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Hedging with options on commodity futures contracts: a safety-first versus expected utility approach Gaspar, Victor J. 1994

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HEDGING WITH OPTIONS ON COMMODITY FUTURES CONTRACTS:A SAFETY-FIRST VERSUS EXPECTED UTILITY APPROACHbyVICTOR J. GASPARB.A. (Econ.), Simon Fraser University, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department of Agricultural Economics)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 1994© Victor J. Gaspar, 1994In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(Signature)_________________Department of______________________The University of British ColumbiaVancouver, CanadaDate-‘\. 4ABSTRACTThis study evaluates how a decision-maker (such as a farmer) facingoutput price risk might use futures contracts and or option contracts on thosefutures to hedge against any potential financial risk attributed to volatile outputprices. Two behavioral models are assumed in this study. One where thedecision-maker behaves as an expected utility maximizer and one where thedecisions made are based upon safety-first rules. The expected utility model inthis study is based on the general utility function defined in an article by Lapan,Moschini, and Hanson (1991). The safety-first model is essentially that of Telser(1955), but enhanced to include option contracts as an additional hedging tool.Both decision-making processes have a single-period time horizon. At thebeginning of the period an agent enters the futures and option markets andplaces a hedge. At the end of the period, the agent offsets his/her futuresposition and sells the commodity in the spot market. The single-period model isformulated such that a hedger can speculate on the futures price bias, but notthe volatility of the option price.Results from the two competing models were derived from parameterscalculated using a forecast error method on canola data spanning a ten yearperiod (1981-90) obtained from the Winnipeg Commodity Exchange. Optimalhedging results for the two models were derived under varying levels of basisrisk, futures and spot price volatilities, and risk aversion.In general, results from the expected utility model suggest that underincreased volatility, uncertainty, or aversion to risk leads to a reduced openspeculative position when a positive futures price bias exists. Most interestingly,unlike the comparative static results derived by Lapan, Moschini, and Hansonsuggesting that if a speculative motive exists then options are used, the resultsfrom this study’s simulations suggest that the use of options are negligible.Results from assuming a safety-first decision-maker indicate that optionsare always used when speculating on the direction of futures price bias. Whenpositive futures price biases increase in size, so do the futures and optionspositions. The opposite occurs when the bias is decreased or downward. Twomajor conclusions can be drawn from the safety-first results. Firstly, optimalhedging positions seem quite sensitive to “small” variations in the parameterslevels. Secondly, due to the multitude of revenue distributions available fromcombining futures and option (which were unobtainable from using only futures)there is a possibility of very extreme outcomes even though the expected oraverage outcome meets the decision-maker’s “safety” requirements.IIITABLE OF CONTENTSAbstract iiList of Tables viList of Figures viiAcknowledgement ViiiChapter 1 Introduction 11.1 Price Risk and Commodity Markets 11 .2 Options versus Futures 41.3 Problem Statement 61 .4 Study Objectives 81 .5 Thesis Outline 8Chapter 2 The Hedging Contract 102.1 TheTheoryof Hedging 102.2.1 Hedgingwith Futures 102.1.2 Hedging with Options 14Chapter 3 Review of Hedging Literature 183.1 Optimal Hedging with Futures and Options: Expected Utility Model 183.1.1 Futures Contract 183.1 .2 Commodity Options 203.2 Optimal Hedging with Futures: Safety-first Approach 243.3 Expected Utility versus Safety-first 25Chapter 4 Model Specification 304.1 Single-period Representation of Revenue 304.2 Decision Rules 324.2.1 Generalized Expected Utility Model 334.2.2 Safety-first Model 344.3 Evaluating a Safety-first Model 35Chapter 5 Empirical Models 405.1 Empirical Model Derivation 405.1.1 Expected Utility Model 405.1.2 Safety-first Model 415.2 Model Parameter Requirements 435.2.1 Review of Techniques 435.2.2 The Forecast Error Method 445.2.3 Parameter Estimation 45ivChapter 6 Simulation Results and Implications 496.1 Expected Utility Model Results 496.2 Results from the Safety-first Model 516.2.1 Base Case 516.2.2 Sensitivity of the Probability Threshold 576.2.3 Variation of the Volatility 586.2.4 Variation of the Basis Risk 606.2.5 Alternating Strike Prices 626.3 Summary of Safety-first Results 63Chapter 7 Summary and Conclusions 657.1 Summary 657.2 Conclusions 667.3 Restrictions and Further Research 67References 69Appendix 1 The Derivation of Unconstrained Optimal Hedge 73Appendix 2 The Derivation of Equation Al .20 81Appendix 3 Deriving the Value of a Single-period Option 84Appendix 4 Derivation of the Safety-first Probability Constraint 87Appendix 5 Derivation of the Safety-first Model Objective Function 95vLIST OF TABLES1 .1 Futures position assumed after the option is exercised 32.1 Hedging under a constant basis 122.2 Hedging scenario under a weaker-than-expected basis 132.3 Hedging under an unfavorable futures price move 132.4 Hedging with a long put option 155.1 Forecast error statistics for September and October (1981-90) 475.2 The standard deviations for September and October 486.1 Optimal futures and option positions under various scenarios 506.2 Summary table of safety-first model results 64viLIST OF FIGURES1.1 Canola spot prices for 1984 .11.2 Payoff diagram for a short futures and a long put 41.3 Distribution of revenues under alternative hedge scenarios 62.1 Low basis risk 162.2 Moderate basis risk 172.3 High basis risk 173.1 Telser’s figure 1 253.2 Measuring risk by probability of loss versus variance 284.1 Creating a synthetic long call option 314.2 Cummulative density functions under an upward price bias 364.3 Tangency condition for maximization 374.4 Feasible regions under varying bias 384.5 Model summary 396.1 Feasible regions: base case and alternate futures price biases 536.2 Payoff of the base case and alternatives 556.3 Optimal hedge payoffs for varied biases 566.4 Reduce the probability threshold to 10 percent 576.5 Payoff for lower probability threshold at 10 percent 586.6 An increase in the standard deviation to 1.25 596.7 Increase the standard deviation to 1.25 606.8 Higher and lower basis risk 616.9 Payoff diagram for high and low levels of basis risk 626.10 Feasible region under higher and lower strike prices 63viiACKNOWLEDGEMENTI would like to extend a many thanks to my principal advisor, JimVercammen for all his help, guidance and personal interest in my thesis topic. Iwould also like to thank my two other committee members Vasant Naik andGeorge Kennedy for their input. Their time and effort was much appreciated.In addition, I would like to thank the staff, Kathy and Retha. A specialthanks go to Gwynne Sykes for her help with obtaining materials on my researchtopic. I am also grateful to: Susie Latham for being my sounding board at7:30am coffee breaks at the Ponderosa Cafeteria; my brother, Carlos for puttingup with my constant questioning of mathematical and statistical concepts; andVictoria Watson for her immense help in preparing for my thesis defense.Finally, cheers to all my fellow grad students of which I have had thepleasure of interacting (or not interacting) with.“There was a one-lot trader named Fred,who tried to reduce risk with a spread.But the spread was his demise-He overdid position size,trading not one but ten instead.”--Author unknownviiiCHAPTER 1 INTRODUCTIONIi Price Risk and Commodity MarketsHistorically agricultural commodity prices have displayed a more volatilebehavior than the prices of non-agricultural goods and services. This difference isillustrated in the following graph which shows the spot price (per metric tonne) ofcanola traded on the Winnipeg Commodity Exchange [WCE] during 1984. In theperiod between May and July, the spot price was highly volatile. Indeed, pricesrose by approximately 70 percent within that brief period of time.Figure 1.1 Canola spot prices for 1984800a)C=a)a)C.001000Jan ‘ Feb ‘ Mar • Apr • May Jun • Jul • Aug • Sep ‘ Oct ‘ Nov • DecThis high degree of variability can be attributed to two major causes:production uncertainty and stock shifting. Output uncertainty for an agriculturalproduct is due to a variety of effects including adverse weather conditions, andbiological factors such as pests and disease. Stocks are important becauseproduction is normally seasonal and anticipated supply or demand shocks areinstantly reflected in the current price. For example, if during the growing season700.aDo,500200.1a major producing region of soybeans is flooded, then expected soybeanproduction will be lower. Producers, processors, speculators will have anincentive to hold on to their current stocks longer since the future value of thosestocks will increase. Coupling these supply effects with a generally inelasticdemand for agricultural commodities can lead to very large price swings; thus,exposing an agricultural firm (producer, processor) to potentially high levels offinancial risk.The emergence of Chicago commodity markets during the 1800’s allowedan agent to hedge against unfavorable price movements by locking in a price inadvance through a forward contract. However, these forward contracts were notstandardized according to quality or delivery time and agents did not always fulfillthe contract commitments. In 1865, the Chicago Board of Trade (CBOT)alleviated this problem by making available futures contracts which, unlike theforward contracts, were standardized according to quantity, quality, and time andplace of delivery (CBOT, 1989). In Canada, the main commodity exchange is theWinnipeg Commodity Exchange [WCEJ. Following its inception in 1887 as “TheWinnipeg Grain and Produce Exchange”, the WCE introduced futures contracts onwheat, oats, and flaxseed in 1904. Barley futures were offered in 1913, followedby rye in 1917. In 1963, canola (CANadian Oil Low Acid; otherwise known asrapeseed) futures with a Vancouver delivery were introduced (Hore, p.83).A futures contract is simply a standardized legal agreement to make or takea deferred delivery of a specified quantity and quality of particular commodity. Thesettlement is to occur at a pre-determined time and location for a previouslyagreed upon price. An individual entering the futures market with a buy position issaid to have a long position. Alternatively, if the agent enters the market in a sellposition, then she is said to have a short position. If the agent’s futures position isshort (long), then when the contract matures the holder can either make (take)2delivery of the commodity, or she can offset her position by buying (selling) backthe futures contract at the price which the contract is currently trading at. Inpractise, the use of offsetting contracts is the most common occurence with actualdeliveries of the commodity occurring only a small percentage of the time. Thisflexibility is the major advantage of a futures contract over a forward contract.In recent years, commodity options have become available on the majorexchanges and have provided the producers and processors of thesecommodities with an additional risk managing tool.An option on a futures contract is a contractual agreement that is traded oncommodity exchanges through a trading system similar to that of futures.However, unlike futures contracts, an option on a futures contract gives the holderthe right, but not the obligation, to enter the futures market under either a long orshort futures position at a pre-determined price. A call option requires the seller todeliver a long futures position to the purchaser of the option. A put option requiresthe seller to deliver a short futures position to the purchaser of the option.Therefore the buyer of an option, known as having a long option position, has theright to a sell (in the case of a put option) or buy (in the case of a call option) afutures contract at a pre-specified price before the particular futures contract hasmatured. The pre-specified price is known as the strike or exercise price of theoption contract. The futures positions assumed by the contract holder when theoption is exercised are summarized in Table 1 .1 below.Table 1.1 Futures position assumed after the option is exercisedCall Option Put OptionBuyer assumes long futures position short futures positionSeller assumes short futures position long futures position3Both call and put options can be either European or American options.European options only allow the holder to exercise the option when the futurescontract reaches maturity; whereas, the holder of an American option can exerciseit any time while the option is valid.1.2 Options Versus FuturesEven though futures contracts enable a hedger to protect against downsideprice risk, they also prohibit the possibility of gaining from a price rally. Optioncontracts, on the other hand, can provide the agent with similar downside pricerisk protection and allow the agent to benefit from any upside potential. The costof this added flexibility is in the form of an option premium (i.e., the option must bepurchased at a positive price). Figure 1.2 below illustrates more clearly thediffering payoff structure of a futures contract, option contract and an unhedgedposition.Figure 1.2 Payoff diagram for a short futures and a long putNet selling price unhedgedper contract‘t::::tpremium {basis0{ futures price4The hedging scenario in Figure 1.2 is for the specific case of hedger whowishes to lock in a future selling price (e.g., farmer, exporter) under theassumption that the spot and futures price are perfectly correlated.1 The diagramdisplays three potential strategies undertaken by the agent. If the agent decidesnot to hedge any of her spot position, then the price that she receives depends onwhat the futures price is at the time of sale. This relationship is represented by the45 degree line in Figure 1 .2. If the agent were to go short in the futures market,and