 16 0 768 77 0 38.4944 107.926 20% 114 0 0 0 0 0 4 8 15764 49611443 0 661 225 0 74.0746 371.006 30% 78 0 0 0 0 0 3 3 12648 50514988 0 260 662 0 137.755 537.436 40% 74 0 0 0 0 0 3 2 1277 4951741180 207 719 0 165.085 626.538 50% 73 0 0 0 0 0 3 0 84 45 2354301300 156 771 0 175.951 804.253 Table 36. Sensitivity anal ysis in mortgage price # of mortgages pooled into MBS (with coupon rate %) Mortgage Price Increment # of Whole Loan Sale 9.5 9 8.587.576.56 5.55 4.54 3.5 # of Released Servicing # of Retained Servicing Sum of buyup amount (%) Sum of buydown amount (%) Sum of Excess servicing fee amount (%) 0% 155 0 0 0 00 53018342813556 8 0 845 0 7.6741 63.6374 45.8948 0.5% 196 0 0 0 00 51419942682 65 130 804 0 0 55.144 62.6578 1% 277 0 0 0 00 51419840636 52 120 723 0 0 48.773 51.4078 1.5% 313 0 0 0 00 51419738836 35 120 687 0 0 45.1942 49.2382 2% 429 0 0 0 00 51419230132 17 100 571 0 0 30.587 44.2382 PAGE 68 68 Table 37. Sensitivity analysis in MBS price # of mortgages pooled into MBS (with coupon rate %) MBS Price Increment # of Whole Loan Sale 9.5 9 8.5 87.576.56 5.55 4.543.5 # of Released Servicing # of Retained Servicing Sum of buyup amount (%) Sum of buydown amount (%) Sum of Excess servicing fee amount (%) 0% 155 0 0 0 00 53018342813556 80 845 0 7.6746 63.6374 45.8949 1% 70 0 0 0 00 51719642820654 240 858 72 0.25 63.7476 62.0756 2% 43 0 0 0 00 52019342821074 270 864 93 0.25 65.3748 66.4206 3% 29 0 0 0 00 52019342721286 280 866 105 0.25 65.523 68.1438 4% 10 0 0 0 00 520193428212104280 866 124 0.25 66.3766 67.9206 5% 1 0 0 0 00 521192428212113271 867 132 0.25 67.6766 67.4706 PAGE 69 69 CHAPTER 4 MORTGAGE PIPELINE RISK MANAGEMENT 4.1 Introduction Mortgage lenders commit to a mortgage interest rate while the loan is in process, typically for a period of 3060 days. If the in terest rate rises before the loan goes to closing, the value of the loans will decline and the lend er will sell the loans at a lower price. The risk that mortgages in process will fall in value prior to their sale is known as interest rate risk. In addition, the profit of mortgage origination is affect ed by fallout. The loan borrowers ha ve the right to walk away at any time before closing. This right is actually a put option fo r the borrowers of the loan commitments. As interest rates fall, fallout rises because borrowers who have locked in a mortgage rate are more likely to find better rates with another lender. Conversely, as rates rise the percentage of loans that close increases. Fallout affects the requi red size of the hedging instrument because it affects the size of the pipeline position to be hedged. At lower rates, fewer rate loans will close and a smaller position in th e hedging instrument is needed. Those risks in the mortgage underwriting process are known as th e mortgage pipeline risks. Lenders often hedge this exposure by selling forward their expected closing volume or by shorting U.S. Treasury notes or futures contract s. In addition, options on U.S. Treasury note futures are used to hedge against the risk of fall out (Cusatis and Thomas, 2005). Mortgage lenders are concerne d with two types of risk: vol atility and extreme loss. The volatility measures the uncertainty of the valu e of loan commitments. High volatility of the exposure means the mortgage institution has high un certainty in the value of locked loans. The extreme loss measures the worse case scenario. Th e downside risk measure is used to decide the reservation of economic capital prepared for the extreme loss. To reduce the volatility and downside risk of locked loan values, mort gage pipeline managers hedge the exposure by PAGE 70 70 purchasing financial instruments whose values are se nsitive to mortgage rates. A direct way is to sell the MBS forward contracts. However, MBS fo rward contracts are trad ed over the counter. In other words, the contract is not standardiz ed and the lack of liquidity may induce high transaction costs. An alternative approach is to sell an equivalent amount of Treasury futures. As rates go up, the value of the short position will rise, offsetting the losses on the loan commitments in the pipeline. The fa llout is also a critic al source of risk in pipeline management. Mortgage loans provide a put option to borrowers allowing th em to not close the loan commitment. This embedded put option reduces the number of loans that go to closing as rates go down and vise versa. In other words, the fallout effect varies the loan amount that need to be hedged. This phenomenon is called negative conve xity in the fixed income market. Figure 