The Effect of Crop or Revenue Insurance on Optimal Hedging. by Keith H. Coble and Richard Heifner

Transcription

1 The ffet of Crop or Revenue Insurane on Optimal Hedging by Keith H. Coble and Rihard Heifner Suggested itation format: Coble, K. H., and R. Heifner The ffet of Crop or Revenue Insurane on Optimal Hedging. Proeedings of the NCR-134 Conferene on Applied Commodity Prie Analysis, Foreasting, and Market Risk Management. Chiago, IL. [

2 The ffet of Crop or Revenue Insurane on Optimal Hedging Keith H. Coble and Rihard Heifnerl The emergene of new risk management tools suh as revenue insurane has dramatially expanded the tools from whih produers may hoose to manage revenue risk. Little is known regarding how these produts interat with market-based risk management tools suh as futures and options. Our analysis addresses this issue by examining optimal futures and put ratios under inreasing levels of insurane overage. Four alternative insurane designs are examined. Two are yield triggered and two reflet urrently available revenue insurane designs. The analysis is onduted by using a revenue simulation model whih inorporates four random variables; futures prie, basis, ounty yield, and farm-ounty yield differenes. Optimal hedge and at-the-money put options ratios are derived for an expeted utility maximizing orn produer in four distint geographial regions. Revenue insurane tends to result in slightly lower hedging demand than would our given the same level of yield insurane overage. To the extent that produers would swith from yield insurane to revenue insurane there would be a deline in the demand for hedging. If a person were to go from being uninsured to the purhase of one of the insurane designs, we fmd that the revenue produts result in a hedge ratio that is at least as high as the uninsured ase when onsidering the permissible levels of overage. The ontext in whih farm program rop produers make futures marketing deisions has been dramatially altered by government poliy in reent years. The elimination of defiieny payments ended a program whih had many similarities to an option ontrat with fixed and nonrisk responsive prodution flexibility payments. This has been widely reognized as having an effet on the risk management deision environment for program rop produers. What has been less often addressed is the nearly simultaneous and rapid evolution of governmentally subsidized insurane produts. The Federal Crop Insurane Corporation was renamed the Risk Management Ageny (RMA) and given broader authority under the 1996 Farm Bill. At the same time, the ageny began a pilot program offering gross rop revenue insurane and allowing private insurane firms to develop other revenue insurane produts whih were aepted for subsidy and reinsurane. To date, three different forms ofrevenue insurane have been offered. Others are likely to be developed. The aeptane of these revenue insurane produts has been fairly dramati. Iowa orn and soybeans insurane data indiates that revenue produts represented more than 3% of insured ares in Beause these revenue insurane produts subsume both prie and yield risk, it is relevant to ask what impliations these insurane produts have for produer forward ontrating demand. If they do affet the demand for forward ontrating, then in what diretion and magnitude. Further, these revenue insurane designs differ in the portion of the gross revenue distribution whih is proteted. Thus, it is plausible that the various revenue insurane designs ould result in 1The authors are Assistant Professor in the Agriultural onomis Department, Mississippi State University and Agriultural onomist at the onomi Researh Servie, USDA. 17