contract for the amount of stocks held, then she would lock in a net sellingprice equal to the futures selling price less the basis. This is shown as thehorizontal line. In the third strategy, the agent holds a long put option. The longput option provides the agent with the same upside potential as the unhedgedcase (less the premium) once the futures price exceeds the strike (exercise) of theoption (K). Alternatively, when the futures price is below the strike price, then thelong put option locks in a net selling price equivalent to that of the short futuresposition less the value of the premium.An alternative method of differentiating the two hedging tools is comparethe probability density functions [PDF] of revenues for a particular hedgingstrategy. This allows one to explicitly examine the effects of varying amounts ofbasis risk on the distribution of revenues for the hedging agent. Figure 1.3,illustrates the distribution of returns assuming joint normally distributed spot andfutures prices with a less than perfect correlation. Three cases are depicted: theagent (i) remains unhedged, (ii) hedges with only short futures, and (iii) hedgeswith only put options. The agent’s decision to use only futures contracts as ahedging tool allows the agent to essentially lock in a price, thus bringing in the tails1 The difference between the spot and futures price is known as the basis. It can be positive ornegative and represents the cost of storage, insurance, interest on invested capital, andtransportation costs. If the spot and futures price always move in tandem then the basis doesnot change overtime meaning that there is no basis risk5of the revenue distribution, (i.e., implying a small variance of revenues). Thisstrategy is vastly different from the no hedge strategy which leaves the hedgeropen to the full brunt of the potential price risk. As evidenced by the flatter andmore dispersed revenue distribution. If the agent decides to use only commodityoptions then relative to the case of hedging with futures, the PDF skews to theright and shifts to the left.Figurel .3 Distribution of revenues under alternative hedge scenarios1.3 Problem StatementGiven the volatile nature of commodity prices, agricultural firms andproducers have an incentive to insure against price risk. A futures contract is atool which can be used for this purpose. However, with the recent introduction ofoptions on futures contracts, and agent has an additional and more flexible riskmanaging tool at his disposal. It is important to determine the conditions whenGpticns heccnIyhec cn)yRevenues6one hedging tool will be chosen over another and when both will be usedsimultaneously.There exists substantial literature which determines optimal hedgingstrategies using futures contracts. Conversely, relatively few studies haveevaluated the same problem by incorporating both futures and options. Onereason is that from a theoretical perspective, when futures are available optionsare redundant (i.e., will not be used) under the standard assumptions of no pricebiases, no output uncertainty and conventional expected utility maximization,(Lapan, Moschini, and Hanson [LMH], 1991). Thus, when a futures price bias(incentive for speculation) does not exist, an agent will chose the hedging toolwhich is the most efficient at reducing the variability of the expected payoff. Giventhe linear relationship between the spot and futures price and the linearity of thepayoff of a futures contract (as a function of its price), then a futures contract ismore effective at reducing the variance of expected returns than is an optioncontract (which has a non-linear payoff--as is illustrated in Figure 1 .3).This result is unusual given the evidence of a recent survey of Iowa farmerssuggesting that the number of producers hedging with options is equal to that offutures (Sapp, 1990). A more recent study of Montana farmers found that 14% ofcrop farmers use futures and 19% use options; alternatively, 6% of livestockfarmers used futures and 11% use options (Sakong et al, 1993).One explanation as to the popularity of options over futures is that farmersmay not be behaving as expected utility maximizers. In fact, results from a recentproducer survey indicate that the use of safety-first decision rules may important toagricultural firms.2 The survey performed by Patrick et al. (as cited in Atwood et2 Safety-first decision rules are part of the lexicographic family (sequential ordering of multiplegoals) of utility functions. A safety-first rule specifies that a decision maker will act in mannersuch that a preference for safety is followed. Once this safety objective is met, the decisionmaker’s goal involves a profit-oriented course of action.7al., 1988) involved 149 agricultural producers in twelve states and many producersresponses “indicated what could be interpreted as substantial ‘safety-first’considerations in their decision making”. The emergence of safety-first decisionrules in economic literature is not a recent one. Telser (1 955) used a safety-firstapproach to evaluate hedging strategies using futures contracts.1.4 Study ObjectivesWith producer surveys showing results of both the higher use of optionsthan futures and presence of safety-first decision making behavior. The majorobjectives of this study are:1. To develope a one-period hedging model which incorporates safety-firstdecision rules and hedging with both futures and options. The safety-firstmodel is similar to one developed by Telser.2. Contrast the safety-first model to the stamdard expected utility hedgingmodel.3. Compare optimal hedging results from both models through simulations usingparameters estimated from canola price data obtained from the WinnipegCommodity Exchange.1.5 Thesis OutlineChapter Two will describe a typical scenario of a farmer who faces pricerisk and must decide which is the most beneficial marketing objective for hisoperation. Some of the relevant literature involving hedging with both futures andoptions will be discussed in Chapter Three. In Chapter Four the theoreticalmodels are defined, and some of the basic safety-first theory is introduced. Thederivation of the empirical models and the various parameters required for themodels is included in Chapter Five. Chapter Six discusses the simulation results8derived from the empirical models. Lastly, Chapter Seven summarizes the results,provides conclusions and offers suggestions for further research and the potentialdrawbacks of the study.9CHAPTER 2 THE HEDGING CONTRACTThis chapter begins with a description of the motivation for hedging usingfutures and/or options via an analysis of a case farm. Following this, theimplications of basis risk for the effectiveness of the hedge is examined.2.1 The Theory of HedgingThe concept of hedging is based on the principle that prices in the cashmarket and the futures market tend to move together. Although this relationshipmay not be a perfect, it is usually sufficiently close such that a farmer can reducehis price risk in the cash market by taking an opposite position in the futuresmarket. By pursuing such a strategy, losses in one market are countered bygains in the other market. This is better explained through an example.2.1.1 Hedging with Futures1Let us assume that a canola farmer has seeded her crop in May and isexpecting to harvest it in September. In July, the farmer examines her crop andexpects to have a crop yield of 1000 tonnes of #1 canola. Satisfied with thecurrent price of a January futures contract trading on the WCE (i.e., aftersubtracting the basis, the price that remains will be sufficient to cover the pertonne cost of production) she decides to hedge her entire crop against price riskby selling (going short) 50 twenty-tonne contracts of January canola which shewill offset after harvest. The following discussion provides a detailed descriptionof the hedging procedure:1 Much of this section indirectly relies on material published in the Commodity Trading Manualpublished by the Chicago Board of Trade.10STEP 1: The farmer must open an account with a Futures CommodityMerchant. This involves the filling in and signing of: a new clientcommodity application form, a margin form, a risk disclosure form, and inthe case of a hedger, a hedging agreement. The hedging agreementconfirms that all the trades made by the farmer will be for the solepurpose of hedging. Speculative trades must be done through a differentaccount where they will be margined at the full speculative margin.STEP 2: In July, the farmer places a hedge order for 50 twenty-tonne #1canola futures contracts with a January delivery which are trading at $350per tonne. The farmer expects the cash price at the time of delivery to thelocal elevator to be $325 per tonne. The farmer has based thisexpectation upon the expected value of the basis which she believes willequal the current basis. Once the hedge order is placed, the farmer mustprovide an initial margin. The initial margin required by the WCE in thiscase would be (50 contracts @ $450) $22,500 or $22.50 per tonne.Margins set by the Futures Commodity Merchant are sometimes morestringent than those of the WCE.STEP 3: In October, the farmer offsets her position by purchasing 50twenty-tonne January canola futures contracts at $375 per tonne, and shesells her canola to the local oilseed elevator for $350 per tonne.Table 2.1 below summarizes the farmer’s transactions and derives her netselling price for a single contract when the basis is constant. Notice that, byentering the futures market the farmer has locked in a net selling priceequivalent to the cash price she suspected would exist in October at the time ofthe delivery to the elevator. This is known as a “perfect hedge”. Had the farmernot entered the futures market she would have received $325 per tonne, whichis an inferior strategy.11Table 2.1 Hedaina scenario under a constant basiscash market futures market basisJuly expected price of sell Jan. canola -$25canola is $350/t futures @ $375/tOctober sell harvested buy Jan. canola -$25canola @ $325/t futures @ $350/tChange $25/t loss $25/t gain 0.0Amount received from cash sale of canola $325/tGain from futures market transaction $25/tNet selling price $350/tOf course, to assume that the basis remains constant may be overlysimplistic, especially when time and/or local supply and demand factors cangreatly affect the volatility of futures prices. Table 2.2 shows how the farmer willdo when the basis is not constant. A weaker-than-expected (widening) basiscan reduce the effectiveness of the short hedge. Alternatively, if the basis wasstronger-than-expected (narrowing) then the short hedge would have been moreeffective. A simplification made in both Table 2.1 and Table 2.2 scenarios is thatthe futures price movement is favorable, If it was not favorable, then not onlymight the farmer be required to post additional margin money, but she would bebetter off not entering the futures market. This scenario is shown in Table 2.3below.12Table 2.2 Hedging scenario under a weaker-than-exgected basiscash market futures market basisJuly expected price of sell Jan. canola -$25canola is $350/t futures @ $375/tOctober sell harvested buy Jan. canola-$35canola @ $315/t futures @ $350/tChange $35/t loss $25/t gain -$10Amount received from cash sale of canola $3151tGain from futures market transaction $25/tNet selling price $3401tTable 2.3 Hedging under an unfavorable futures price movecash market futures market basisJuly expected price of sell Jan. canola -$25canola is $325/t futures @ $3501tOctober sell harvested buy Jan. canola-$25canola @ $350/t futures @ $375/tChange $25/t gain $25/t loss $0.0Amount received from cash sale of canola $350/tLoss from futures market transaction $25/tNet selling price $325/tIn Table 2.3 the futures price has increased unfavorably from $350 pertonne to $375 per tonne. The farmer has lost $25 per tonne by entering into the13futures market. Fortunately, this loss is compensated by an equivalent gain inthe cash market. This ensures a net selling price equal to the price expected bythe farmer in July. However, in this scenario the farmer would have been betteroft not to have entered the futures market because he could have received a netselling price of $350 per tonne. This returns us back to the discussion inchapter one where it was said that one of the drawbacks of futures contracts wasthat they “prohibit the possibility of gaining from a price rally.” In this case, theflexibility of an option on the futures contract becomes beneficial.2.1.2 Hedging with OptionsBecause farmers who hold options are not required to exercise them, theycan benefit from the gain in the cash market and simultaneously avoid theeffects of any unfavorable movement in the futures price. The cost of this addedflexibility is reflected in the purchase price of the option, referred to as the optionpremium. Table 2.4 describes the potential benefits from purchasing (goinglong) a put option with a $375 per tonne strike price, a $15 per tonne premium,and an expected basis of $25 per tonne under various potential futures pricerealizations.14Table 2.4 Hedging with a long put optionJan. futures Actual Cash Put option Net sellingprice in Oct. basis price gain/loss price($/t) ($/t) ($/t) ($/t) ($/t)300 -25 275 +60 335325 -25 300 ÷35 335350 -25 325 ÷10 335375 -25 350 -15 335400 -25 375 -15 360425 -25 400 -15 385One can see from the example in Table 2.4 that under a constant basis,no matter how far the futures price is below the strike price, the farmer willalways lock in a floor price of $335 per tonne.2 If the futures price exceeds thestrike price of $375, then the farmer can receive a net selling price equivalent tothe current cash price for canola less the option premium. In this respect, theoption contract is similar to conventional insurance.The discussion above evaluated the effects of hedging under specificstrategies and under various basis relationships. Figures 2.1, 2.2, and 2.3 --whichare similar to Figure 1 .3-- provide an alternative means to illustrate hedgingscenarios under increasing levels of basis risk. As the basis risk increases, therevenue PDF for each hedge type becomes flatter, more elongated, and closer tothe no hedge PDF. This implies that when the basis risk is high, the use of eitherfutures or options is not really as effective for hedging against price