Hedging Financial and Business Risks in Agriculture with Commodity ...

Hedging Financial and Business Risks in Agriculture with Commodity-Linked Loans Yufei Jin and Calum G. Turvey

Abstract One of the particular problems facing agribusiness firms is the relationship between commodity price risk (a source of business risk) and debt repayment ability (a source of financial risk). This study examines the use of commodity-linked loans applied to agricultural credits. A commodity-linked loan is a credit instrument whose payoff is contingent on the value of an underlying commodity or portfolio of commodities. The payoff structure includes an option (call or put) rider that provides a payoff if the commodity price rises above or drops below a preset strike price. The payoff is applied directly to the loan. This study introduces the general concept, reviews the literature, and develops and applies a particular model. Simulation results illustrate the interrelationship between options payoffs, strike prices, volatility, and downside financial risk reduction. Key words: business risk, commoditylinked loans, debt repayment, financial risk, hedging

Yufei Jin is a doctoral candidate, and Calum Turvey is a professor in the Department of Agricultural Economics and Business and the Faculty of Management at the University of Guelph.

The recent declines in many commodity prices, combined with the inherent risk of markets, have had a significant negative impact on the agricultural economy, with the most significant impact being in primary production and the input supply sectors. The extent of this uncertainty can be observed in Table 1, which illustrates the range of commodity volatility for a number of futures contracts. The data show that throughout the 1990s, the average annual change in prices decreased for all but live cattle prices. Wheat and corn prices decreased by approximately 15%/year, and the price of soybeans fell on average by over 9% per year. The range of prices was high. Hog prices ranged from $90.12/cwt to $25.22/cwt with a standard deviation of $14.86/cwt. The high price was just slightly under four times the low price, and the annualized standard deviation of the percentage change in prices was 0.426. This implies that in any given year there was a 67% chance the price of hogs could increase or decrease by 42.6%. Likewise, the percentage changes in wheat, oats, corn, and soybeans were 40.1%, 37.7%, 37.3%, and 27.7%. Only live cattle showed an annual increase in prices, but its volatility was still significant at 21.1% (Table 1). Amid reductions in public support programs, a number of financial and insurance instruments have emerged for primary producers. These include revenue insurance contracts, crop yields futures contracts, and weather insurance. The

42 Hedging Risks in Agriculture with Commodity-Linked Loans

Table 1. Recent Trends and Volatility in Commodity Futures Prices, 3/21/1996S S2/9/2000 Commodity

High

Low

Mean

Std. Dev.

Mean Annual Change

Annualized Volatility

Lean Hogs ($/cwt) Wheat ($/bu.) Oats ($/bu.) Corn ($/bu.) Soybeans ($/bu.) Live Cattle ($/cwt)

90.12 7.16 2.86 5.48 8.94 73.63

25.22 2.24 0.99 1.78 4.10 54.80

60.25 3.45 1.49 2.72 6.37 65.61

14.86 0.95 0.42 0.77 1.26 3.24

!0.050 !0.156 !0.197 !0.146 !0.095 0.023

0.426 0.401 0.377 0.373 0.277 0.211

Source: Data purchased from Technical Tools, Inc., Chicago, IL.

goals of such products are to reduce the downside risk or variance of prices, volume, or both. Furthermore, these products have generally been marketed as a means to stabilize business risk, and academic assessments of the problem have provided only passive guidance on how these products interact with financial risk. The relationship between business risk and financial risk is an important one. However, most of the academic and applied research on risk management treats them separately. Turvey and Baker contend the two components should be viewed simultaneously and develop a model which optimally selects a debt/equity ratio and an optimum hedge position. They argue that a financial strategy which increases debt use would optimally require an increase in the hedge ratio. The increased financial risk due to financial leverage is offset by the reduction in business risk from the commodity hedge. This study examines a different aspect of the relationship between business risk and financial risk by evaluating commoditylinked loans. Commodity-linked loans are debt instruments whose payoff structure is intrinsically tied to the movement in commodity futures prices. In essence, the loan is linked to an underlying commodity through a put or call option. In either case, if the option expires in-the-money, the principal and/or accrued interest on the loan is offset by the specific amount. A loan with an attached put option would be attractive to a producer who faces the risk of a price decline, while a loan with an attached call option would be attractive

to a processor who faces a risk of price increases.

