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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 APPROACH by VICTOR J. GASPAR B.A. (Econ.), Simon Fraser University, 1989  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Agricultural Economics)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA July 1994 © Victor J. Gaspar, 1994  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted the head by of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of The University of British Columbia Vancouver, Canada Date  -‘\.  4  ABSTRACT This study evaluates how a decision-maker (such as a farmer) facing output price risk might use futures contracts and or option contracts on those futures to hedge against any potential financial risk attributed to volatile output prices.  Two behavioral models are assumed in this study.  One where the  decision-maker behaves as an expected utility maximizer and one where the decisions made are based upon safety-first rules. The expected utility model in this 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 the  beginning of the period an agent enters the futures and option markets and places a hedge.  At the end of the period, the agent offsets his/her futures  position and sells the commodity in the spot market. The single-period model is formulated such that a hedger can speculate on the futures price bias, but not the volatility of the option price. Results from the two competing models were derived from parameters calculated using a forecast error method on canola data spanning a ten year period (1981-90) obtained from the Winnipeg Commodity Exchange. Optimal  hedging results for the two models were derived under varying levels of basis risk, futures and spot price volatilities, and risk aversion. In general, results from the expected utility model suggest that under increased volatility, uncertainty, or aversion to risk leads to a reduced open speculative position when a positive futures price bias exists. Most interestingly, unlike the comparative static results derived by Lapan, Moschini, and Hanson suggesting that if a speculative motive exists then options are used, the results from this study’s simulations suggest that the use of options are negligible.  Results from assuming a safety-first decision-maker indicate that options are always used when speculating on the direction of futures price bias. When positive futures price biases increase in size, so do the futures and options positions. The opposite occurs when the bias is decreased or downward. Two major conclusions can be drawn from the safety-first results.  Firstly, optimal  hedging positions seem quite sensitive to “small” variations in the parameters levels. Secondly, due to the multitude of revenue distributions available from combining futures and option (which were unobtainable from using only futures) there is a possibility of very extreme outcomes even though the expected or average outcome meets the decision-maker’s “safety” requirements.  III  TABLE OF CONTENTS Abstract  ii  List of Tables  vi  List of Figures  vii  Acknowledgement  Viii  Chapter 1 1.1 1 .2 1.3 1 .4 1 .5  Introduction Price Risk and Commodity Markets Options versus Futures Problem Statement Study Objectives Thesis Outline  Chapter 2 The Hedging Contract 2.1 TheTheoryof Hedging 2.2.1 Hedgingwith Futures 2.1.2 Hedging with Options  1 1 4 6 8 8 10 10 10 14  Chapter 3 Review of Hedging Literature 18 3.1 Optimal Hedging with Futures and Options: Expected Utility Model 18 3.1.1 Futures Contract 18 3.1 .2 Commodity Options 20 3.2 Optimal Hedging with Futures: Safety-first Approach 24 3.3 Expected Utility versus Safety-first 25 Chapter 4 Model Specification 4.1 Single-period Representation of Revenue 4.2 Decision Rules 4.2.1 Generalized Expected Utility Model 4.2.2 Safety-first Model 4.3 Evaluating a Safety-first Model  30 30 32 33 34 35  Chapter 5 Empirical Models 5.1 Empirical Model Derivation 5.1.1 Expected Utility Model 5.1.2 Safety-first Model 5.2 Model Parameter Requirements 5.2.1 Review of Techniques 5.2.2 The Forecast Error Method 5.2.3 Parameter Estimation  40 40 40 41 43 43 44 45  iv  Chapter 6 Simulation Results and Implications 6.1 Expected Utility Model Results 6.2 Results from the Safety-first Model 6.2.1 Base Case 6.2.2 Sensitivity of the Probability Threshold 6.2.3 Variation of the Volatility 6.2.4 Variation of the Basis Risk 6.2.5 Alternating Strike Prices 6.3 Summary of Safety-first Results  49 49 51 51 57 58 60 62 63  Chapter 7 7.1 7.2 7.3  65 65 66 67  Summary and Conclusions Summary Conclusions Restrictions and Further Research  References  69  Appendix 1 The Derivation of Unconstrained Optimal Hedge  73  Appendix 2 The Derivation of Equation Al .20  81  Appendix 3 Deriving the Value of a Single-period Option  84  Appendix 4 Derivation of the Safety-first Probability Constraint  87  Appendix 5 Derivation of the Safety-first Model Objective Function  95  v  LIST OF TABLES  1 .1 2.1 2.2 2.3 2.4 5.1 5.2 6.1 6.2  Futures position assumed after the option is exercised Hedging under a constant basis Hedging scenario under a weaker-than-expected basis Hedging under an unfavorable futures price move Hedging with a long put option Forecast error statistics for September and October (1981-90) The standard deviations for September and October Optimal futures and option positions under various scenarios Summary table of safety-first model results  3 12 13 13 15 47 48 50 64  vi  LIST OF FIGURES 1.1 1.2 1.3 2.1 2.2 2.3 3.1 3.2 4.1 4.2 4.3 4.4 4.5 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10  Canola spot prices for 1984 Payoff diagram for a short futures and a long put Distribution of revenues under alternative hedge scenarios Low basis risk Moderate basis risk High basis risk Telser’s figure 1 Measuring risk by probability of loss versus variance Creating a synthetic long call option Cummulative density functions under an upward price bias Tangency condition for maximization Feasible regions under varying bias Model summary Feasible regions: base case and alternate futures price biases Payoff of the base case and alternatives Optimal hedge payoffs for varied biases Reduce the probability threshold to 10 percent Payoff for lower probability threshold at 10 percent An increase in the standard deviation to 1.25 Increase the standard deviation to 1.25 Higher and lower basis risk Payoff diagram for high and low levels of basis risk Feasible region under higher and lower strike prices  .1 4 6 16 17 17 25 28 31 36 37 38 39 53 55 56 57 58 59 60 61 62 63  vii  ACKNOWLEDGEMENT  I would like to extend a many thanks to my principal advisor, Jim Vercammen for all his help, guidance and personal interest in my thesis topic. I would also like to thank my two other committee members Vasant Naik and George Kennedy for their input. Their time and effort was much appreciated. In addition, I would like to thank the staff, Kathy and Retha. A special thanks go to Gwynne Sykes for her help with obtaining materials on my research topic. I am also grateful to: Susie Latham for being my sounding board at 7:30am coffee breaks at the Ponderosa Cafeteria; my brother, Carlos for putting up with my constant questioning of mathematical and statistical concepts; and Victoria Watson for her immense help in preparing for my thesis defense. Finally, cheers to all my fellow grad students of which I have had the pleasure 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 demiseHe overdid position size, trading not one but ten instead.” --Author unknown  viii  CHAPTER 1 INTRODUCTION Ii Price Risk and Commodity Markets Historically agricultural commodity prices have displayed a more volatile behavior than the prices of non-agricultural goods and services. This difference is illustrated in the following graph which shows the spot price (per metric tonne) of canola traded on the Winnipeg Commodity Exchange [WCE] during 1984. In the period between May and July, the spot price was highly volatile. Indeed, prices rose by approximately 70 percent within that brief period of time.  Figure 1.1 Canola spot prices for 1984 800 700.  aDo, a) C =  500  a) a) C. 0 0  200.  100 0 Jan  ‘  Feb  ‘  Mar  •  Apr  •  May  Jun  •  Jul  •  Aug  •  Sep  ‘  Oct  ‘  Nov  •  Dec  This high degree of variability can be attributed to two major causes: production uncertainty and stock shifting. Output uncertainty for an agricultural product is due to a variety of effects including adverse weather conditions, and biological factors such as pests and disease.  Stocks are important because  production is normally seasonal and anticipated supply or demand shocks are instantly reflected in the current price. For example, if during the growing season  1  a major producing region of soybeans is flooded, then expected soybean production will be lower.  Producers, processors, speculators will have an  incentive to hold on to their current stocks longer since the future value of those stocks will increase.  Coupling these supply effects with a generally inelastic  demand for agricultural commodities can lead to very large price swings; thus, exposing an agricultural firm (producer, processor) to potentially high levels of financial risk. The emergence of Chicago commodity markets during the 1800’s allowed an agent to hedge against unfavorable price movements by locking in a price in advance through a forward contract. However, these forward contracts were not standardized according to quality or delivery time and agents did not always fulfill the contract commitments.  In 1865, the Chicago Board of Trade (CBOT)  alleviated this problem by making available futures contracts which, unlike the forward contracts, were standardized according to quantity, quality, and time and place of delivery (CBOT, 1989). In Canada, the main commodity exchange is the Winnipeg Commodity Exchange [WCEJ. Following its inception in 1887 as “The Winnipeg Grain and Produce Exchange”, the WCE introduced futures contracts on wheat, oats, and flaxseed in 1904. Barley futures were offered in 1913, followed by rye in 1917.  In 1963, canola (CANadian Oil Low Acid; otherwise known as  rapeseed) futures with a Vancouver delivery were introduced (Hore, p.83). A futures contract is simply a standardized legal agreement to make or take a deferred delivery of a specified quantity and quality of particular commodity. The settlement is to occur at a pre-determined time and location for a previously agreed upon price. An individual entering the futures market with a buy position is said to have a long position. Alternatively, if the agent enters the market in a sell position, then she is said to have a short position. If the agent’s futures position is short (long), then when the contract matures the holder can either make (take) 2  delivery of the commodity, or she can offset her position by buying (selling) back the futures contract at the price which the contract is currently trading at.  In  practise, the use of offsetting contracts is the most common occurence with actual deliveries of the commodity occurring only a small percentage of the time. This flexibility is the major advantage of a futures contract over a forward contract. In recent years, commodity options have become available on the major exchanges and have provided the producers and processors of these commodities with an additional risk managing tool. An option on a futures contract is a contractual agreement that is traded on commodity exchanges through a trading system similar to that of futures. However, unlike futures contracts, an option on a futures contract gives the holder the right, but not the obligation, to enter the futures market under either a long or short futures position at a pre-determined price. A call option requires the seller to deliver a long futures position to the purchaser of the option. A put option requires the 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 the right to a sell (in the case of a put option) or buy (in the case of a call option) a futures contract at a pre-specified price before the particular futures contract has matured. The pre-specified price is known as the strike or exercise price of the option contract. The futures positions assumed by the contract holder when the option is exercised are summarized in Table 1 .1 below.  Table 1.1 Futures position assumed after the option is exercised Call Option  Put Option  Buyer assumes  long futures position  short futures position  Seller assumes  short futures position  long futures position  3  Both call and put options can be either European or American options. European options only allow the holder to exercise the option when the futures contract reaches maturity; whereas, the holder of an American option can exercise it any time while the option is valid.  1.2 Options Versus Futures Even though futures contracts enable a hedger to protect against downside price risk, they also prohibit the possibility of gaining from a price rally. Option contracts, on the other hand, can provide the agent with similar downside price risk protection and allow the agent to benefit from any upside potential. The cost of this added flexibility is in the form of an option premium (i.e., the option must be purchased at a positive price).  Figure 1.2 below illustrates more clearly the  differing payoff structure of a futures contract, option contract and an unhedged position.  Figure 1.2 Payoff diagram for a short futures and a long put  Net selling price per contract  premium  { 0{  unhedged  ‘t  ::::t  futures price  basis  4  The hedging scenario in Figure 1.2 is for the specific case of hedger who wishes to lock in a future selling price (e.g., farmer, exporter) under the assumption that the spot and futures price are perfectly correlated. 