risk. Withrelatively low basis risk, the futures and option contracts provide the hedger with2 Notice that this floor price is lower than that established by entering the futures market, but thisstrategy allows the farmer to gain if prices move upwards. Also, note how the scenarios of table2.4 translate into the payoff diagram (figure 1.2) of chapter one.15very different outcomes. These differences disappear as the basis risk becomeslarge.Figure 2.1 Low basis riskF utur hec cnlyOia,sno heceR evenues16Figure 12 Moderate basis riskFigure 2.3 High basis riskcØicns hecs crayFutures hececnIyfutures hedge cnlynohecRevenuescpncns heccnIyno hedgeRevenues17CHAPTER 3 REVIEW OF HEDGING LITERATUREThis chapter begins by examining the evolution of optimal hedging theorywhich is primarily concerned with futures contracts only. Two competinghypotheses (standard expected utility maximization and the safety-firstapproach) are discussed. The second section reviews the literature whichincorporates both futures and options on futures into the hedging model.3.1 Optimal Hedging with Futures and Options: Expected Utility Model3.1.1 Futures ContractsThe original notion of an optimal hedge referred to the case where thedecision maker [DMJ would take an equal but opposite position in the futuresmarket as that taken in the cash or spot market. In other words, if you expectedto have (or held in a warehouse in the case of an exporter) Xtonnes of canolathen you would hedge an equal amount in the futures market. Johnson (1959-60) and Stein (1961) were amongst the first studies to use modern portfoliotheory as a means of deriving optimal hedging strategies.1 Their studies showedthat under certain situations the optimal hedge could be less than unity (i.e., the‘traditional hedge’). More recently, studies by Rolfo (1980), Chavas and Pope(1982), KahI (1983), Anderson and Danthine (1983), Meyer and Robinson(1988), and others have continued the use of the mean-variance [MV] frameworkof portfolio theory when determining the optimal hedging decisions using futurescontracts.The results obtained by Johnson and Stein do not necessarily followthose of expected utility maximization except under special conditions. This1 Stein’s method of optimization was based on previous work done by Tobin (1958). In addition,Markowitz (1959) had done research in the area of portfolio selection in a mean-varianceframework.18relationship can be shown by deriving the optimal hedge expected utilityframework. Under the assumption of either a quadratic utility function (Tobin,1958), a normally distributed attribute (Samuelson, 1970), or a random attributewhich is an affine transformation of a single random variable (Meyer, 1987), thenthe mean-variance objective function can be expessed as,U(7t) = E(t)+?Var(7t) (31)where ‘jt is the expected return from the portfolio of assets, and ? represents theArrow-Pratt measure of absolute risk aversion (a positive value denoting a riskaverse individual). For the case of two risky assets, the agent’s cash positionand the corresponding futures position, then the expected return from theportfolio can be expressed as (where the superscripted bars denote expectedvalues),E(t)=XS(2—sl)+XfCf—f) (3.2)where X and X are the spot and futures positions respectively; 2’ f2 the end ofperiod spot and futures prices; s1 , f1 are the spot and futures prices at thebeginning of the period. Similarily, the variance of returns is,Var(t) = Xa + Xo. +2XXo (3.3)where represent the variance of the end-of-period spot price, thevariance of the end-of-period futures price, and the covariance of the end-ofperiod spot price and futures price.19Now substituting equations (3.2) and (3.3) into (3.1) and differentiatingwith respect to X1 (assuming that the spot position is given; hence, known by theagent) then solving out for the futures positions one obtains the followingrelationship,(3.4)X o 2XaIf the second (speculative) component of equation (3.4) equals zero due to noprice bias (i.e., fj =4) then the optimal hedge equals the first component. TheJohnson-Stein mean-variance technique of minimizing risk via the variance ofthe portfolio is equivalent to an expected utility maximization only under thespecial circumstance of no price biases.One benefit of this approach is that the optimal hedge can be derivedempirically by regressing futures prices on spot prices and using classical linearregression to derive the slope coefficient (covariance of spot and futures relativeto the variance of futures) which is equivalent to the ‘optimal hedge’ from MVanalysis.23.1.2 Commodity OptionsWith the introduction of commodity options on the major U.S. exchangesin the early 1980’s, there was a need to examine how one could use options as2 Considerable controversy has arisen around this technique because of the debate as towhether the data used should be in the form of price levels, price changes, or percentagechanges. Both theoretical and statistical arguments have been forwarded as support to either ofthe three forms (e.g., Bond et al., 1987; Benninga et aL, 1984; Witt et al., 1987; Shafer, 1993). Inaddition, statistical detection of serial correlation (e.g., Herbst et al., 1989) and autoregressiveconditional heterosckedastic errors (e.g., Sephton, 1993, or for a bibliography: Bollerslev, 1992)have meant the use of more complex techniques of deriving the risk minimizing optimal hedge.20an additional risk management tool. Ritchken (1985) investigated how singleperiod MV utility maximizers might respond when they include Black-Scholes[BS] priced options in their investment opportunity set. What he found was thatportfolios which lie on the MV efficient set include primarily short rather than longpositions in the options market; thus, making the joint assumption of mean-variance and Black-Scholes pricing theoretically inconsistent. One study whichincorporated the use of both hedging tools was undertaken by Wolf (1987). Heconsidered an optimal hedging strategy in the case of a linear MV model and alogarithmic utility function. Within the MV framework Wolf derived comparativestatic results both with and without basis risk. The logarithmic utility function wasused in his simulation analysis of optimal portfolios. Wolf’s initial assumptionswere that the DM has a fixed position in the physical commodity and that thespot price was nonstochastic. The results under these assumptions were:• A DM which has a long futures position and faces fairly priced options will sella call or purchase a put. The opposite was true for a short futures position.• A DM will purchase a call (put) option to guard against any adverse pricemovements when he holds a short (long) futures position. He will sell the call(put) in order to generate income against an adverse price move for a long(short) futures postion.Once the initial assumptions are relaxed, Wolf concludes that:• In the absence of basis risk and with fairly priced hedging instruments, theDM uses only futures since options are redundant.Although Wolf’s analysis provides useful insights, the validity of his resultsare questionable because the MV model may be inappropriate in his context.This is because the distribution of profits become truncated when options are21introduced into the portfolio.3 Realizing this weakness in the standard MVmodel, Ladd and Hanson, (1991) provide an alternative method that is based onthe generalized expected utility analysis. Their model assumes that there are notransaction costs, basis risk, margin calls, and that the DM has an exponentialutility function which displays constant absolute risk aversion (CARA). Firstly,they derive an income density function which is the sum of two truncated normaldistributions. This is to account for the truncation which occurs when options areincluded in the hedging process. Secondly, Ladd and Hanson build a factorialmodel which is used to determine the input variable levels required for use assubstitutes for market factors in their income density function. Finally, theynumerically integrate and optimize their generalized expected utility model foreach set of estimated market factors. The following are a few of Ladd andHanson’s major results:• If either the futures or options markets are considered to be biased then theDM will speculate by increasing their position in the options market anddecreasing (increasing) their position in short (long) futures.• The availability of an options market in addition to a cash and futures markethas no or little value when the markets are unbiased or biased --within a$0.04 deviation.Ladd and Hanson conclude by saying:From a speculative standpoint, futures and options contracts some whatoffset one another, and allowing the DM to speculate in both marketsdoes not add much value to the DM over allowing him to speculate in justone market.”Including options in a DM’s portfolio can violate any three sufficiency conditions of a MVrepresentation of expected utility maximization: (i) a DM exhibiting a concave utility function witha normally distributed random variable; (ii) the utility attribute is an affine transformation of asingle random variable; (iii) the DM has a quadratic utility function, (Ladd and Hanson, 1991).22Following Ladd and Hanson, is an article by Lapan, Moschini, and Hanson(LMH, 1991). They use an expected utility maximization approach andthrough comparitive statics determine a number of results under thepresence and absence of price biases. In addition, their single-period modelallows for basis risk and assumes that the DM has an exponential utilityfunction exhibiting CARA characteristics. Some of their more relevant resultswere:• If both futures and options prices are unbiased and spot and futures pricesare correlated, then the DM will hedge a proportion () of their output in thefutures market, and no options will be used.4• If futures prices are unbiased, and the relationship between the spot andfutures prices is linear, then a futures contract provides a superior hedge toan option because of the futures linear payoff versus the options non-linearpayoff. Thus, in this case, the use of options is redundant.• If there are perceived biases in the futures prices and/or options premiumsand it is also assumed that there is no basis risk and that the hedger is fullyhedged, then options are used in combination with futures to speculate onthe perceived biases.Bullock and Hayes (1992) modify Wolf’s original model by endogenatingthe variance-covariance matrix of portfolio returns. In their model, theyexamined both scenarios, with and without basis risk. Unlike Wolf, they do notallow for the use of both call and put options since a call option can be createdsynthetically via a combination of a futures contract and a put option.5 Fromtheir model without basis risk, they derived the following observations:Where the proportion () is equal to Cov[b,p]Nar(p) where b is the spot price and p is thefutures price at the end of the period.For more information on this combination and other strategies, see Cox and Rubinstein (1985).23• The futures contract is the preferred instrument for hedging a fixed inventoryor spot position and the option is the preferred instrument when speculating.• The futures contract is always the main speculative instrument whenspeculating on the mean spot price.• The put option is the main speculative tool for information on the variance.With the introduction of basis risk, Bullock and Hayes found that the aboveresults still hold. In addition, they found that increases in the mean basis levelwould induce a DM to hold shorter inventory positions and increases in thevariance of the basis impel the DM to reduce his/her inventory position. Bullockand Hayes also found that the basis information had no impact on the optimaloption use level and that the futures contract is still the most ideal tool to usewhen hedging.63.2 Optimal Hedging with Futures: Safety-first ApproachOne decision rule which has mostly overlooked in hedging models, is thatof the safety-first model. Initially developed by Roy (1952) for the case of assetholding, it was later modified by Telser (1955-6) to include futures. Telser’ssafety-first rule simply stated that a DM maximizes expected returns subject toensuring that returns do not fall below some pre-determined disaster level morefrequently than an accepted likelihood.Similar to the MV studies, Telser found that it was optimal for a DM tohold some combination of hedged and unhedged stocks depending on a DM’sexpectation of spot and futures price movements and their inter-relationship.6Alternatively, Hauser and Eales (1987) use a target-deviation model to evaluate the risk andreturn associated with nine most “commonly” used option strategies when the hedger holds ashort futures position. They find that a put option when the hedger is risk averse over outcomeswhich fall below the expected hedge price, and risk preferring with outcomes above the expectedhedge price. Two other articles include: Hauser and Andersen, Hauser and Eales, 1986.24Telser’s Figure 1 is re-created below (Figure 3.1). Telser states that when gainsare expected from holding both a short hedge and an unhedged stock then theexpected net income will have a negative slope like line (A)and the optimalcombination of hedged and unhedged stocks is given by the tangency of line Ato the elliptic-shaped constraint --point (A’). Also, if a gain is expected from ashort hedge, but not from the unhedged position, then the income line is (B) andthe optimal point becomes (B’). As a third alternative, when gains are expectedfrom a long hedge position, and losses are expected from holding unhedgedstocks then (C) represents the income line and (C’) is the optimal point. In all ofthe cases the expected net income lines increase in the direction of the arrows.Figure 3.1 Telser’s figure 13.3 Expected Utility versus Safety-firstUnlike the expected utility model, which is a commonly accepted means ofmodelling the behavior of agents under uncertainty via the axioms of vonNeumann-Morgenstern, the use of safety-first models is less common.X2 Unhedged Stocks A—vCBX 1 Hedged Stocks25Safety-first models are part of the lexicographic family of utility modelswhich, unlike the expected utility model, have no theoretical base or set ofaxioms. Instead, as the word lexicographic suggests, the utility functions of thesecan be thought of as a representation of sequential goals. The goal of highestpriority must be met first before a decision maker is allowed to consider thesecond goal. The main concept is that a decision maker is first concerned withsatisfying some safety measure, and then once having attained that measure hecan follow a profit maximization objective.Three main forms of safety-first rules exist according to Pyle andTurnovsky (1970). However, Anderson (1979) provides an alternativecatagorization which is adopted in this discussion. In the first catagory,Anderson defines what he calls a safety principle. One of the more familiarsafety principles is that of Roy (1952) which states that a decision maker’sobjective is choosing an action which minimizes the probability of some attribute,usually profits(it), falling below some specified “disaster” level (d*). In otherwords,MinPr(1u.