4.1 shows the negative convexity in the pricerate diagram. Curve A shows the convexity of fixed income security value with respec t to interest rate. For locked mo rtgages in pipeline, when rates drop below the locked rate, the value of loans in pipeline is increasi ng in a decreasing rate because of fallout. Therefore, the value of locked mortgages with respect to rates has a negative convexity property shown in Curve B. Figure 41. Negative convexity. When rates drop below the locked rate, the value of loans in pipeline is increasing in a decr easing rate because of fallout. Rate s Value of loans in pipeline Locked Rate A B PAGE 71 71 Put options on Treasury futures is a preferred in strument for hedging the fallout risk. It is well known that the 10year Treasu ry notes are the preferred c hoice for hedging pipeline risk, mainly because the duration more closely matc hes the loan portfolio (Taflia, 2003). This research considers using the put options on 10year Treasury futures a nd the 10year Treasury futures for hedging the mortgage pipeline risk. Th e other advantage in se lecting these two hedge instruments is that those instruments are traded in exchange with high liquidity. So the hedge portfolio could be rebalanced without unexpected transaction costs. It is worth mentioning that the spread between mortgage rates and Treasury rates creates basis risk. So an optimal cross hedge model for the pipeline risk manage ment is developed in next section. This research investigates the best risk measures for mortgage pipeline risk hedging strategy. Five deviation risk measures and tw o downside risk measures are selected for the optimal pipeline risk hedge model. We test ed the hedging performance by optimizing the hedging strategy associated with each risk measur e based on an (insample ) historical dataset. Then, the outofsample performance of th e optimal hedging strategy is examined. 4.2 Model 4.2.1 Locked Loan Amount Evaluation To hedge the pipeline risk, we first model th e value of locked loan amount with respected to the change in mortgage rate. The fallout of mortgage loans varies for different regions. A general way to estimate the fallout is using a regression model based on th e regional historical data. This research defined the fallout as a func tion of the price change in the first generic 10year Treasury futures given by ) 1 exp( TY b a Fallout (41) where a and b are coefficients, and 1T Y is the price change on the mortgage backed security. Although the fallout effect increases when rates drop, there are always some loans that do not go PAGE 72 72 to closing even the rates go up. To model this effect, the fallout is assumed to be greater than a minimum fallout level Fallout Fallout (42) where Fallout is the lower bound of the fallout, wh ich is also regional dependent. The value change of the locked loan amount can be formulated as the value change of the fallout adjusted loan amount. We assume that the value change of mortgage loans can be approximately equivalent to the value change of 10year Treasury futures. 1 ) 1 (TY Fallout LV (43) where LVis the value change of locked loan amount. 4.2.2 Pipeline Risk Hedge Agenda Assume a pipeline risk manager has 100 million dollars loan amount flowing into pipeline every two weeks. The lock period of the mortgage loans is four weeks. The risk manager hedges the loan exposure twice during the lock peri od, one at the time point when the loans are originated, i.e., in the start date of the first week of the lock period, and the other in the middle of the lock period, i.e., on the beginning of the thir d week. In this hedge agenda, the risk manager hedges the pipeline risk every two weeks with tw o different locked loan amounts. The first part of the loans was locked in the current mortgage rate with a loan am ount of 100 million dollars. The second part of the loans was locked in the ra te two weeks earlier. The loan amount is fallout adjusted. Therefore, the total value change is given by t t tLV LV LV2 1 (44) where tLV1 is the value change of the loans origin ated at time t1 during [t1,t], and tLV2 is the value change of the loans originated at time t2 during [t1,t]. tLV1 is given by PAGE 73 73 1 100 / ) 1 1 ( )), 1 1 ( exp( max 1 1 100 / ) 1 1 ( ) 1 ( 1 1 11 1 1 1 1 1 t t t t t t t t t t t tLV TY TY Fallout TY TY b a LV TY TY Fallout LV LV LV (45) where tFallout is the fallout during [t1,t], and 11 tLVis the loan amount originated at time t1. tLV2 is given by 1 1 1 12 100 / ) 1 1 ( ) 1 ( 2 2 2 t t t t t t t tLV TY TY Fallout LV LV LV (46) The