1 The diagram displays three potential strategies undertaken by the agent. If the agent decides not to hedge any of her spot position, then the price that she receives depends on what the futures price is at the time of sale. This relationship is represented by the 45 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 selling price equal to the futures selling price less the basis. This is shown as the horizontal line. In the third strategy, the agent holds a long put option. The long put option provides the agent with the same upside potential as the unhedged case (less the premium) once the futures price exceeds the strike (exercise) of the option (K). Alternatively, when the futures price is below the strike price, then the long put option locks in a net selling price equivalent to that of the short futures position less the value of the premium. An alternative method of differentiating the two hedging tools is compare the probability density functions [PDF] of revenues for a particular hedging strategy. This allows one to explicitly examine the effects of varying amounts of basis risk on the distribution of revenues for the hedging agent.  Figure 1.3,  illustrates the distribution of returns assuming joint normally distributed spot and futures prices with a less than perfect correlation. Three cases are depicted: the agent (i) remains unhedged, (ii) hedges with only short futures, and (iii) hedges with only put options.  The agent’s decision to use only futures contracts as a  hedging tool allows the agent to essentially lock in a price, thus bringing in the tails 1  The difference between the spot and futures price is known as the basis. It can be positive or negative and represents the cost of storage, insurance, interest on invested capital, and transportation costs. If the spot and futures price always move in tandem then the basis does not change overtime meaning that there is no basis risk  5  of the revenue distribution, (i.e., implying a small variance of revenues). This strategy is vastly different from the no hedge strategy which leaves the hedger open to the full brunt of the potential price risk. As evidenced by the flatter and more dispersed revenue distribution. If the agent decides to use only commodity options then relative to the case of hedging with futures, the PDF skews to the right and shifts to the left.  Figurel .3 Distribution of revenues under alternative hedge scenarios  hec cn)y Gpticns heccnIy  Revenues  1.3 Problem Statement Given the volatile nature of commodity prices, agricultural firms and producers have an incentive to insure against price risk. A futures contract is a tool which can be used for this purpose. However, with the recent introduction of options on futures contracts, and agent has an additional and more flexible risk managing tool at his disposal. It is important to determine the conditions when  6  one hedging tool will be chosen over another and when both will be used simultaneously. There exists substantial literature which determines optimal hedging strategies using futures contracts.  Conversely, relatively few studies have  evaluated the same problem by incorporating both futures and options.  One  reason is that from a theoretical perspective, when futures are available options are redundant (i.e., will not be used) under the standard assumptions of no price biases, 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 tool which is the most efficient at reducing the variability of the expected payoff. Given the linear relationship between the spot and futures price and the linearity of the payoff of a futures contract (as a function of its price), then a futures contract is more effective at reducing the variance of expected returns than is an option contract (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 farmers suggesting that the number of producers hedging with options is equal to that of futures (Sapp, 1990). A more recent study of Montana farmers found that 14% of crop farmers use futures and 19% use options; alternatively, 6% of livestock farmers used futures and 11% use options (Sakong et al, 1993). One explanation as to the popularity of options over futures is that farmers may not be behaving as expected utility maximizers. In fact, results from a recent producer survey indicate that the use of safety-first decision rules may important to agricultural firms. 2 The survey performed by Patrick et al. (as cited in Atwood et 2  Safety-first decision rules are part of the lexicographic family (sequential ordering of multiple goals) of utility functions. A safety-first rule specifies that a decision maker will act in manner such that a preference for safety is followed. Once this safety objective is met, the decision maker’s goal involves a profit-oriented course of action.  7  al., 1988) involved 149 agricultural producers in twelve states and many producers responses “indicated what could be interpreted as substantial ‘safety-first’ considerations in their decision making”.  The emergence of safety-first decision  rules in economic literature is not a recent one. Telser (1 955) used a safety-first approach to evaluate hedging strategies using futures contracts.  1.4 Study Objectives  With producer surveys showing results of both the higher use of options than futures and presence of safety-first decision making behavior. The major objectives of this study are: 1.  To develope a one-period hedging model which incorporates safety-first decision rules and hedging with both futures and options. The safety-first model is similar to one developed by Telser.  2.  Contrast the safety-first model to the stamdard expected utility hedging model.  3.  Compare optimal hedging results from both models through simulations using parameters estimated from canola price data obtained from the Winnipeg Commodity Exchange.  1.5 Thesis Outline  Chapter Two will describe a typical scenario of a farmer who faces price risk and must decide which is the most beneficial marketing objective for his operation. Some of the relevant literature involving hedging with both futures and options will be discussed in Chapter Three.  In Chapter Four the theoretical  models are defined, and some of the basic safety-first theory is introduced.  The  derivation of the empirical models and the various parameters required for the models is included in Chapter Five. Chapter Six discusses the simulation results 8  derived from the empirical models. Lastly, Chapter Seven summarizes the results, provides conclusions and offers suggestions for further research and the potential drawbacks of the study.  9  CHAPTER 2 THE HEDGING CONTRACT  This chapter begins with a description of the motivation for hedging using futures and/or options via an analysis of a case farm.  Following this, the  implications of basis risk for the effectiveness of the hedge is examined.  2.1 The Theory of Hedging The concept of hedging is based on the principle that prices in the cash market and the futures market tend to move together. Although this relationship may not be a perfect, it is usually sufficiently close such that a farmer can reduce his price risk in the cash market by taking an opposite position in the futures market. By pursuing such a strategy, losses in one market are countered by gains in the other market. This is better explained through an example.  2.1.1 Hedging with Futures 1 Let us assume that a canola farmer has seeded her crop in May and is expecting to harvest it in September. In July, the farmer examines her crop and expects to have a crop yield of 1000 tonnes of #1 canola. Satisfied with the current price of a January futures contract trading on the WCE (i.e., after subtracting the basis, the price that remains will be sufficient to cover the per tonne cost of production) she decides to hedge her entire crop against price risk by selling (going short) 50 twenty-tonne contracts of January canola which she will offset after harvest. The following discussion provides a detailed description of the hedging procedure:  1  Much of this section indirectly relies on material published in the Commodity Trading Manual published by the Chicago Board of Trade.  10  STEP 1: The farmer must open an account with a Futures Commodity Merchant.  This involves the filling in and signing of: a new client  commodity application form, a margin form, a risk disclosure form, and in the case of a hedger, a hedging agreement.  The hedging agreement  confirms that all the trades made by the farmer will be for the sole purpose of hedging. Speculative trades must be done through a different account where they will be margined at the full speculative margin.  STEP 2: In July, the farmer places a hedge order for 50 twenty-tonne #1 canola futures contracts with a January delivery which are trading at $350 per tonne. The farmer expects the cash price at the time of delivery to the local elevator to be $325 per tonne.  The farmer has based this  expectation upon the expected value of the basis which she believes will equal the current basis. Once the hedge order is placed, the farmer must provide an initial margin. The initial margin required by the WCE in this case would be (50 contracts @ $450) $22,500 or $22.50 per tonne. Margins set by the Futures Commodity Merchant are sometimes more stringent than those of the WCE.  STEP 3:  In October, the farmer offsets her position by purchasing 50  twenty-tonne January canola futures contracts at $375 per tonne, and she sells her canola to the local oilseed elevator for $350 per tonne. Table 2.1 below summarizes the farmer’s transactions and derives her net selling price for a single contract when the basis is constant. Notice that, by entering the futures market the farmer has locked in a net selling price equivalent to the cash price she suspected would exist in October at the time of the delivery to the elevator. This is known as a “perfect hedge”. Had the farmer not entered the futures market she would have received $325 per tonne, which is an inferior strategy. 11  Table 2.1 Hedaina scenario under a constant basis cash market  futures market  basis  expected price of sell Jan. canola -$25  July  canola is $350/t October  sell canola  Change  futures  harvested buy Jan. canola -$25  @ $325/t  $25/t loss  futures  Gain from futures market transaction  @ $350/t  $25/t gain  Amount received from cash sale of canola  Net selling price  @ $375/t  0.0 $325/t $25/t $350/t  Of course, to assume that the basis remains constant may be overly simplistic, especially when time and/or local supply and demand factors can greatly affect the volatility of futures prices. Table 2.2 shows how the farmer will do when the basis is not constant. A weaker-than-expected (widening) basis can reduce the effectiveness of the short hedge. Alternatively, if the basis was stronger-than-expected (narrowing) then the short hedge would have been more effective. A simplification made in both Table 2.1 and Table 2.2 scenarios is that the futures price movement is favorable, If it was not favorable, then not only might the farmer be required to post additional margin money, but she would be better off not entering the futures market. This scenario is shown in Table 2.3 below.  12  Table 2.2 Hedging scenario under a weaker-than-exgected basis cash market July  futures market  basis  expected price of sell Jan. canola -$25 canola is $350/t  October  Change  sell  futures @ $375/t  harvested buy Jan. canola -$35  canola @ $315/t  futures @ $350/t  $35/t loss  $25/t gain  Amount received from cash sale of canola Gain from futures market transaction  -$10  $3151t $25/t  Net selling price  $3401t  Table 2.3 Hedging under an unfavorable futures price move cash market July  futures market  basis  expected price of sell Jan. canola -$25 canola is $325/t  October  Change  sell  futures @ $3501t  harvested buy Jan. canola -$25  canola @ $350/t  futures @ $375/t  $25/t gain  $25/t loss  Amount received from cash sale of canola Loss from futures market transaction Net selling price  $0.0 $350/t $25/t $325/t  In Table 2.3 the futures price has increased unfavorably from $350 per tonne to $375 per tonne. The farmer has lost $25 per tonne by entering into the  13  futures market. Fortunately, this loss is compensated by an equivalent gain in the cash market. This ensures a net selling price equal to the price expected by the farmer in July. However, in this scenario the farmer would have been better oft not to have entered the futures market because he could have received a net selling  price of $350 per tonne.  This returns us back to the discussion in  chapter one where it was said that one of the drawbacks of futures contracts was that they “prohibit the possibility of gaining from a price rally.” In this case, the flexibility of an option on the futures contract becomes beneficial.  2.1.2 Hedging with Options Because farmers who hold options are not required to exercise them, they can benefit from the gain in the cash market and simultaneously avoid the effects of any unfavorable movement in the futures price. The cost of this added flexibility is reflected in the purchase price of the option, referred to as the option premium.  Table 2.4 describes the potential benefits from purchasing (going  long) a put option with a $375 per tonne strike price, a $15 per tonne premium, and an expected basis of $25 per tonne under various potential futures price realizations.  14  Table 2.4 Hedging with a long put option Jan. futures price in Oct. ($/t)  Actual basis ($/t)  Cash price ($/t)  Put option gain/loss ($/t)  Net selling price ($/t)  300  -25  275  +60  335  325  -25  300  ÷35  335  350  -25  325  ÷10  335  375  -25  350  -15  335  400  -25  375  -15  360  425  -25  400  -15  385  One 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 will always lock in a floor price of $335 per tonne. 