3 substantially differing hedging demand. In fat, this issue has beome a matter of publi debate between the futures industry and rop insurane ompanies in reent months. These issues demand onsideration of the interations of prie and yield risk management and the joint optimization of insurane and hedging deisions. Muh of the literature on produer hedging deisions has assumed non-stohasti yields and a single risk market -futures (Myers and Thompson). Hanson and Ladd, as well as Lapan and Moshini have addressed the market for options assuming deterministi yields. Lapan and Moshini generalized from the assumptions of variane minimization to an expeted utility framework and allow both futures and options to exist. Under their assumptions, options are dominated by hedging. MKinnon and Grant have shown, in a variane minimization framework, that orrelation between prie and yield signifiantly affet the hedging deision. Where negative orrelation exists then a natural hedge results in an optimal hedge lower than expeted output. Sakong, Hayes, and Hallam extended the Lapan, Moshini, and Hanson work by introduing yield risk. In this ontext they found that options are no longer dominated in the hoie set, partiularly when expeted orrelation between yield and prie is non-zero. A natural extension of the previous work is to allow for both yield and prie risk markets in a model of optimal hedging. Reent literature has started to address the omponents of revenue variability and inorporate the instruments whih are hoie variables in a preseason risk management deision proess. Poitras has addressed the analytial issues assoiated with the likely asymmetries of terminal wealth when ensoring instruments suh as options or insurane are available. The ombination of prie and yield futures hedging has been addressed by Li and Vukina for orn in North Carolina, Tirupattur et al. for soybeans in Illinois, and by Heifner and Coble (1996) for orn aross the United States. Combinations of forward priing and rop yield insurane have been examined by Heifner and Coble (1997). Dhuyvetter and Kastens examined ombinations of hedging with yield insurane and with a partiular form of revenue insurane - Crop Revenue Coverage (CRC). However, they do not diretly address hedging levels, but'rather show omparisons of mean and variane of returns. There are several hallenges to modeling the interations of insurane with forward priing instruments. Insurane and futures options ensor the underlying distributions. As will be shown later, some of the insurane designs now being offered to produers are of a mixed type whih ombine omponents of revenue and prie guarantees or prie and yield guarantees. Further, the underlying distributions may be inherently asymmetri, whih leads one away from the tratable ase of joint normality.nelson and Prekel have given strong indiation that yields often appear to be non-gaussian. When modeling the joint distribution of prie times yield, are must also be given to the potential for orrelation of prie and yield to influene outomes. Babok and Hennessy, as well as Heifner and Coble (1996) have shown empirially that ovariane between farm-level prie and yields may be non-trivial. In this paper, we analyze the relationship of four insurane designs at various overage levels to the optimal hedge ratio of a risk averse orn produer. In partiular, we show the sensitivity of optimal hedge and put ratios to varying quantities of insurane. Numerial proedures are used to allow simultaneous evaluation of the insurane designs under realisti empirial representations. In partiular, we are able to inorporate farm-ievel yield information whih more aurately haraterizes yield variability and its effet on optimal deisions The analysis is repliated aross four regionally diverse representative farms. This allows 18

4 omparisons of how differenes in yield variability and yield-prie orrelation affets outomes and shows the diversity of outomes aross regions. The Insurane Tools xamined Four insurane produts are modeled to reflet the insurane produts that are now appearing in the rop insurane market. Two of the designs examined are yield triggered, while the seond two are revenue triggered. A brief explanation of eah instrument follows. Multi-peril rop insurane (MPCI), is the 'traditional' rop insurane program whih is generally available for major rops in most states. MPCI indemnifies yield losses when an insured areage's yield falls below the guaranteed level. These losses are valued at a preseason prie seleted at sign up time. Thus, variation in market pries during the season are not taken into aount. The indemnity equation for MPCI may be written as follows: NI = 1 * Max [y * yo -Yl' ] -p where NI is the net return to insurane purhase,fo is the preseason prie for a harvest month futures ontrat, y is the insurane overage level, Yo and YI are respetively the expeted fann yield at planting and realized yield at harvest? The insurane premium (P) reflets the produer paid insurane premium ost for the poliy. The Market Value Protetion (MVP) design, shown in equation 2, is also yield triggered. However in this ase losses are valued at the maximum of either springtime expeted prie or the atual harvest time prie,.h.prie is multiplied by.95 to reflet basis from the futures market. - (2) NI =.95 * Ma.xlfo, I.] * Ma.x [y * yo -Yl' ] -p In 1997 and 1998, three types ofrevenue insurane have been offered to U.S. produers - Crop Revenue Coverage (CRC), Inome protetion (IP), and Revenue Assurane (RA). All three of these produts insure the gross revenue of the insured rop. The produts differ in rate setting proedures and loation where they are offered. All three are reinsured and subsidized by the USDA and use harvest month futures pries at sign up and at harvest to ompute losses. Beause of similarities in design, IP and RA are treated as a single insurane type designated as RI. quation 3 shows the net return from RI. Here, shortfalls in harvest revenue ifi * YI) trigger losses rather than Y I' as in the ase of yield insurane. (3' NI = Max [y * 1 * yo -it * YI' ] p The fourth insurane design is Crop Revenue Coverage (CRC). This insurane design ombines the revenue insurane protetion ofri with the 'upside' prie protetion ofmvp. Ninety-five perent of the maximum of preseason prie expetations or the atual harvest time futures are used to ompute the overage. 2 The MPCI prie guarantee used for MPCI is based on internal USDA foreasts rather than diretly tied to the futures markets. However, preseason futures pries are used here. 19