Total Risk, Business Risk, and Financial Risk The total risk facing any business is defined as the variability in the returns to equity. For corporate firms this will be reflected in the variability in share prices, and for proprietors and partnerships this will be reflected in the variability of owners’ equity or net worth. Regardless of the nature of the firm (e.g., corporate or farm), the sources of total risk are universally equated to the sum of business risk and financial risk. Business risk refers to the variability in the firm’s return on assets and is due to volatility in the markets for its goods and services as well as volatility in the costs of goods sold. In standard deviation form, the financial risk facing the firm is related to the degree of financial leverage employed by the firm, and is a linear multiple of business risk (Collins).1 1 By defining the return on assets as the weighted average cost of capital, the basic model as presented by Collins is ROA ' ROE ( (E/A) % r ( (D/A), where ROE is the return on equity, E; r is the cost of debt, D; and A = D + E is assets. Rewriting the equation in terms of equity, using A = D + E, defining * = D/E, and taking expectations gives E [ROE ] = E [ROA ](1 + *) – r *, with 2 2 variance (total risk) FROE ' FROA (1 % *)2 and r being 2 deterministic. The term FROA is the variance of the return on assets, defined as business risk. Taking the standard deviation and differentiating gives MFROE /MROA ' (1 % *) and MFROE /M* ' FROA . Hence, an increase in business risk increases total risk by a factor (1 + *), and an increase in financial leverage increases total risk by the factor FROA , everything else held constant.

Agricultural Finance Review, Spring 2002

The problem of optimal capital structure represents an ongoing debate as both academics and practitioners attempt to balance the benefits of financial leverage through increased expected returns to equity against the incremental risk, or variability, in the returns on equity that arise from increased debt. How financial risk interacts with business risk is an important aspect of this problem. As business risk increases, so will total risk; but if new debt is issued or borrowed for operations, capital expansion, mergers, or acquisitions, the total risk will increase even further. This is problematic for agribusiness firms. Commodity price fluctuations are exogenous to the firm, and therefore much of the business risk facing the firms cannot easily be mitigated. Indeed, most agricultural business firms can seek out forward contract initiatives or hedge with commodity futures and options. Forward contracts and options can reduce downside risk very effectively for some direct or indirect costs, whereas futures contracts, when used in an optimal hedging strategy, can reduce the overall variance of prices. The use of financial hedging instruments is often regarded independently of the financing function of the firm. One approach, called risk balancing, hypothesizes that shareholders are comfortable with a given level of total risk, and therefore any strategy which reduces business risk can be used to leverage an increase in financial risk through increased borrowings or debt issuance so total risk remains relatively constant (Collins; Featherstone, Moss, Baker, and Preckel).2 2 From footnote 1, we can set E [ROE ] = 0. Solving for ROA gives ROA* = r */(1 + *). ROA* is the critical return on assets. If actual ROA < ROA*, the business cannot cover its interest payments from operating revenue. If f (ROA) describes the probability distribution function for ROA, then IROA*f (ROA) dROA is the probability ROA < ROA*. Since MROA*/M* > 0, an increase in financial leverage will increase the probability of default. This is the definition of financial risk.

Jin and Turvey 43

An alternative view, at least in the case of commodity futures hedging, suggests the optimal hedge ratio should not be regarded independently of financial structure and, in fact, a firm with no or little debt should not have to hedge at all. The symmetrical argument is that if a firm is increasing its use of debt, some form of hedging strategy should be considered in order to reduce business risk and financial risk (Turvey and Baker). Myers and Thompson, and Anderson, Gilbert, and Powell develop similar arguments for using commodity-linked loans to hedge repayment of national debts of developing countries, when repayment of debt is contingent on commodity-based export earnings. They conclude that issuing loans with repayment linked to major export earnings can avert such debt crises as those observed in the 1980s and periodically since then. The interrelationship between financial leverage, the return on assets, and the return on equity is a complex one, and under conditions of risk it serves little purpose to investigate each component independently. Rather, it may be useful for many agribusiness firms to examine alternative forms of debt whose payoff schedule is directly, and inversely, related to the very sources of uncertainty observed to cause financial distress in the first place. It is this aspect of risk management that motivates our investigation into commodity-linked debt. The purpose of this analysis is to examine the applicability of commodity-linked debt in agriculture. The term “debt” can be taken to mean a bond issued by a corporation and purchased by investors, or a loan made by a commercial lender to a farmer. There are two client groups considered. The first group is comprised of agribusiness firms who buy or sell commodities. These are large firms possessing the economic strength to issue bonds as part of their financing strategies.


Farmers represent the second client group in this research. One can imagine a financial product such as an operating loan or the current portion of a mortgage having an option rider that provides the holder with a payment on the loan if commodity prices fall below a trigger (strike) price. This relieves financial distress when commodity prices fall. If commodity prices rise, then the face value of the loan plus interest is paid in full. The remainder of the article proceeds as follows. In the next section we review commodity-linked debt in other (nonagriculture) industries. A bond pricing model, based on the earlier work of Schwartz, and of Carr, is then developed and its comparative static properties described through a number of propositions. Models are illustrated using Monte Carlo simulations of debt linked to the price of wheat. The values of the contracts are estimated and the cumulative probability distribution functions for the debt payments are reported and discussed. Some thoughts on how commodity-linked loans and mortgages can be applied to the farm sector are then presented, followed by our concluding comments.