<d*) (3.5)The second category is that of safety first rule. This rule was put forth by Telser(1955-56) and assumes that a decision maker maximizes expected returns E(it),usually profits, subject to a constraint of the probability of returns not fallingbelow some crucial probability (y).Max E(t) s.t. Pr(t d*) ‘y (3.6)26Anderson defines the third category as that of a safety fixed rule. Initiallyintroduced by Kataoka (1963), it involves the maximization of some minimumreturn (d*) such that the probability of returns falling below this minimum level islower than some crucial value (y).Max d* s.t. Pr(it d*) ‘ (3.7)Unlike the expected utility model where risk is commonly (in the simpletwo moment case) represented by a distribution’s second moment, eithervariance or standard deviation, decisions made using safety-first rules can differgreatly from variance-based decisions. Figure 3.2 is a good example of how thetwo methods will have two different outcomes depending on which the decisionmaker is using.In the first graph of Figure 3.2 there are two distributions of revenue, Aand B. Distribution A has a lower mean revenue (Pa) than distribution B (tb);however, distribution B has a larger variance (o) than distribution A (a). Inaddition, if one is to define (d) as the “disaster” level of revenue (perhaps thecost of production or the mortgage payment on the farm) the action that resultsin distribution A will have a greater likelihood (Ya) of revenues falling below thecritical (d). Alternatively, action B has a distribution with a lower probability(almost negligible) of revenues exceeding the critical point (d). In other words, ifa decision maker were to use Telser’s safety-first rule (equation 3.6) he wouldrank distribution A as a more risky venture than distribution B; whereas, anexpected utility maximizer who is very risk averse (i.e., is only concerned withminimizing variance) would rank distribution B as more risky than A.27Figure 3.2.b: y >y, ()X <(3)z distribution’s skewness/ 5: \In graph 3.2.b., there are again two distributions (X and Z) of differingshape. In this case, both distributions have equal means (i.t=) and equalvariances (a = at), but they are skewed in different directions. Distribution X isDiagrams are based upon those depicted in Young (1984 p.33).Figure 3.2 Measuring risk by probability of loss versus variance7Figure 3.2.a: Ya >Tb, a<o, d = “disaster” level of revenueDistnbution AI / \/I Distribution BI /Ii-.I, \\ ,‘ :\A :/1/ I---5-.---2’0&2.000.d Revenue JibI\S\SDistribution X ///Distribution ZSSd = Revenues28negatively skewed ([J3]< 0) and distribution Z is positively skewed ([l’3]> 0). Itis useful to once again make a comparison on how a decision maker mayevaluate the riskiness of either action based upon behaving in either a safetyfirst fashion or an expected utility manner. One would expect that a safety-firstagent would rank the action responsible for distribution X as riskier venture> y. whereas the risk or variance minimizer would be indifferent between thetwo actions.29CHAPTER 4 MODEL SPECIFICATIONIn this chapter a model of a hedger who wishes to lock in a futuresposition then offsets that position at a later date is developed. This is followedby a discussion of the behavior of two hedging agents who follow two differentdecision making rules. One agent decides his optimal hedging strategy by themaximizing the expected utility of the revenues derived from the futures and/oroptions position. The other agent bases his decision upon the maximization of asafety-first rule.4.1 Single-Period Representation of RevenueAt the beginning of the period (t0), the agent decides to hedge a tractionof his operation’s output by taking a short position in the futures market andeither purchasing or selling a put option. At end of the period (t1), the agent liftsthe hedge by offsetting his position in the futures market and then exercising theoption if it has value.This study will consider the use of put options as an agent’s hedging tooland not call options because calls are redundant when in the presence of bothfutures and put options. In other words, one can create a call optionsynthetically via the combination of a put option and a futures contract. Thisinter-relationship is best exemplified using a diagram. In Figure 4.1, the longfutures and long put option positions (designated by the dark solid lines) arecombined to artificially create the long call (dotted line) with a strike price Kand apremium of r.30Figure 4.1 Creating a synthetic long call optionProfit percontractFutures priceGiven these assumptions, one can define (similarly to Lapan, Moschini,and Hanson) the random end-of-period revenue function as:Ibv+(f—p)x÷(K—p—r)z f p<K (4.1)by+(f—p)x—rz if pKwhere:t — end-of-period revenueb end-of-period cash pricey end-of-period output (exogenous)f — futures price at the beginning of the periodp — futures price at the end of the periodx proportion of output hedged in futures (x >0 buying , x<0 selling)r the price of the put option (premium)z proportion of output hedged in options (z>0 buying, z.<0 writing)K — strike (exercise) priceIn the two-state revenue equation above, the revenue expression in the first-state occurs when the futures price (p) is below the option contract strike price+Long0- Long callLong put31(K). The second-state occurs when the futures price exceeds the strike orexercise price.There are a number of assumptions made regarding the generalrepresentation of the above end-of-period revenue:1. There are no costs of production (or profits are stated as net of productioncosts), and production (y) is exogenous and equals one.12. The premium of the put option (r), is assumed to be the compounded value(using the market or riskless rate of interest) of the premium forgone at thetime the hedge is placed (t0).3. There are of no transaction/brokerage costs, no transportation costs, and nomargins required.4. There is no constraint on the agent’s ability to borrow to finance the cost ofthe futures and options contract(s), and he does not face any borrowingcosts.5. The exercise price of the option is exogenous.6. The agent produces a single output.7. The futures and option contract units are perfectly divisible.4.2 Decision RulesThis section discusses the general model used to compare hedging withoptions versus futures. The general model allows for either standard expectedutility maximization or a safety-first approach (maximizing expected revenuesubject to a safety-first constraint).1 Although output uncertainty is an important factor, it is ignored here due to the addedcomplexity to the model. Some studies which touch upon the topic of hedging when both pricerisk and output risk are allowed for include: Rolfo (1980), Chavas and Pope (1982), Grant (1985)and more recently Sakong, Hayes, and Hallam (1993).324.2.1 Generalized Expected Utility ModelIf an agent has a revenue function similar to equation (4.1), and he/she isan expected utility maximizer, then the agent will choose his/her futures and/oroptions position so as to satisfy the following objective:max E[U(t)] =E[U(t1)]+E[UOt2]K (4.2)=JJU(7tV..K...p) f(b,p)dpdb + S 5U(rc..0)f(b,p)dpdbwherethe utiltiy of revenue when the option has value (v),(i.e., thefutures price is below the strike price.U(ico) the utility of revenue when the option has no value (v), (i.e., thefutures price is above the strike price.f(b,p) joint distribution of cash price (b) and futures price (p).U(t)-exp(-Ait) where A Arrow-Pratt level of risk aversion.Equation (4.2) states that the agent will maximize expected utility of revenue;where E[U(t1)] and E[U(it2)} are the expected utility of revenues when thefutures price falls below and above the option strike price, respectively.Similarly to LMH, results from the maximization of equation (4.2) aredependent on an agent’s perception of future prices and option values. Lapan,Moschini, and Hanson found that without price biases there was no need foroptions, and that options were only useful to speculate on biased values offutures prices and options.A price bias is defined as a situation where an agent’s expectation of thefutures price at the end of the period differs from the price of the futures contract(i.e., E(j3)*f). Since the premium of the option is dependent on the expectedvalue of the futures price at the end of the period (viz., f), and the expected33value of the option (as perceived by the agent) depends on the agent’sexpectation of the end of period futures price (fl), then, as in Lapan, Moschini,and Hanson, we can express the value of a single-period put option as follows:E()= $(K—p) h(p;=,a)dp andpremium r = f(K_p) g(p; =f a)dp (4.3)where g(p) and h(p) are the respective marginal distributions of the futures pricewith a mean of and a variance of °‘. Inkeeping with LMH, note that bothdistributions share the same variance, thus any speculative motive based ondiffering perceptions of expected volatility or variance is precluded.4.2.2 Safety-first ModelSimilar to the general expected utility model, if the agent has a revenuefunction equal to that of equation (4.1), then his/her objective function can bedescribed as below:max E[it; x, z] = ff1ty.Kp f(b, p) dpdb + S fv=O f(b, p) dp db (44)subjectto Pr(t itt) yThe agent maximizes his expected revenue via his futures and option contractposition but is subject to a probability constraint. The constraint states that theprobability of an agent’s revenue falling below a predetermined level (‘L) cannotexceed a preset probability denoted above as y. This model is essentiallyTelser’s safety-first model, except that it has been enhanced by introducingoptions in the decision making process.34As in (4.2), equation (4.3) represents the expected revenue depending onwhether futures prices fall above or below the exercise price of the put option.Also similar to the expected utility model above, futures price biases areincorporated into the model.4.3 Evaluating a Safety-first ModelImportant to the use of any model is the establishment of the intuitionbehind the theory through some geometric interpretation. Whereas the expectedutility model involves a straight-forward maximization problem of a concaveobjective function, the safety-first model has a less obvious interpretation. Onemeans of evaluating the safety-first model is to use the common method ofstochastic dominance.2 First-degree stochastic dominance (FSD) states that ifaction F, with a cummulative distribution function (CDF) of F(it) is preferred toaction G with a CDF G(it) ifF(rc)G(it) Vit, where ic€ [a,b] (4.5)Figures 4.2 illustrates the CDF’s for three different hedging strategies for thecase of a ten percent upward futures price bias: (i) short futures and long putcombination, (ii) short futures only, (iii) short futures and short put optioncombination. Using the FSD method to evaluate these variousstrategies, one can see that they all satisfy the safety-first “disaster” level ofrevenue (denoted here as itL) 10 percent of the time. Hence, given that theconstraint is met by all three of the strategies, the optimal strategy must be theCDF with the largest amount of area above it. Visually, this is difficult decision tomake, since different strategies dominate at differing levels of revenue.2 For an indepth discussion of various levels of stochastic dominance see either Anderson (1979,pp. 51-53) or King and Robison (1984, pp.69-72).35Figure 4.2 Cummulative density functions under an upward price bias0.90.80.7. 