OPTIMIZATION APPROACHES IN RISK MANAGEMENT: APPLICATIONS IN FINANCE AND AGRICULTURE

Transcription

1 OPTIMIZATION APPROACHES IN RISK MANAGEMENT: APPLICATIONS IN FINANCE AND AGRICULTURE By CHUNG-JUI WANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

2 2007 Chung-Jui Wang 2

3 To y faily 3

4 ACKNOWLEDGMENTS I thank y supervisory coittee chair (Dr. Stan Uryasev) and supervisory coittee ebers (Dr. Farid AitSahlia, Dr. Liqing Yan, and Dr. Jason Karceski) for their support, guidance, and encourageent. I thank Dr. Victor E. Cabrera and Dr. Clyde W. Fraisse for their guidance. I a grateful to Mr. Philip Laren for sharing his knowledge of the ortgage secondary arket and providing us with the dataset used in the case study. 4

5 TABLE OF CONTENTS ACKNOWLEDGMENTS...4 LIST OF TABLES...7 LIST OF FIGURES...8 ABSTRACT...9 CHAPTER INTRODUCTION... 2 OPTIMAL CROP PLANTING SCHEDULE AND HEDGING STRATEGY UNDER ENSO-BASED CLIMATE FORECAST...6 page 2. Introduction Model Rando Yield and Price Siulation Mean-CVaR Model Model Ipleentation Proble Solving and Decoposition Case Study Results and Discussion Optial Production with Crop Insurance Coverage Hedging with Crop Insurance and Unbiased Futures Biased Futures Market Conclusion EFFCIENT EXECUTION IN THE SECONDARY MORTGAGE MARKET Introduction Mortgage Securitization Model Risk Measure Model Developent Case Study Input Data Result Sensitivity Analysis Conclusion MORTGAGE PIPELINE RISK MANAGEMENT Introduction

6 4.2 Model Locked Loan Aount Evaluation Pipeline Risk Hedge Agenda Model Developent Case Study Dataset and Experient Design Analyses and Results Conclusion CONCLUSION...82 APPENDIX A EFFICIENT EXECUTION MODEL FORMULATION...84 LIST OF REFERENCES...87 BIOGRAPHICAL SKETCH

7 LIST OF TABLES Table page Table 2-. Historical years associated with ENSO phases fro 960 to Table 2-2. Marginal distributions and rank correlation coefficient atrix of yields of four planting dates and futures price for the three ENSO phases...3 Table 2-3. Paraeters of crop insurance (2004) used in the far odel analysis...32 Table 2-4. Optial insurance and production strategies for each cliate scenario under the 90% CVaR tolerance ranged fro -$20,000 to -$2,000 with increent of $ Table 2-5. Optial solutions of planting schedule, crop insurance coverage, and futures hedge ratio with various 90% CVaR upper bounds ranged fro -$24,000 to $0 with increent of $4,000 for the three ENSO phases...35 Table 2-6. Optial insurance policy and futures hedge ratio under biased futures prices...36 Table 2-7. Optial planting schedule for different biases of futures price in ENSO phases...39 Table 3-. Suary of data on ortgages...6 Table 3-2. Suary of data on MBS prices of MBS pools...6 Table 3-3. Guarantee fee buy-up and buy-down and expected retained servicing ultipliers...6 Table 3-4. Suary of efficient execution solution under different risk preferences...65 Table 3-5 Sensitivity analysis in servicing fee ultiplier...67 Table 3-6. Sensitivity analysis in ortgage price...67 Table 3-7. Sensitivity analysis in MBS price...68 Table 4- Mean value, standard deviation, and axiu loss of the 50 out-of-saple losses of hedged position based on rolling window approach...8 Table 4-2 Mean value, standard deviation, and axiu loss of the 50 out-of-saple losses of hedged position based on growing window approach...8 7

8 LIST OF FIGURES Figure page Figure 2-. Definition of VaR and CVaR associated with a loss distribution...23 Figure 2-2. Bias of futures price versus the optial hedge ratio curves associated with different 90% CVaR upper bounds in the La Niña phase...38 Figure 2-3. The efficient frontiers under various biased futures price. (A) El Niño year. (B) Neutral year. (C) La Niña year....4 Figure 3-. The relationship between participants in the pass-through MBS arket Figure 3-2. Guarantee fee buy-down Figure 3-3. Guarantee fee buy-up Figure 3-4: Efficient Frontiers Figure 4-. Negative convexity...70 Figure 4-2 Value of naked pipeline position and hedged pipeline positions associated with different risk easures...79 Figure 4-3 Out-of-saple hedge errors associated with eight risk easures using rolling window approach...80 Figure 4-4. Out-of-saple hedge errors associated with eight risk easures using growing window approach