5 NI = Max [ y *.95 * Max ' I.] * yo -/ (4) * YI' ] -p The returns from futures marketing are modeled in equation 5. As shown, futures hedging protets against prie risk on a given quantity hedged. The futures marketing hedge ratio is represented by a and represents the proportion of the expeted yield whih is proteted. In this ase the ost of risk protetion, P, reflets ommissions and interest harges to arry out the hedging transation. (5) NF = a * yo * ifo -1;) -p The returns from a put option ontrat are shown in equation 6. In this ase, the put option ratio is represented by a and represents the proportion of the expeted yield whih is prie hedged. The option strike prie relative to the futures prie is y. The ost of a put option, P, inludes the option premium and ommissions and interest harges. NF = a * yo (6) * Max[ y ifo -),O] -p The Behavioral Model To analyze the effet of revenue insurane produts on the demand for hedging, we examine the planting time optimization behavior of a produer offered ombinations of insurane and hedging strategies. The produer is assumed to maximize expeted utility aording to a von Neumann-Morgenstem utility funtion defmed over end of season wealth (W) and whih is stritly inreasing, onave, and twie ontinuously differentiable. The omponents of gross revenue, harvest time prie and rop yield, are assumed stohasti and potentially orrelated. All other parameters of the deision are assumed non-stohasti. For ease of illustration, prie basis and basis risk are omitted in the following equation, but inorporated in the later empirial simulation. The wealth dynami for a produer evaluating growing season risk management strategies may then be written as: (7) Wo + A[f.*yl-C+ NI(y,P,j,y)+NF(a,yo'/)] where the insurane and forward priing deisions are respetively y and a. Initial wealth is represented by W. Crop ares is A and prodution osts are denoted as C. The funtion NI( y j, p J,y) represents the net return to insurane and is generally onditional upon random prie and yield, the premium harged, and the quantity of insurane, y hosen. The funtion NF( a,yo..i) represents the generalized net returns to either futures or put 'option purhase.3 Stohasti Speifiation 3 Produers obviously have a hoie among insurane produts whih is a straightforward generalization of this model. We have hosen to examine eah insurane design separately to onentrate on the relationship of hedging with eah insurane design rather than finding the optimal design for a partiular produer. 1

6 The numerial integration model used in the analysis is a funtion of four random variables, ounty yield deviation from expetation, deviation of farm yield from ounty yield, prie hange from planting to harvest and harvest time basis risk. At deision time, expeted yield, urrent futures prie for the harvest month ontrat fo, and the historial harvest time basis are assumed known. Harvest time futures pries are generated assuming a multipliative shok suh that (8) fl = fo * l, where l is the relative futures prie movement from planting to harvest time and assumed to follow a log-normal distribution. Loal harvest time pries are generated as follows, (9) PI = fo * l + bo + 2, here bo reflets the expeted harvest time basis and 2 equals deviations in the realized basis from the expeted basis. Basis risk, 2' is assumed normally distributed. Farm yields are generated assuming that farm yield may be deomposed into a systemati portion orrelated with ounty yields and non-systemati idiosynrati individual variation..this approah is taken to augment fairly short available farm yield series with the added information available at the ounty level. This relationship may be written as, (9) Yl = YO+PI3+4 where Yo is the expeted faml yield, 131 reflets the systemati relationship between the individual and ounty yields as shown in Miranda, 3 is the deviation in ounty yield from expetation, and 4 is the non-systemati variation in faml yields. Given that we are onstruting a representative faml for a partiular ounty from data for several individuals, the are-weighted average of all faml-ounty yield differenes will equal zero. Miranda also shows that the are-weighted average of all13's within a ounty equals 1. Thus equation 9 may be rewritten as, (1) y 1 = ' where is the expeted ounty yield. The potential non-normality of yields is assumed to be aptured by 3 Following Miranda again, we assume that the non-systemati variation offarmounty yields, 4, are normally distributed. Parameterization of the Model The joint probability distribution of the four random shoks, l' 2' 3' and 4 determine the density of gross revenue and thus, the effets of risk management tools. Given the nonnormal marginal density of l and 3' transformations to approximate normality are used to allow the speifiation of a multi-variate normal distribution whih is used to estimate the probabilities of different outomes. Produt moment orrelations between the four transformed random variables are estimated following the suggestion of Fakler (p. 193). Transformation of the prie distribution is ahieved by using the logarithms of prie.