Background An emerging market for structured financial products has in recent years included commodity-linked bonds. Commodity-linked bonds are derivative debt instruments whose payoff structure is contingent on the outcomes of one or more underlying commodities. To introduce these products, consider a simple coupon bond that pays a fixed interest amount on an annual or semiannual basis on the principal of the bond. Suppose the bond has attached to it a covenant that eliminates or reduces the interest payment or coupon if the price of an underlying commodity (e.g., corn) falls below, or rises above, a specific price at, or over, some specified period of time. This bond will not be priced using conventional bond yield

estimates, but will have to take into account the value of the underlying option value if the bond is to be priced in the market. The use of commodity-linked debt dates back at least to the 19th century. In 1863, the Confederate States of America took the unusual step of issuing bonds payable in bales of cotton. In 1980, the Sunshine Mining Company issued bonds payable in ounces of silver; both Mexico and the British Oil and Gas Corporation have issued petroleum bonds, and the Reagan administration had proposed issuing oil-indexed bonds to finance the strategic Oil Reserve. France has experimented with gold bonds, as have several private companies (Priovolos and Duncan).3 The French gold bond was known as the “Giscard” bond. In 1973, the French government appealed to investors with a gold-indexed bond issue. The “Giscard,” which carried a 7% nominal coupon rate and redemption value indexed to the price of a 1 kg. bar of gold, raised 6.5 million French Francs (FFr). The bondholders were protected by a safeguard clause under which interest and principal payments would be linked to gold in the event the French Franc lost its parity with gold. With the floating of the French Franc and the changes in gold prices since 1977, the safeguard clause was triggered. As a result, in 1980, the government of France paid 333 Francs as interest payments for each single bond issued instead of the 70 Francs originally called for. Moreover, the “Giscard,” which traded at par in 1977, matured in January 1988 with a redemption value of 8,981 Francs—an increase of about 700% in just over 10 years.

3 Commodity-indexed debt also has appeared outside of organized bond markets. Shared Appreciation Mortgages, for example, represent a debt indexed to the price of housing. Even bank loans can be indexed to underlying commodities, as evidenced by the “Oil-Indexed” loans between Japanese banks and the Soviet Union.


As reported by Privolos and Duncan, Inco, the world’s largest nickel producer and an important producer of copper, silver, cobalt, and platinum, in 1984 raised Can$90 million on the financial market through the issuance of bonds linked to the price of nickel or copper. With this issue, the company was able to raise funds at a cost substantially below what it would have had to pay otherwise, and was able to resolve financial difficulties it faced at that time. The Inco bonds matured in 1991, and paid a coupon rate of 10% per annum. The bondholders had the option to receive the face value or the monetary value of a specified amount of nickel or copper at the maturity date. In 1987, Cominco Ltd., another Canadian mining company in the copper and zinc business, raised US$54 million by issuing preferred shares and commodityindexed warrants for the financing of its investment program. The bondholder had the right to exchange each warrant on or before August 1992 for a number of common shares of the corporation based on the average market price of zinc or copper and on the market value of common stocks on the date of the exercise. The bonds discussed in this section attach a call option rider to the value of the bonds. These are typically used as incentives to encourage investors to buy the bonds. The investor benefits from the option only if price movements are favorable to the bond issuer. In contrast, the loans/bonds considered in this study focus on risk mitigation if prices move against the issuer/borrower.

Commodity-Linked Debt in Agriculture Very few farming operations would have the ability to issue commodity-linked bonds to raise capital, and most farmers borrow money through conventional lending institutions. To these farmers, the most appropriate product would be a

Jin and Turvey 45

commodity-linked mortgage or commoditylinked loan. Commodity-linked mortgages, while uncommon, are not new for agriculture. Canada’s Farm Credit Corporation (FCC), for example, offered a mortgage instrument with a payoff structure tied to commodity markets. When specified commodity prices fell, the amount of principal and interest paid by the farmer was lowered; if commodity prices rose, then the farmer would pay a larger amount of principal and interest, with accrued interest being paid first. While such products offer farmers benefits in buffering liquidity reserves when commodity prices fall, the optionlike qualities are only transitory and, over time, the complete mortgage is required to be paid in full. In fact, most farmers participating in the FCC commodity loan program converted to conventional mortgages when, after a few low years of commodity prices, the loan balance neared or exceeded the original mortgage amount. Research by Myers and Thompson, and by Anderson, Gilbert, and Powell, deals with a similar scheme. In Myers and Thompson, the financial payment of the bond at maturity is equal to the price of the underlying commodity, so that an increase in the commodity price represents a higher than par repayment while a decrease in commodity prices represents a lower than par repayment. Anderson, Gilbert, and Powell’s model also includes forward price contracts attached to the loans. A commoditylinked mortgage or loan with option qualities is quite different than one with a forward contract. If the commodity covenant stipulates a payoff structure when the commodity prices fall (i.e., a put option), then the farmer would be forgiven the total principal and interest of the mortgage up to the intrinsic value of the option.