9 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillent of the Requireents for the Degree of Doctor of Philosophy OPTIMIZATION APPROACHES IN RISK MANAGEMENT: APPLICATIONS IN FINANCE AND AGRICULTURE Chair: Stanislav Uryasev Maor: Industrial and Systes Engineering By Chung-Jui Wang Deceber 2007 Along with the fast developent of the financial industry in recent decades, novel financial products, such as swaps, derivatives, and structure financial instruents, have been invented and traded in financial arkets. Practitioners have faced uch ore coplicated probles in aking profit and hedging risks. Financial engineering and risk anageent has becoe a new discipline applying optiization approaches to deal with the challenging financial probles. This dissertation proposes a novel optiization approach using the downside risk easure, conditional value-at-risk (CVaR), in the reward versus risk fraework for odeling stochastic optiization probles. The approach is applied to the optial crop production and risk anageent proble and two critical probles in the secondary ortgage arket: the efficient execution and pipeline risk anageent probles. In the optial crop planting schedule and hedging strategy proble, crop insurance products and coodity futures contracts were considered for hedging against yield and price risks. The ipact of the ENSO-based cliate forecast on the optial production and hedge decision was also exained. The Gaussian copula function was applied in siulating the scenarios of correlated non-noral rando yields and prices. 9

10 Efficient execution is a significant task faced by ortgage bankers attepting to profit fro the secondary arket. The challenge of efficient execution is to sell or securitize a large nuber of heterogeneous ortgages in the secondary arket in order to axiize expected revenue under a certain risk tolerance. We developed a stochastic optiization odel to perfor efficient execution that considers secondary arketing functionality including loan-level efficient execution, guarantee fee buy-up or buy-down, servicing retain or release, and excess servicing fee. The efficient execution odel balances between the reward and downside risk by axiizing expected return under a CVaR constraint. The ortgage pipeline risk anageent proble investigated the optial ortgage pipeline risk hedging strategy using 0-year Treasury futures and put options on 0-year Treasury futures as hedge instruents. The out-of-saple hedge perforances were tested for five deviation easures, Standard Deviation, Mean Absolute Deviation, CVaR Deviation, VaR Deviation, and two-tailed VaR Deviation, as well as two downside risk easures, VaR and CVaR. 0

11 CHAPTER INTRODUCTION Operations Research, which originated fro World War II for optiizing the ilitary supply chain, applies atheatic prograing techniques in solving and iproving syste optiization probles in any areas, including engineering, anageent, transportation, and health care. Along with the fast developent of the financial industry in recent decades, novel financial products, such as swaps, derivatives, and structural finance instruents, have been invented and traded in the financial arkets. Practitioners have faced uch ore coplicated probles in aking profit and hedging risks. Financial engineering and risk anageent has becoe a new discipline applying operations research in dealing with the sophisticated financial probles. This dissertation proposes a novel approach using conditional value-at-risk in the reward versus risk fraework for odeling stochastic optiization probles. We apply the approach in the optial crop planting schedule and risk hedging strategy proble as well as two critical probles in the secondary ortgage arket: the efficient execution proble, and the ortgage pipeline risk anageent proble. Since Markowitz (952) proposed the ean-variance fraework in portfolio optiization, variance/covariance has becoe the predoinant risk easure in finance. However, this risk easure is suited only to elliptic distributions, such as noral or t-distributions, with finite variances (Szegö 2002). The other drawback of variance risk easure is that it easures both upside and downside risks. In practice, however, finance risk anageent is concerned ostly with the downside risk. A popular downside risk easure in econoics and finance is Value-at- Risk (VaR) (Jorion 2000), which easures α percentile of loss distribution. However, as was shown by Artzner et al. (999), VaR is ill-behaved and non-convex for general distribution. The

12 disadvantage of VaR is that it only considers risk at α percentile of loss distribution and does not consider the agnitude of the losses in the α-tail (i.e., the worst -α percentage of scenarios). To address this issue, Rockafellar and Uryasev (2000, 2002) proposed Conditional Valueat-Risk, which easures the ean value of the α-tail of loss distribution. It has been shown that CVaR satisfies the axios of coherent risk easures proposed by Artzner et al. (999) and has desirable properties. Most iportantly, Rockafellar and Uryasev (2000) showed that CVaR constraints in optiization probles can be forulated as a set of linear constraints and incorporated into probles of optiization. This linear property is crucial in forulating the odel as a linear prograing proble that can be efficiently solved. This dissertation proposes a ean-cvar odel which, like a ean-variance odel, provides an efficient frontier consisting of points that axiize expected return under various risk budgets easured by CVaR. Since CVaR is defined in onetary units, decision akers are able to decide their risk tolerance uch ore 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 easure has its drawbacks, it has been the proxy of risk easure for odeling the stochastic optiization proble in financial industry. The ain reason is that the portfolio variance can be easily calculated given individual variances and a covariance atrix. However, the linear correlation is a siplified odel and is liited in capturing the association between rando variables. Therefore, a ore general approach is needed to odel the ore coplicated relationship between ultivariate rando variables, e.g., tail dependent. More iportantly, the approach should provide a convenient way to create the portfolio loss distribution fro arginal ones, which is the ost iportant input data in the 2