value change of the loans originated at time t2 during [t1,t] considered fallouts in two consecutive periods. The second fallout, 1 ,t tFallout, is calculated based on the value change of 10year Treasury futures during [t 2,t] and adjusted by the first one, 1tFallout, Fallout Fallout TY TY b a Falloutt t t t t, )) 1 1 ( exp( max1 2 1 (47) and the time t1 value of the lock ed loans originated at time t2,12tLV, is given by t t tLV Fallout LV2 ) 1 ( 21 1 (48) 4.2.3 Model Development An optimization model for the mortgage pipe line risk hedge was developed. The model considers shorting the first generic 10year Tr easury futures (TY1) and buying the put options on 10year Treasury futures, such as TYZ7P 118, fo r pipeline risk hedge. As mortgage rates rise, the locked loan values as well as the value of 10year Treasury futures drop. The increasing value in the short position of 10year Treasury futures offsets the losses in the locked loan value. On the other hand, the decreasing value of the 10year Treasury futures increases the value of long position of the put option on 10year Treasury futures. To investigate the performance of different risk measures on the pipeline hedging strategies, five deviation measures: standard deviation, mean absolute devi ation, CVaR deviation, VaR deviation, and twotailed VaR PAGE 74 74 deviation, and two downside risk measures: CVaR and VaR, were tested in the pipeline hedge model The notations used in the m odel formulation are listed as follows. x = vector ] [1 1x xof decision variables (hedge positions), 1x = hedge position of 10year Treasury futures, 2x = hedge position of put options on 10year Treasury futures, = random vector ] [2 1 0 of parameters (prices), j = the jth scenario of random vector ] [2 1 0 j0 = the jth scenario of value change in the locked loan amount in one analysis period, j1 = the jth scenario of price change in 10y ear Treasury futures in one analysis period, j2 = the jth scenario of price change in put options on 10year Treasury futures in one analysis period, 0 = value change in the locked lo an amount in one analysis period, 1 = price change in 10year Treasur y futures in one analysis period, 2 = price change in put options on 10year Treasury futures in one analysis period,J = number of scenarios, jx L = the jth scenario of loss based on hedge position x and jth scenario of prices j The model formulation associated with each risk measure is described as follows. The objective function (49) of the minimum standard deviation model is equivalent to the standard deviation since the value of the sum of the square is larger than one. A draw back of the standard deviation measure is that the squared term am plifies the effect of outliers. The formulation of minimum standa rd deviation model is given by J j J j j jJ x L x L Min1 2 1, (49) j x x x Lj j j j 2 2 1 1 0, s.t. To overcome the drawback of st andard deviation mentioned above the mean absolute deviation takes the absolute value instead of the s quared value in calculating dispersion. PAGE 75 75 The formulation of minimum mean absolute standard deviation model is given by J j J j j jJ x L x L Min1 1, j x x x Lj j j j 2 2 1 1 0, s.t. The nonlinear programming problem was formulat ed as an equivale nt linear programming problem s.t. 2 2 1 1 0 1 1j x x x L J x, L x, L z J x, L x, L z z Minj j j j J j j j j J j j j j j Conditional ValueatRisk (CVaR ) deviation measures the distance between the mean value and CVaR of a distribution. In other words, the minimum CVaR deviation model minimizes the difference between 50% percentile and mean of tail of a distribution. The minimum CVaR deviation model can be formulated as a linear programming problem j z j x x x L j x L z J x L z J Minj j j j j j j J j j J j j 0 s.t. ) 1 ( 1 2 2 1 1 0 1 1 where is a real number variable representing the percentile of a distribution. PAGE 76 76 ValueatRisk (VaR ) deviation measures the distan ce between the mean value and VaR of a distribution. The minimum CVaR deviation model minimizes the difference between 50% percentile and percentile of a distribution. The minimum VaR deviation model can be formulated as a 01 mixed linear programming problem j y j x x x L J y j My x L J x L Minj j j j j J j j j j J j j 1 0 1 s.t. 2 2 1 1 0 1 1 where M is a big number used to enforce 1 jy when 0 jx L The twotailed VaR deviation measures the distance between two VaR values in two tails of a distribution. This meas ure ignores th e outliers beyond and 1tails of a distribution. The property can be used in forecast and robust regression. j z y j x