2 If the futures price exceeds the strike price of $375, then the farmer can receive a net selling price equivalent to the current cash price for canola less the option premium. In this respect, the option contract is similar to conventional insurance. The discussion above evaluated the effects of hedging under specific strategies and under various basis relationships. Figures 2.1, 2.2, and 2.3 --which are similar to Figure 1 .3-- provide an alternative means to illustrate hedging scenarios under increasing levels of basis risk. As the basis risk increases, the revenue PDF for each hedge type becomes flatter, more elongated, and closer to the no hedge PDF. This implies that when the basis risk is high, the use of either futures or options is not really as effective for hedging against price risk.  With  relatively low basis risk, the futures and option contracts provide the hedger with  2  Notice that this floor price is lower than that established by entering the futures market, but this strategy allows the farmer to gain if prices move upwards. Also, note how the scenarios of table 2.4 translate into the payoff diagram (figure 1.2) of chapter one.  15  very different outcomes. These differences disappear as the basis risk becomes large.  Figure 2.1 Low basis risk  F utur hec cnly  Oia,s no hece  R even ues  16  Figure 12 Moderate basis risk  cØicns hecs cray futures hedge cnly nohec  Revenues  Figure 2.3 High basis risk  Futures hececnIy cpncns heccnIy no hedge  Revenues  17  CHAPTER 3 REVIEW OF HEDGING LITERATURE  This chapter begins by examining the evolution of optimal hedging theory which is primarily concerned with futures contracts only.  Two competing  hypotheses (standard expected utility maximization and the safety-first approach) are discussed.  The second section reviews the literature which  incorporates both futures and options on futures into the hedging model.  3.1 Optimal Hedging with Futures and Options: Expected Utility Model 3.1.1 Futures Contracts The original notion of an optimal hedge referred to the case where the decision maker [DMJ would take an equal but opposite position in the futures market as that taken in the cash or spot market. In other words, if you expected to have (or held in a warehouse in the case of an exporter) Xtonnes of canola then you would hedge an equal amount in the futures market. Johnson (195960) and Stein (1961) were amongst the first studies to use modern portfolio theory as a means of deriving optimal hedging strategies. 1 Their studies showed that 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] framework of portfolio theory when determining the optimal hedging decisions using futures contracts. The results obtained by Johnson and Stein do not necessarily follow those of expected utility maximization except under special conditions.  This  1  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-variance framework.  18  relationship can be shown by deriving the optimal hedge expected utility framework. Under the assumption of either a quadratic utility function (Tobin, 1958), a normally distributed attribute (Samuelson, 1970), or a random attribute which is an affine transformation of a single random variable (Meyer, 1987), then the mean-variance objective function can be expessed as,  U(7t)  where  ‘jt  =  E(t)+?Var(7t)  (31)  is the expected return from the portfolio of assets, and ? represents the  Arrow-Pratt measure of absolute risk aversion (a positive value denoting a risk averse individual). For the case of two risky assets, the agent’s cash position and the corresponding futures position, then the expected return from the portfolio can be expressed as (where the superscripted bars denote expected values),  (3.2)  E(t)=XS( — 2 sl)+XfCf—f)  where X and X are the spot and futures positions respectively; period spot and futures prices; s 1  ,  2’  2 the end of f  f1 are the spot and futures prices at the  beginning of the period. Similarily, the variance of returns is,  Var(t)  where  =  Xa + Xo. + 2 XXo  (3.3)  represent the variance of the end-of-period spot price, the  variance of the end-of-period futures price, and the covariance of the end-of period spot price and futures price.  19  Now substituting equations (3.2) and (3.3) into (3.1) and differentiating with respect to X 1 (assuming that the spot position is given; hence, known by the agent) then solving out for the futures positions one obtains the following relationship,  X  o  2Xa  (3.4)  If the second (speculative) component of equation (3.4) equals zero due to no price bias (i.e., fj  =4) then the optimal  hedge equals the first component. The  Johnson-Stein mean-variance technique of minimizing risk via the variance of the portfolio is equivalent to an expected utility maximization only under the special circumstance of no price biases. One benefit of this approach is that the optimal hedge can be derived empirically by regressing futures prices on spot prices and using classical linear regression to derive the slope coefficient (covariance of spot and futures relative to the variance of futures) which is equivalent to the ‘optimal hedge’ from MV 2 analysis.  3.1.2 Commodity Options With the introduction of commodity options on the major U.S. exchanges in the early 1980’s, there was a need to examine how one could use options as  2 Considerable controversy has arisen around this technique because of the debate as to whether the data used should be in the form of price levels, price changes, or percentage changes. Both theoretical and statistical arguments have been forwarded as support to either of the three forms (e.g., Bond et al., 1987; Benninga et aL, 1984; Witt et al., 1987; Shafer, 1993). In addition, statistical detection of serial correlation (e.g., Herbst et al., 1989) and autoregressive conditional 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.  20  an additional risk management tool.  Ritchken (1985) investigated how single  period MV utility maximizers might respond when they include Black-Scholes [BS] priced options in their investment opportunity set. What he found was that portfolios which lie on the MV efficient set include primarily short rather than long positions in the options market; thus, making the joint assumption of meanvariance and Black-Scholes pricing theoretically inconsistent. One study which incorporated the use of both hedging tools was undertaken by Wolf (1987). He considered an optimal hedging strategy in the case of a linear MV model and a logarithmic utility function. Within the MV framework Wolf derived comparative static results both with and without basis risk. The logarithmic utility function was used in his simulation analysis of optimal portfolios. Wolf’s initial assumptions were that the DM has a fixed position in the physical commodity and that the spot price was nonstochastic. The results under these assumptions were: •  A DM which has a long futures position and faces fairly priced options will sell a 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 price movements 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, the DM uses only futures since options are redundant. Although Wolf’s analysis provides useful insights, the validity of his results  are questionable because the MV model may be inappropriate in his context. This is because the distribution of profits become truncated when options are  21  introduced into the portfolio. 3  Realizing this weakness in the standard MV  model, Ladd and Hanson, (1991) provide an alternative method that is based on the generalized expected utility analysis. Their model assumes that there are no transaction costs, basis risk, margin calls, and that the DM has an exponential utility function which displays constant absolute risk aversion (CARA). Firstly, they derive an income density function which is the sum of two truncated normal distributions. This is to account for the truncation which occurs when options are included in the hedging process. Secondly, Ladd and Hanson build a factorial model which is used to determine the input variable levels required for use as substitutes for market factors in their income density function.  Finally, they  numerically integrate and optimize their generalized expected utility model for each set of estimated market factors.  The following are a few of Ladd and  Hanson’s major results: •  If either the futures or options markets are considered to be biased then the DM will speculate by increasing their position in the options market and decreasing (increasing) their position in short (long) futures.  •  The availability of an options market in addition to a cash and futures market has 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 what offset one another, and allowing the DM to speculate in both markets does not add much value to the DM over allowing him to speculate in just one market.”  Including options in a DM’s portfolio can violate any three sufficiency conditions of a MV representation of expected utility maximization: (i) a DM exhibiting a concave utility function with a normally distributed random variable; (ii) the utility attribute is an affine transformation of a single random variable; (iii) the DM has a quadratic utility function, (Ladd and Hanson, 1991).  22  Following Ladd and Hanson, is an article by Lapan, Moschini, and Hanson (LMH, 1991). They use an expected utility maximization approach and through comparitive statics determine a number of results under the presence and absence of price biases. In addition, their single-period model allows for basis risk and assumes that the DM has an exponential utility function exhibiting CARA characteristics. Some of their more relevant results were: •  If both futures and options prices are unbiased and spot and futures prices are correlated, then the DM will hedge a proportion  () of their output in the  futures market, and no options will be used. 4 •  If futures prices are unbiased, and the relationship between the spot and futures prices is linear, then a futures contract provides a superior hedge to an option because of the futures linear payoff versus the options non-linear payoff. Thus, in this case, the use of options is redundant.  •  If there are perceived biases in the futures prices and/or options premiums and it is also assumed that there is no basis risk and that the hedger is fully hedged, then options are used in combination with futures to speculate on the perceived biases. Bullock and Hayes (1992) modify Wolf’s original model by endogenating  the variance-covariance matrix of portfolio returns.  In  their model, they  examined both scenarios, with and without basis risk. Unlike Wolf, they do not allow for the use of both call and put options since a call option can be created synthetically via a combination of a futures contract and a put 5 option. From their 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 the futures 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 inventory or spot position and the option is the preferred instrument when speculating.  •  The futures contract is always the main speculative instrument when speculating 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 above results still hold. In addition, they found that increases in the mean basis level would induce a DM to hold shorter inventory positions and increases in the variance of the basis impel the DM to reduce his/her inventory position. Bullock and Hayes also found that the basis information had no impact on the optimal option use level and that the futures contract is still the most ideal tool to use when hedging. 6  3.2 Optimal Hedging with Futures: Safety-first Approach  One decision rule which has mostly overlooked in hedging models, is that of the safety-first model. Initially developed by Roy (1952) for the case of asset holding, it was later modified by Telser (1955-6) to include futures. Telser’s safety-first rule simply stated that a DM maximizes expected returns subject to ensuring that returns do not fall below some pre-determined disaster level more frequently than an accepted likelihood. Similar to the MV studies, Telser found that it was optimal for a DM to hold some combination of hedged and unhedged stocks depending on a DM’s expectation of spot and futures price movements and their inter-relationship.  6 A lternatively, Hauser and Eales (1987) use a target-deviation model to evaluate the risk and return associated with nine most “commonly” used option strategies when the hedger holds a short futures position. They find that a put option when the hedger is risk averse over outcomes which fall below the expected hedge price, and risk preferring with outcomes above the expected hedge price. Two other articles include: Hauser and Andersen, Hauser and Eales, 1986.  24  Telser’s Figure 1 is re-created below (Figure 3.1). Telser states that when gains are expected from holding both a short hedge and an unhedged stock then the expected net income will have a negative slope like line (A)and the optimal combination of hedged and unhedged stocks is given by the tangency of line A to the elliptic-shaped constraint --point (A’). Also, if a gain is expected from a short hedge, but not from the unhedged position, then the income line is (B) and the optimal point becomes (B’). As a third alternative, when gains are expected from a long hedge position, and losses are expected from holding unhedged stocks then (C) represents the income line and (C’) is the optimal point. In all of the cases the expected net income lines increase in the direction of the arrows.  Figure 3.1 Telser’s figure 1 2 X  Unhedged Stocks  A  —v  C B  X 1  Hedged Stocks  3.3 Expected Utility versus Safety-first  Unlike the expected utility model, which is a commonly accepted means of modelling the behavior of agents under uncertainty via the axioms of von Neumann-Morgenstern, the use of safety-first models is less common.  