13 ean-cvar optiization odel. This dissertation applies the copula function to odel the correlation between rando variables. In Chapter 2, the use of copulas to generate scenarios of dependent ultivariate rando variables is discussed. Furtherore, the siulated scenarios are incorporated into the ean-cvar odel. Chapter 2 investigates the optial crop planting schedule and hedging strategy. Crop insurance products and futures contracts are available for hedging against yield and price risks. The ipact of the ENSO-based cliate forecast on the optial production and hedging decision is exained. Gaussian copula function is applied in siulating the scenarios of correlated nonnoral rando 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 ENSO-based cliate phases. Chapters 3 and 4 are devoted to two optiization probles in the secondary ortgage arket. In Chapter 3, the efficient execution proble was investigated. Efficient execution is a significant task faced by ortgage bankers attepting to profit fro the secondary arket. The challenge of efficient execution is to sell or securitize a large nuber of heterogeneous ortgages in the secondary arket in order to axiize expected revenue under a certain risk tolerance. A stochastic optiization odel was developed to perfor efficient execution that considers secondary arketing functionality including loan-level efficient execution, guarantee fee buy-up or buy-down, servicing retain or release, and excess servicing fee. Since efficient execution involves rando cash flows, lenders ust balance between expected revenue and risk. We eploy a CVaR risk easure in this efficient execution odel that axiizes expected revenue under a CVaR constraint. By solving the efficient execution proble under different risk tolerances specified by a CVaR constraint, an efficient frontier was found, which provides 3

14 secondary arket anagers the best execution strategy associated with different risk budgets. The odel was forulated as a ixed 0- linear prograing proble. A case study was conducted and the optiization proble was efficiently solved by the CPLEX optiizer. Chapter 4 exaines the optial ortgage pipeline risk hedging strategy. Mortgage lenders coit to a ortgage rate while the borrowers enter the loan transaction process. The process is typically for a period of days. While the ortgage 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 ortgages still being processed prior to their sale is known as ortgage pipeline risk. Lenders often hedge this exposure by selling forward their expected closing volue or by shorting U.S. Treasury notes or futures contracts. Mortgage pipeline risk is affected by fallout. Fallout refers to the percentage of loan coitents that do not go to closing. As interest rates fall, fallout rises since borrowers who have locked in a ortgage rate are ore 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 instruent because it changes the size of risky pipeline positions. At lower rates, fewer loans will close and a saller position in the hedging instruent is needed. Lenders often use options on U.S. Treasury futures to hedge against the risk of fallout (Cusatis and Thoas, 2005). A odel was proposed for the optial ortgage pipeline hedging strategy that iniizes the pipeline risks. A case study considered two hedging instruents for hedging the ortgage pipeline risks: the 0 year Treasury futures and put options on 0-year Treasury futures. To investigate the ipact of different risk easureent practices on the optial hedging strategies, we tested five deviation easures, standard deviation, ean absolute deviation, CVaR deviation, VaR deviation, and two-tailed VaR deviation, as well as two downside risk easures, VaR and 4

15 CVaR, in the iniu ortgage pipeline risk odel. The out-of-saple perforances of the five deviation easures and two downside risk easures were exained. 5

16 CHAPTER 2 OPTIMAL CROP PLANTING SCHEDULE AND HEDGING STRATEGY UNDER ENSO- BASED CLIMATE FORECAST 2. Introduction A risk-averse farer preferring higher profit fro growing crops faces uncertainty in the crop yields and harvest price. To anage uncertainty, a farer ay 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 cliate variability (i.e., the El Niño Southern Oscillation, ENSO), gives the opportunity to forecast crop yields in different planting dates. With the flexibility in planting tiing, the profit can be axiized by selecting the best planting schedule according to cliate forecast. Risk averse is another critical factor when farers ake the decision. Farers ay hedge the yield and price risks by purchasing crop insurance products or financial instruents. Two aor financial instruents for farers to hedge against crop risks are crop insurances and futures contracts. The Risk Manageent Agency (RMA) of the United States Departent of Agricultural (USDA) offers crop insurance policies for various crops, which could be categorized into three types: the yield-based insurance, revenue-based insurance, and policy endorseent. The ost popular yield-based insurance policy, Actual Production History (APH), or Multiple Peril Crop Insurance (MPCI), is available for ost crops. The policies insure producers against yield losses due to natural causes. An insured farer 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 indenity is paid based on the shortfall at the election price. The ost popular revenue-based insurance policy, Crop Revenue Coverage (CRC), provides revenue protection. An insured farer selects a coverage level of the guarantee revenue. If the realized revenue is below the guarantee revenue, 6

17 the insured farer is paid an indenity to cover the difference between the actual and guaranteed revenue. Catastrophic Coverage (CAT), a policy endorseent, pays 55% of the established price of the coodity on crop yield shortfall in excess of 50%. The cost of crop insurances includes a preiu and an adinistration fee. The preius on APH and CRC both depend on the crop type, county, practice (i.e., irrigated or non-irrigated), acres, and average yield. In addition, the APH preiu depends on price election and yield coverage, and the CRC depends on revenue coverage. The preiu on CAT coverage is paid by the Federal Governent; however, producers pay the adinistrative fee for each crop insured in each county regardless of the area planted 2. In addition to crop insurance coverage, farers ay anage coodity price risk by a traditional hedge instruent such as futures contract. A futures contract is an agreeent between two parties to buy or sell a coodity at a certain tie in the future, for a specific aount, at a certain price. Futures contracts are highly standardized and are traded by exchange. The cost of futures contract includes coissions and interest foregone on argin deposit. A risk-averse producer ay consider using insurance products in conunction with futures contacts for best possible outcoes. El Niño Southern Oscillation refers to interrelated atospheric and oceanic phenoena. The baroetric pressure difference between the eastern and western equatorial Pacific is frequently changed. The phenoenon is known as the Southern Oscillation. When the pressure over the western Pacific is above noral and eastern Pacific pressure is below noral, it creates abnorally war sea surface teperature (SST) known as El Niño. On the other hand, when the The adinistration fee is $30 for each APH and CRC contract and $00 for each CAT contract. 2 Source: 7