0.60.52O_ Figure 4.3, the optimization problem is represented in two-dimensionalfutures (X) and options (Z) space for when a price bias exists. The elliptical areaof the graph is defined as the iso-probability frontier, which represents the locusof X and Z combinations that strictly satisfy the safety-first constraint. Any pointwithin this closed set satisfies the constraint, thus comprising the feasible region.The objective function of the safety-first model is depicted as the iso-revenue linein Figure 4•33The slope of the iso-revenue line can be derived by totally differentiatingequation (4.1) as follows:1tdt =—dx+—dz= 0az0=(f—p)dx+(7—r)dz where Y=K—7Re-arranging termsdx (V—r)slope=—=— (4.6)dz (f—p)Figure 4.3 is similar to that of Telser (1955) Figure 1 except that he plots out hedged andunhedged stocks.Shorttilures cnd Long Put Combnaflonulures Only36To maximize expected revenue, the agent would choose combinations of X andZ that push the iso-revenue line in the direction of the two arrows illustrated inFigure 4.3. However, the agent must remain within the feasible region of the iso-probability frontier; therefore, the optimal hedging combination is at a pointwhere the two are tangent to each other.Figure 4.3 Tangency condition for maximizationIn Figure 4.4 the relationship between the size of the feasible region andthe size of the price bias is illustrated. One can observe two major effects.Firstly, as the size of the bias increases (not in absolute terms) the feasibleregion increases in size, and therefore the number of option and futurescombinations satisfying the iso-probability frontier also increase. Secondly, asthe price bias increases, the absolute value of the slope decreases. An increasein the price bias causes the denominator (f—fl) to increase at a quicker ratethan the numerator (—r). Therefore, if the slope diminishes as the biasF utures (short)linelso-Probcbllhiy F roner(i.e.,scteiyconslrdn1Options (long)37becomes larger, the optimal strategy may vary from option/futures combinationsA, B, C, or D. In other words, the agent will normally take a short futuresposition, but may alternate between a long or short option position depending onthe price bias.Figure 4.4 Feasible regions under varying biasIn summary, as highlighted in Figure 4.5, this study evaluates twodecision making rules as applied to a single-period framework. In both modelsthe DM, who has a given quantity of a commodity, can hedge against a potentialloss of revenue by entering into a futures contract and/or an option contract.The DM of the first model is an expected utility maximizer with an exponentialutility function exhibiting constant absolute risk aversion (CARA). The DM of thesecond model makes decisions based on the safety-first rule of Telser. He(shalLcrPi1veBiS rrdlcpms (shatpiDcplcns (lcng put)38maximizes expected revenue subject to the constraint that the revenue does notfall below a pre-specified level of revenue with more than a given likelihood.Both of the models allow for upward (positive) and downward (negative) futuresprice biases. This means that the agents can speculate on the expected valueof the futures price, but not the volatility.Figure 4.5 Model summarySingle-period hedging modelwith futures and optionsunder two alternativedecision-making rulesMax E (it)subject to safety-first probabilityconstraintPr (2t ) ‘Y39CHAPTER 5 EMPIRICAL MODELSIn this chapter, the theoretical models of Chapter 4 are re-specified for thecase where the cash and futures prices are joint normally distributed.Expressions are derived for: the probability constraint; the expected utilityexpression (for both the exponential and linear utility functions); and the value ofa single period option contract. Also, the parameter requirements of the modelsare discussed and several methods of obtaining them are analyzed.5.1 Empirical Model Derivation5.1.1 Expected Utility ModelIf the cash price b and the futures price p are joint normally distributed,then equation (4.2) can be restated as (see Appendix 1):E[UJ = —exp{_A[fr + (K —r)z]}I / — —\ A2r22 22*explAj(x + z)p — yb)÷-y[y a,, — 2py(x + z)a,,o +(x + z) aEK—+A(pya,,a—(x+z)a)*____________________________— expt—A[ — rz][*exp{A(x_yb)+4i[y2a—2pyxa,,a +x2a]}* 1—(K — + A(pya,,o — xo) (5.1)op )In (5.1) p, is the mean futures price and 1 is the mean spot price, and arethe respective variances of p and b. The correlation coefficient between p and bis denoted p. The parameter A represents the Arrow-Pratt measure of risk40aversion and I(.) is the cumulative density function (CD F) of a standard normaldistribution.5.1.2 Safety-first ModelUnlike the expected utility model above, the safety-first model as depictedin equation (4.4) requires two functions to be constructed: (i) the objectivefunction, and (ii) the constraint. The objective function in the safety-first model issimply an expression for expected profits. The constraint limits the probability offalling below a specified level. To specify the objective function for the case ofnormally distributed prices, substitute the appropriate functions represented bythe two states of single-period revenue, (i.e., v=K-p and and integrate.The resulting expression can be written as (see Appendix 5):E(;x,z)=by+(f—)x+z (K_)* K-p K—pG) O)-z (K_f)*1K_+p1Kap (5.2)apJ ajwhere cp(.) denotes the PDF of the standard normal distribution.All of the other variables in equation (5.2) are as defined in Chapter 4.Equation (5.2) allows for the case where futures price biases exist. If no biasesexist, (i.e., f=) then the z[.] terms will cancel each other, and the (f—)xterm will equal zero. Further simplification leaves, E(it;x,z)=bywhich is the expected spot price multiplied by the agents production.To derive an expression for the constraint, it is necessary to transform thedistribution function of the random variable it to a distribution function composedof the random variables b and p. One technique which can be used for this41purpose is the cumulative-distribution-function technique.1 Using this methodone can perform the conversion of the cummulative function of revenue (FJI[ltL])as follows:°° 1k—PFL)Pr(t1tL)=Pr(b+p7tL)= J$f(b,p)dbdp=J $f(b,p)dbdp (5.3)—However, for the specific case of this paper’s two-state revenue function,it is necessary to incorporate the fact that the upper limit of integration (the valueof b) is dependent upon whether the futures price p is above or below the strikeprice of the put option K. Therefore equation (5.3) can be redefined as:(1ki2P> (lk—P1 P2P)’Pr(7uTUL)= $ $ f(b,p)dbdp+J $f(p)dbdp (5.4)where the above parameters are defined as follows:& =fx+(K—r)z a2 =—(x+z) forp<K13=fx—rz for pKEquation (5.4) is still a general expression which must be specified forempirical use. When b, p are joint normally distributed with meansvariances and are correlated by a factor which we will denote as p, thenan expression for (5.4) can be derived and written as (see Appendix 4):F(itL; X,Z) = ff (m) (A1[m])d7u + - (5.5)where =bY—(X+Z)+j’X-i-(K—r)Z____ _____________2 _bYXp+JXrZ= (X+Z)2o+Y2a+2p(X÷Z)YaPob= X2a+Y2a+2PXYGPab1 For a detailed exposition of this technique, please refer to Mood, Graybill, and Boes, pp. 181-188. A more basic analysis is provided by Freund and Wapole, pp. 226-230.K-.(YPab a)@—142f1(it) is the pdf of a univariate normal distribution with mean,and variance a,a1=X÷Z, anda2=X respectively.5.2 Model Parameter RequirementsThis section discusses alternative techniques for obtainging empricalestimates of the parameters specified above. Statistical estimates of the means,p, variances and the correlation coefficient p will allow one to conductcommodity specific analysis; in the case of this paper, canola trading at theWinnipeg Commodity Exchange.5.2.1 Review of TechniquesSince the introduction of mean-variance models for use with commodityfutures by both Johnson (1960) and Stein (1961), there have been a plethora ofstudies which provide econometric methods to derive the parameters needed fora “risk or variance minimizing” optimal hedge ratio from time series data. Aspreviously mentioned, the optimal hedge, as derived using portfolio theory, isdefined as the covariance of the spot and futures prices divided by the varianceof the futures price (let it be denoted as H*=abp/o). Ederington (1979) was oneof the first to use an ordinary least squares technique to estimate the optimalhedge in the case of Government National Mortgage Association (GNMA) and Tbill futures contracts. He suggested that one could derive H* through the simpleregression of spot price levels (B) on futures price levels (P) as follows,B=a+bP÷e (5.6)Where the b (the slope coefficient) is defined in classical linear regression to beequal to or in this case, the risk minimizing hedge ratio.43With the advent of this estimation method there later evolved a myriad ofquestions and debate over such issues as: should equation (5.6) be estimatedusing price levels, price changes, or the ratio of spot market returns to futuresmarket returns; should the data be daily, weekly, monthly etc.; should the databe corrected for heteroscedasticity and/or serial correlation?Myers and Thompson (1989) suggest that the simple regression is just aspecial case of a more generalized linear, reduced form, equilibrium modelwhich they define as follows:b.=X.1a+u, (5.7)p, =X1..j1+v, (5.8)where is a vector of variables at time (t-1), which can be used to forecast thespot price (b )and the futures price (Pt.) Possible variables to be included arespot and futures prices, production, storage, exports, consumer income, andother factors. The a and f3 are vectors of unkown parameters, and u and v arerandom shocks with a mean of zero and serially uncorrelated.By estimating equations (5.7) and (5.8) and collecting their respectiveerror term vectors (u and vi), one can create the necessary variance-covariancematrix. Unfortunately, due to the lack of data available on factors other than spotand futures prices, this method was not used in this study.5.2.2 The Forecast Error MethodThe method used in this study to determine the correlation between thespot and futures prices, so as to be able to use this parameter as a measure ofbasis risk, follows that of Peck (1975) and Rolfo (1980). In Rolfo’s article onderiving optimal hedge ratios for the case of cocoa producing countries, heemploys a method of forecast errors as a means of capturing a measure foruncertainty in both price and quantity. He defines a forecast error as being “the44difference between realized and forecast prices divided by the forecast price.”This differs slightly from Peck, who does not divide the forecasting error by theforecast price. Rolfo suggests that by doing this one allows for differential ratesof historical inflation. Hence, using Rolfo’s method one can define theforecast errors for the spot price and the futures price as follows,Eb=,8Pf (59)What equation (5.9) suggests is that the best one-step ahead predictor of bothspot prices (E[b+1]) and futures prices (E1o+]) are the current values of eachvariable.5.2.3 Parameter EstimationAs indicated above, both models require parameter estimates for themean spot and futures prices (b,), the variance (a ,o ) of both price seriesand in addition, the covariance (ap) of the two prices in order to derive thecorrelation coefficient (p ). Therefore, using the Rolfo’s forecast method onecan define the variance of spot and futures prices as follows,= Var(Eb) (5.10)a =Var(,,)where 8 b’ and 8 represent the forecast errors of spot and futures pricesrespectively. Assuming that the forecast price is an unbiased predictor of therealized price2, then the variance of the forecast errors is simply equal to thesum of the squared forecast errors and the covariance is the sum of thedot-products of the forecast errors (8,,.) divided by the number ofobservations less one (N-i) for correction purposes.2 By assuming that the forecast price is an unbiased predictor of the realized price, it is impliedthat, on average, the forecast error will have a mean of zero.45In this study, the necessary forecast errors are calculated for canolatrading at the Winnipeg Commodity Exchange [WCE]. The data span a ten yearperiod during the years 1981-1990. The spot price data are the daily closingcash prices of Number One Canada canola with the basis in-store Vancouver.The futures price data are the daily futures pices as recorded at the WOE forNumber One Canada canola.As described in chapter 2, it is assumed that in July, the farmer enters thefutures market with a contract which matures in January. After harvest inSeptember, the farmer offsets his position in the futures market and sells hiscrop to the local elevator. Hence, following the Rolfo model, one can re-defineequation 5.9 for this particular scenario as,— bsept—f1 — Psept —8b,sept—p ‘ Ep,sept—pJjul JjuITable 5.1 below, shows the variances, co-variances, and correlation ofspot and futures forecast errors for an individual year and the entire time series(1981-90) for the months of September and October. Note how most of thecorrelational values hinge around the mid to high 0.9 values. Two outliers occurduring the 1984 production year. If one assumes that the farmer has theopportunity to offset his/her position in either September or October, then thisstudy will use an average of the September and October correlation values asestimates of the basis relationship between spot and futures prices after harvest.Thus following this procedure, the base case scenario will have a basis risk of pequal to 0.95. The lower basis risk values can be used as a means of testing themodels’ sensitivity. Having the basis risk value, one still requires an estimate ofthe standard deviation of the spot and futures price at the time of the offset.46Table 5.1 Forecast error statistics for Seotember and October (1981 -90)September OctoberYear 2E 8pb P€ 5p P€81 0.0051 0.0139 0.0082 0.9737 0.0057 0.0084 0.0069 0.999282 0.0022 0.0204 0.0062 0.9191 0.0054 0.0241 0.0113 0.990983 0.11760.0911 0.1035 0.9991 0.1024 0.0776 0.0890 0.998584 0.0008 0.0033 0.0014 0.8252 0.0006 0.0008 0.0004 0.637585 0.0075 0.0097 0.0083 0.9785 0.0235 0.0249 0.0242 0.999586 0.0090 0.0182 0.0128 0.9973 0.0055 0.0106 0.0076 0.996787 0.0037 0.0077 0.0051 0.9549 0.0017 0.0016 0.0012 0.766588 0.0066 0.0110 0.0082 0.9616 0.0205 0.0283 0.0240 0.998289 0.0028 0.0090 0.0049 0.9695 0.0054 0.0114 0.0078 0.993990 0.0036 0.0109 0.0061 0.9761 0.0035 0.0088 0.0055 0.992181-85 0.0267 0.0277 0.0255 0.9391 0.0275 0.0272 0.0264 0.925186-90 0.0052 0.0114 0.0074 0.9719 0.0073 0.0121 0.0092 0.949581-90 0.0151 0.0157 0.0186 0.9346 0.0166 0.017 0.0187 0.9626To obtain the standard deviation of the spot and futures prices one firstfinds the variance of each price as a function of its forecast error. Thus rearranging equation (5.9),b = f(1 +eb), and p = f(1 + e) (5.12)Treating b, p, e,, and e as random variables, one can derive the variances ofthe expressions in (5.12) as,var(b) =f2*v(e)var(p) =f2*v(e) (5.13)If f the mean price in July, is assumed to equal 5.0 then the standard deviationsas displayed in table 5.2 can be calculated.3 It would seem that the price datafrom the period 1981 to 1985 is responsible for a greater proportion of thevolatility for the entire time series spanning 1981-90.Due to stability problems when models with exponential functions have ‘large’ exponents, fwasassigned a value of 5.47Table 5.2 The standard deviations for September and October.Time Period September OctoberSD of futures SD of spot SD of futures SD of spot81-85 0.81 70 0.8325 0.8300 0.824886-90 0.3606 0.5347 0.4288 0.551981-90 0.61 44 0.6264 0.6442 0.6519As a simplification, this study will, in the base case, assume that thestandard deviation of the futures and spot prices are equal and that they bothequal a value of 0.8. Although this is a higher value than that derived for theperiod 1981-90, and lower than 1981 -85, it is felt that the value of 0.8 provides asufficient balance of the two periods, and that any further analysis can involve aninvestigation of the sensitivity of the models to values around the base value.48CHAPTER 6 SIMULATION RESULTS AND IMPLICATIONSIn Chapter 5 the data requirements for the empirical models werediscussed. This chapter involves the evaluation of both the safety-first andexpected utility models using the base case parameters derived in Chapter 5and additional sensitivity analysis. This is then followed by a comparison of theresults