25  Safety-first models are part of the lexicographic family of utility models which, unlike the expected utility model, have no theoretical base or set of axioms. Instead, as the word lexicographic suggests, the utility functions of these can be thought of as a representation of sequential goals. The goal of highest priority must be met first before a decision maker is allowed to consider the second goal. The main concept is that a decision maker is first concerned with satisfying some safety measure, and then once having attained that measure he can follow a profit maximization objective. Three main forms of safety-first rules exist according to Pyle and Turnovsky (1970).  However, Anderson (1979) provides an alternative  catagorization which is adopted in this discussion. Anderson defines what he calls a safety principle.  In the first catagory,  One of the more familiar  safety principles is that of Roy (1952) which states that a decision maker’s objective is choosing an action which minimizes the probability of some attribute, usually profits(it), falling below some specified “disaster” level (d*).  In other  words,  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 falling below some crucial probability (y).  Max E(t) s.t. Pr(t  d*) ‘y  (3.6)  26  Anderson defines the third category as that of a safety fixed rule.  Initially  introduced by Kataoka (1963), it involves the maximization of some minimum return (d*) such that the probability of returns falling below this minimum level is lower 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 simple two moment case) represented by a distribution’s second moment, either variance or standard deviation, decisions made using safety-first rules can differ greatly from variance-based decisions. Figure 3.2 is a good example of how the  two methods will have two different outcomes depending on which the decision maker is using. In the first graph of Figure 3.2 there are two distributions of revenue, A and B. Distribution A has a lower mean revenue  (Pa)  than distribution B  however, distribution B has a larger variance (o) than distribution A (a).  (tb);  In  addition, if one is to define (d) as the “disaster” level of revenue (perhaps the cost of production or the mortgage payment on the farm) the action that results in distribution A will have a greater likelihood (Ya) of revenues falling below the critical (d).  Alternatively, action B has a distribution with a lower probability  (almost negligible) of revenues exceeding the critical point (d). In other words, if a decision maker were to use Telser’s safety-first rule (equation 3.6) he would rank distribution A as a more risky venture than distribution B; whereas, an expected utility maximizer who is very risk averse (i.e., is only concerned with minimizing variance) would rank distribution B as more risky than A.  27  Figure 3.2 Measuring risk by probability of loss versus variance 7 Figure 3.2.a: Ya >Tb, a<o, d = “disaster” level of revenue  I I I / Ii I,  2’ 0  &  Distnbution A  /  \/ Distribution B  -.  \ \  :  A /1 / I  :  ,‘  \  -5-.--  -  --  d  Revenue  Figure 3.2.b: y >y,  ()X  I  Jib  distribution’s skewness  <(3)z  \S  \S  2  Distribution X  /  Distribution Z  .0 0  0.  / /  \  5:  /  d  S S  =  Revenues  In graph 3.2.b., there are again two distributions (X and Z) of differing shape.  In this case, both distributions have equal means  variances  (a  =  at),  (i.t=)  and equal  but they are skewed in different directions. Distribution X is  Diagrams are based upon those depicted in Young (1984 p.33).  28  negatively skewed  ([J3]<  0) and distribution Z is positively skewed  ([l’3]>  0). It  is useful to once again make a comparison on how a decision maker may evaluate the riskiness of either action based upon behaving in either a safety first fashion or an expected utility manner. One would expect that a safety-first agent would rank the action responsible for distribution X as riskier venture >  y. whereas the risk or variance minimizer would be indifferent between the  two actions.  29  CHAPTER 4 MODEL SPECIFICATION In this chapter a model of a hedger who wishes to lock in a futures position then offsets that position at a later date is developed. This is followed by a discussion of the behavior of two hedging agents who follow two different decision making rules. One agent decides his optimal hedging strategy by the maximizing the expected utility of the revenues derived from the futures and/or options position. The other agent bases his decision upon the maximization of a safety-first rule.  4.1 Single-Period Representation of Revenue At the beginning of the period 0 (t ) , the agent decides to hedge a traction of his operation’s output by taking a short position in the futures market and either purchasing or selling a put option. At end of the period 1 (t ) , the agent lifts the hedge by offsetting his position in the futures market and then exercising the option if it has value. This study will consider the use of put options as an agent’s hedging tool and not call options because calls are redundant when in the presence of both futures and put options.  In other words, one can create a call option  synthetically via the combination of a put option and a futures contract. This inter-relationship is best exemplified using a diagram. In Figure 4.1, the long futures and long put option positions (designated by the dark solid lines) are combined to artificially create the long call (dotted line) with a strike price Kand a premium of r.  30  Figure 4.1 Creating a synthetic long call option  Long +  -  Long call  Profit per contract 0 Futures price Long put  Given 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 by+(f—p)x—rz  f p<K if pK  (4.1)  where: t —  end-of-period revenue  b  end-of-period cash price  y  end-of-period output (exogenous)  f — futures price at the beginning of the period p  —  futures price at the end of the period  x  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) price In the two-state revenue equation above, the revenue expression in the firststate occurs when the futures price (p) is below the option contract strike price  31  (K).  The second-state occurs when the futures price exceeds the strike or  exercise price. There are a number of assumptions made regarding the general representation of the above end-of-period revenue: 1. There are no costs of production (or profits are stated as net of production costs), and production (y) is exogenous and equals one. 1 2. 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 the time the hedge is placed 0 (t ) . 3. There are of no transaction/brokerage costs, no transportation costs, and no margins required. 4. There is no constraint on the agent’s ability to borrow to finance the cost of the futures and options contract(s), and he does not face any borrowing costs. 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 Rules This section discusses the general model used to compare hedging with options versus futures. The general model allows for either standard expected utility maximization or a safety-first approach (maximizing expected revenue subject to a safety-first constraint).  1 Although output uncertainty is an important factor, it is ignored here due to the added complexity to the model. Some studies which touch upon the topic of hedging when both price risk and output risk are allowed for include: Rolfo (1980), Chavas and Pope (1982), Grant (1985) and more recently Sakong, Hayes, and Hallam (1993).  32  4.2.1 Generalized Expected Utility Model If an agent has a revenue function similar to equation (4.1), and he/she is an expected utility maximizer, then the agent will choose his/her futures and/or options position so as to satisfy the following objective: max E[U(t)] = 2 )]+E[UOt 1 E[U(t ) ]  (4.2)  K  =JJU(7tV..K...p) f(b,p)dpdb  +  S  )f(b,p)dpdb 0 5U(rc..  where the utiltiy of revenue when the option has value (v),(i.e., the futures price is below the strike price. U(ico)  the utility of revenue when the option has no value (v), (i.e., the  futures 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 1 E[U(t ) ] and 2 E[U(it ) } are the expected utility of revenues when the futures price falls below and above the option strike price, respectively. Similarly to LMH, results from the maximization of equation (4.2) are dependent on an agent’s perception of future prices and option values. Lapan, Moschini, and Hanson found that without price biases there was no need for options, and that options were only useful to speculate on biased values of futures prices and options. A price bias is defined as a situation where an agent’s expectation of the futures 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 expected value of the futures price at the end of the period (viz., f), and the expected  33  value of the option (as perceived by the agent) depends on the agent’s expectation 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  premium r  =  f(K_p) g(p;  and  =f a)dp  (4.3)  where g(p) and h(p) are the respective marginal distributions of the futures price with a mean of  and a variance of  °‘.  Inkeeping with LMH, note that both  distributions share the same variance, thus any speculative motive based on differing perceptions of expected volatility or variance is precluded.  4.2.2 Safety-first Model Similar to the general expected utility model, if the agent has a revenue function equal to that of equation (4.1), then his/her objective function can be described as below:  max E[it; x, z] =  ff1ty.Kp  f(b, p) dpdb +  subjectto Pr(t  itt)  S  fv=O  f(b, p) dp db  (44)  y  The agent maximizes his expected revenue via his futures and option contract position but is subject to a probability constraint. The constraint states that the probability of an agent’s revenue falling below a predetermined level exceed a preset probability denoted above as y.  (‘L)  cannot  This model is essentially  Telser’s safety-first model, except that it has been enhanced by introducing options in the decision making process.  34  As in (4.2), equation (4.3) represents the expected revenue depending on whether futures prices fall above or below the exercise price of the put option. Also similar to the expected utility model above, futures price biases are incorporated into the model.  4.3 Evaluating a Safety-first Model  Important to the use of any model is the establishment of the intuition behind the theory through some geometric interpretation. Whereas the expected utility model involves a straight-forward maximization problem of a concave objective function, the safety-first model has a less obvious interpretation. One means of evaluating the safety-first model is to use the common method of stochastic dominance. 2 First-degree stochastic dominance (FSD) states that if action F, with a cummulative distribution function (CDF) of F(it) is preferred to action G with a CDF G(it) if F(rc)G(it)  Vit,  where  ic€  [a,b]  (4.5)  Figures 4.2 illustrates the CDF’s for three different hedging strategies for the case of a ten percent upward futures price bias: (i) short futures and long put combination, (ii) short futures only, (iii) short futures and short put option combination. Using the FSD method to evaluate these various strategies, one can see that they all satisfy the safety-first “disaster” level of revenue (denoted here as  itL)  10 percent of the time.  Hence, given that the  constraint is met by all three of the strategies, the optimal strategy must be the CDF with the largest amount of area above it. Visually, this is difficult decision to make, 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).  35  Figure 4.2 Cummulative density functions under an upward price bias  0.9 0.8  Shorttilures cnd Long Put Combnaflon  0.7 .  0.6  0.5  2  O_  0.4 0.3  ulures Only  0.2 0.1 0  In Figure 4.3, the optimization problem is represented in two-dimensional futures (X) and options (Z) space for when a price bias exists. The elliptical area of the graph is defined as the iso-probability frontier, which represents the locus of X and Z combinations that strictly satisfy the safety-first constraint. Any point within 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 line in Figure 4•33 The slope of the iso-revenue line can be derived by totally differentiating equation (4.1) as follows: 1t  dt =—dx+—dz= 0 az 0=(f—p)dx+(7—r)dz where Y=K—7  Re-arranging terms dx (V—r) slope=—=— dz (f—p)  (4.6)  Figure 4.3 is similar to that of Telser (1955) Figure 1 except that he plots out hedged and unhedged stocks.  36  To maximize expected revenue, the agent would choose combinations of X and Z that push the iso-revenue line in the direction of the two arrows illustrated in Figure 4.3. However, the agent must remain within the feasible region of the isoprobability frontier; therefore, the optimal hedging combination is at a point where the two are tangent to each other.  Figure 4.3 Tangency condition for maximization F utures (short)  line  lso-Probcbllhiy F roner (i.e.,scteiyconslrdn1  Options (long)  In Figure 4.4 the relationship between the size of the feasible region and the 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 feasible region increases in size, and therefore the number of option and futures combinations satisfying the iso-probability frontier also increase. Secondly, as the price bias increases, the absolute value of the slope decreases. An increase in the price bias causes the denominator (f—fl) to increase at a quicker rate than the numerator (—r).  Therefore, if the slope diminishes as the bias  37  becomes larger, the optimal strategy may vary from option/futures combinations A, B, C, or D.  In other words, the agent will normally take a short futures  position, but may alternate between a long or short option position depending on the price bias.  