18 east-west baroetric pressure gradient is reversed, it creates abnorally cold SST known as La Niña. The ter neutral is used to indicate SSTs within a noral teperature range. These equatorial Pacific conditions known as ENSO phases refer to different seasonal cliatic conditions. Since the Pacific SSTs are predictable, ENSO becoes an index for forecasting cliate and consequently crop yields. A great deal of research has been done on the connection between the ENSO-based cliate prediction and crop yields since early 990s. Cane et al. (994) found the long-ter forecasts of the SSTs could be used to anticipate Zibabwean aize yield. Hansen et al. (998) showed that El Niño Southern Oscillation is a strong driver of seasonal cliate variability that ipact crop yields in the southeastern U.S. Hansen (2002) and Jones et al. (2000) concluded that ENSObased cliate forecasts ight help reduce crop risks. Many studies have focused on the crop risk hedging with crop insurance and other derivative securities. Poitras (993) studied farers optial hedging proble when both futures and crop insurance are available to hedge the uncertainty of price and production. Chabers et al. (2002) exained optial producer behavior in the presence of area-yield insurance. Mahul (2003) investigated the deand 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 progras on deand for futures contract. Soe researchers have studied the ipacts of the ENSO-based cliate inforation on the selection of optial crop insurance policies. Cabrera et al. (2006) exained the ipact of ENSO-based cliate forecast on reducing far risk with optial crop insurance strategy. Lui et al. (2006), following Cabrera et al. (2006), studied the application of Conditional Value-at-Risk (CVaR) in the crop insurance industry under cliate variability. Cabrera et al. (2007) included 8

19 the interference of far governent progras on crop insurance hedge under ENSO cliate forecast. The purpose of this research is two-fold. First, a ean-cvar optiization odel was proposed for investigating the optial crop planting schedule and hedging strategy. The odel axiizes the expected profit with a CVaR constraint for specifying producer s downside risk tolerance. Second, the ipact of the ENSO-based cliate forecast on the optial decisions of crop planting schedule and hedging strategy was exained. To this end, we generate the scenarios of correlated rando yields and prices by Monte Carlo siulation 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 ean-cvar odel is solved for the optial production and hedging strategy for the specified ENSO phase. The reainder of this article is organized as follows. The proposed odel for optial 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 optial planting schedule, crop insurance policy selection, and hedging position of futures contract. Finally, section 2.5 presents the conclusions. 2.2 Model 2.2. Rando Yield and Price Siulation To investigate the ipact of ENSO-based cliate forecast on the optial production and risk anageent 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, 99). Then, rando yield and price scenarios associated with the ENSO phase are generated by Monte Carlo siulation. 9

20 We assue a farer ay plant crops in a nuber 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 rando production and rando price is crucial in risk anageent since negative correlated production and price provide a natural hedge that will affect the optial hedging strategy (McKinnon, 967). 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 noral distributed, a ethod to siulate correlated ultivariate non-noral rando yield and price was needed. Copulas are functions that describe dependencies aong variables, and provide a way to create distributions to odel correlated ultivariate data. Copula function was first proposed by Sklar (959). The Sklar theore states that given a oint distribution function F on n R with arginal distribution F i, there is a copula function C such that for all x,..., x n in R, F ( x,..., xn ) = C( F ( x ),..., Fn ( xn )). (2-) Furtherore, if F i are continuous then C is unique. Conversely, if C is a copula and F i are distribution functions, then F, as defined by the previous expression, is a oint distribution function with argins F i. We apply the Gaussian copula function to generate the correlated nonnoral ultivariate distribution. The Gaussian copula is given by: C ρ ( F ( x ),..., F n ( xn )) = Φ n, ρ ( Φ ( F ( x )),..., Φ ( Fn ( xn ))), (2-2) which transfers the observed variable x i, i.e. yield or price, into a new variable y i using the transforation y i = Φ [ F ( x )] i i, (2-3) 20