The Pricing of CommodityLinked Loans In this section we review and apply results from the bond models of Schwartz, and Carr, to develop a simple pricing model for commodity-linked loans. The model assumes the interest rate (r ) is fixed for the life of the loan,4 and the price of the reference commodity, P, follows a continuous-time diffusion process: (1) dP/P = "dt + FdZ, where P is the price of the commodity to which the loans are indexed, " is the instantaneous rate of change in the value of P, F is the standard deviation of the percentage change in P, and dZ denotes Gauss-Wiener processes.5 Other assumptions for pricing options are the same as in Black; Black and Scholes; and Merton.6 Given these assumptions, the total market value of the commodity-linked loan issued by the lender is expressed as A(P, t ). A(P, t ) can be viewed as a bond or 4 This is not a trivial assumption. In actuality, the value of a bond tends to decrease when market interest rates rise above the coupon rate as investors sell the bond and reinvest in securities with the higher market return. The opposite is true if interest rates fall. Carr develops a model with stochastic interest rates that would be more appropriate if interest rates and/or bond yields were random. 5 The assumption of Brownian motion is a common feature of most commodity option pricing models. Whether or not the random walk assumption is satisfied is debated in the literature. For example, Cromwell, Labys, and Kouassi found there is persistent memory in commodity prices, but Turvey, in a direct examination of the Brownian motion assumption, could not reject the null hypothesis of Brownian motion for 14 of 17 commodity price series. 6 This assumption should also be scrutinized if the option is indexed to a commodity that is not continuously traded. The pricing of options on nontraded assets is discussed by Stokes, Nayda, and English, and by Yin and Turvey, who appeal to the capital asset pricing model to achieve equilibrium conditions. Other types of bonds such as weatherlinked bonds would require pricing the option portion using equilibrium rather than Black-Merton-Scholes conditions.

a loan depending on the time horizon. The drift and the diffusion of this loan are determined by the application of Ito’s lemma, which gives: (2) dA ' APdP % ½ APP (dP )2 & At dt. Substituting dP in equation (1) for dP in equation (2) gives: (3) dA ' "PAP % ½ F2APP & At dt % APFPdZ. Equation (2) can be expressed in compact form as: (4) dA/A ' "A dt % 0P dZ, where "A ' "PAP % ½ F2APP & At

A

and 0P ' FPAP /A. By constructing a hedge portfolio and using a no-arbitrage argument, the partial differential equation for valuing the commodity-linked loan is specified as: (5)

½ F2P 2APP % rPAP & At & rA ' 0.

Note that this differential equation, and hence the value of the loan, is independent of the expected returns on the commodity ("). Further, this equation is the same as that derived from the arbitrage arguments in Black-Scholes. Assume a borrower requires, at time t = 0, a loan amount B. In addition to the base loan amount, the borrower wants to insure loan repayment by tying to the loan an option rider, which pays in the case of downside price movements. In addition to an advancement of the loan amount B, the lender sells the option at a price W. Assume this amount is also borrowed, so the total loan advanced is B + W. At time T, the total loan amount is repaid with interest. The amount to be repaid is F = (B + W )e r T. At time T, the lender


Jin and Turvey 47

receives the amount F, less the intrinsic value of the option. For example, if the option is a call with strike price X, the boundary condition is:

commodity-linked loans. Furthermore, because the notional value of the option is equivalent to the notional value of the bond, end users need not be concerned with specific contract sizes or brokerage fees, even in the presence of market-traded instruments.

(6)

A(P, T ) = F ! Max(0, P ! X ).

Equation (6) states that at expiration the loan value is simply the face value of the loan plus the payoff from the contingent claim. In contrast, the boundary condition for a conventional loan without the linked commodity option is: (7)

A(P, T ) = F.

Combining (5) and (6), the pricing solution for a commodity-linked loan requires satisfying the following boundary conditions: (8)

½ F2P 2APP % rPAP & At & rA ' 0; A(P, T ) ' F & Max(0, P & X ).

For the second condition to hold, it must be true that the value of the loan at any t < T must be (9)

A(P, t ) ' F & E[ Max(0, P & X ] e &r(T&t),

and when t = 0, (10) A(P, 0) ' Fe &rT & E[Max(0, P & X )]e &rT, where E [·] denotes an expected value. For a put option, the boundary condition Max(0, P ! X ) is replaced by the boundary condition Max(0, X ! P). Equation (10) states that the present value of the loan instrument is equal to the present value of its face value F plus the expected present value of the commodity option. An immediate implication of this result is that farmers (for example) need not buy a commodity-linked loan directly since they can achieve the same result by simply buying options and using the payoffs from the options to repay the loan. However, when commodities are not traded, there are no market-oriented alternatives available to replicate the terms of the