Figure 4.4 Feasible regions under varying bias  LcrPi1veBi  (shal  S rrdl  cpms (shatpiD cplcns (lcng put)  In summary, as highlighted in Figure 4.5, this study evaluates two decision making rules as applied to a single-period framework. In both models the DM, who has a given quantity of a commodity, can hedge against a potential loss 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 exponential utility function exhibiting constant absolute risk aversion (CARA). The DM of the second model makes decisions based on the safety-first rule of Telser.  He  38  maximizes expected revenue subject to the constraint that the revenue does not fall below a pre-specified level of revenue with more than a given likelihood. Both of the models allow for upward (positive) and downward (negative) futures price biases. This means that the agents can speculate on the expected value of the futures price, but not the volatility.  Figure 4.5 Model summary  Single-period hedging model with futures and options under two alternative decision-making rules  Max E (it) subject to safety-first probability constraint Pr (2t ) ‘Y  39  CHAPTER 5 EMPIRICAL MODELS  In this chapter, the theoretical models of Chapter 4 are re-specified for the case where the cash and futures prices are joint normally distributed. Expressions are derived for: the probability constraint; the expected utility expression (for both the exponential and linear utility functions); and the value of a single period option contract. Also, the parameter requirements of the models are discussed and several methods of obtaining them are analyzed.  5.1 Empirical Model Derivation 5.1.1 Expected Utility Model  If 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  /  —  *explAj(x + z)p *  —\  —  A2r22  yb)÷-y[y a,,  2py(x + z)a,,o  —  EK—+A(pya,,a—(x+z)a) —  [  expt—A[  *exp{A(x_yb)+4i[y2a —2pyxa,,a *  1—  (  K  —  + A(pya,,o  op  —  22  +(x + z) a  —  rz]  +x2a]}  (5.1)  xo)  )  In (5.1) p, is the mean futures price and 1 is the mean spot price, and  are  the respective variances of p and b. The correlation coefficient between p and b is denoted  p. The parameter A represents the Arrow-Pratt measure of risk  40  aversion and I(.) is the cumulative density function (CD F) of a standard normal distribution.  5.1.2 Safety-first Model Unlike the expected utility model above, the safety-first model as depicted in equation (4.4) requires two functions to be constructed: (i) the objective function, and (ii) the constraint. The objective function in the safety-first model is simply an expression for expected profits. The constraint limits the probability of falling below a specified level. To specify the objective function for the case of normally distributed prices, substitute the appropriate functions represented by the two states of single-period revenue, (i.e., v=K-p and  and integrate.  The resulting expression can be written as (see Appendix 5):  K-p E(;x,z)=by+(f—)x+z (K_)*  K—p  G)  O)  -z (K_f)*1K_+p1Kap apJ aj  (5.2)  where 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 biases exist, (i.e., f=) then the z[.] terms will cancel each other, and the (f—)x term will equal zero. Further simplification leaves, E(it;x,z)=by which is the expected spot price multiplied by the agents production. To derive an expression for the constraint, it is necessary to transform the distribution function of the random variable  it  to a distribution function composed  of the random variables b and p. One technique which can be used for this  41  purpose is the cumulative-distribution-function 1 technique. Using this method one can perform the conversion of the cummulative function of revenue (FJI[ltL]) as follows: °°  1k—P  FL)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 value of b) is dependent upon whether the futures price p is above or below the strike price of the put option K. Therefore equation (5.3) can be redefined as: (lk—P1 P2P)’  (1k i 2P>  $ $  Pr(7uTUL)=  f(b,p)dbdp+J  $f(p)dbdp  (5.4)  where the above parameters are defined as follows: & =fx+(K—r)z  2 =—(x+z) a  forp<K for pK  13 = 1 fx—rz  Equation (5.4) is still a general expression which must be specified for empirical use.  When b, p are joint normally distributed with means  variances  and are correlated by a factor which we will denote as p, then  an expression for (5.4) can be derived and written as (see Appendix 4): F(itL; X,Z) =  where  =bY—(X+Z)+j’X-i-(K—r)Z 2  =  1 [m])d7u + ff (m) (A  (5.5)  -  K-.(YPab  ) 1 a)@—  _bYXp+JXrZ  o +Y 2 (X+Z) a +2p(X÷Z)YaPob 2 = X a +Y 2 a +2PXYGPab 2  1  For a detailed exposition of this technique, please refer to Mood, Graybill, and Boes, pp. 181188. A more basic analysis is provided by Freund and Wapole, pp. 226-230.  42  f ( 1 it)  is the pdf of a univariate normal distribution with mean  ,  and variance  a,  a = 1 X÷Z, and a =X respectively. 2  5.2 Model Parameter Requirements This section discusses alternative techniques for obtainging emprical estimates of the parameters specified above. Statistical estimates of the means and the correlation coefficient p will allow one to conduct ,p, variances commodity specific analysis; in the case of this paper, canola trading at the Winnipeg Commodity Exchange.  5.2.1 Review of Techniques Since the introduction of mean-variance models for use with commodity futures by both Johnson (1960) and Stein (1961), there have been a plethora of studies which provide econometric methods to derive the parameters needed for a “risk or variance minimizing” optimal hedge ratio from time series data. As previously mentioned, the optimal hedge, as derived using portfolio theory, is defined as the covariance of the spot and futures prices divided by the variance of the futures price (let it be denoted as H*=abp/o). Ederington (1979) was one of the first to use an ordinary least squares technique to estimate the optimal hedge in the case of Government National Mortgage Association (GNMA) and T bill futures contracts. He suggested that one could derive H* through the simple regression 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 be equal to or in this case, the risk minimizing hedge ratio.  43  With the advent of this estimation method there later evolved a myriad of questions and debate over such issues as: should equation (5.6) be estimated using price levels, price changes, or the ratio of spot market returns to futures market returns; should the data be daily, weekly, monthly etc.; should the data be corrected for heteroscedasticity and/or serial correlation? Myers and Thompson (1989) suggest that the simple regression is just a special case of a more generalized linear, reduced form, equilibrium model which they define as follows:  b.=X. a 1 +u,  (5.7)  .=X 1 .j1+v,  (5.8)  p,  where  is a vector of variables at time (t-1), which can be used to forecast the  spot price (b )and the futures price (Pt.) Possible variables to be included are spot and futures prices, production, storage, exports, consumer income, and other factors. The a and f3 are vectors of unkown parameters, and u and v are random shocks with a mean of zero and serially uncorrelated. By estimating equations (5.7) and (5.8) and collecting their respective error term vectors (u and vi), one can create the necessary variance-covariance matrix. Unfortunately, due to the lack of data available on factors other than spot and futures prices, this method was not used in this study.  5.2.2 The Forecast Error Method  The method used in this study to determine the correlation between the spot and futures prices, so as to be able to use this parameter as a measure of basis risk, follows that of Peck (1975) and Rolfo (1980).  In Rolfo’s article on  deriving optimal hedge ratios for the case of cocoa producing countries, he employs a method of forecast errors as a means of capturing a measure for uncertainty in both price and quantity. He defines a forecast error as being “the 44  difference between realized and forecast prices divided by the forecast price.” This differs slightly from Peck, who does not divide the forecasting error by the forecast price. Rolfo suggests that by doing this one allows for differential rates of historical inflation.  Hence, using Rolfo’s method one can define the  forecast errors for the spot price and the futures price as follows, 8 P f  Eb=,  (59)  What equation (5.9) suggests is that the best one-step ahead predictor of both spot prices 1 (E[b+ ] ) and futures prices ] o+ 1 (E ) are the current values of each variable.  5.2.3 Parameter Estimation  As indicated above, both models require parameter estimates for the mean spot and futures prices (b,), the variance (a ,o and in addition, the covariance correlation coefficient (p  ).  (ap)  )  of both price series  of the two prices in order to derive the  Therefore, using the Rolfo’s forecast method one  can 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 prices  respectively. Assuming that the forecast price is an unbiased predictor of the , then the variance of the forecast errors is simply equal to the 2 realized price sum of the squared forecast errors and the covariance is the sum of the dot-products of the forecast errors  (8,,  .) divided by the number of  observations less one (N-i) for correction purposes. 2  By assuming that the forecast price is an unbiased predictor of the realized price, it is implied that, on average, the forecast error will have a mean of zero.  45  In this study, the necessary forecast errors are calculated for canola trading at the Winnipeg Commodity Exchange [WCE]. The data span a ten year period during the years 1981-1990. The spot price data are the daily closing cash 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 for Number One Canada canola. As described in chapter 2, it is assumed that in July, the farmer enters the futures market with a contract which matures in January.  After harvest in  September, the farmer offsets his position in the futures market and sells his crop to the local elevator. Hence, following the Rolfo model, one can re-define equation 5.9 for this particular scenario as, —  b,sept 8  —  bsept  —  p  1 f  —  ‘  Ep,sept —  Jjul  Psept p  —  JjuI  Table 5.1 below, shows the variances, co-variances, and correlation of spot 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 the  correlational values hinge around the mid to high 0.9 values. Two outliers occur during the 1984 production year.  If one assumes that the farmer has the  opportunity to offset his/her position in either September or October, then this study will use an average of the September and October correlation values as estimates 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 p equal to 0.95. The lower basis risk values can be used as a means of testing the models’ sensitivity. Having the basis risk value, one still requires an estimate of the standard deviation of the spot and futures price at the time of the offset.  46  Table 5.1 Forecast error statistics for Seotember and October (1981 -90)  September Year 81 82 83 84 85 86 87 88 89 90 81-85 86-90 81-90  2  October  E  b p 8  P€  0.0051 0.0139 0.0022 0.0204 0.11760.0911 0.0008 0.0033 0.0075 0.0097 0.0090 0.0182 0.0037 0.0077 0.0066 0.0110 0.0028 0.0090 0.0036 0.0109 0.0267 0.0277 0.0052 0.0114 0.0151 0.0157  0.0082 0.0062 0.1035 0.0014 0.0083 0.0128 0.0051 0.0082 0.0049 0.0061 0.0255 0.0074 0.0186  0.9737 0.9191 0.9991 0.8252 0.9785 0.9973 0.9549 0.9616 0.9695 0.9761 0.9391 0.9719 0.9346  0.0057 0.0054 0.1024 0.0006 0.0235 0.0055 0.0017 0.0205 0.0054 0.0035 0.0275 0.0073 0.0166  0.0084 0.0241 0.0776 0.0008 0.0249 0.0106 0.0016 0.0283 0.0114 0.0088 0.0272 0.0121 0.017  p 5  P€  0.0069 0.0113 0.0890 0.0004 0.0242 0.0076 0.0012 0.0240 0.0078 0.0055 0.0264 0.0092 0.0187  0.9992 0.9909 0.9985 0.6375 0.9995 0.9967 0.7665 0.9982 0.9939 0.9921 0.9251 0.9495 0.9626  To obtain the standard deviation of the spot and futures prices one first finds the variance of each price as a function of its forecast error.  Thus re  arranging equation (5.9),  b = f(1 +eb), and p = f(1 + e) Treating b, p,  e,,  (5.12)  and e as random variables, one can derive the variances of  the expressions in (5.12) as,  *v(e) 2 var(b) = f *v(e) 2 var(p) = f  (5.13)  If f the mean price in July, is assumed to equal 5.0 then the standard deviations as displayed in table 5.2 can be calculated. 3 It would seem that the price data from the period 1981 to 1985 is responsible for a greater proportion of the volatility for the entire time series spanning 1981-90.  Due to stability problems when models with exponential functions have ‘large’ exponents, f was assigned a value of 5.  47  Table 5.2 The standard deviations for September and October.  Time Period  September  October  SD of futures  SD of spot  SD of futures  SD of spot  81-85  0.81 70  0.8325  0.8300  0.8248  86-90  0.3606  0.5347  0.4288  0.5519  81-90  0.61 44  0.6264  0.6442  0.6519  As a simplification, this study will, in the base case, assume that the standard deviation of the futures and spot prices are equal and that they both equal a value of 0.8. Although this is a higher value than that derived for the period 1981-90, and lower than 1981 -85, it is felt that the value of 0.8 provides a sufficient balance of the two periods, and that any further analysis can involve an investigation of the sensitivity of the models to values around the base value.  48  CHAPTER 6 SIMULATION RESULTS AND IMPLICATIONS  In Chapter 5 the data requirements for the empirical models were discussed.  This chapter involves the evaluation of both the safety-first and  expected utility models using the base case parameters derived in Chapter 5 and additional sensitivity analysis. This is then followed by a comparison of the results of the two different decision making beliefs.  6.1 Expected Utility Model Results  Using the parameter estimation results from Chapter 5 one can define the following base case scenario: Y=1, b=5, Y= , 5  , 8 Ob=Op=O.  