of the two different decision making beliefs.6.1 Expected Utility Model ResultsUsing the parameter estimation results from Chapter 5 one can define thefollowing base case scenario:Y=1, b=5, Y=5, Ob=Op=O.8,K=5, f=5.2, p=O.95, A=O.5This base case implies a small futures price bias of 4 percent and a fairly lowlevel of basis risk (p=O.95). The level of risk aversion is obtained via sensitivityanalysis of the model to varying levels of A, and roughly represents a medianpoint.Table 6.1 below provides a summary of the results of the EU model underalternative scenarios. Beginning with the standard no futures price bias case(i.e., no speculative motive) the farmer will hedge a proportion of productionequal to the ratio of the co-variance between the spot and futures price dividedby the variance of the futures price. Using values from the base case, theoptimal hedge involves using 0.95 futures and no options contracts.1 This resultis the standard hedging rule as discussed by Johnson (1959-60) and Stein(1961), where the optimal hedge is equal to the ratio of the covariance of thespot and futures prices and the variance of the futures price.1 Note that the optimal futures X*=O.95 is equivalent to the basis risk (p=O.95). This is due to thefact that the standard deviation of the futures price and spot price are equivalent.49Now, if an upward futures price bias of 4 percent is introduced, theexpected utility maximizer will “overhedge” , (i.e., hedge more than his cashposition), in the futures market, hence X*=1 .58. As one would expect, when theparameters of the model are varied, the optimal futures positions also change.Increasing the base cases’standard deviation (volatility) of the spot and futuresprices from 0.8 to 1 .25 ceteris paribus induces a fall in the optimal futuresposition from 1.58 to 1.21. This reflects the added risk involved holding agreater unhedged position (i.e., speculating with an the open position), when thejoint distribution of prices has a greater volatility or variance.Table 6.1 Optimal futures and option positions under various scenarios.Futures position (X*) Option position (Z*)No bias 0% 0.95Base case 1.58Bias -4% 0.33Higher SD (aD=ab=l .25) 1.21Higher basis risk (p=O.82) 1.45Lower basis risk (p=O.99) 1.62Lower risk aversion (A=0.1) 4.08Higher risk aversion (A=1 .0) 1.26A similar relationship between the optimal futures position and theparameter being varied can be found for the instances of a higher basis risk or ahigher risk aversion. By increasing the basis risk to a value of 0.82 from thebase case value of 0.95, the hedge ratio falls to 1 .45 futures contracts. The fallin the amount hedged is a result of the reduced certainty of the expected change50in the spot price over time as the futures price changes. Recalling chapter one,as the basis risk increased (lower p) then the resulting revenue distributionswere more volatile.An increase in A (Arrow-Pratt risk aversion) from 0.5 to 1.0 means that theagent is more averse to risk and derives a both lower and negative marginalutility (U”(it)<0). Alternatively, if A is lowered to 0.1 or if p is raised to 0.99, theagents optimal futures position increases. However, note that all the optimalhedge postions in Table 6.1 share one characteristic. That is, all the optimaloption positions are approximately equal to zero, (actual values are very smallpositive and negative values). This result is in contrast to the comparative staticresults derived by Lapan, Moschini, and Hanson (LMH, 1991), where they foundthat:The optimal hedging strategy involves using only futures, and the amountof futures is determined by the covariance of cash and futures prices.However, if futures prices and/or options premiums are perceived asbiased, options are typically used along with futures. Thus, in this modeloptions are appealing more as speculative tools to exploit privateinformation on the price distribution, and less so as an alternative hedginginstrument. (p.74.)Thus, the LMH results are not very significant or valuable in this particular case.6.2 Results from the Safety-first Model6.2.1 Base CaseSimilar to the expected utility model, the base case will be a function ofthe same initial parameter values:Y=1, b=5, =5, ab=crP=O.8K=5, f=5.2, p=O.95In addition, the base case assumes that the agent has a limiting revenue (EL)equal to 4.0, and a probability threshold (y) of 0.15. That is, the farmer wants theassurance that his revenue does not fall below the limiting revenue (e.g., a51mortgage payment on the farm, the cost of production) of 4.0 more than 15percent of the time.Figure 6.1 is an illustration of the two-dimensional analysis techniquedescribed in Chapter 4, where the iso-probability frontiers are de-lineated in afutures (X) and options (Z) space (i.e., points on this curve show combinations offutures and options positions that give rise to an equal probability of falling belowthe revenue threshold). An initial observation is that as the magnitude of thepositive futures price bias increases, then the potential combinations of X and Zwhich satisfy the feasible region also increase. As discussed in Chapter 3 andillustrated in that chapter’s Figure 3.1, this study’s interpretation of therelationship between the safety-first’s objective function and its constraint issimilar to that of Telser (1955) except that he examines the behavior of a hedgerwho has a choice between or a combination of hedged (futures) and unhedged(spot) stocks.A second, important observation is that the bottom loci of the isoprobability frontiers tend to share the same (X, Z) co-ordinates. This can beexplained by examining the expected revenue relationship as the futures position(X) approaches zero. As X approaches zero, the expected revenue isdependent primarily on expected spot prices as small changes in the futuresprice f has little impact on it.52Figure 6.1 Feasible regions: base case and alternate futures price biasesFutures (short)5-3As introduced in Chapter 4, the objective of the agent is to maximizerevenue subject to the probability constraint. Thus, the agent would like to pushout the iso-revenue line in the direction of the arrows shown in Figure 6.1, yetremaining within the feasible region of the constraint. The optimal point orhedging combination for the hedger is at the point where the objective functionand the constraint are tangent. In the base case scenario, the optimal point isthat of 1.31 short futures contracts and 3.83 long put options. However, iffutures prices are biased in a downward direction (f < fl’), the optimal valuesoccur on the underside of the feasible region even though the slope of the iso-revenue line has not changed. The reason for this result is that the agent feelsthat the futures contract is underpriced, then, in order maximize expectedrevenues the agent would take a long futures position or certainly reduce his/hershort position from the unbiased case. In Figure 6.1, the point of tangency forBase CaseBias +0.4%(0.91,2.91)-o -4 -2Options(short put)-2aBias -4%(-1.55. 2.12)Options(iong put)53when a 4% downward bias exists is where the optimal position is a long futuresof 1 .55 contracts and 2.12 long put option position.In Figure 6.2 , the payoff of the optimal X, Z combination for the basecase scenario of Figure 6.1 is contrasted with other combinations which,although are not the optimal points, do comprise points on the base cases iso-probability frontier. Of the three strategies, the short futures and short put optionposition of (3.62, -2.2) is the least desirable. Even though the strategy gives theagent the largest net selling price for a futures price realization of 5 (point A), itfails to provide the agent with higher payoffs than the other two when the futuresprice realizations are below or above their expected value. Also, if the futuresprice was to rise to a value above 7.5 then the net selling price would benegative and the agent potentially could suffer infinite losses. It is for this reasonthat an implicit constraint is included along with the safety-first model’sconstraints. The additional implicity constraint is one which precludes the desireof the agent to choose an infinite combination of futures and options that will onaverage satisfy the probability constraint and maximize expected revenue, butpotentially make the hedger susceptable to potentially infinite losses.54Figure 6.2 Payoff of the base case and alternatives3025Average NetSelling Pric0151050-5In contrast, the optimal base case strategy of X=1.31 and Z=3.83, whichhas the lowest average net selling price when the futures price realization equals5, enables the agent to have very high net selling price when the futures pricefalls below 5.0 into the “low risk” region.2 In addition, this strategy provides theagent with protection against any potentially low net selling price realizations inthe “high risk” regions. In other words, the base case scenario is the ideal tacticwhether the realized futures prices are above or below the expected value.2 difference between the net selling price of the base case strategy and the futures onlystrategy when the futures price equals 5 is the amount of the option premium which shifts thepayoff downward.Futures only(2.11,0)Base Case(1.31,3,83)0.5Short FuturesShort Put Options (3.62. -2.2)4.5 5 5,5 6 6.5 7 7.5 8Realized Futures price55Figure 6.3 Optimal hedge payoffs for varied biases3025AverageSng Bose CasePrice 20 (1.31,3,83)Realized Futures PriceFigure 6.3, illustrates the payoffs for the optimal hedges under the basecase scenario, an upward bias of 0.4% and a downward price bias of 4%. Asshown, the base case of (1.31, 3.83) involves the use of a larger quantity of bothfutures and put options as compared to the case of the upward bias of 0.4%.However, the base case also comprises a greater proportion of options than the0.4% upward bias case. This is exemplified by the increased slope of the basecase s payoff function for futures price realizations below 5.0. The increase inthe use of options relative to futures as the size of the upward bias increases,suggests that an option is a more efficient tool to use when both “safety” andspeculative gains are a concern of the agent. Notice how the payoff function forthe case where there is a downward bias of 4% is essentially a mirrored imageof the base case. This is because with a downward price bias of 4% the agentfeels that the price of the futures contract is under valued; thereby, the agent is56wishing to capitalize on the instance that the futures price rises before thefutures contract is offset.6.2.2 Sensitivity of the Probability ThresholdFigure 6.4 Reduce the probability threshold to 10 percent- ‘Put Options(long)A number of results are expected when the probability threshold isreduced to 10 percent as displayed in figure 6.4. Firstly, the magnitude of the Xand Z combinations which satisfy the constraint should decrease; which they dowhen comparing the positive bias of 4% scenario to the base case. Secondly,one would expect the behavior of the agent to be more “risk averse”; thereby,witness a reduction in his optimal futures position. This response does occur;however, in addition to a reduced futures position the agent takes an increasedoptions position. One might expect the opposite to occur. Figure 6.5 belowillustrates why the optimal hedging strategy of (1.05, 4.04) under a 4 percent54 Bias ÷4%Base Case-6 -4 -2Put Options(short) —1-2657upward bias is superior to the strategy of (1.23, 3.6) which involves the use offewer futures and options contracts, yet satisfies the probability constraint. Onceagain, the payoff for the (1.05, 4.04) hedge gives a higher payoff when futuresprice realizations are below 5.0 and creates a superior floor price to that of thealternative (1.23, 3.6) strategy when prices are above 5.0. As before in figure6.3, the optimal strategy under a downward price bias of 4 percent (-1.50,2.33),involves a payoff which is a mirrored image of the upward bias case.Figure 6.5 Payoff for lower probability threshold of 10 percent25Bias +4%( /Average N t :“:N. / B’ 4°!Selling Piic ‘as -Bias +4%(1.23,36)5.—— —0 I I I I I I I I I I I I0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8Realized Futures Price6.2.3 Variation of the VolatilityFigures 6.6 and 6.7 represent the optimal hedge strategies and theirrespective payoffs when the standard deviation of futures and spot prices isincreased from 0.8 to 1.25. One can observe from Figure 6.6 that with anexpansion of the standard deviation, which implies a higher level of volatility, thesize of the feasible region declines from the base case. In conjuction with the58smaller feasible region, the optimal hedge of X* and Z* is also reduced.However, note that the relative point on the iso-probability frontier has notchanged from the base case. This is because the slope of iso-revenue line hasnot changed with the increased volatility of prices. Also, unlike Figure 6.1, theiso-probability frontiers do not share the same lower locus of points.Figure 6.6 An increase in the standard deviation to 1.25-24Put Options(long)When a downward bias of 4 percent exists, the result contrasts the resultsof the previous scenarios. Instead of purchasing put options, the agent sellsthem. In other words, with an increased volatility of futures and spot prices theagent feels that there is less risk involved in speculating with options (shortposition) than with futures (long position). However, this result is obtained byimplicitly constraining the X and Z to combinations which do not give rise topayoffs which fall below zero. This can be seen in the payoff diagram of Figure5Futures (short)4 Bias +4%(1.17, 1.92)Base Case(1.31, 3.83)--I-6 -4 -2Put Option (short)6596.7. The payoff line for the bias -4% case remains above zero for all futuresprice realizations.Figure 6.7 Increase the standard deviation to 1.25Base Case(1,31,3.83)Bias -4%-.. (1,73,-1.89)oI 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8‘-5Realized Futures PriceAlso note that when the standard deviation is increased from the basecase, the optimal payoff strategy is less steep in the “low risk” region and flatterin the “high risk” area. The optimal strategy must have a flatter payoff than thatof the base case because the distribution of the net selling price around themean is more volatile.6.2.4 Varation of the Basis RiskFigure 6.8 shows how the