7 The transformation of 3 is made using the hyperboli tangent transformation proposed by Taylor. The transformation to normality involves first expressing the umulative density as a hyperboli tangent funtion of yield, F(Y) =.5+.5 *tanh(po+pl y +P2 y2+p3 y3) where the Pk are estimated with maximum likelihood, and then finding the standard nonnal value with umulative probability of F(Y). Prior to estimating the Pk, the yields are detrended by regressing on a 2nd order polynomial of time. Heteroskedastiity is orreted by using weighted least squares and by assuming the variane of the t is also a 2nd degree polynomial of time. The expeted futures prie was set at $2.8 and prie variability over the growing season at 19% respetively, based on Marh 1998 observations in the Chiago Board of Trade orn futures and impliit volatility from the options markets. xpeted basis and basis risk measures were onstruted by omparing differenes in NASS data reporting monthly state pries reeived by farmers and the monthly average futures prie at harvest over the period. NASS ounty yield data over the years were used to estimate eah ounty yield distribution.4 The varianes offarm-ounty yield differenes were estimated by ombining farm yield observations provided by the RMA with orresponding ounty yield observations and pooling all farms in the ounty.the orrelations between ounty yield, futures prie, and basis were estimated using transformed data over the period. Correlations between the farm-ounty yield differenes and other variables were set to zero, as they must be on average for all farms in the ounty (Heifner and Coble, 1997). Commission and other trading osts are set at $5 per ontrat, margin deposits at 8%, the farmers interest on margin deposits and options premiums at 8% and the interest rate for priing options at 6%. The insurane subsidy for eah type of insurane is assumed to be equal to 23.5% of the premium for 75% MPCI overage. Representativ Farms The underlying prie, basis, ounty and farm yields were available for a large number of U.S. ounties (Heifner and Coble, 1996). Beause prie variability tends to differ little among farms and basis risk is small relative to prie risk, regional differenes are most apparent in yield variability and yield-prie orrelation. Four ounties were hosen to represent farms from areas with differing levels of yield variability and yield-prie orrelation. Statistis for these ounties are reported in Table 1. Iroquois County in east entrali1linois, was hosen to represent the typial Corn belt ase of relatively low yield variability and yield-prie orrelation that is strongly negative. Shawnee County in east entral Kansas, represents an area with relatively high yield variability and high yield-prie orrelation. Linoln County in west entral Nebraska, is an irrigated area with low yield variability and low yield prie orrelation. Pitt County in east prior to Jerry Skees of the University of Kentuk;y assembled the ounty yield observations 12