21 where Φ n, ρ is the oint distribution function of a ultivariate Gaussian vector with ean zero and correlation atrix ρ. Φ is the distribution function of a standard Gaussian rando variable. In oving fro x i to y i we are apping observation fro the assued distribution F i into a standard noral distribution Φ on a percentile to percentile basis. We use the rank correlation coefficient Spearan s rho ρ s to calibrate the Gaussian copula to the historical data. For n pairs of bivariate rando saples ( X i, X ), define R = rank X ) and R = rank X ). Spearan s saple rho (Cherubini, 2004) is given by i ( i ( n ( R R ) ik k k = ρ s = 6. 2 n( n ) (2-4) Spearan s rho easures the association only in ters of ranks. The rank correlation is preserved under the onotonic transforation in equation 3. Furtherore, there is a one-to-one apping between rank correlation coefficient, Spearan s rho ρ s, and linear correlation coefficient ρ for the bivariate noral rando variables y, ) (Kruksal, 958) ( y 2 6 ρ( y, y2 ) ρ s ( y, y2 ) = arcsin (2-5) π 2 To generate correlated ultivariate non-noral rando variables with argins F i and Spearan s rank correlation ρ s, we generate the rando variables y i s fro the ultivariate noral distribution Φ n, ρ with linear correlation πρ ρ = 2sin s 6, (2-6) by Monte Carlo siulation. The actual outcoes x i s can be apped fro y i s using the transforation 2

22 x i F [ Φ( y )] = i i. (2-7) Mean-CVaR Model Since Markowitz (952) proposed the ean-variance fraework in portfolio optiization, variance/covariance has becoe the predoinant risk easure in finance. However, the risk easure is suited only to the case of elliptic distributions, like noral or t-distributions with finite variances (Szegö, 2002). The other drawback of variance risk easure is that it easures both upside and downside risks. In practice, finance risk anageent is concerned only with the downside risk in ost cases. A popular downside risk easure in econoics and finance is Value-at-Risk (VaR) (Jorion, 2000), which easures α percentile of loss distribution. However, as was shown by Artzner et al. (999), VaR is ill-behaved and non-convex for general distribution. The other disadvantage of VaR is that it only considers risk at α percentile of loss distribution and does not consider the agnitude of the losses in the α-tail (the worst -α percentage of scenarios). To address this issue, Rockafellar and Uryasev (2000, 2002) proposed Conditional Value-at-Risk, which easures the ean value of α-tail of loss distribution. Figure 2- shows the definition of CVaR and the relation between CVaR and VaR. It has been shown that CVaR satisfies the axios of coherent risk easures proposed by Artzner et al. (999) and has desirable properties. Most iportantly, Rockafellar and Uryasev (2000) showed that CVaR constraints in optiization probles can be forulated as a set of linear constraints and incorporated into the probles of optiization. The linear property is crucial to forulate the odel as a ixed 0- linear prograing proble that could be solved efficiently by the CPLEX solver. This research proposes a ean-cvar odel that inherits advantages of the return versus risk fraework fro the ean-variance odel proposed by Markowitz (952). More iportantly, the odel utilizes the CVaR risk easure instead of variance to take the 22

23 advantages of CVaR. Like ean-variance odel, the ean-cvar odel provides an efficient frontier consisting of points that axiize expected return under various tolerances of CVaR losses. Since CVaR is defined in onetary units, farers are able to decide their risk tolerance uch ore intuitively copared 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 exaple, a -$20,000 90% CVaR eans the average of the worst 0% scenarios should provide a profit equal to $20,000. Frequency VaR Maxial loss α-tail (with probability - α) CVaR Figure 2-. Definition of VaR and CVaR associated with a loss distribution Model Ipleentation Assue a farer who plans to grow crops in a farland of Q acres. There are K possible types of crops and ore than one crop can be planted. For each crop k, there are T k potential planting dates that give different yield distributions based on the predicted ENSO phase, as well as I k available insurance policies for the crop. The decision variables x kti and η 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 23

24 The randoness of crop yield and harvest price in a specific ENSO phase is anaged by the oint distribution corresponding to the ENSO phase. We saple J scenarios fro the oint distribution by Monte Carlo siulation with Gaussian copula, and each scenario has equal probability. Let Ykt denote the th realized yield (pound per acre) of crop k planted on date t, and P k denote the th realized cash price (dollar per pound) for crop k at the tie the crop will be sold. The obective function of the odel, shown in (2-8), is to axiize the expectation of rando profit f (, η ) that consists of the rando profit fro production f P ( x ) x kti k I F insurance f ( x ), and fro futures contract ( ) kti ax Ef f η. P I F ( x, η ) = ax Ef ( x ) + Ef ( x ) + Ef ( η ) kti k k [ ] kti kti k kti, fro crop, (2-8) The profit fro production of crop k in scenario is equal to the incoe fro selling the crop, Y t T k I k P kt k = i= x kti T k I k, inus the production cost, C k x kti t= i=, and plus the subsidy, T k I k S k x kti t = i=., where Ck and Sk are unit production cost and subsidy, respectively. Consequently Equation (2-9) expresses the expected profit fro production. Ef P J K Tk I k ( xkti ) = ( Ykt Pk Ck + S k ) J = k = t= i= x kti. (2-9) Three types of crop insurance policies are considered in the odel, including Actual Production History (APH), Crop Revenue Coverage (CRC), and Catastrophic Coverage (CAT). For APH farers select the insured yield, a percentage α i fro 50 to 75 percent with five percent increents of average yieldy k, as well as the election price, a percentage β i, between 24