Since the notional value of the loan F is deterministic, the dynamics in the first part of (8) must be attributable to the contingent payoff function. That is, the solution to the first part of (8) requires only Max(0, P ! X ) as a boundary condition, and its solution is identical to the BlackScholes solution for the pricing of a call option. The solution, which is consistent with that derived by Schwartz, is: (11) A(P, t ) = Fe!r t ! W(P, t ). If, for example, W(P, t ) is a call option, the Black-Scholes solution to the value of a call option with exercise price X can be used: (12) W(P, t) ' PN(d ) & Xe &rt N d & F t , where d'

ln(P/X ) % rt F t

% ½F t ,

and N(·) is the cumulative normal density function. Likewise, if W(P, t ) is a put option, the Black-Scholes solution to the put can be used instead of equation (12). The solution (11) is very general and can be applied to bonds or bank loans. In fact, noting the option price (W ) charged the borrower will equal the present value of the expected payout on the option, in the absence of transactions costs it is easy to see that A(P, 0) = B. Solution (11) is an example of a premium bond since the value of the loan extended (B + W ) exceeds, or is at a premium to, the actual amount required (B). For example, suppose a farmer requires a one-year $10,000 operating loan at 10%, with an attached put option rider at a value of


$1,000. At harvest, the price of the commodity has fallen below the strike price and the intrinsic value of the option rider is $2,500. The operating loan extended is $11,000, and accrued interest is $1,100. If prices did not fall below the strike level, the farmer would repay $12,100 in principal and interest. However, with $2,500 in intrinsic value, the farmer would repay only $9,600. Equation (11) can also be written as a discount loan. In this case, the borrower does not buy the option up front, but promises to pay a fixed amount F at some future date. Its value is also given by (13) A(P, t ) = Fe!r T ! W(P, t ). In (13), the value of the loan is decreased by the value of the option. A discount bond would more likely be used by agribusinesses issuing bonds as marketable securities with a stated face value and coupon structure. The redemption value at expiration is reduced by the intrinsic value of the bond. For example, if W(·) is a put option (e.g., the present value of the expected value of [Max(0, X !P)] ), its value increases if commodity prices fall. If so, then the option part expires in-the-money and the face value of the bond is reduced. If prices rise, the option expires worthless and the face value of the bond is paid in full. To illustrate, suppose a $10,000 par bond is issued and the present value of the bond is $9,900 given the coupon rate. If the expected value of a put option on an underlying commodity were $500, then the issuance price of the bond would be $9,400. If at expiration the intrinsic value of the put option is $2,000, then the issuer is only required to repay $8,000 of the face value, but if the option expires worthless, then the issuer will have to pay the $10,000 in full. Hence the value of the bond will always be less than the expected present value of its face value. Mathematically, there is no fundamental difference between what we refer to as a

discount loan and a premium loan (or bond). The qualitative difference is in how the option rider is priced to the borrower. For commercial agriculture loans, the option rider is explicitly purchased in advance, and in addition to, a specified loan amount B. A corporation would more likely issue the discount loan as a bond, such that the issuance price of the bond is implicitly discounted by the option’s present value. Finally, in equation (13), we subtract the option value because our concern is with a reduction in financial risk if prices move against the cash position. The Giscard, Inco, and other bonds discussed in the “background” section of this article describe bonds where the option rider was provided as an incentive to invest. Specifically, the option payout would correspond to a favorable movement in commodity prices. For these bonds, the right-hand side of equation (13) would have a plus (+ ) rather than a minus (!). In fact, the models developed by Schwartz, and by Carr, assume an incentive-type bond, so their solutions include a plus rather than a minus. This is a key, and significant, difference between our model and theirs.

Assessing CommodityLinked Loans In this section, commodity-linked loans are assessed from both a pricing and risk management perspective using the formulations developed in the previous section. Numerical analysis is done using Monte Carlo simulations of the pricing equations directly. The context is a $10,000 loan held for a period of 30 days, 90 days, and one year, securitized by an option on the price of wheat. All four linked-loan types are considered (puts and calls on premium and discount loans). To price the commodity option, daily observations on the nearby Chicago Board of Trade (CBOT) wheat futures contracts from 11/18/1980 to 2/8/2000 were used.


Jin and Turvey 49

Table 2. Summary Statistics for Wheat Futures and Cash Prices, 11/18/1980S S2/8/2000 Kansas City Cash Price

Chicago Wheat Futures

Basis

Log Cash Price Change

Log Futures Price Change

Average

3.733

3.499

0.234

!0.032

!0.037

Std. Dev.

0.739

0.647

0.243

0.224

0.243

Minimum

2.355

2.240

!0.515

n.a.

n.a.

Maximum

7.500

7.165

1.443

n.a.

n.a.