8 entral North Carolina, is representative of an area with high yield variability and low yield prie orrelation. Numerial Integration The numerial integration proedure used allows unequal sampling intensity from eah of the four random variates. The range of eah normal variable from -4 to +4 standard deviations is divided into mi equal length intervals. The numbers of intervals used for the results reported here were 39, 1,2, and 2 respetively for futures prie, basis, ounty yield, and farm-ounty yield differene. This gives approximately equal attention to farm-level prie and yield dispersion and reflets the smaller variation in basis than in futures pries. Midpoints of the intervals, Zih' i=i,2,...4, h=1,2,...mj, are determined. ah ombination is assigned a probability proportional to its multi-variate normal density, rn (A>J.= -./Cz.), m J j=1,2,...m. where /m is the proportion of the total probability spae represented by eah set of midpoints, andf(z} is its probability density, I-IIRI-1/2e -1/2e'R-1e where Oj is the standard deviation of the ith normalized variable, n is the number of variables, R is the orrelation matrix, jj = (Zjj -Jlj)/Oj, i = 1, 2,...,n, and Jlj is the mean for the ith normalized variable. Farm-ounty yield differenes are assumed to be unorrelated with the other variables. Inverse transformations are applied at eah midpoint to obtain orresponding values in original units. xpeted Utility Assumptions Certainty equivalent gains are estimated for two ombinations of initial wealth and risk aversion using onstant relative risk aversion (CRRA) utility funtions. Initial wealth levels for a farm with 5 ares of orn is set at $4,. Relative risk aversion is set at 2 to represent moderate risk aversion and 4 to represent high risk aversion. The ertainty equivalent measures show how the individual's initial wealth and degree of risk aversion affet the gains from alternative strategies. However, the estimated ertainty equivalent dollar gains are not neessarily representative beause they rest on assumptions about wealth and risk aversion. Two rounds of numerial integration are performed. The first round estimates the insurane premium, expeted values of rop sales, pries and yields. In the seond round, insurane osts are applied and deviations from respetive means are omputed. Our assumption is that insurane rates are atuarially fair before subsidy and overhead osts are not loaded into the rates. The optimal hedge and put ratio is found by a grid searh of forward priing ratios ranging from to 1 in.1 inrements. At eah grid point the ertainty equivalent gain is evaluated and the optimal hedge is hosen based on the hedge ratio whih maximizes the ertainty equivalent gain. 13

9 Results We begin by examining the optimal hedge ratio without insurane for eah of the four representative farms. Table 2 shows the optimal planting time hedge ratio and ertainty equivalent gain from hedging in eah loation. The differenes in the underlying yield variability and yield-prie orrelation result in optimal hedge ratios ranging from no hedging up to hedging 6% of the expeted rop under both moderate and strong risk aversion. The highest hedge ratio results in the loation where yield-prie orrelation is low and yield variability is relatively low as ompared to other regions due to the predominane of irrigation in this ounty.conversely, the lowest optimal hedge ratio ours in the loation where yield-prie orrelation is strongly negative and yield variability is relatively large. These two loations bear out that the demand for hedging is negatively orrelated with yield variability and yield-prie orrelation. Certainty equivalent gains, whih reflet the inreased produer welfare from risk redution, are also reported. They reveal a generally small gain relative to the per are rop value. However, the greatest gain does ome in Nebraska where the hedge appears most effetive. The two other loations are representative of areas where yield-prie orrelation and yield variability produe a mixed effet. The Iroquois County, Illinois, farm has a ten perent hedge ratio whih is held low by the strongly negative yield prie orrelation, in spite of a relatively low yield variability. The North Carolina farm's base ase hedge ratio is 3%. Here the natural hedge does not exist to limit the optimal hedge ratio, but the relatively large yield variability appears to be a more signifiant fator in revenue variability. To address the effet of various insurane designs, the model was estimated with insurane overage varied from zero to 1% of expeted yield in 12.5% inrements. This allowed us to examine the potentially nonlinear response of the optimal hedge as insurane quantities were inreased. Although insurane overage above the 75% overage is not allowed in any of the programs investigated here, the analysis was arried to the 1% level to more fully reveal the relationship between a partiular insurane program and the optimal hedge ratio. Figure 1 shows the relationships found for eah insurane design for eah of the four loations. First, very low levels of insurane protetion had no effet on the optional hedge ratio in all four representative farms. The Pitt County, North Carolina farm saw the earliest hange in the optimal hedge ratio at 12.5% insurane overage. The Linoln County, Nebraska, optimal hedge was unaffeted by insurane overage until overage reahed the 62.5% level. Given the relevant range of insurane overages offered in the U.S. is the 5% to 75% levels, little hange in observed hedging demand would be expeted from any of the designs in the Nebraska ase. As the insurane overage is inreased for eah of the four loations, a onsistent relationship is found as one ompares aross the four insurane designs. For eah of the loations, at higher overage levels, MVP is always assoiated with the highest optimal hedge. MPCI is the seond highest, with revenue produts, CRC, and RI ranked third and fourth respetively. The two yield insurane designs, when they do ause a hange in the optimal hedge, always result in an inreased optimal hedge. Thus, it appears that the yield insurane designs are found to be purely omplementary to hedging in all four ases. It would appear that the MVP omponent whih indemnifies produers at the greater of preseason prie or harvest time prie does provide a slightly greater optimal hedge than MPCI. This is most obvious in the Anderson County, Kansas, ase. MVP results in a 1% inrease in the optimal hedge as ompared to MPCI when overage is 62.5% and higher. 14