25 55 and 00 percent, of the of the established price Pk established annually by RMA. If the harvest is less than the yield insured, the farer is paid an indenity based on the difference T k ( iyk Ykt ) t = α x at price β i Pk. The indenity of APH insurance policy i I APH for crop k in kti the th scenario is given by D ki Tk = ax t= ( α iyk Ykt ) xkti,0 β i Pk i I APH. (2-0) For CRC, producers elect a percentage of coverage level γ i between 50 and 75 percent. The guaranteed revenue is equal to the coverage level γ i ties the product of Y x k kti T k t= and the higher of the base price (early-season price) of crop k, b P k and the realized harvest price in the th scenario h P k. 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 T k t= Y kt x kti P k is less than the guaranteed one, the insured will be paid the difference. Equation (2- ) shows the indenity of CRC insurance policy i I CRC for crop k in the th scenario. D ki = ax Tk i Yk xkti ax k k t= t= Tk b h [ P, P ] Y x P,0 i I CRC γ kt kti k, (2-) The CAT insurance pays 55 % of the established price of the coodity on crop losses in excess of 50 %. The indenity of CAT insurance policy given by i ICAT for crop k in the th scenario is D ki Tk = ax t= ( 0.5Y Y ) k kt x kti,0 0.55Pk. (2-2) 25

26 The cost of insurance policy i for crop k is denoted by R, which includes a preiu and an adinistration fee. For the case of CAT, the preiu is paid by the Federal Governent. Therefore, the cost of CAT only contains a $ 00 adinistrative fee for each crop insured in each county. The expected total profit fro insurance is equal to the indenity fro the insurance coverage inus the cost of the insurance is given by Ef I J J K k T k ( xkti ) = Dki Rki I = k = i= t= x kti ki. (2-3) The payoff of a futures contract for crop k in scenario for a seller is given by π = ( F f ) η, (2-4) F k k k k where F k is the futures price of crop k in the planting tie, f k is the th realized futures price of crop k in the harvest tie, and η k is the hedge position (in pounds) of crop k in futures contract. It is worth noting that the futures price f k is not exactly the sae as the local cash price P k at harvest tie. Basis, defined in (2-5), refers to the difference that induces the uncertainty of futures hedging known as the basis risk. The rando basis can be estiated fro coparing the historical cash prices and futures prices. The cost of a futures contract, Basis = Cash Price Futures Price. (2-5) F C k, includes coissions and interest foregone on argin deposit. Equation (2-6) expresses the expected profit fro futures contract. Ef F J K F F ( ) = ( π C ) η k k k (2-6) J = k= 26

27 We introduce binary variables z ki in constraint (2-7) and (2-8) to ensure only one insurance policy can be selected for each crop k. where Tk t= x kti I k i= Q z z ki = ki i, k k (2-7) (2-8) z ki if crop k is insured by policy i, = 0 otherwise. Constraint (2-9) restricts the total planting area to a given planting acreage Q. The equality in this constraint can be replaced by an inequality ( ) to represent farers choosing not to grow the crops when the production is not profitable. K Tk I k k = t= i= x kti = Q (2-9) To odel producer s risk tolerance, we ipose the CVaR constraint where L(, η ) is a rando loss equal to the negative rando profit f ( η ) x kti k The definition of α CVaR L(, η )) is given by ( x kti k where ζ α ( L, η )) is the α-quintile of the distribution of L( η ) ( x kti k α CVaR( L( x, η )) U. (2-20) 20) enforces the conditional expectation of the rando loss L( η ) kti k [ L( x, η ) L( x, η ) ζ ( L( x, η ))] x kti x kti k x kti, defined in (8). αcvar( L( xkti, ηk )) = E kti k kti k α kti k, (2-2) k,. Therefore, constraint (2-, given that the rando loss exceeds α-quintile to be less than or equal to U. In other words, the expected loss of α-tail, i.e. (- α)00% worst scenarios, is upper liited by an acceptable CVaR upper bound U. Rockafellar k 27

28 & Uryasev (2000) showed that CVaR constraint (2-20) in optiization probles can be expressed by linear constraints (2-22), (2-23), and (2-24) z J ( L( x )) + ζ α kti, η k z U, (2-22) J ( α) K Tk Ik k = t= i= L = ( x,η ) ζ α ( L( x, η )) kti k kti k, (2-23) z 0, (2-24) where z are artificial variables introduced for the linear forulation of CVaR constraint. Note that the axiu obective function contains indenities D that include a ax ter shown in equation (2-0), (2-), and (2-2). To ipleent the odel as a ix 0- linear proble, we transfor the equations to an equivalent linear forulation by disunctive constraints (Nehauser and Wolsey, 999). For exaple, equation (2-0), ki D ki Tk = ax t= ( α iyk Ykt ) xkti,0 β i Pk, can be represented by a set of ix 0- linear constraints D ki Tk Dki t= Tk i k t= D D 0, ki ( α iyk Ykt ) xkti β i Pk ( α Y Y ), kt xkti β i Pk + MZ M( -Z ), ki ( Y Y ) x β P M( -Z ) Tk α i k kt kti i k ki, t= Tk ( α iyk Ykt ) xkti β i Pk MZ ki. t= ki ki, (2-25) where M is a big nuber and Z ki is a 0- variable. Siilarly, equation (2-) and (2-2) can be transfored into a set of ix 0- linear constraints in the sae way. Consequently, the optial crop production and hedging proble has been forulated as a ix 0- linear prograing proble. 28