x x L J z j Mz x L J y j My x L Minj j j j j j J j j j j J j j j j 1 0 1 1 s.t. 2 2 1 1 0 1 1 1 1 In addition to the deviation m easures, two downside risk measures are considered in the mortgage pipeline risk models. PAGE 77 77 Conditional ValueatRisk (CVaR ) measures the mean of the tail of a distribution. The minimum CVaR model minimizes the mean of the worst (1) *100 % loss scenarios. The minimum CVaR model can be formulated as a linear programming problem. j z j x x x L j x L z z J Minj j j j j j j J j j 0 s.t. ) 1 ( 1 2 2 1 1 0 1 ValueatRisk (VaR ) measures the percentile of a distribution. The minimum VaR model minimizes the loss value at percentile of a distribution. The minimum VaR model can be formulated as a 01 mixe d linear programming problem 1 0 1 s.t. 2 2 1 1 0 1j y j x x x L J y j My x L Minj j j j j J j j j j 4.3 Case Study 4.3.1 Dataset and Experiment Design The dataset for the case study incl udes biweekly data of the pr ice of the generic first 10year Treasury furthers (TY1)7, the price of put options on the 10year Treasury futures8, and the 7 The TY1 is the near contract generic TY future. A genetic is constructed by pasting together successive Nth contract prices from the primary months of March, June, September, and December. 8 The strike prices of the put options are selected based on the price of TY1. PAGE 78 78 mortgage rate index MTGEFNCL9. All prices and index are the close price on Wednesday from 8/13/2003 to 10/31/2007. The dataset includes 110 scenarios. To test th e performance of six de viation measures and two downside risk measures in the pipeline risk management, we used the first 60 time series data as known historical data and solve the op timal hedge positions associated with each risk measure. Next, the optimal hedge positions associated with different risk measures are applied in hedging the next (the 61st) scenario, and a hedge error can be calculated for each model. Then, the window is rolled one step forward. The datase t from the second to the 61st scenarios are used to compute the optimal hedging positions, and th e one step forward outofsample hedge error can be get from the 62nd scenario. This procedure was repeated for the remaining 50 scenarios, and 50 outofsample hedge errors were used to ev aluate the performances of eight risk measures in pipeline risk hedge. To investigate whether more historical data provides better solution in outofsample performance, an alternative approach is to fix the window start date. The growing window approach keeps all the known hist orical data for solving the optim al hedge positions. The results from two approaches are compared. 4.3.2 Analyses and Results The experiment was conducted in a UNIX work station with 2 Pentium4 3.2 GHz processor and 6GB of memory. The optimization models were solved using CPLEX optimizer. Figure 42 shows the values of naked (unhedge d) and hedged pipeline positions. It shows that no matter which risk measure is selected, the pipeline hedge reduces the volatility of the value of locked loans dramatically. Figure 43 and Figure 44 show the outofsample hedge 9 The MTGEFNCL index represents the 30 Year FNMA current coupon, which has been used as an index for the 30 year mortgage rate. PAGE 79 79 performances associated with risk measures using rolling window approach and growing window approach, respectively. The risk meas ures selected for the optimal hedging model includes: standard deviation (STD), mean abso lute deviation (MAD), CVaR deviation with 90% confident interval (CVaR90 Dev), VaR deviation with 90% confident interval (VaR90 Dev), twotailed VaR deviation with 90% confident interval (2TVaR90), twotailed VaR deviation with 99% confident interval (2TVaR90), CVaR90, and VaR90. The Figures show that the standard deviation has the worst outofsample hedge perfor mance among all risk measures. Table 41 and Table 42 show the mean value, standard deviation, and maximum loss of the 50 outofsample losses of hedged position base d on rolling window and growing window approach, respectively. Comparing these two tables we can see that the two approaches are not significantly different. For the rolling window approach, the two tailed 99% VaR has the best hedge performance among all risk measures. In ad dition, the mean standard deviation has the Figure 42 Value of naked pipe line position and hedged pipeline positions associated with different risk measures. V alue of loans in pipeline 4000000 3000000 2000000 1000000 0 1000000 2000000 3000000 4000000 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 