Description

Assuming a 250-day trading year, the annualized futures price volatility was calculated as F 250 , where F is the standard deviation of the (logarithmic) daily price changes. The wheat futures price statistics are summarized in Table 2. For comparison, the cash prices in Kansas City are also presented. The annualized volatility is about 22%/year for the futures contracts and about 24% for the cash prices. Figure 1 depicts the wheat cash and futures prices over this period. The data series ended on February 8, 2000, with a wheat price of $2.67/bu. This was used as the current price P0 for pricing the options. The risk-free rate was assumed to be 6.5%/year. From this initial position, European put and call option premiums were calculated using 5,000 Monte Carlo draws under the assumption of a risk-neutral valuation (the futures price grows at the risk-free rate rather than the natural rate). Assuming one calendar year equaled 250 trading days, a random price series was generated using the following: (14) Pt ' Pt&1 exp (r & 0.5F2 )(1/250) % FN(0, 1) 1/250 , where N(0, 1) is a normally distributed random deviate with mean 0 and standard deviation of 1 (Winston). Each 250-day sequence is referred to as an iteration, and the three probability distributions of futures prices, g(PT |T = 30, 90, 250), were defined over 5,000 iterations. Across all

iterations, the boundary conditions for put and call options were calculated as Max(0, X ! PT ) and Max(0, PT ! X ) for T = 30, 90, and 250 trading days. Three strike prices were used for each option and each T : (a) at-the-money puts and calls at $2.698/bu., (b) in-the-money puts and out-of-the-money calls at $2.158/bu. (80% of $2.698/bu.), and (c) out-of-the-money puts and in-themoney calls at $3.237/bu. (120% of $2.698/bu.). Option premiums were calculated as the expected present values (discounted at 6.5%) of the various boundary conditions. The results are summarized in Table 3. The top portion of Table 3 gives the various option premia on a $/bushel basis. As anticipated, the option premia increase with time. For example, the $2.698 call is $0.35/bu. for a 250trading-day (one calendar year) horizon, but is only $0.10/bu. for a 30-trading-day horizon. With respect to strike prices, the 250-day call option decreased from $0.71/bu. to $0.15/bu. as X increased from $2.158 to $3.237, while the put option increased from $0.03/bu. to $0.48/bu. The values in the lower half of Table 3 are the option premiums for a $10,000 loan. The loan is converted into 3,707 bushel equivalents [$10,000/($2.698/bu.)]. The loan option value is obtained by multiplying 3,707 bushels by the per bushel option prices. For example, the 250-day at-the-money loan option price is $1,305 (3,707 bu. × $0.35/bu., with an allowance for rounding error).


Table 3. Call and Put Option Values (250, 90, and 30 trading days) Strike Price ($/bu.) 2.158 2.698 3.237 2.158 2.698 3.237

Call 250 Days

90 Days

Put 30 Days

250 Days

90 Days

30 Days

!!!!!!!!!!!!!!!!!!!!!!!!!!!! 0.71 0.60 0.56 0.03 0.01 0.00 0.35 0.19 0.10 0.18 0.12 0.08 0.15 0.03 0.00 0.48 0.49 0.52 !!!!!!!!!!!!!!!!!!!!!!! 2,643.79 2,207.11 2,062.33 128.88 23.32 0.65 1,305.23 691.20 369.67 664.45 461.15 292.45 538.60 109.39 5.54 1,771.96 1,833.09 1,912.79

Table 4. Expected Value of Loans at Expiration (250, 90, and 30 trading days) Strike Price ($/bu.) 2.158 2.698 3.237 2.158 2.698 3.237

Call 250 Days

90 Days

Put 30 Days

250 Days

90 Days

30 Days

!!!!!!!!!!!!!!!!!! 10,672 10,237 10,078 10,672 10,237 10,078 10,672 10,237 10,078 10,672 10,237 10,078 10,672 10,237 10,078 10,672 10,237 10,078 !!!!!!!!!!!!!!!!!! 7,850 7,977 8,000 10,534 10,213 10,078 9,279 9,529 9,706 9,963 9,765 9,784 10,097 10,125 10,073 8,781 8,360 8,151


The expected payoff values for the premium and discount bonds were calculated as the means from the 5,000 Monte Carlo iterations and are summarized in Table 4. The payoff values for each iteration for the premium loans were calculated from (15) Payoff ' [10,000 % Premium]e 0.065T/250 & Max(0, X & PT ) for the put, and from (16) Payoff ' [10,000 % Premium]e

0.065T/250

& Max(0, PT & X ) for the call. Premium refers to the values in the lower half of Table 3. In essence, the lender is selling the option to the borrower, and this is added to the loan amount. For the discount loans, the payoff values at expiration were calculated from each iteration as (17) Payoff ' (10,000e 0.065T/250 ) & Max(0, X & PT ) for the put, and (18) Payoff ' (10,000e 0.065T/250 ) & Max(0, PT & X ) for the call.7 In Table 4, the expected payout values of the premium loans are $10,672, $10,237, and $10,078 for T = 250, 90, and 30 trading days, respectively. Discounting these values at 6.5% verifies the initial loan principal amount of $10,000.8

7 Again, if the options’ payoffs in (17) and (18) were added to the loan amount (or face value) F, the loan would take on the characteristics of the incentive bonds (e.g., the French Giscard bond) discussed in the “background” section of this article. 8 We have used 6.5% as the risk-neutral rate and the loan rate for convenience. There is no requirement that the loan and risk-neutral rate equal each other.