29 2.2.4 Proble Solving and Decoposition Although the ix 0- linear prograing proble can be solved with optiization software, the solving tie increases exponentially when the proble becoes large. To iprove the solving efficiency, we ay decopose the original proble into sub-probles that could be solved ore efficiently than the original proble. Since only one insurance policy could be selected for each crop, we decoposed the original proble into sub-probles in which each crop is insured by a specific insurance policy. The original proble contains K types of crops, and for the k th type of crop there are I k eligible insurance policies. Therefore, the nuber of the sub-probles is equal to the nuber of all possible insurance cobinations of the K crops, K I k k =. The forulation of the sub-proble is the sae as the original proble except that the index i s are fixed and the equation (2-7) and (2-8) are reoved. Solving sub-probles gives the optial production strategy and futures hedge aount under a specific cobination of insurance policies for K crops. The solution of the sub-proble with the highest optial expected profit aong all sub-probles gives the optial solution of the original proble in which the optial production strategy and futures hedge position are provided fro the subproble solution and the optial insurance coverage is the specific insurance cobination of the sub-proble. 2.3 Case Study Following the case study in Cabrera et al. (2006), we consider a representative farer who grows cotton on a non-irrigated far of 00 acres in Jackson County, Florida. Dothan Loay Sand, a doinant soil type in the region, is assued. The farer ay trade futures contracts fro the New York Board of Trade and/or purchase crop insurance to hedge the crop yield and 29

30 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 farer ay select only one eligible insurance policy to hedge against the risk or opt for none. For APH, the eligible coverage levels of yield are fro 65% to 75% with 5% increents, and the election price is assued to be 00% of the established price. In addition, the available coverage levels of revenue for CRC are fro 65% to 85% with 5% increents. To investigate the ipact of ENSO-based cliate forecast in the optial decisions of production and hedging strategy, we select historical cliate data fro 960 to 2003 for the nuerical ipleentation. ENSO phases during this period included years of El Niño, 9 years of La Niña, and the reaining 25 years of Neutral, according to the Japan Meteorological Index (Table 2-). Table 2-. Historical years associated with ENSO phases fro 960 to 2003 EL Niño Neutral La Niña The cotton yields during the period of were siulated using the CROPGRO- Cotton odel (Messina et al., 2005) in the Decision Support Syste for Agrotechnology Transfer (DSSAT) v4.0 (Jones et al., 2003) based on the historical cliate data collected at Chipley weather station. The input for the siulation odel followed the current anageent practices of variety, fertilization and planting dates in the region. More specifically, a ediu to full season Delta & Pine Land variety (DP55), 0 kg/ha Nitrogen fertilization in two applications, and four planting dates, 6 Apr, 23 Apr, May, and 8 May, were included in the 30

31 yield siulation, which was further stochastically resapled to produce series of synthetically generated yields following the historical distributions (for ore details see Cabrera et al., 2006). Assue cotton would be harvested and sold in Deceber. The Deceber cotton futures contact was used to hedge the price risk. In addition, assue the farer will settle the futures contract on the last trading date, i.e. seventeen days fro the end of Deceber. The historical settleent prices of the Deceber futures contract on the last trading date fro 960 to 2003 were collected fro the New York Board of Trade. The statistics and the rank correlation coefficient Spearan s rho atrix of yields and futures price are suarized in Table 2-2, 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 Niño and Neutral phases, but not in La Niña. We assued the rando yields and futures price follow the epirical distributions of yields and futures price. Table 2-2. Marginal distributions and rank correlation coefficient atrix of yields of four planting dates and futures price for the three ENSO phases Statistics of Marginal Distribution Rank Correlation Coefficient Matrix Spearan s rho ENSO Variable Yield Yield Standard Yield Yield Future Mean on on Deviation on 5/ on 5/8 s Price 4/6 4/23 Yield on 4/6 (lb) Yield on 4/23 (lb) El Niño Yield on 5/ (lb) Yield on 5/8 (lb) Futures Price ($/lb) Yield on 4/6 (lb) Yield on 4/23 (lb) Neutral Yield on 5/ (lb) Yield on 5/8 (lb) Futures Price ($/lb) Yield on 4/6 (lb) Yield on 4/23 (lb) La Niña Yield on 5/ (lb) Yield on 5/8 (lb) Futures Price ($/lb)