10 The revenue insurane designs show a more omplex relationship with the optimal hedge. The "MVP like" omponent of CRC results in an optimal hedge for CRC that is always equal to or greater than that of RI. In fat, CRC is found to always inrease the optimal hedge over the uninsured ase in three of the four loations. The Linoln County, Nebraska, ase is the exeption with CRC resulting in lower than uninsured hedging levels when CRC overage reahes the 62.5% of expeted revenue. Interestingly, results for RI reveal a distintly nonlinear relationship with hedging in Illinois and North Carolina. Here, an inrease in hedging ours over the mid-range of overages, but as RI overage inreases, the optimal hedge ratio begins to fall. It appears that RI has the strongest substitution effet on hedging of the four insurane designs. The relationship between insurane overage and at-the-money put option ratio is explored in Figure 2. Analyzed over the same range of insurane overages as for futures, the put option perentages tend to follow a similar pattern. Comparing between the hedge and put ratios shows that, in general, the put ratio is higher. We surmise that the higher option ratios our beause options hedgers are not subjet to suh large losses in low yield-high prie years as are futures hedgers who may have to buy bak their ontrats at a high prie. As for hedge ratios, the put ratios ompared aross insurane designs show that when differenes appear, MVP results in the highest put ratio with MPCI, CRC and RI following respetively. The effets of purhasing insurane on the optimal put tends to not beome pronouned until higher levels of overage. In Nebraska, there is no hange until insurane overage reahes 75%. A different relationship is observed between CRC and put option levels than was found in the relationship of CRC and futures hadging. In Figure 2, it an be seen that CRC tends to be more ompetitive with puts than with hedging. For example, in Illinois,. inreasing CRC tended to inrease futures hedging. However, it auses redued put perentages at higher levels of overage. This likely results from CRC being a lower-bounding ativity, whih ompetes more diretly with puts, whih are also lower- bounding. This is in ontrast with futures, whih are lower and upper bounding. In other words, the upside prie protetion provided by CRC is similar to a all option in that the payoff inreases as the prie rises. This omplements a futures hedge given a net position similar to a syntheti put. Suh strong omplementarity is absent when CRC is ombined with a put option. Conlusions The proliferation of new insurane produts greatly hanges the ontext in whih hedging deisions are made. This study was onduted to provide empirial analysis of optimal futures and put hedging levels when produers have yield, prie, and revenue risk management markets available to them. Beause of the diffiulty of jointly onsidering omplex insurane designs and hedging we have taken a numerial approah to addressing the effet of alternative insurane programs on the optimal hedge ratio. In general, revenue insurane tends to result in lower hedging demand than would our given the same level of yield insurane overage. However, the differenes tend to be small (no more than a 1% ) over the relevant range of insurane overage. We also fmd a onsistent pattern that the upside prie protetion of MVP and CRC design tends to be more omplementary to hedging than the RI design. To the extent that produers would swith from yield insurane to revenue insurane, there would be a deline in the demand for hedging. One may also onsider the hedging levels observed with the various insurane produts as ompared to the uninsured 115