Jin and Turvey 51

Recall that with the premium loan, the principal advanced equals $10,000 plus the linked-option premia. For the discount loan, the actual amount advanced is $10,000 less the option premia. The terminal values in Table 4 give the expected values of the principal and interest to be paid at expiration. For the premium loans in the top half of Table 4, the present values at 6.5%/year exactly equal $10,000 in all cases. For the discount loan (bond) values in the lower half of Table 4, consider the 250-day at-the-money call. The expected repayment is $9,279, and for the equivalent put it is $9,963. The actual loan advanced is therefore the present value of these amounts. It can immediately be verified that the loan amounts advanced are equal to exactly $10,000 less the option premiums in Table 3. For example, the present values of the 250-day at-the-money discount loans are $8,696 and $9,337 for the call and put, respectively. Adding to these the option premia of $1,305 and $664 from Table 3 gives precisely $10,000.

The Loan Repayment Distribution The economic principle motivating this analysis was that increased debt use in the face of business risk leads to an increase in total risk (the variability in the return on equity). As operating margins decrease due to price volatility, the likelihood of not meeting fixed financial obligations increases. By linking the value of an option to the repayment of loan principal and/or interest, the total risk facing the firm can be reduced substantially. To illustrate the extent of risk reduction, the cumulative distribution functions (CDFs) for repayment are presented in Tables 5S8. For a strike price of $2.158/bu., the call option is out-of-themoney and the put option is in-the-money. For the premium loan (Table 5), the


probability of paying less than the accrued amount at expiration is about 85%. Of a maximum repayment value of $13,493, there is a 50% chance of having to repay $11,111 or less and a 5% chance of paying $5,882 or less. For the put option (Table 6), there is only a 15% chance of paying less than the maximum amount and only a 5% chance of paying $9,752 or less. For a strike price of $3.237/bu., there is a 25% chance of paying less than the maximum amount for the 250-day call (Table 5) and a 70% chance of paying less than the maximum amount for the 250day put (Table 6). For the call, there is a 5% chance of paying $7,635 or less, and for the put there is a 5% chance of paying $7,506 or less. Similar results are found for the discount loans (Tables 7 and 8). At a strike price of $2.158/bu., the 250-day call has an 85% chance of paying less than the full amount of $10,672, and a 5% chance of paying $3,060 or less. For the out-of-the-money put, there is only a 10% chance of paying less than $10,672, and a 5% chance of paying $9,615 or less. For a $3.237/bu. strike, the out-of-the-money call has only a 25% chance of paying less than $10,672, and only a 5% chance of paying $7,060 or less. In contrast, the $3.237 put option has a 70% chance of paying less than $10,672, and a 5% chance of repaying $5,615 or less. The results show for 90- and 30-tradingday horizons that the cumulative distribution functions are tighter. The possible reduction in repayment due to commodity risk increases with time (as does the cost of the linked loan). The results also indicate that the risk reduction benefits increase, as one would expect, as the options are issued in-the-money. The cost of risk reduction naturally increases with these benefits since the price of an option increases the further it goes into-the-money. The benefits at-the-money are in the middle of the two extremes, with the probability of

paying less than the maximum amount being approximately 50% in all cases. Based on our findings, there are numerous structures of commodity-linked loans which could be made available to farmers and agricultural businesses. The results show the relationship between the parameters of the model that directly affect the risk structure of the payoff and the costs to the end users for financial risk protection. The particular structure, costs, and benefits of the commoditylinked bonds will depend on the specific needs of the end user.

Commodity-Linked Loans and the Effective Cost of Capital From a lender’s perspective, the derivative value of the option can be incorporated directly into the interest rate of the loan. The purpose of this section is to illustrate how the risk premium above common interest rates can be calculated in such a way that the end user would be indifferent to paying the option premium directly through an additional charge to the loan or indirectly through a higher interest charge on the loan. In this regard, the repayment structure of a commodity-linked loan can also be cast as an effective (pre-tax) interest charge. This can be achieved by dividing the values in Tables 5S8 by the original loan principal amount. The effective rates will have the same distribution characteristics as the repayment amounts and will increase as put strike prices increase or call strike prices fall, all other things held constant. For example, in Tables 5 and 6, a 250trading-day premium loan with a strike price of $2.158/bu. will have a maximum (at 100% on the CDF) effective rate of 34.93% (13,493/10,000) for the call option and 8.09% (10,809/10,000) for the put option. At a strike option price of $2.698, the rates are 20.64% for the call and 13.81% for the put. At the highest strike price of $3.237, the respective call and put rates are 12.46% and 25.63%.