32 We further estiated the local basis defined in Equation (2-5). The onthly historical data on average cotton prices received by Florida farers fro the USDA National Agricultural Statistical Service were collected (979 to 2003) as the cotton local cash prices. By subtracting the futures price fro the local cash price, we estiated the historical local basis. Using the Input Analyzer in the siulation software Arena, the best fitted distribution based on iniu square error ethod was a beta distribution with probability density function BETA (2.76, 2.38). We calibrated the Gaussian copula based on the saple rank correlation coefficient Spearan s rho atrix for the three ENSO phases. For each ENSO phase, we sapled 2,000 scenarios of correlated rando yields and futures price based on the Gaussian copula and the epirical distributions of yields and futures price by Monte Carlo siulation. Furtherore, we siulated the basis and calculated the local cash price fro the futures price and basis. We assued the futures coission and opportunity cost of argin 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 paraeters of crop insurance are listed in Table 2-3. Table 2-3. Paraeters of crop insurance (2004) used in the far odel analysis Crop Insurance Paraeters Values APH preiu 65%~75% $9.5/acre ~$38/acre CRC preiu 65%~85% $24.8/acre~$6.9/acre Established Price for APH $0.6/lb Average yield 84 lb/acre Source: Results and Discussion This section reports the results of optial planting schedule and hedging strategy with crop insurance and futures contract for the three predicted ENSO phases. In section 2.4. we assued crop insurances were the only risk anageent tool for crop yield and price risk together with an 32

33 unbiased futures arket 3. In section we considered both insurance and futures contracts were available and assued the future arket being unbiased. In section we investigated the optial decision under biased futures arkets Optial Production with Crop Insurance Coverage This section considers crop insurance as the only crop risk anageent tool. Since the indenity of CRC depends on the futures price, we assue the futures arket is unbiased, i.e., F = Ef where F is the futures price in planting tie and f is the rando futures price in harvest tie. Table 2-4 shows that the optial insurance and production strategies for each ENSO phase with various 90%CVaR upper bounds ranged fro -$20,000 to -$2,000 with increents of $2000. Rearks in Table 2-4 are suarized 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 Niña year has lowest expected profit and highest downside loss. Second, the 65%CRC and 70%CRC crop insurance policies are desirable to the optial 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 anageent can be conducted through changing the planting schedule. The last two rows associated with the Neutral phase shows that planting 00 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 5 acres in date 4. Last, changing the insurance coverage together with the planting schedule ay reduce the downside risk. In the La Niña phase, planting 00 acres in date 4 provides a 90%CVaR of - 3 Although only crop insurance contracts were considered, the unbiased futures arket assuption is need since the indenity of CRC depends on the futures price. 33

34 $4,000 that can be reduced to -$0,000 by purchasing a 65%CRC insurance policy and shifting the planting date fro date 4 to date. Table 2-4. Optial insurance and production strategies for each cliate scenario under the 90% CVaR tolerance ranged fro -$20,000 to -$2,000 with increent of $2000 Optial ENSO 90%CVaR Upper Optial Expected Optial Planting Schedule Insurance Phases Bound Profit Strategy Date Date2 Date3 Date4 El Niño <-8000 infeasible CRC70% to CRC65% > No Neutral < infeasible CRC70% to CRC65% No > No La Niña <-2000 infeasible to CRC65% > No No = no insurance. Planting dates: Date = April 6, Date2 = April 23, Date3 = May, Date4 = May 8. Negative CVaR upper bounds represent profits Hedging with Crop Insurance and Unbiased Futures In this section, we consider anaging the yield and price risk with crop insurance policies and futures contracts when the futures arket is unbiased. Since the crop yield is rando, we define the hedge ratio of the futures contract as the hedge position in the futures contract divided by the expected production. The optial solutions of the planting schedule, crop insurance coverage, and futures hedge ratio with various 90%CVaR upper bounds ranged fro -$24,000 to $0 with increent of $4,000 for the three ENSO phases (Table 2-5). Fro Table 2-5, when the futures arket is unbiased, the futures contract doinating all crop insurance policies is the only desirable risk anageent tool. The optial hedge ratio increases when the upper bound of 90%CVaR decreases. This eans that to achieve lower downside risk, higher hedge ratio is needed. Next, we copare the hedge ratio in different ENSO phases with the sae CVaR upper bound, the La Niña phase has the highest optial hedge ratio 34

35 Table 2-5. Optial solutions of planting schedule, crop insurance coverage, and futures hedge ratio with various 90% CVaR upper bounds ranged fro -$24,000 to $0 with increent of $4,000 for the three ENSO phases ENSO 90%CVaR Upper Optial Expected Optial Insurance Expected Optial Hedge Optial Hedge Optial Planting Schedule Phases Bound Profit Strategy Production Aount Ratio Date Date2 Date3 Date infeasible No No El Niño No No No < No infeasible No Neutral No No No No infeasible La Niña No No No No = no insurance. Planting dates: Date= April 6, Date2= April 23, Date3= May, Date4= May 8. Negative CVaR upper bounds represent profits. and the Neutral phase has the lowest one. Siilar to the result in Table 2-4, the Neutral phase has the highest expected profit and the lowest feasible downside loss. In contrast, the La Niña phase has the lowest expected profit and the highest feasible downside loss. Furtherore, we copare two risk anageent tools: insurance (in Table 2-4) and futures (in Table 2-5). The futures contract provides higher expected profit under the sae CVaR upper bound, as well as a larger feasible region associated with the CVaR constraint. Finally, the optial production strategy with futures hedge is to plant 00 acres in date for the El Niño phase, in date 3 for the Neutral phase, and in date 4 for the La Niña phase Biased Futures Market In the section we assued the futures arket is unbiased. However, the futures prices observed fro futures arket in the planting tie ay be higher or lower than the expected 35