K=5, f=5.2, p=O.95, A=O.5  This base case implies a small futures price bias of 4 percent and a fairly low level of basis risk  (p=O.95).  The level of risk aversion is obtained via sensitivity  analysis of the model to varying levels of A, and roughly represents a median point. Table 6.1 below provides a summary of the results of the EU model under alternative scenarios. Beginning with the standard no futures price bias case (i.e., no speculative motive) the farmer will hedge a proportion of production equal to the ratio of the co-variance between the spot and futures price divided by the variance of the futures price.  Using values from the base case, the  optimal hedge involves using 0.95 futures and no options 1 contracts. This result is 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 the spot and futures prices and the variance of the futures price.  Note that the optimal futures X*=O.95 is equivalent to the basis risk (p=O.95). This is due to the fact that the standard deviation of the futures price and spot price are equivalent. 1  49  Now, if an upward futures price bias of 4 percent is introduced, the expected utility maximizer will “overhedge”  ,  (i.e., hedge more than his cash  position), in the futures market, hence X*=1 .58. As one would expect, when the parameters of the model are varied, the optimal futures positions also change. Increasing the base cases’standard deviation (volatility) of the spot and futures prices from 0.8 to 1 .25 ceteris paribus induces a fall in the optimal futures position from 1.58 to 1.21.  This reflects the added risk involved holding a  greater unhedged position (i.e., speculating with an the open position), when the joint distribution of prices has a greater volatility or variance.  Table 6.1 Optimal futures and option positions under various scenarios. Futures position (X*) No bias 0%  0.95  Base case  1.58  Bias -4%  0.33  Higher SD (aD=ab=l .25)  1.21  Higher basis risk (p=O.82)  1.45  Lower basis risk (p=O.99)  1.62  Lower risk aversion (A=0.1)  4.08  Higher risk aversion (A=1 .0)  1.26  Option position (Z*)  A similar relationship between the optimal futures position and the parameter being varied can be found for the instances of a higher basis risk or a higher risk aversion.  By increasing the basis risk to a value of 0.82 from the  base case value of 0.95, the hedge ratio falls to 1 .45 futures contracts. The fall in the amount hedged is a result of the reduced certainty of the expected change  50  in the spot price over time as the futures price changes. Recalling chapter one, as the basis risk increased (lower p) then the resulting revenue distributions were more volatile. An increase in A (Arrow-Pratt risk aversion) from 0.5 to 1.0 means that the agent is more averse to risk and derives a both lower and negative marginal utility (U”(it)<0). Alternatively, if A is lowered to 0.1 or if p is raised to 0.99, the agents optimal futures position increases.  However, note that all the optimal  hedge postions in Table 6.1 share one characteristic. That is, all the optimal option positions are approximately equal to zero, (actual values are very small positive and negative values). This result is in contrast to the comparative static results derived by Lapan, Moschini, and Hanson (LMH, 1991), where they found that: The optimal hedging strategy involves using only futures, and the amount of futures is determined by the covariance of cash and futures prices. However, if futures prices and/or options premiums are perceived as biased, options are typically used along with futures. Thus, in this model options are appealing more as speculative tools to exploit private information on the price distribution, and less so as an alternative hedging instrument. (p.74.) Thus, the LMH results are not very significant or valuable in this particular case.  6.2 Results from the Safety-first Model 6.2.1 Base Case Similar to the expected utility model, the base case will be a function of the same initial parameter values: Y=1, b=5, =5, ab=crP=O. 8 K=5, f=5.2,  95 p=O.  In 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 the assurance that his revenue does not fall below the limiting revenue (e.g.,  a  51  mortgage payment on the farm, the cost of production) of 4.0 more than 15 percent of the time. Figure 6.1 is an illustration of the two-dimensional analysis technique described in Chapter 4, where the iso-probability frontiers are de-lineated in a futures (X) and options (Z) space (i.e., points on this curve show combinations of futures and options positions that give rise to an equal probability of falling below the revenue threshold). An initial observation is that as the magnitude of the positive futures price bias increases, then the potential combinations of X and Z which satisfy the feasible region also increase. As discussed in Chapter 3 and illustrated in that chapter’s Figure 3.1, this study’s interpretation of the relationship between the safety-first’s objective function and its constraint is similar to that of Telser (1955) except that he examines the behavior of a hedger who 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 iso probability frontiers tend to share the same (X, Z) co-ordinates. This can be explained by examining the expected revenue relationship as the futures position (X) approaches zero.  As X approaches zero, the expected revenue is  dependent primarily on expected spot prices as small changes in the futures price f has little impact on it.  52  Figure 6.1 Feasible regions: base case and alternate futures price biases Futures (short) 5  Base Case Bias +0.4% (0.91,2.91)  -o  -4  Options (short put)  -2  a Options (iong put)  -2 -3  Bias -4% (-1.55. 2.12)  As introduced in Chapter 4, the objective of the agent is to maximize revenue subject to the probability constraint. Thus, the agent would like to push out the iso-revenue line in the direction of the arrows shown in Figure 6.1, yet remaining within the feasible region of the constraint.  The optimal point or  hedging combination for the hedger is at the point where the objective function and the constraint are tangent. In the base case scenario, the optimal point is that of 1.31 short futures contracts and 3.83 long put options.  However, if  futures prices are biased in a downward direction (f < fl’), the optimal values occur on the underside of the feasible region even though the slope of the isorevenue line has not changed. The reason for this result is that the agent feels that the futures contract is underpriced, then, in order maximize expected revenues the agent would take a long futures position or certainly reduce his/her short position from the unbiased case. In Figure 6.1, the point of tangency for  53  when a 4% downward bias exists is where the optimal position is a long futures of 1 .55 contracts and 2.12 long put option position. In Figure 6.2  ,  the payoff of the optimal X, Z combination for the base  case scenario of Figure 6.1 is contrasted with other combinations which, although are not the optimal points, do comprise points on the base cases isoprobability frontier. Of the three strategies, the short futures and short put option position of (3.62, -2.2) is the least desirable. Even though the strategy gives the agent the largest net selling price for a futures price realization of 5 (point A), it fails to provide the agent with higher payoffs than the other two when the futures price realizations are below or above their expected value. Also, if the futures price was to rise to a value above 7.5 then the net selling price would be negative and the agent potentially could suffer infinite losses. It is for this reason that an implicit constraint is included along with the safety-first model’s constraints. The additional implicity constraint is one which precludes the desire of the agent to choose an infinite combination of futures and options that will on average satisfy the probability constraint and maximize expected revenue, but potentially make the hedger susceptable to potentially infinite losses.  54  Figure 6.2 Payoff of the base case and alternatives 30  Futures only (2.11,0)  25  Average Net Selling Pric 0  Base Case (1.31,3,83)  15  10  5  0 0.5 -5  4.5  5  5,5  6  6.5  7  7.5  8  Realized Futures price Short Futures Short Put Options (3.62. -2.2)  In contrast, the optimal base case strategy of X=1.31 and Z=3.83, which has the lowest average net selling price when the futures price realization equals 5, enables the agent to have very high net selling price when the futures price falls below 5.0 into the “low risk” region. 2 In addition, this strategy provides the agent with protection against any potentially low net selling price realizations in the “high risk” regions. In other words, the base case scenario is the ideal tactic whether 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 only strategy when the futures price equals 5 is the amount of the option premium which shifts the payoff downward.  55  Figure 6.3 Optimal hedge payoffs for varied biases 30  25 Average Sng Price 20  Bose Case (1.31,3,83)  Realized Futures Price  Figure 6.3, illustrates the payoffs for the optimal hedges under the base case scenario, an upward bias of 0.4% and a downward price bias of 4%. As shown, the base case of (1.31, 3.83) involves the use of a larger quantity of both futures 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 the 0.4% upward bias case. This is exemplified by the increased slope of the base case s payoff function for futures price realizations below 5.0. The increase in the 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” and speculative gains are a concern of the agent. Notice how the payoff function for the case where there is a downward bias of 4% is essentially a mirrored image of the base case. This is because with a downward price bias of 4% the agent feels that the price of the futures contract is under valued; thereby, the agent is  56  wishing to capitalize on the instance that the futures price rises before the futures contract is offset.  6.2.2 Sensitivity of the Probability Threshold  Figure 6.4 Reduce the probability threshold to 10 percent 5  4  Bias ÷4%  Base Case  -6  -4 Put Options (short)  -2  6 —1 -  -2  ‘Put Options (long)  A number of results are expected when the probability threshold is reduced to 10 percent as displayed in figure 6.4. Firstly, the magnitude of the X and Z combinations which satisfy the constraint should decrease; which they do when 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 increased options position.  One might expect the opposite to occur.  Figure 6.5 below  illustrates why the optimal hedging strategy of (1.05, 4.04) under a 4 percent  57  upward bias is superior to the strategy of (1.23, 3.6) which involves the use of fewer futures and options contracts, yet satisfies the probability constraint. Once again, the payoff for the (1.05, 4.04) hedge gives a higher payoff when futures price realizations are below 5.0 and creates a superior floor price to that of the alternative (1.23, 3.6) strategy when prices are above 5.0. As before in figure 6.3, 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 percent 25 Bias +4%  ( 20 Average N t Selling Piic  /  /  :“:N.  B’‘as 4°! -  Bias +4% (1.23,36) 5.  0 0  —  ——  0.5  1  1.5  I 2  I 2.5  I 3  I 3.5  I 4  I 4.5  I 5  I 5.5  6  I  I  I  I  6.5  7  7.5  8  Realized Futures Price  6.2.3 Variation of the Volatility Figures 6.6 and 6.7 represent the optimal hedge strategies and their respective payoffs when the standard deviation of futures and spot prices is increased from 0.8 to 1.25.  One can observe from Figure 6.6 that with an  expansion of the standard deviation, which implies a higher level of volatility, the size of the feasible region declines from the base case. In conjuction with the  58  smaller feasible region, the optimal hedge of X* and Z* is also reduced. However, note that the relative point on the iso-probability frontier has not changed from the base case. This is because the slope of iso-revenue line has not changed with the increased volatility of prices. Also, unlike Figure 6.1, the iso-probability frontiers do not share the same lower locus of points.  Figure 6.6 An increase in the standard deviation to 1.25 5  Futures (short) 4  Bias +4% (1.17, 1.92) Base Case (1.31, 3.83)  -I -6  -4 Put Option (short)  -2  4  6  Put Options (long) -2  When a downward bias of 4 percent exists, the result contrasts the results of the previous scenarios.  Instead of purchasing put options, the agent sells  them. In other words, with an increased volatility of futures and spot prices the agent feels that there is less risk involved in speculating with options (short position) than with futures (long position). However, this result is obtained by implicitly constraining the X and Z to combinations which do not give rise to payoffs which fall below zero. This can be seen in the payoff diagram of Figure  59  6.7. The payoff line for the bias -4% case remains above zero for all futures price realizations.  Figure 6.7 Increase the standard deviation to 1.25 30  Base Case  Bias +4% (1,17, 1.93)  25  Average Net 20 Selling Price 15  (1,31,3.83) Bias -4% (1,73,-1.89)  ..  o I  0.5  1  1.5  2  2.5  3  3.5  4  4.5  5  5.5  6  6.5  7  7.5  8  ‘-5  Realized Futures Price Also note that when the standard deviation is increased from the base case, the optimal payoff strategy is less steep in the “low risk” region and flatter in the “high risk” area. The optimal strategy must have a flatter payoff than that of the base case because the distribution of the net selling price around the mean is more volatile.  6.2.4 Varation of the Basis Risk Figure 6.8 shows how the optimal hedging strategies vary from the base case when the amount of basis risk is increased or decreased. As one might expect, when the basis risk is decreased will satisfy the feasible region.  (p=O.99)  a larger quantity of X and Z  The opposite occurs when the basis risk is  increased (p=O.82). Not only do the available combinations of X and Z satisfying  60  the safety-first model’s probability constraint decrease under increased levels of basis risk, but the optimal strategy is to use fewer long puts and more short futures contracts.  Figure 6.8 Higher and lower basis risk Futures (short) 5 High Basis Risk (1.50,2.52, p=O. ) 82  Low Basis Risk  ) 99 p=o.  -6  -4 Put Options (short)  -2  6  Figure 6.9 below shows how the agent will choose a strategy which sacrifices higher net selling prices at low futures price realizations (reflected in the reduced slope of the payoff in this region) in order to obtain overall higher net selling prices when futures prices exceed 5.0.  This is a necessary tactic  because the distribution of the net selling price is more disperse when the basis risk is higher; thus, an increased probability of the net selling price dipping below zero exists.  61  Figure 6.9 Payoff diagram for high and low levels of basis risk 30  Low Basis Risk (1.18, 4.54, ) 99 . 0 P  High Basis Risk (1.5,2.52, p=O. ) 82  Selling Price 20  10  (1.31,3.83)  5  0 0  0,5  1  1.5  2  2.5  3  3.5  4 4,5 5.5 5 Reaized Futures Price  6  6.5  7  7.5  8  6.2.5 Alternating Strike Prices Finally, Figure 6.10 illustrates how the feasible region are modified when the strike price of the put option is alternated. The most obvious observation that can be made is that the feasible regions pivot around two points on the futures (X) axis. This is not a surprising result, given that changes in the strike price of the option should have no effect on the level of futures contracts which make the probability constraint binding. As the strike price is raised by 4 percent, the optimal X increases from the Base Case, but the optimal Z decreases. The opposite occurs when there is a downward bias of 4 percent.  62  Figure 6.10 Feasible region under higher and lower strike prices 6  Futures (short) 5  strike price =5.2 (1.74, 2.02)  Base Case (1.31,3.83)  I’  -8  -6 Put Options (short)  strike price=4.8 (1,24,5.71)  / -4  -2  8  -1 -2  Put Options (long)  -3  6.3 Summary of Safety-first Results Table 6.2 provides a summary of the results shown graphically above. As before, positive (negative) futures value corresponds to a short (long) position. The opposite is true for the put option values where a positive (negative) values implies a long (short) put option position. One conclusion that can be made about the results derived from the safety-first model is that the optimal hedging positions tend to be fairly sensitive to the parameter combination. One instance in particular is where the volatility of the futures price and cash price is raised to a standard deviation value of 1.25 from 0.8 (this raises the coefficient of variation from 0.16 to 0.25). Under the assumptions of higher futures price volatility and a downward futures price bias of 4 percent exists, the optimal hedging position involves taking a short futures position combined with a short put position. As mentioned previously, this contradicts the general results of a short futures and long put position as obtained from varying the other model parameters from the 63  base case. varying  In closing, the extreme behavior of the safety-first model under  parameter  values  prevents  one  from  making  any  general  recommendations for an agent such as a farmer.  Table 6.2 Summary table of safety-first model results Level of Futures  Put option  bias  position (X*) position (Z*)  +4%  1.31  3.83  -4%  -1.55  2.12  +4%  1.05  4.04  A 0/  -1.50  2.33  +4%  1.17  1.92  -4%  1.73  -1.89  Higher basis level (p=O.99)  +4%  1.50  2.52  Lower basis level (p=O.82)  +4%  1.18  4.54  Higher strike price (k=5.2)  +4%  1.74  2.02  Lower strike Drice (k=4.8)  +4%  1.24  5.71  Base case  Lower probability threshold (y=O.1)  /0  Increased volatility (a=1 .25)  64  CHAPTER 7 SUMMARY AND CONCLUSIONS  7.1 Summary  Agricultural commodity prices have through history behaved in a potentially highly volatile manner; thus, leading to the placement of high levels of financial risk on producers, processors, exporters, and other agents involved in the procurement and/or sale of these commodities. For some time, the availability of futures contracts has provided these agents with a risk management tool. Basically, by entering the futures market via a futures contract the agent is able to substitute price risk for basis risk The argument being that basis risk is generally less than price risk.  In the early  1980’s option contracts on commodity futures were made available on some the U.S. commodity exchanges. Options provided an agent with an additional risk management tool.  Options can be used with futures or on their own.  The  advantage of using option contracts is that an option allows for a variety of revenue distributions not previouly available with only futures. This is because the option is like insurance coverage, the holder of the contract can either exercise the option or not. The drawback of this arrangement is, like insurance coverage, that the holder of the option must forego a premium even if he does exercise the option. This study has attempted to evaluate how a decision maker, such as a farmer, might use futures contracts and options on those contracts as a means of managing price risk. Two contrasting models were used for this purpose. An expected utility model and a safety-first model. The safety-first model was that of Telser, except enhanced to include not only futures but options as well. Both models are based upon a single-period framework where at the beginning of the period the agent enters the futures and option markets and places a hedge. At 65  the end of the period the agent offsets his futures position and sells his commodity in the spot market. The single-period model was formulated such that the agent could speculate on the futures price bias and not the volatility of the option price. Once the empirical models were defined, then the parameters were derived for the models using canola data obtained from the Winnipeg Commodity Exchange. A technique utilized by Rolfo (1980) was used to obtain the necessary parameters from the data. The data series spanned a ten year period froml98l to 1990. Optimal hedging results from the two models were derived under alternative parameter scenarios which involved varying levels of basis risk, futures and spot price volatilities, and risk aversion.  7.2 Conclusions The results from the maximization of the expected utility model suggests that an agent will (when futures or option price biases exist) hedge an amount equal to that of the co-variance of the spot and futures prices divided by the variance of the futures price. If a price bias exists in the futures market, then the agent takes the same futures position as the no bias case and then speculates on 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 a reduced open speculative position when in the presence of a positive futures price bias. Of course, the opposite occurs when these factors are lower than the  base case.  The most interesting result of this model is that unlike the  comparative static result of Lapan, Moschini, and Hanson (1991), put options do not figure into the optimal hedge even when there is a speculative component to the hedge. 66  An opposite result is obtained under the safety-first model.  With this  model, options are always used as a means of speculating on the upward and downward future price biases. Other results are similiar to the expected utility model’s. In general, as the price bias increases, so do the overall futures and put option 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 some  cases, the short futures positions may change to a long positions. When the risk level is reduced (in the safety-first model this is represented by the level of the probability threshold) the hedging agent increases his long put position and lowers his optimal futures position. When the volatility or standard deviation of futures prices is increased, the agent’s response is to reduce his futures and options positions. However, when the bias is negative, the agent increases his futures position and writes put options instead of purchasing them. This illustrates the rather sensitive nature of the safety-first model to parameter changes. As the amount of basis risk increases (decreases) the agent takes a larger (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 have on occasion a very extreme outcome which could involve a large financial loss.  7.3 Restrictions and Further Research A number of assumptions made in this study may prohibit the comprehensive generalization of the hedging results.  In the case of the  expected utility model, the results could differ significantly if one assumed a different utility function from that of the exponential and its constant absolute risk aversion.  Also, the assuming away of transaction costs and margins 67  requirements may be a strong deterant for agents, since this may involve the need for capital of which they do not have. Certainly, if borrowing is required then it is not obtained at a free rate.  In the safety-first model there was an  instance 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 the hedger in the futures market. Another drawback of the study involves the perfect divisibility of the contracts. Of course, in the real world this is not the case and the lumpiness of the contract sizes may lead to increased vulnerability to the hedger 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 farmer could modify the results because of the farmer’s hesitation to hedge Xof his crop in case producer fell below X and the farmer would have to payout the deficiency. This uncertainty could lead to an increased use of options because of its contingent exercise aspect. 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Hayenga, (1987): “Comparison of Analytical Approaches for Estimating Hedge Ratios for Agricultural Commodities.” Journal of Futures Markets, 7(2): 135-146. Wolf, A., (1987): “Optimal Hedging with Futures Options.” Journal of Economics and 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 State University Press.  72  APPENDIX 1 THE DERIVATION OF THE UNCONSTRAINED OPTIMAL HEDGE  Case 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’s profits are defined as follows. It =  py  +  (f  -  p)x  +  (v r)z -  1K-i  where v  fp<K K  =  Alternatively, 1 = It =  (y-x-z)p  +  (y-x)p  fx -rz  +  fx  +  (K-r)z  Equivalently,  1 t + N p 1 2 2 = t it p + N 2 wheret (y-x-z) = 1 =(y-x) 2 t N= fx+(K-r)z 2 N = fx-rz =  Deriving Expected Utility of Profits Assumptions: (i) futures price is normally distributed with a mean value of  and a  variance of a. (ii) the decision maker’s utility function is as follows: U=eAit  where A  >  0  A <0  risk averse risk prefering  73  Given the information above, one can derive an expression which denotes a decision maker’s expected utility of profit as a function of his/her position in both the 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 when the futures price lies below its strike price 1 (E[U 1 ) and the expected utility of profits when the futures price lies above its strike price (E[U ]. From this point 2 onward we will work with the E[U ] portion of equation (Al .2) since analogous 1 statements can be made for the later half, 2 (E[U ] .  We can redefine E[U ] as follows: 1 1 E[U  J  =  f  1 e’ f(p) dp C  where  1 =  —  e’  (A1.3)  13 =—Ati 1  Since f(p)=  e  2a)  ‘2ita then one can one can restate (Al .3) in the following manner, K  ]=C 1 E[U  fe J27to  2 d p  (AlA)  -  Similarly,  74  K  1  J=C 1 E[U  fe  (Al.5)  2 ’ ’dp  Re-arranging (Al .5), v  1  1=C 1 E[U  -  i(P—P)——i —i—  e”Je  dp  (Al.6)  2ta Re-arranging the exponent of the natural exponential function, _J[(_)2  _2a13 ( 1 p_)]  (Al.7)  We want to re-organize the above expression (Al.7) in such a manner that we can derive the moment generating function (m.g.f.) of a single, normally distributed 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  J’2’°’ 1 E[U e =C  K  1  Jq2ita e  “  dp  (Al.lO)  Note that the integrand is the probability density function (p.d.f.) of a normally distributed random variable with a mean of -i-3 a and a variance of a. 1 Therefore,  75  ]=C e1 1 E[U  1K0  (A1.1 1)  Op  Where I(.) is the cumulative density function of the standard normal distribution. Analogously,  $  ]=C 2 E[U  2  c3  e  ‘  dp  K  —c —  where C 2  =  e2P2P*  2  —  2 ( 1 a p_(+ )”i (A1.12)  e2  and  12 =  —  2 At  Case 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 (exercise option versus not exercising) as follows: =  2 = it  (y-x-z)p  (y-x)p  +  ÷ fx fx  We can re-define it  2 it  where  +  (K-r)z  -rz  it  and  2 it  ifp<  K  ifp  K  as,  1 p-i-N 12 b+t 11 =t ’+t 2 t 2 p+N 22 1  11 t  =y  t  121  =3’  22 t  =—(x+z) = —x  1 =fx-i-(K—r)z N 2 N  =  ft  —  rz  Now we can define the decision maker’s expected utility of profits in the following manner, E[U] =  —  J $ e f(b,p)dp db —$ $  K-°  =  2 ] 1 E[U ] -i-E[U  e2f(b, p)dp  db (A1.13)  76  Working with E[U ] we have, 1 ]=C 1 E[U 1 where C 1  13  =  J  Jeh1b12P  f(b,p)dpdb  —e -Aj’  1 =—At  1312 =  (A1.14)  12 —At  Manipulating the integral expression,  2ltabaP..SJ1—p  where hb  =  2 exP{13iib+1 P _ 3i  2  (  b—b  and h  P  [2 h_2PhbhP+h]}dPdb  =  (A1.15)  Re-working the exponent of the natural exponential function by multiplying and  1312  by a ratio of  subtracting f3 b and 11  ab/ab  1312  and  a/a  we have,  13llab(a13llb+131PP[a]+1312P_  =  respectively. Then by both adding and  oh + 1312k 12 13 ii 1 + 13Hli + 13 —  2(1  )[hP+2PhbhP 1 ( 2 2 +hP]  2 [ hp  + 2phbhP +  h]  (A1.16)  (Al .17)  Given equation (Al .17), we can re-write (Al.l5) as,  eh1b412P  27ObOg1  2 —  aboPJ  oh 12 SexP{13llabhb + 13 —  2(1 P 2  [h  —  2phbhP +  (Al.18)  77  Since we defined h,,  b-b =  and h  then by differentiating we have  =  dhb=-’-db, dh=-i-dp ap  Now we can substitute the above differentials into equation (Al.18) and re express it as, ePhh12PobaPf$  exP{llabhb  ah 1 + 2  12 [h —2phbhP  —  dhPdhb  (Al.19) Re-working the exponent of the integrand in (Al .19)  —  2(1  — —  2phbhP + h  —  2(1 p )llabhP 2 —  —  2(1  —  p2)i a 2 php]  (Al .20)  We can re-write (Al .20) as follows 1 )+(h _Kpi)2J+(ia 1 p(hb Kbj)(hP —K 2 2 — —Kb!)  