STD MAD CVAR90Dev VAR90Dev 2TVAR90 CVaR90 VAR90 2TVAR99 UnHedge PAGE 80 80 Figure 43 Outofsample hedge errors associated with eight risk measur es using rolling window approach Figure 44. Outofsample hedge errors associat ed with eight risk measures using growing window approach 300000 200000 100000 0 100000 200000 300000 400000 500000 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 STD MAD CVAR90Dev VAR90Dev 2TVAR90 CVaR90 VAR90 2TVAR99Value of hedged position in pipeline Value of hedged position in pipeline 300000 200000 100000 0 100000 200000 300000 400000 500000 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 STD MAD CVAR90Dev VAR90Dev 2TVAR90 CVAR90 VAR90 2TVAR99 PAGE 81 81 best performance in reducing the volatility. Fo r the growing window approach, CVaR deviation has best performance in mean value and sta ndard deviation, and th e CVaR has the best performance in maximum loss. Table 41 Mean value, standard deviation, and maximum loss of the 50 outofsample losses of hedged position based on rolling window approach STD MAD CVaR90 Dev VaR90 Dev 2TVaR90 2TVaR99 CVaR90 VaR90 Mean 45260 36648 31863 42667 37571 30178 31317 38491 StDev 94182 54407 71278 66014 62836 55409 77217 62209 Max Loss 394484 273282 270711 295226 290365 261490 278274 295226 Table 42 Mean value, standard deviation, and maximum loss of the 50 outofsample losses of hedged position based on growing window approach STD MAD CVaR90 Dev VaR90 Dev 2TVaR90 2TVAR99 CVaR90 VaR90 Mean 45260 42028 35546 47363 37571 38296 35944 44566 StDev 94182 59247 54723 76232 62836 59040 60523 65864 Max Loss 394484 276070 278462 314817 290365 296030 252958 314817 4.4 Conclusion This chapter studies the optimal mortgage pipeline risk management strategy. We developed an optimization model to minimize the pi peline risk under five di fferent deviation risk measures and two downside risk measures. A case study using 10year Treasury futures and options on the 10year Treasury futures as the he dging instruments shows th at under the growing window approach, the CVaR deviation has a bette r performance for hedgi ng the volatility and CVaR is better for hedging the downside risk. On the other hand, when the rolling window approach is used, the two tailed 99% VaR has a better performance in general and the mean absolute deviation performs well in hedging volatili ty. It is worth mentioning that the result of the case study shows the standard deviation is the worst risk measure for the outofsample pipeline risk hedge. PAGE 82 82 CHAPTER 5 CONCLUSION This dissertation has shown a novel stochastic optimization model using conditional valueatrisk as the risk measure in the reward versus risk framework. This optimization approach was applied in modeling the optimal crop production and risk hedging strategy. To get a better performance in capturing the correlation between marginal distributions of multivariate random variables, the copula function was used to generate the portfolio loss distri bution. Incorporating the copula based loss distribution within the CVaR optimization mode l produces a powerful model for solving the optimal planting schedule and hedging strategy problem. In addition, the ENSObased climate forecast information was used to get a better prediction of the crop yield in the coming season. A case study showed that 65%CRC or 70%CRC would be the optimal insurance coverage when the CVaR constraint re aches a strict level. Fu rthermore, the optimal hedging strategy with crop insurance products and futures contracts was examined. When futures price are unbiased or positive biased, the optimal hedging strategy only contains futures contracts and no crop insurance policies are de sirable. However, when the futures price is negative biased, the optimal hedging strategy depends on the ENSO phases. The optimal futures hedge ratio increases with the increasing CVaR upper bound wh en the insurance strategy is unchanged. For a fixed CVaR upper bound, the optimal hedge ratio increases when the positive bias of futures price increases. However, when the futures pric e is negative biased, the optimal hedge ratio depends on the value of CVaR upper bound. For the secondary mortgage market efficient execution problem, a st ochastic optimization model is proposed to perform the efficient ex ecution analysis. The model considers secondary marketing functionalities, including the loanl evel execution for an MBS/whole loan, guarantee fee buyup/buydown, servicing retain/release, and excess servicing f ee. Since secondary PAGE 83 83 marketing involves random cash