11 ase. That is, if a person were to go from being uninsured to the purhase of one of the insurane designs, we find that the revenue produts result in a hedge ratio that is at least as high as the uninsured ase when onsidering the permissible levels of overage. Beause some of the insurane tools examined here are so new to produers and suffiiently distint in thir design, produers at this point may have diffiulty evaluating the deisions modeled here. One might expet that as produers beome more familiar with the impliations of these alternatives, there will be an evolution in how produers utilize the ombinations of insurane and forward priing instruments. We see several natural extensions to this work. Obviously, other rops and regions may be examined. It would also be useful to onsider the joint optimization of areage, insurane and forward priing deisions. Possibilities of further risk redutions by ombining insurane with the joint use of futures and options (or ombinations of options at different strike pries) deserve exploration in light of the Sakong, Hayes and Hallam artile, where insurane was not inluded. 16

12 Referenes Babok, B.A. and D.A. Hennessy. "Input Demand Under Yield and Revenue Insurane.' Amerian Journal of Agriultural onomis 78(1996): Dhuyvetter, K.C. and T.L. Kastens. "Crop Insurane and Forward Priing Linkages: ffets. on Mean Inome and Variane." Paper presented at the NCR-134 Conferene, Fakler, p.l. "Modeling Interdepene: An Approah to Simulation and liitation." Amerian Journal of AQ:riultural onomis 73(1991): Grant, D. "Theory of the Firm with Joint Prie and Output Risk and a Forward Market." Amerian Journal of Agriultural onomis 67985): Hanson, S.D. and G.W. Ladd. "Robustness of the Mean-Variane Model with Trunated Probability Distributions" Amerian Journal of AQ:riultural onomis 73(1991): Heifner, Rihard and K.H. Coble,. "Optimizing Farmers' Joint Use of Forward Priing and Crop Insurane by omparing Revenue Distributions stimated with Numerial Integration" Presented atncr-134 Conferene, Chiago, April 21-22, Heifner, Rihard and K.H. Coble. "Yield-Prie Correlation at the Farm Level: stimation Methods, Findings, and Impliations for Risk Management." Researh Svmosium Proeedings, Chiago Board of Trade, Spring 1996, pages Lapan, Harvey and Gianarlo Moshini. "Futures Hedging Under Prie, Basis, and Produti'on Risk." Amerian Journal of Agriultural onomis 76(Aug. 1994): Li, Dong-feng and Tomislav Vukina. "Crop Yield Futures and Optimal Hedging Strategy for North Carolina Corn Produers." Paper presented at the Chiago Board of Trade Spring Researh Seminar,1996. MKinnon, Ronald I. "Futures Markets, Buffer Stoks, and Inome Stability for Primary Produers." Journal of Politial onomy, 6(1967): Meyers, R.J. and S.R. Thompson. "Generalized Optimal Hedge Ratio stimation" Amerian Journal of Agriultural onomis 71(1989): Miranda, Mario J. "Area- Yield Insurane Reonsidered" Amerian Journal of Agriultural onomis 73(1991 ): Nelson, C.H. and P.V. Prekel. "The Conditional Beta Distribution as a stohasti Prodution Funtion" Amerian Journal of Agriultural onomis 71(1989): Poitras, G. "Hedging and Crop Insurane." Journal of Futures Markets 13(1993):

13 Sakong, Y., D.J. Hayes, and A. Hallam. "Hedging Prodution Risk with Options." Journal of A!riultural onomis 75(1993): Amerian Taylor, C.R. " A Flexible method for mpirially stimating Probability Funtions." Western Journal of A(!riultural onomis 9(1984): Tirupattur, Visvanath, Robert J. Hauser, and Nabil M. Chaherli. Crop Yield and Prie Distributional ffets on Revenue Hedging." Paper presented at the Chiago Board of Trade Spring Researh Seminar,

14 Table 1 --stimated parameters for ounties inluded in the numerial analysis - Parameter Counties Basis, $/bu Mean Std. dev Futures prie-basis Correl County yield-basis Correl Farm prie Mean v Fann yield Mean v Revenue Mean v Fann yield-prie Correl Table 2 -Optimal hedge ratio without insurane. Loation CRRA=2 CRRA=4 Wealth=$4,OOO Iroquios County, IL 2% ($.42) Anderson County, KS % Wealth=$4,OO 3% ($1.21) % ($.) Linoln County, NB 6% ($2.49) ($.) 6% ($6.35)

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