P11l2Gbap 2 +  a) 12 -i-  where  Kbl 1 K  llGb+PI I 3 2GP =  Gb 1 3 PI 1  a 12 +l3  (A1.21) By replacing the exponent of the integrand in (Al .19) with (Al .21) we arrive at  2  ]=C 1 E[U  * 2(i_p2)  1  exP{  3 P + a 2 l baP llf +a)  [(hb —KbI) 2 2 —2p(hb  Kbl)(hP  )2]}dpdb 1 —K ) 1 +(h _K  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 normal distribution with the means  Kbl  ,  K and variances of one.  Therefore we are  able to write equation (Al .22) as 1I E[U  =  }*  C *ex{ 1  (K—(-i-plloboP l2Gp  where  is a special case of Pl  (Al .23) Analogously, 1  —  22 b P+13 21 13 2 + l 2 ab 23 P 2 + 2 2 2GGp 1I l +1 220p 3 —  ]=C E[U * 2 exp  where C 2  =  121  =  122  =  ‘  *  1  —e  21 At  (Al .24) Therefore,  +  [K_  i + 22 2 2 *exp{ c P+  +  ]] 2 K  +  —  (Al .25)  79  Now substituting back in for the 1 C s T , 1s and the K s. 1 E[U] = —exp{_A[fx + (K * exp{_Ay  b  +  —  r)z]}  A(x +  pAy app  + +[A2y2o + A(x+  —  y(x + z)abaP + A 2 2pA 2 (x + z) 2a  J}  Z)O)l_exp{_A[ft _rzj}  [ J * exp{_Ay b + Ax + [A2y2a 2pA y 2 x abap + A2x2a]} —  *  l_1K_(_PAYabap+Axa  )  (A1.26)  Gathering the A’s E[U] = _exp{_A[fr + (K r)z}} —  1• / — —\ A2r2 *expAj(x+ z)p —yb)+---[y + A(pyaba —(x  *  —  L  +  2 0 b  22  —  2py(x+z)oboP +(x+ z) a  z)a)1  exp{_A[fr —rz]}  ]  *exp{A(x_y)÷i[y2a — pyxabaP +x2a]} 2 *  l1K(pyab.px0 a”  )  (A1.27)  80  APPENDIX 2 THE DERIVATION OF EQUATION A1.20  Beginning with (Al .20) and then completing the square on both  hb, lz  (see Mood  et al, p.165).  ) 2 2(1—p  [h:  —  2PhbhP +  —  2(1  —  P  jlGbhb 3  —  2(1  o + 2(1 r 1 )13 + (1— p ah 12 2 )13 I3 + (1—p 1 )P13t abaP 2 )3 a] 2 2 12  —  —  + 2pI3Ill abaP + 2  (A2.1) Now we can re-arrange the term in square brackets into the form of —2puv+v First let, 2 u . ii =  Jib + Kb  v  h +K  =  uv = hbhP h  =  + hbKP + hPKb + KbKP  +2 h1b’<b + K + 2hK + K  =  (A2.2)  Then placing the above expressions into the u 2 form we have, 2 —2puv-i-v h,  +2 hbKb + K, +  + 2hK + K  —  2phh  —  2phbKP  —  2phPKb  —  2pKbKP  (A2.3)  }  (A2.4)  Re-arranging (A2.3), —  2 b 1 P ”p  working with the  +  +  f2[h,K,,  [.1 term  + hK  —  phbKP  —  phpKbj + K  +  K  —  2 p KbKP  in (A2.4) we can re-write it as, (A2.5)  hb[Kb—PKPj+hP[KP —pKb]  Using equation’s (A2.5) format, we can make a similar arrangement using the  [.]  term in (A2.l). _2[(1  —  2 )13 lab/lb + p  2 )13 (1—p oh 12  J  (A2.6)  From the relationship described in (A2.5) and applying it to (A2.6) we can state the following,  81  Kb —pK  =  K —pKb  =  2 )I —(1—p 110b 3  (A2.7)  Taking the above system of equations in (A2.7) we can solve for Kb, K. Kb 2 =pKP—(1— ) f3llab p  K  =  pKb  (A2.8)  —(1— 12 )13 2 p a  through substitution we can solve the equations in (A2.8) such that, Kb  =  (1  P ) 2 I3llab +  —  12 ) 2 p a f3 +pKb]  = —(llab  K  (A2.9) = —(12oP  +p11Ob)  Given (A2.9) we can define the rest of the terms in K  f3 +p a p 2 + 1 3 1 2ab0P 1f 12  =  =  {.} of equation (A2.4).  p1 l2ObOp 2 + 1 1i2°p 3  Pi1b  (A2.1O)  +P 131112abaP + 13 2 lll2aboP + PIl 2  =  Substituting the terms in (A2.1O) into the remaining terms in the  {.} of (A2.4)  then, K, +K —2pKbKP  =  f3 3 2 +p a p 2 + l j pbap lI f2 13° 3 r3 1 +p o P 2 i i PzPp i1 I ÷1 iP 3  o +P 1 _2p[pf3 131113120b0p + 11I 2 120b0p + pI3 3 aJ 2  (A2.11) Gathering the similar terms, f l 3 lab  13 2 —p a —P I 2 3lab 3 13ll13l 3 —2p a baP (A2.12) j p 2 i ppp if +1 1Pp 2 3  Factoring out the term (1-p ) we have 2 2 )13lab + (1—p (1—p 2 )132° + (1—p 2) abap 2 1 l1l 3 P 2 (1— p2)[lab + 2Plllpbap + 13120p]  (A2.1 3)  82  Note that (A2.13) is essentially the term added to [.] in (A2.l). Therefore we have now derived an expression which is equivalent to (Al .20) in a quadratic form.  83  APPENDIX 3 DERIVING THE VALUE OF A SINGLE-PERIOD OPTION  First we let G=Jpf(p)dp  (A3.1) where f(p) is the probability density function for a normally distributed random variable p (price). We can re-write equation (A3.1) in the following manner,  G=ff(p)dp $P -  K  dp  $f(p)dp (A3.2)  K  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) is  the probability density function (p.d.f.) of a truncated normal distribution with p restricted between  -oo and K. Therefore, the term [.] becomes the expected  value of the random variable p when p’s distribution is  a truncated normal  distribution. Johnson and Kotz (p.81) describe how one can transform the expected value of a random value --whose distribution is a truncated normal distributioninto an expression which is a function of the p.d.f. and c.d.f. of a standard normal distribution. The expected value of p is given by  84  E(p)=p.  +  0 -  a)  \a where P(.) c.d.f. of the standard normal cp p.d.f. of the standard normal the lower and upper limits of the integral A, B  (A3.3)  Substituting the upper and lower limits from equation (A3.2) into (A3.3), and noting 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 the strike price (k) and the price of the underlying asset summed over all potential prices. Alternatively, E()= J(K-p)f(p)dp (A3.7) wherep-’ N(i.i,  02)  Equivalently, 85  v= K Jf(p)dp -  Sf() dp  Given that the first integral in equation (A3.8) is equal to F(.)  (A3.8)  --  the c.d.f. of the  standard 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, its variance and the strike price.  86  APPENDIX 4 DERIVATION OF THE SAFETY-FIRST PROBABILITY CONSTRAI NT  Deriving an expression for Pr(it) when prices are joint normally  distributed via the cumulative-distribution-function technique.  (i) Consider the following profit function for when the price (p) of an underlying asset is below and then above the strike price (K). p-i-& 2 +& b Jâ 3  p<K  3 + 1 ‘j3 p 2 b +13 13  pK  where b and p have a joint normal distribution of f(b,p) with means  ,  j5  variances 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,  (—1  Pr(7t)=J  where  t’  =  1 a  =  2 a  =  define U(b)  2P)’  3 p  a3  f(b,p)dbdp  +  J  Jf(b,p)dbdp  (A4.1)  3 =W l3 =131/133 3 / 2 a cc =  132  (A4.2)  132/133  p+b)—b 2 1 +a (a ab  and dU  =  a,,  (A4.3)  The substitution of (A4.3) into equation (A4.1) requires a change in the limits of integration; as follows, 1_ U[b]=U[(_& 1 P)r]U{ —a 2 =  ( -)/a,,  when p<K  1 ÷a a 1 p 2 +(n_a 2 —a p )—  (A45)  87  Analogously, U[bj =  —  =  —  —  5)/a  (when p K)  (A4.6)  Re-arranging (A4.3) into the following form, b-  U(b)  =  (A4.7)  -  Now lets define q=PP  dq=  ,  ap  where the limit of integration is c(K)  =  Kap  (A4.8)  Re-arranging (A4.8) with q as the argument and p as the value of the function and then substituting into equation (A4.7) we can define the following relation ships, b—b  +c ( 2 qap  =U(b)_[1  1 0 =U-ã q ã 1  where  =  —  +  b 0  °2—  p  °b  a °1  2  Substituting (A4.8), (A4.5) into (A4.1) we have the following expression for when (p.K)  $ $  1 2icl—p 2  —1  ,  [(u_ao_ajq)2_2p(u_ao_a q 1 )q+q2]  ’ 2 e  dUab dqo (A4.1O)  Making analogous substitutions for when (pK)  88  I  2n1  —  2  —1  i  dUob dqa  (A41 1)  2 abap p  K—  Letting k=  -2 (u_o_.iq)q+q2]  ** ,  ,  —  1 t  and substituting them into (A4.1O)  =  and (A4.11) respectively, we have, 1  1  +  2  1 02 q) —2p(U—& 1 —& 0 _aiq)q÷q2]}dudq 2 {(u_& 2(1—p)  expl  2 { [ 2 (U 1 exp 1  —  —  —  2p(U  —  —  1 q )q  +q2]}dudq  (A4.1 2) Expanding the expression within the exponent fo the first half of (A4.12) we can re-define it as follows: —1 U2 1 2(1_p2)[  0 +0 u(a q)+(& + 2 1 à q) —2pUq+2p(& 1 & 0 + aiq)q+q2J [.] of (A4.13):  Gathering the terms in  [(U2  ) 2 2(1p where U 2  —  0 2Uc(  —  —  —  0+ 2Uä  (A4.1 3)  +  2)  -2 =  —  cx + 1 p& + i) 2q(Uã 2 q 21—2 —  —  —  pã + pU)l  (A4.14)  (U  —  If we define the following relationships: (U—a 2 ) 0 2 1—p  1+  =  U—ã 2 ) 0 1 1+ä+2pà  + 2p& 1  [i+a +2p& 1 _(1_p2)l + 2 p (l+a+ ) (UO) al_p  ) = (& 2 —(1—p  2 + p)  and (Ià  —  —  pà  + pU =  (U  (A4.15)  (A4.16) —  & )(ã  + p)  Now we can re-write (A4.14) as:  89  r(, 0 (U-& +p) 2 ) +(1+& 2 )q —2(& 1 +2pà  —(U—& 2 ) 0 —1 )2(1_p2)[ 1 2(1+&÷2p&  1 1+&÷2p&  (A4.17) Re-arranging (A4.17) into the form of a perfect square in the bracketed 2 ) 0 -(U-a  +  -(1÷&+2p& ) 1 E 2 [q  [.] term:  2(& + 1 p) (U- ã )q+I 0 pã 2 (l+&+ ) 1 J 1 [1+a+2pa  0)2]  (A4.1 8) We can simplify (A4.18): —(U—a 2 ) 0 ) 1 —(1+&+2pä + ) 1 2(1+&+2pã ) 2 2(1—p  r  0 + 1 (& ) p)U—& l ) 1 (1+ã+2pã  2  (A419)  j  Alternatively, 1 -(U—a 2 ) 0 ) 1 2(1-i-ä-i-2pä  )1 0 ÷p)(U-& 2 Lq- (&(1+ä+2p& ) j 1 (1—p ) 2 (1+& -i-2p& ) 1  (A4.20)  Substituting (A4.20) for the exponent in the first half of (A4.12) then we can re write --due to the second half being analogous-- then entire expression of (A4.12) as follows:  itr  k  1  Pr( =  +  —1  ll2(1 +  1  -i-p)(U—ä 1 (ä ) 0 1 L1[ 2  2 ) 0 (U—a  I  + 2p&)  JJ g2t(1÷r3 +2pf3 ) 1  —1  1+?+2Pj  1  e ( 2 121) *  dqdU  2(1p) ) 1 (1+a+2pd  F  _ [ 1 q  2 ) 0 (U-  1  e2(121) *  I  2n(1p2)  ) 1 (1++2p  j  e2(1_p 1 2 21)  dqdU  (1-i.+2p)  (A4.21) where  90  1  -1 (U-a 2 ) 0 2(1+&+2p&,)  e  (A4.22)  /2t(1+ä +2pà ) 1 —1  1  e  2 ) 0 (U2(1++2p)  (A4.23)  ) 1 /it(1+13 +2p13  are the p.d.f.’s of the normal distribution with means a 13 and variances ) and (1÷f3 +2pf3 1 (1+&÷2p& ) respectively. Also, 1 I 0 +p)(U—ä 1 (ä ) t  ]  1  e  2n(1_p2)  I  (1+ã+2pa) T(1_p2)/(1+a?÷2pa ) 1  (A4.24)  ) 1 (1+ã+2pã  I  1  I  (÷p)(U4 ) 0 1  —IL’ e  j  (1?÷2pi) 2(1_p ) 2 /(1++2p,)  ) 2 2n(1—p  (A4.25)  (1.i-+2p)  ) 0 +p)(U—& ) 0 +p)(U—3 are the p.d.f.’s of a normal with means (& and (13k and ) 1 (1+ã+2pä (1-i-f3+2p13 ) 1 variances of  (1—p ) 2 ) 1 (1+&+2p&  ‘  (1—p ) 2 respectively. ) 1 (1-i-3+2p3  Recall from (A4.2), (A4.9), and (A4.1 1) the following relationships:  1 2 + 1 =—(a , a &)  *  0  —  **  1  (A4.26)  =  0,  where a  =  =  3 / 2 a a  =  ,  . 3 7t/c(  Aggregating these terms through  substitution, we have: ao  —  (& +  ** ,  —  1 It  —  —  b 3 a  a 0 3 J,  3 a  cx2ap  -  ,  —  1 a  =  (A4.27)  cx a 3 b  Given the relationships in (A4.27) we can define: 0 a -  —  3 + 1 It—(cX p 2 b -I-cX cL ) =  a a 3 b  (A4.28)  91  Substituting & from (A4.27) into the constant term of (A4.22),then:  427t(1 +  + 2pã)  +  2p  +2  =  cYb 3 a  =  +  +2 P a Pbap a  cL a 3 b  1  =  cxc  =  427t(&cY +âa +2pâ abaP) 3 à 2  g2it(ac +âcY +2p& &pbcYP) 2  (A4.29) Making the appropriate substitutions from (A4.27) into the exponent of the natural exponential of (A4.22): r  1  ,-,.  lu—Ij  2 + 1 cx p cx  L  2 ) 0 (U-&  2 1+ã + 1 P 2 &  apbU—(cl  )]  a3a  21  —  [ 2 &o +&o +2p&2&3aboP  —  X G 3 b  ÷a p 2 )  cc a 3 b  -I-cL p 2 )}  2  (A4.30)  “2222  +3 X G b +  Now if we define: tht=& o 3 bdU  If we substitute  it  (A4.31)  into the integral, we need to change the limits of integration in  (A4.21) to the following:  =  ÷b]  =  ]+b]=  (**)  =  =  3it2 =  (A432)  (A4.33)  Solving it(U) for U:  u=r_-—i_-= L3 Jab  ab a a 3 b  then substituting (A4.34) into (A4.30) and combining it with (A4.29) we have: 92  bu—(al÷a 3 [a p 2 ]  e  2&4+2pâ à 2 3baP  j2t(&a  +  (A4.35)  + 2pâ abaP) 3 & 2  Now, making similar sets of substitutions for (A4.24) as with (A4.22) then: 2 P (& G 3 ÷ 2 ) b)(—(ala +pa ) a 22 22 a o p 2 o 3 a +a ,,o + a  —1 2  1  I  e  2it(1—p ) 2 &a ‘2 2 a -I-a 2 cx o3 3 p + c 2 O x cx POb  If we define  =  +  =  )&a 2 (1-.p  (A4.36)  b as the mean level of profits when p<K 3 & +â5+&  +  ,  and  2p&2&3oPob as the variance of profits when p<K then we can  re-write the expression (A4.21) --when making analogous statements for when p K-- as:  (2  k  Pr(t =  o T LL2  e  2  *  dqtht  1 JM  (  2 a 2 ( a 3 ) b)—t p+p  Iq—— ii  M2  _{it  +ff where  1 M  =  e  2  afl2  *  [(1_ Pa ab] 3 ,  I  e  2 M  =  —j  dqdit  [(1_ (A4.37)  Alternatively, if we integrate with respect to q, then we obtain:  93  K—15  —  Ob)(t 3 (&2aP +PG(  n  Pr(t)=JJt)  K—  (2aP+P3ob)(it—i2)  +ff ( 2 ) where  dic  M  d  M  Jj (t)  p.d.f. of the normal distribution with mean 1 and variance  (7t) 2 f  p.d.f. of the normal distribution with mean 2 and variance c c. d.f. of the standard normal distribution  (A4.38)  Substituting for the cx’s, 13’s, and Ms, we obtain the following: F(ltL; X,Z)= where  =  J  bY—(X+Z)-i-JX+(K-r)Z =  =  [ii)tht + 1 )4(A  bY  -  X+ JX rZ -  a +Y 2 (X+Z) a +2p(X+Z)Yoa 2 =  a +Y 2 X a + 2pXY aPab 2  (A4.39)  -  (Ypa, 1 a)(t _C( )  K-p ap  F(1_p2)Y2al2  [  2  j  94  APPENDIX 5 DERIVATION OF THE SAFETY-FIRST MODEL OBJECTIVE FUNCTION  Beginning with the standard two state dependent relationship of the single-period revenue model defined in section 4.1 of this paper and in an article by Lapan, Moschini, and Hanson, (shown below), Ibv÷(f—p)x+(K—p—r)z by+(f—p)x—rz  7C  if p<K if  pK  one can define the expected profits as the expectation of the two states as is shown in equation (A5.1) below. E(n)=$f[by+(f—p)x+(K_p)z—,]g(b,p)dbdp +JJ[by+(f—p)x_rz]g(b,p)dbdp where g(b, p) is the joint normal density function  (A5.1)  If one intergrates over the random variable b then one obtains, 1 E()=Jyh(p)dp+J[(f—p)x+(K—p)z—rz]h(p)dp +$iyh(p)dp+ K  f[(f— p)x_rz]h(p)dp K  14’here h(p) is the marginal density function ofg(b, p).  (A5.2)  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, then their marginal distributions X and Y are univariate normal distributions.  95  Since 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 which is 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 redefine equation (A5.4) as, E()=y+(f-)x+z (K_1K+p1Kap a} k\a’)  -z where cp[.]  (K_f)1K_+p(Kap i%\a)  apj  PDF of the standard normal density. CDF of the standard normal density.  (A55)  96  


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