optimal hedging strategies vary from the basecase when the amount of basis risk is increased or decreased. As one mightexpect, when the basis risk is decreased (p=O.99) a larger quantity of X and Zwill satisfy the feasible region. The opposite occurs when the basis risk isincreased (p=O.82). Not only do the available combinations of X and Z satisfyingBias +4%(1,17, 1.93)3025AverageNet 20SellingPrice 1560the safety-first model’s probability constraint decrease under increased levels ofbasis risk, but the optimal strategy is to use fewer long puts and more shortfutures contracts.Figure 6.8 Higher and lower basis riskFutures (short)Figure 6.9 below shows how the agent will choose a strategy whichsacrifices higher net selling prices at low futures price realizations (reflected inthe reduced slope of the payoff in this region) in order to obtain overall higher netselling prices when futures prices exceed 5.0. This is a necessary tacticbecause the distribution of the net selling price is more disperse when the basisrisk is higher; thus, an increased probability of the net selling price dipping belowzero exists.5High Basis Risk(1.50,2.52, p=O.82)Low Basis Riskp=o.99)-6 -4 -2Put Options (short)661Figure 6.9 Payoff diagram for high and low levels of basis risk3050 0,5 1 1.5 2 2.5 3 3.5 4 4,5 5Reaized Futures Price6.2.5 Alternating Strike PricesFinally, Figure 6.10 illustrates how the feasible region are modified whenthe strike price of the put option is alternated. The most obvious observationthat can be made is that the feasible regions pivot around two points on thefutures (X) axis. This is not a surprising result, given that changes in the strikeprice of the option should have no effect on the level of futures contracts whichmake the probability constraint binding.As the strike price is raised by 4 percent, the optimal X increases from theBase Case, but the optimal Z decreases. The opposite occurs when there is adownward bias of 4 percent.Selling Price20Low Basis Risk(1.18, 4.54,P0.99) High Basis Risk(1.5,2.52, p=O.82)10(1.31,3.83)05.5 6 6.5 7 7.5 862Figure 6.10 Feasible region under higher and lower strike prices6.3 Summary of Safety-first ResultsTable 6.2 provides a summary of the results shown graphically above. Asbefore, positive (negative) futures value corresponds to a short (long) position.The opposite is true for the put option values where a positive (negative) valuesimplies a long (short) put option position. One conclusion that can be madeabout the results derived from the safety-first model is that the optimal hedgingpositions tend to be fairly sensitive to the parameter combination. One instancein particular is where the volatility of the futures price and cash price is raised toa standard deviation value of 1.25 from 0.8 (this raises the coefficient of variationfrom 0.16 to 0.25). Under the assumptions of higher futures price volatility and adownward futures price bias of 4 percent exists, the optimal hedging positioninvolves taking a short futures position combined with a short put position. Asmentioned previously, this contradicts the general results of a short futures andlong put position as obtained from varying the other model parameters from the6Futures (short)5I’strike price =5.2(1.74, 2.02) Base Case(1.31,3.83) strike price=4.8(1,24,5.71)/-8 -6 -4 -2Put Options-1(short)-2-38Put Options(long)63base case. In closing, the extreme behavior of the safety-first model undervarying parameter values prevents one from making any generalrecommendations for an agent such as a farmer.Table 6.2 Summary table of safety-first model resultsLevel of Futures Put optionbias position (X*) position (Z*)+4% 1.31 3.83-4% -1.55 2.121.05 4.04-1.50 2.331.17 1.92+4%A 0//0Base caseLower probability threshold (y=O.1)Increased volatility (a=1 .25)Higher basis level (p=O.99)Lower basis level (p=O.82)Higher strike price (k=5.2)Lower strike Drice (k=4.8)+4%-4% 1.73 -1.89+4%+4%+4%+4%1.501.181.741.242.524.542.025.7164CHAPTER 7 SUMMARY AND CONCLUSIONS7.1 SummaryAgricultural commodity prices have through history behaved in apotentially highly volatile manner; thus, leading to the placement of high levels offinancial risk on producers, processors, exporters, and other agents involved inthe procurement and/or sale of these commodities.For some time, the availability of futures contracts has provided theseagents with a risk management tool. Basically, by entering the futures marketvia a futures contract the agent is able to substitute price risk for basis risk Theargument being that basis risk is generally less than price risk. In the early1980’s option contracts on commodity futures were made available on some theU.S. commodity exchanges. Options provided an agent with an additional riskmanagement tool. Options can be used with futures or on their own. Theadvantage of using option contracts is that an option allows for a variety ofrevenue distributions not previouly available with only futures. This is becausethe option is like insurance coverage, the holder of the contract can eitherexercise the option or not. The drawback of this arrangement is, like insurancecoverage, that the holder of the option must forego a premium even if he doesexercise the option.This study has attempted to evaluate how a decision maker, such as afarmer, might use futures contracts and options on those contracts as a meansof managing price risk. Two contrasting models were used for this purpose. Anexpected utility model and a safety-first model. The safety-first model was that ofTelser, except enhanced to include not only futures but options as well. Bothmodels are based upon a single-period framework where at the beginning of theperiod the agent enters the futures and option markets and places a hedge. At65the end of the period the agent offsets his futures position and sells hiscommodity in the spot market. The single-period model was formulated suchthat the agent could speculate on the futures price bias and not the volatility ofthe option price.Once the empirical models were defined, then the parameters werederived for the models using canola data obtained from the WinnipegCommodity Exchange. A technique utilized by Rolfo (1980) was used to obtainthe necessary parameters from the data. The data series spanned a ten yearperiod froml98l to 1990. Optimal hedging results from the two models werederived under alternative parameter scenarios which involved varying levels ofbasis risk, futures and spot price volatilities, and risk aversion.7.2 ConclusionsThe results from the maximization of the expected utility model suggeststhat an agent will (when futures or option price biases exist) hedge an amountequal to that of the co-variance of the spot and futures prices divided by thevariance of the futures price. If a price bias exists in the futures market, then theagent takes the same futures position as the no bias case and then speculateson the bias by either taking an additional position or reducing his existing one;this depends on the direction of the bias.In general, increased volatility, uncertainty, or aversion to risk leads to areduced open speculative position when in the presence of a positive futuresprice bias. Of course, the opposite occurs when these factors are lower than thebase case. The most interesting result of this model is that unlike thecomparative static result of Lapan, Moschini, and Hanson (1991), put options donot figure into the optimal hedge even when there is a speculative component tothe hedge.66An opposite result is obtained under the safety-first model. With thismodel, options are always used as a means of speculating on the upward anddownward future price biases. Other results are similiar to the expected utilitymodel’s.In general, as the price bias increases, so do the overall futures and putoption positions. When the price bias is reduced or is in the downward direction,then the short futures is reduced as is the long put option position. In somecases, the short futures positions may change to a long positions. When the risklevel is reduced (in the safety-first model this is represented by the level of theprobability threshold) the hedging agent increases his long put position andlowers his optimal futures position. When the volatility or standard deviation offutures prices is increased, the agent’s response is to reduce his futures andoptions positions. However, when the bias is negative, the agent increases hisfutures position and writes put options instead of purchasing them. Thisillustrates the rather sensitive nature of the safety-first model to parameterchanges. As the amount of basis risk increases (decreases) the agent takes alarger (smaller) futures position and lowers (raises) his put option position.One weakness of the safety-first approach is that even though the “safety”requirement is being met, the return of a given hedging combination may haveon occasion a very extreme outcome which could involve a large financial loss.7.3 Restrictions and Further ResearchA number of assumptions made in this study may prohibit thecomprehensive generalization of the hedging results. In the case of theexpected utility model, the results could differ significantly if one assumed adifferent utility function from that of the exponential and its constant absolute riskaversion. Also, the assuming away of transaction costs and margins67requirements may be a strong deterant for agents, since this may involve theneed for capital of which they do not have. Certainly, if borrowing is requiredthen it is not obtained at a free rate. In the safety-first model there was aninstance where the hedger choose to write put options instead of purchase them.When writing the option the seller is required to place a margin same as thehedger in the futures market. Another drawback of the study involves the perfectdivisibility of the contracts. Of course, in the real world this is not the case andthe lumpiness of the contract sizes may lead to increased vulnerability to thehedger if the amount he desires to hedge is in-between the two contract sizes.By relaxing the assumption of no output uncertainty for the case of a farmercould modify the results because of the farmer’s hesitation to hedge Xof his cropin case producer fell below X and the farmer would have to payout thedeficiency. This uncertainty could lead to an increased use of options becauseof its contingent exercise aspect.Further enhancements to the methods used here could include theallowance for a multi-period model (perhaps a stochastic dynamic programmingmodel) and further research on the expected utility model to identify why theresults of this study contradict those given by Lapan, Moshini, and Hanson(1991).68REFERENCESAnderson, Jock R., (1979): “Perspectives on Models of Uncertain Decisions.” InRisk, Uncertainty, and Agricultural Developement. J. Roumasset et al.,eds. New York: Agricultural Development Council.Anderson, R.W. and J.P. Danthine, (1983): “Hedger Diversity in FuturesMarkets.” Economic Journal, 93: 370-389.Atwood, J.A., M.J. Watts, G.A. Helmers, and L.J. Held, (1988): “IncorporatingSafety-First Constraints in Linear Programming Production Models.”Western Journal Agricultural Economics, 13(1): 29-36.Benninga, S., R. Eldor, and I. 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Ames, IA: Iowa State University.71Sephton, Peter S., (1993): “Optimal Hedge Ratios at the Winnipeg CommodityExchange.” Canadian Journal of Economics, 1: 175-193.Shafer, Carl E., (1993): “Hedge Ratios and Basis Behaviour: An IntuitiveInsight?” Journal of Futures Markets, 13(8): 837-847.Stein, J.L., (1961): “The Simultaneous Determination of Spot and FuturesPrices.” American Economic Review, 51: 1012-25.Telser, L., (1955-56): “Safety-First and Hedging.” Review of Economic Studies,23: 1-16.Tobin, J., (1958): “Liquidity Preference as Behavior Towards Risk.” Review ofEconomic Studies, 26: 65-86.Witt, Harvey J., Ted C. Schroeder, and Marvin L. Hayenga, (1987): “Comparisonof Analytical Approaches for Estimating Hedge Ratios for AgriculturalCommodities.” Journal of Futures Markets, 7(2): 135-146.Wolf, A., (1987): “Optimal Hedging with Futures Options.” Journal of Economicsand Business, 39: 141-58.Young, Douglas L., (1984): “Risk Concepts and Measures for Decision Analysis.”In Risk Management in Agriculture. Peter J. Barry, ed., Iowa: Iowa StateUniversity Press.72APPENDIX 1 THE DERIVATION OF THE UNCONSTRAINED OPTIMALHEDGECase I:Assume that the cash and futures prices are perfectly correlated (i.e.,b=p). From before we know that the general representation of a decision maker’sprofits are defined as follows.1K-i fp<KIt= py + (f - p)x + (v - r)z where v= KAlternatively,It1 = (y-x-z)p + fx + (K-r)z= (y-x)p + fx -rzEquivalently,= t1p + N1it2 = t2p + N2wheret1= (y-x-z) t2=(y-x)N= fx+(K-r)z N2 = fx-rzDeriving Expected Utility of ProfitsAssumptions:(i) futures price is normally distributed with a mean value of and avariance of a.