flows, lenders mu st balance between exp ected revenue and risk. The advantage of the CVaR risk measure wa s introduced and employed in the model that maximizes expected revenue under a CVaR cons traint. By solving the efficient execution problem under different risk to lerances, efficient frontiers could be found, which provides optimal execution strategies associated with diffe rent risk budgets. A sensitivity analysis was conducted in parameters of expected retained servicing fee multipliers, mortgage prices, and MBS prices. In the optimal mortgage pipeline risk ma nagement strategy problem, we developed an optimization model to minimize the mortgage pipe line risks under five diffe rent deviation risk measures and two downside risk measures. A case study using 10year Treasury futures and out options on the 10year Treasury futures as the hedging instruments was conducted. The result shows that the CVaR Deviation and CVaR have better outofsample performances in controlling the volatility and maximum loss, respectively, u nder the growing window approach. In addition, the twotailed 99% VaR deviation performs better on mean loss and maximum loss for the rolling window approach. In contrast, st andard deviation performed the worst. PAGE 84 84 APPENDIX A EFFICIENT EXECUTION MODEL FORMULATION Notations: Indices: m = index of mortgages (1,2,,M) M = total number of mortgages, j = index of mortgage groups (1,2,,J) J = total number of groups, k = index of scenarios (1,2,,K) K = total number of scenarios, c = index of MBS coupon rate (1,2,,Cm) mC = number of possible MBS coupon rates of mortgage m Decision Variables: m cz = otherwise, 0, index rate coupon with MBS into pooled is mortgage if 1 c m m wz = otherwise, 0, loan, a whole as sold is mortgage if 1 m m sboz = otherwise, 0 retained, is mortgage of servicing the if 1 m m sbrz = otherwise, 0 sold, is mortgage of servicing the if 1 m m gur = guarantee fee buyup spread of mortgage m m gdr = guarantee fee buydown spread of mortgage m m serr = retained excess servicing fee spread of mortgage m kz = real variables used in CV aR constraint formulation, = real variables used in CV aR constraint formulation. Input Data: mL = loan amount of mortgage m m t cP = price of MBS with coupon rate index c and maturity mt, mt = maturity of mortgage m m wP = whole loan sale price of mortgage m cR = MBS coupon rate related to index c m uK = guarantee fee buyup multiplier of mortgage m m dK = guarantee fee buydown mu ltiplier of mortgage m PAGE 85 85 m nR = note rate of mortgage m m sbR = base servicing fee of mortgage m m gbR = base guarantee fee of mortgage m m B = base servicing value of mortgage m m sc = servicing cost of mortgage m kp = the probability of scenario k mk srK = retained servicing fee multiplier of mortgage m under scenario k m guU = upper bound of guarantee fee buyup spread of mortgage m m gdU = upper bound of guarantee fee buydown spread of mortgage m m seU = upper bound of retained excess servicing fee of mortgage m a seU = upper bound of average retained exce ss servicing fee of all mortgages, j seU = upper bound of average retained excess servicing fee of mortgages in group j U = upper bound of CVaR losses, = percentile of CVaR. Model Formulation: M m m w m w m C c m c m t c mz P z P L11 revenue from MBSs or whole loan sale M m m sbr m s K k mk sr m sb m k M m m sbo m mz c K R L p z B L11 1 revenue from the base servicing fee of securitized mortgages 11 MK mmsm ksrser mkpLKr revenue from the excess servicing fee of securitized mortgages Max 11 MM mmmmmm ugudgd mm K LrKLr revenue from the guarantee fee buyup/buydown of securitized mortgages s.t. 111,2,...,mC mm cw czzmM mortgage must be either securitized as an MBS or sold as a whole loanm 11,2,...,mmm wsbosbrzzzmM if mortgage is sold, there is no servicing; otherwise, servicing either be sold or be retained m 1 1,2,...,mmmm wgugdserzrrrmM if mortgage is sold, there is no guarantee fee and excess servicing fee m 11,2,...,mC mmmmmmm ccgugdsernsbgb cRzrrrRRRmM if mortgage m is securitized, note rate = MBS coupon rate + servicing fee rate + guarantee fee rate 1 1(1)K kk kpzU 1,...,kkzLkK 0,1,...,kzkK (): the expected loss (negative revenue) of the worst 1percent of senarios should be less than or equal to U CVaRlossU PAGE 86 86 111 11 11 mMCM mmmmmmmm kccwholewsbo mcm MM mmmkmmmkm sbsrsbrsrser mm MM mmmmmm ugudgd mmLLPzPzLBz LRKzLKr KLrKLr 11(1)MM mmamm sesew mmLrULz the average excess servicing fee of all securitized mortgages is restricted to an upper bound. 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