(ii) the decision maker’s utility function is as follows:U=eAit where A > 0 risk averseA <0 risk prefering73Given the information above, one can derive an expression which denotes adecision maker’s expected utility of profit as a function of his/her position in boththe futures and options markets.E[U]=E{_e} (A1.1)E[U]=_Je1 f(p)dp — Je2f(p)dp (At2)Equation (Al .2) is equal to the sum of the expected utilty of profits whenthe futures price lies below its strike price (E[U1) and the expected utility ofprofits when the futures price lies above its strike price (E[U2]. From this pointonward we will work with the E[U1] portion of equation (Al .2) since analogousstatements can be made for the later half, (E[U2].We can redefine E[U1] as follows:E[U1J = C1 f e’ f(p) dp where 1 = — e’(A1.3)13 =—Ati1 2a)Since f(p)= e‘2itathen one can one can restate (Al .3) in the following manner,KE[U1]=C fe 2dp (AlA)J27to -Similarly,741 KE[U1J=C fe 2’’dp (Al.5)Re-arranging (Al .5),v -1 i(P—P)——i —i—E[U1=C e”Je dp (Al.6)2taRe-arranging the exponent of the natural exponential function,_J[(_)2_2a131(p_)] (Al.7)We want to re-organize the above expression (Al.7) in such a mannerthat we can derive the moment generating function (m.g.f.) of a single, normallydistributed random variable.By completing the square on (p—fl) in (Al .7) we have,‘[(p )2 —2o13 (p— )÷o—a] (Al .8)Simplifying (Al .8),1(.2)2122 (Al.9)Substituting (Al .9) into (Al .6),1 KE[U1J=Ce’2’°’ J 1 e “ dp (Al.lO)q2itaNote that the integrand is the probability density function (p.d.f.) of a normallydistributed random variable with a mean of -i-31a and a variance ofa.Therefore,75E[U1]=Ce11K0 (A1.1 1)OpWhere I(.) is the cumulative density function of the standard normal distribution.Analogously,E[U2]=C $ 2 e c3 ‘ dpK—c e2P2P*1(p_(+a)”i— 2 (A1.12)where C2 = — e2 and 12 = — At2Case II: When the spot price and futures price are not perfectly correlated.Once again we can define the profit function for the two possible states (exerciseoption versus not exercising) as follows:= (y-x-z)p ÷ fx + (K-r)z ifp< Kit2 = (y-x)p + fx -rz ifp KWe can re-define it and it2 as,it=t11b+t2p-i-Nit2 21’+t2p Nwhere t11 =y t =—(x+z) N1 =fx-i-(K—r)z121 =3’ t22 = —x N2 = ft — rzNow we can define the decision maker’s expected utility of profits in the followingmanner,E[U] = — J $e f(b,p)dp db —$ $ e2f(b, p)dp dbK-° (A1.13)= E[U1]-i-E[U276Working with E[U1]we have,E[U1]= C1 J Jeh1b12P f(b,p)dpdb-Aj’where C1 = —e13 =—At1 (A1.14)1312 = —At12Manipulating the integral expression,2exP{13iib+13i_ 2[h_2PhbhP+h]}dPdb2ltabaP..SJ1—p ( Pwhere hb= b—band h = (A1.15)Re-working the exponent of the natural exponential function by multiplyingand 1312 by a ratio of ab/ab and a/a respectively. Then by both adding andsubtracting f311b and 1312 we have,13llab(a13llb+131PP[a]+1312P_(12)[hP+2PhbhP+hP] (A1.16)= 13ii + 13Hli +132oh + 1312k— 2(1 2[hp + 2phbhP + h] (Al .17)Given equation (Al .17), we can re-write (Al.l5) as,27ObOg1—2eh1b412PaboPJ SexP{13llabhb +132oh— 2(1 P2[h— 2phbhP +(Al.18)77b-b___Since we defined h,, = and h = then by differentiating we havedhb=-’-db, dh=-i-dpapNow we can substitute the above differentials into equation (Al.18) and reexpress it as,ePhh12PobaPf$ exP{llabhb+12ah — 12 [h—2phbhP dhPdhb(Al.19)Re-working the exponent of the integrand in (Al .19)— 2(1 — — 2phbhP + h — 2(1 —p2)llabhP — 2(1— p2)iaphp] (Al .20)We can re-write (Al .20) as follows1—Kb!)2—2p(hb Kbj)(hP —K1)+(h _Kpi)2J+(ia+2P11l2Gbap -i-12a)whereKbl llGb+PI3I2GK1 = PI311Gb +l312a(A1.21)By replacing the exponent of the integrand in (Al .19) with (Al .21) we arrive at+2Pllf3labaP+a)E[U1]=C 2*2(i_p2)exP{ 2[(hb —KbI)2—2p(hb Kbl)(hP —K1)+(h_K1)2]}dpdb1 See Appendix 2 for a more thorough derivation.78(Al .22)Note that the integrand in (Al .22) is the p.d.f. of a bivariate joint normaldistribution with the means Kbl , K and variances of one. Therefore we areable to write equation (Al .22) asE[U1 I = C1 *ex{ }*(K—(-i-plloboP l2Gpwhere is a special case of Pl(Al .23)Analogously,—— 1‘ 221321b+22P+13lab+2Pl1I2GGp +13220pE[U2]=C*exp * 1where C2 = —e121 = At21122 =(Al .24)+c2 *exp{i+22P+ + +—[K_K2]](Al .25)Therefore,79Now substituting back in for the C1Ts, 1s and the K1s.E[U] = —exp{_A[fx + (K — r)z]}* exp{_Ay b + A(x + + +[A2y2o— 2pAy(x + z)abaP + A2(x + z)2aJ}pAy app + A(x+ Z)O)l_exp{_A[ft_rzj}[ J* exp{_Ay b + Ax + [A2y2a— 2pAyxabap + A2x2a]}* l_1K_(_PAYabap+Axa) (A1.26)Gathering the A’sE[U] = _exp{_A[fr + (K— r)z}}1• / — —\ A2r2 2 22*expAj(x+ z)p —yb)+---[y 0b — 2py(x+z)oboP +(x+ z) a* —+ A(pyaba —(x+ z)a)1 exp{_A[fr —rz]}L ]*exp{A(x_y)÷i[y2a—2pyxabaP +x2a]}* l1K(pyab.px0a” ) (A1.27)80APPENDIX 2 THE DERIVATION OF EQUATION A1.20Beginning with (Al .20) and then completing the square on both hb, lz (see Moodet al, p.165).2(1—p)[h: — 2PhbhP + — 2(1 — P 3jlGbhb — 2(1 — )1312ah + (1— p2)31o+ 2(1 — r2)P13t1I32abaP + (1—p2)132a]+ 2pI3IllabaP +(A2.1)Now we can re-arrange the term in square brackets into the form ofu2—2puv+v. First let,ii = Jib + Kbv = h + Kuv = hbhP + hbKP + hPKb + KbKP= h +2h1b’<b + K= + 2hK + K (A2.2)Then placing the above expressions into the u2 —2puv-i-v form we have,h, +2hbKb + K, + + 2hK + K — 2phh—2phbKP— 2phPKb—2pKbKP (A2.3)Re-arranging (A2.3),—2P1b”p + + f2[h,K,, + hK — phbKP — phpKbj + K + K —2pKbKP } (A2.4)working with the [.1 term in (A2.4) we can re-write it as,hb[Kb—PKPj+hP[KP —pKb] (A2.5)Using equation’s (A2.5) format, we can make a similar arrangement using the [.]term in (A2.l)._2[(1—p2 )13 lab/lb + (1—p2)1312ohJ (A2.6)From the relationship described in (A2.5) and applying it to (A2.6) we can statethe following,81Kb —pK = —(1—p2)I3110b (A2.7)K—pKb =Taking the above system of equations in (A2.7) we can solve for Kb, K.Kb =pKP—(1—p2)f3llab (A2.8)K = pKb —(1—p2)1312athrough substitution we can solve the equations in (A2.8) such that,Kb = (1 P2)I3llab + —p2)f312a+pKb]= —(llab(A2.9)K= —(12oP +p11Ob)Given (A2.9) we can define the rest of the terms in {.} of equation (A2.4).K = +2p131 f2ab0P+p2f3a= 1i2°p+2p131l2ObOp Pi1b (A2.1O)= +P2131112abaP +13lll2aboP + PIl2Substituting the terms in (A2.1O) into the remaining terms in the {.} of (A2.4)then,K, +K—2pKbKP= 13°+2pfllIjpbap+p2r3o+p2f31a2PI3ii1PzPp÷13iP_2p[pf31o+P2131113120b0p +11I320b0p + pI32aJ(A2.11)Gathering the similar terms,f3llab —P2I3lab2p3jifippp+131Pp —p213a—2p313ll13l2abaP (A2.12)Factoring out the term (1-p2)we have(1—p2)13lab + (1—p2)132° + (1—p2)2P31l1l2abap(1— p2)[lab + 2Plllpbap + 13120p] (A2.1 3)82Note that (A2.13) is essentially the term added to [.] in (A2.l). Therefore wehave now derived an expression which is equivalent to (Al .20) in a quadraticform.83APPENDIX 3 DERIVING THE VALUE OF A SINGLE-PERIOD OPTIONFirst we letG=Jpf(p)dp(A3.1)where f(p) is the probability density function for a normally distributed randomvariable p (price). We can re-write equation (A3.1) in the following manner,G=ff(p)dp $P K dp-- $f(p)dpK (A3.2)where $ f(p) dp is the CDF of the normal distribution evaluated at k.According to Mood et al. (p.124), the expression (.) in equation (A3.2) isthe probability density function (p.d.f.) of a truncated normal distribution with prestricted between -oo and K. Therefore, the term [.] becomes the expectedvalue of the random variable p when p’s distribution is a truncated normaldistribution.Johnson and Kotz (p.81) describe how one can transform the expectedvalue of a random value --whose distribution is a truncated normal distribution-into an expression which is a function of the p.d.f. and c.d.f. of a standard normaldistribution. The expected value of p is given by84E(p)=p. + 0-___a) \awhere P(.) c.d.f. of the standard normalcp p.d.f. of the standard normal (A3.3)A, B the lower and upper limits of the integralSubstituting the upper and lower limits from equation (A3.2) into (A3.3), andnoting that the lower bound --equalling infinity-- reduces (A3.3) to(K-u)E(p) =—(A3A)Therefore,G=(A3.5)G - (A3.6)We can define the value of a single period option as the difference between thestrike price (k) and the price of the underlying asset summed over all potentialprices. Alternatively,E()= J(K-p)f(p)dp(A3.7)wherep-’ N(i.i, 02)Equivalently,85v= K Jf(p)dp- Sf() dp (A3.8)Given that the first integral in equation (A3.8) is equal to F(.) -- the c.d.f. of thestandard normal distribution-- we can then substitute both F(.) and equation(A3.6) into (A3.8) resulting in,v = KF(K)_ K t)a (A3.9)where F(K)Therefore,v = (K_)4(”J + cp(1<Ja (A3.1O)is the value of a single period option as a function of the mean futures price, itsvariance and the strike price.86APPENDIX 4 DERIVATION OF THE SAFETY-FIRST PROBABILITYCONSTRAI NTDeriving an expression for Pr(it) when prices are joint normallydistributed via the cumulative-distribution-function technique.(i) Consider the following profit function for when the price (p) of an underlyingasset is below and then above the strike price (K).where b and p have a joint normal distribution of f(b,p) with means , j5variances of a , a and are correlated by a value of p.(ii) Therefore, for a given and p, we can define the following expression,(—â,Pr(7t)=Ja3(—1 2P)’p3f(b,p)dbdp+ J Jf(b,p)dbdp (A4.1)(A4.2)and dU = (A4.3)a,,The substitution of (A4.3) into equation (A4.1) requires a change in the limits ofintegration; as follows,U[b]=U[(_&12P)r]U{ —a1a1÷a2p+(n_a—a2p)—= ( -)/a,,Jâ+&2p-i-&3b‘j31+13p+1p<KpKwhere t’ =a1 =a2 =a2/cc3=W3l3 =131/133132 132/133define U(b) = (a1+a2p+b)—babwhen p<K(A45)87Analogously,U[bj = ——=— 5)/a (when p K) (A4.6)Re-arranging (A4.3) into the following form,b-= U(b) - (A4.7)Now lets defineq=PP, dq=apwhere the limit of integration is c(K) = K- (A4.8)apRe-arranging (A4.8) with q as the argument and p as the value of the functionand then substituting into equation (A4.7) we can define the followingrelation ships,b—b=U(b)_[1+c2(qap=U-ã0-ã1q1 °2-where=— +—p0b °ba°1 2Substituting (A4.8), (A4.5) into (A4.1) we have the following expression for when(p.K)1 —1 , [(u_ao_ajq)2_2p(u_ao_aq)q+q2]$ $ 2 e2’ dUab dqo (A4.1O)2icl—pMaking analogous substitutions for when (pK)88i —12 (u_o_.iq)q+q2]-2dUob dqa (A41 1)I 2n1—p2 abapK— **____Letting k= , — , t1 = and substituting them into (A4.1O)and (A4.11) respectively, we have,1expl12 {(u_&0—&1q)2—2p(U—&0_aiq)q÷q2]}dudq2(1—p)12{211[(U——— 2p(U——1q)q +q2]}dudq+ exp(A4.1 2)Expanding the expression within the exponent fo the first half of (A4.12) we canre-define it as follows:—1 1U2 u(a0+à1q)+(&0+&1q)2—2pUq+2p(&0+ aiq)q+q2J (A4.1 3)2(1_p2)[Gathering the terms in [.] of (A4.13):_________— —2)21—22(1p)[(U2—2Uc(0 + — q cx +2p&1+ i) — 2q(Uã — — pã + pU)l (A4.14)where U2 — 2Uä0+ -2 = (U—If we define the following relationships:(U—a0)2 U—ã0)2 [i+a +2p&1_(1_p2)l1—p2 = 1+ä+2pà + (l+a+pal_p)(UO) (A4.15)1 + + 2p&1 —(1—p2)= (& + p)2and (A4.16)(Ià—— pà + pU = (U—& )(ã + p)Now we can re-write (A4.14) as:89—(U—&0)2 —1 r(,+p)2(U-&0)+(1+& +2pà1)q—2(&2(1+&÷2p&)2(1_p2)[ 1+&÷2p&(A4.17)Re-arranging (A4.17) into the form of a perfect square in the bracketed [.] term:-(U-a0)2 -(1÷&+2p&)E 2 2(&1+p) (U-ã0)q+I+ [q (l+&+pã) [1+a+2paJ 0)2](A4.1 8)We can simplify (A4.18):2—(U—a0)2 —(1+&+2pä)r (&1+p)U—&0)l (A419)+2(1+&+2pã 2(1—p) (1+ã+2pã jAlternatively,1 (& ÷p)(U-&0)12-(U—a0)2 Lq- (1+ä+2p& j2(1-i-ä-i-2pä (1—p2) (A4.20)(1+& -i-2p&1)Substituting (A4.20) for the exponent in the first half of (A4.12) then we can rewrite --due to the second half being analogous-- then entire expression of(A4.12) as follows:[ (ä1-i-p)(U—ä0)12itr k —1 (U—a0)21L1+?+2Pj1e2(121) * 1 dqdUPr(= ll2(1+ + 2p&) I 2(1p)(1+a+2pd)F—1 (U-0)2 _1[q (1++2p)j+ JJ 1 e2(121) * 1 e2(1_p12 ) dqdUg2t(1÷r3 +2pf31) I 2n(1p2)(1-i.+2p)(A4.21)where90-1 (U-a0)21 2(1+&+2p&,)e/2t(1+ä +2pà1) (A4.22)—1 (U-0)21 2(1++2p)e/it(1+13 +2p131) (A4.23)are the p.d.f.’s of the normal distribution with means a 13 and variances(1+&÷2p&)and (1÷f3 +2pf31) respectively. Also,I (ä1+p)(U—ä0)t(1+ã+2pa) ]1 T(1_p2)/(1+a?÷2paeI 2n(1_p2) (A4.24)(1+ã+2pã)I (÷p)(U40)1—IL’ (1?÷2pi) j1 2(1_p)/(1++2p,)eI 2n(1—p)(1.i-+2p) (A4.25)(& +p)(U—&0and (13k +p)(U—30 andare the p.d.f.’s of a normal with means (1+ã+2pä) (1-i-f3+2p13)(1—p2) (1—p2)variances of respectively.(1+&+2p& ‘ (1-i-3+2p3Recall from (A4.2), (A4.9), and (A4.1 1) the following relationships:* —1 0 **& =—(a+a2),1 = (A4.26)0,where a = =a2/a3 , = 7t/c(3. Aggregating these terms throughsubstitution, we have:—(& + ** ——a3b-cx2apao — , It1 — , a1 = (A4.27)a3 a30J, cx3abGiven the relationships in (A4.27) we can define:-It—(cX1+cL2p-I-c3b)—a0 = (A4.28)a3b91Substituting & from (A4.27) into the constant term of (A4.22),then:427t(1 + + 2pã)=++ 2 2p= + +2PaaPb pa3cYb cL3ab= 1 = cxc427t(&cY +âa+2pâà3abaP) g2it(ac +âcY +2p&&pbcYP)(A4.29)Making the appropriate substitutions from (A4.27) into the exponent of thenatural exponential of (A4.22):r ,-,.lu—I cx1+cx2p apbU—(cl ÷a2p)1 (U-&0)2 L j a3a )] [2 1+ã+2P&1— 21 — 2 &o +&o +2p&2&3aboPX3Gb cc3ab“2222-I-cL2p)}(A4.30)2 + X3Gb +Now if we define:tht=&3obdU (A4.31)If we substitute it into the integral, we need to change the limits of integration in(A4.21) to the following:=÷b] = = (A432)(**)= ]+b]= 3it2 = (A4.33)Solving it(U) for U:u=r_-—i_-= abL3 Jab a3bthen substituting (A4.34) into (A4.30) and combining it with (A4.29) we have:92[a3bu—(al÷a2p]e2&4+2pâà3baPj2t(&a + +2pâ&3abaP) (A4.35)Now, making similar sets of substitutions for (A4.24) as with (A4.22) then:(&2P+pa3Gb)(—(ala÷a))222 22—1 a+aop,,o2 (1-.p2)&a1eI‘2 22it(1—p)&a (A4.36)cxa-I-a3o+2pcxcx3OPObIf we define = & +â5+&3b as the mean level of profits when p<K , and= + + 2p&2&3oPob as the variance of profits when p<K then we canre-write the expression (A4.21) --when making analogous statements for when pK-- as:k(2________Pr(t= LL2To e2*JM1dqtht_________{it_____+ff e 2 afl2 * e dqditwhere M1 = [(1_Pa3ab] , M2 = [(1_ (A4.37)Alternatively, if we integrate with respect to q, then we obtain:2( 2ap+p3b)—t)Iq—— IiiM2—j93nK—15 — (&2aP +PG(3Ob)(tPr(t)=JJt)MdicK— (2aP+P3ob)(it—i2)+ff2()Mdwhere Jj (t) p.d.f. of the normal distribution with mean1 and variancef2(7t) p.d.f. of the normal distribution with mean2 and variance c (A4.38)c. d.f. of the standard normal distributionSubstituting for the cx’s, 13’s, and Ms, we obtain the following:F(ltL; X,Z)= J )4(A1[ii)tht + - (A4.39)where = bY—(X+Z)-i-JX+(K-r)Z K-p (Ypa,_C(1a)(t= bY - X+ JX - rZ ap= (X+Z)2a+Y2a+2p(X+Z)Yoa F(1_p2)Y2al2= X2a+Y2a+ 2pXY aPab [ 2 j94APPENDIX 5 DERIVATION OF THE SAFETY-FIRST MODEL OBJECTIVEFUNCTIONBeginning with the standard two state dependent relationship of thesingle-period revenue model defined in section 4.1 of this paper and in an articleby Lapan, Moschini, and Hanson, (shown below),Ibv÷(f—p)x+(K—p—r)z if p<K7Cby+(f—p)x—rz if pKone can define the expected profits as the expectation of the two states as isshown in equation (A5.1) below.E(n)=$f[by+(f—p)x+(K_p)z—,]g(b,p)dbdp+JJ[by+(f—p)x_rz]g(b,p)dbdp(A5.1)where g(b, p) is the joint normal density functionIf one intergrates over the random variable b then one obtains,1E()=Jyh(p)dp+J[(f—p)x+(K—p)z—rz]h(p)dp+$iyh(p)dp+ f[(f— p)x_rz]h(p)dpK K (A5.2)14’here h(p) is the marginal density function ofg(b, p).Further manipulating expression (A5.2),E()= fyh(p)dp+fvh(p)dp+J(f_p)x h(p)dp+f(f—p)x h(p)dp+zf(K-p) h(p) dp - rz[ Sh(p) dp +J h(p) d] (A5.3)1 According to Mood, Graybill and Boes (p.167), if (X,Y) has a bivariate normal distribution, thentheir marginal distributions X and Y are univariate normal distributions.95Since J(f—p)x h(p)dp-i-f (f—p)x h(p)dp is equivalent to f(f—p)x h(p)dp whichis just the expected value of (f—5)x where denotes the randomness of p.Given this, the bracketed term in -rz[.] equals one. Re-organizing (A5.3),E()= y +(f -)x +zS(K -p) h(p)dp-rz (A5.4)Using the derivation from Appendix 3, (equation (A3.1O)) one can redefineequation (A5.4) as,E()=y+(f-)x+z (K_1K+p1Kapk\a’) a}-z (K_f)1K_+p(Kapi%\a) apjwhere cp[.] PDF of the standard normal density. (A55)CDF of the standard normal density.96


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