Jin and Turvey 53

Table 5. Cumulative Probability Distributions for Premium Loans with Call Options (250, 90, and 30 trading days) Percent Probability 250 days 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100%

5,882 7,274 8,144 8,780 9,313 9,711 10,094 10,482 10,792 11,111 11,433 11,775 12,066 12,382 12,686 13,036 13,451 13,493 13,493 13,493

Strike Price ($/bushel) 2.158

2.698

3.237

90 days

30 days

250 days

90 days

30 days

250 days

90 days

30 days

7,623 8,334 8,705 9,065 9,340 9,569 9,790 10,010 10,193 10,388 10,550 10,729 10,921 11,120 11,310 11,512 11,742 12,065 12,496 12,496

8,599 8,966 9,221 9,400 9,555 9,676 9,797 9,899 10,008 10,115 10,206 10,314 10,416 10,528 10,649 10,789 10,943 11,135 11,416 12,157

6,453 7,845 8,716 9,352 9,884 10,283 10,665 11,054 11,364 11,683 12,004 12,064 12,064 12,064 12,064 12,064 12,064 12,064 12,064 12,064

8,071 8,782 9,153 9,513 9,788 10,018 10,238 10,458 10,641 10,837 10,944 10,944 10,944 10,944 10,944 10,944 10,944 10,944 10,944 10,944

8,893 9,260 9,515 9,694 9,849 9,970 10,091 10,193 10,302 10,409 10,451 10,451 10,451 10,451 10,451 10,451 10,451 10,451 10,451 10,451

7,635 9,027 9,898 10,533 11,066 11,246 11,246 11,246 11,246 11,246 11,246 11,246 11,246 11,246 11,246 11,246 11,246 11,246 11,246 11,246

9,475 10,187 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349 10,349

10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084 10,084

Table 6. Cumulative Probability Distributions for Premium Loans with Put Options (250, 90, and 30 trading days) Percent Probability 250 days 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100%

9,752 10,365 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809 10,809


2.698

3.237

90 days

30 days

250 days

90 days

30 days

250 days

90 days

30 days

10,229 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261 10,261

10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079 10,079

8,324 8,936 9,423 9,836 10,187 10,493 10,808 11,098 11,381 11,381 11,381 11,381 11,381 11,381 11,381 11,381 11,381 11,381 11,381 11,381

8,678 9,137 9,463 9,691 9,893 10,085 10,284 10,476 10,655 10,709 10,709 10,709 10,709 10,709 10,709 10,709 10,709 10,709 10,709 10,709

9,112 9,394 9,586 9,741 9,881 10,002 10,114 10,216 10,323 10,373 10,373 10,373 10,373 10,373 10,373 10,373 10,373 10,373 10,373 10,373

7,506 8,118 8,605 9,018 9,369 9,674 9,989 10,280 10,623 10,941 11,263 11,573 11,962 12,349 12,563 12,563 12,563 12,563 12,563 12,563

8,082 8,542 8,867 9,095 9,297 9,490 9,689 9,880 10,059 10,218 10,413 10,599 10,819 11,040 11,269 11,542 11,904 12,113 12,113 12,113

8,745 9,027 9,219 9,374 9,514 9,635 9,747 9,849 9,956 10,048 10,154 10,264 10,366 10,487 10,607 10,762 10,942 11,195 11,562 12,006


Table 7. Cumulative Probability Distributions for Discount Loans with Call Options (250, 90, and 30 trading days) Percent Probability 250 days 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100%

3,060 4,452 5,323 5,959 6,491 6,890 7,272 7,661 7,971 8,290 8,612 8,954 9,245 9,560 9,865 10,215 10,629 10,672 10,672 10,672


2.698

3.237

90 days

30 days

250 days

90 days

30 days

250 days

90 days

30 days

5,363 6,075 6,446 6,805 7,080 7,310 7,531 7,751 7,934 8,129 8,290 8,470 8,661 8,861 9,050 9,252 9,483 9,805 10,237 10,237

6,521 6,888 7,143 7,322 7,477 7,597 7,719 7,821 7,929 8,037 8,128 8,235 8,338 8,449 8,570 8,711 8,865 9,057 9,338 10,078

5,060 6,452 7,323 7,959 8,491 8,890 9,272 9,661 9,971 10,290 10,612 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672

7,363 8,075 8,446 8,805 9,080 9,310 9,531 9,751 9,934 10,129 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237

8,521 8,888 9,143 9,322 9,477 9,597 9,719 9,821 9,929 10,037 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078

7,060 8,452 9,323 9,959 10,491 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672

9,363 10,075 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237

10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078

Table 8. Cumulative Probability Distributions for Discount Loans with Put Options (250, 90, and 30 trading days) Percent Probability 250 days 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100%

9,615 10,227 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672


2.698

3.237

90 days

30 days

250 days

90 days

30 days

250 days

90 days

30 days

10,206 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237

10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078

7,615 8,227 8,714 9,127 9,478 9,784 10,098 10,389 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672 10,672

8,206 8,665 8,991 9,218 9,421 9,613 9,812 10,004 10,183 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237 10,237

8,817 9,099 9,292 9,446 9,586 9,707 9,819 9,921 10,029 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078 10,078

5,615 6,227 6,714 7,127 7,478 7,784 8,098 8,389 8,732 9,050 9,372 9,682 10,071 10,458 10,672 10,672 10,672 10,672 10,672 10,672

6,206 6,665 6,991 7,218 7,421 7,613 7,812 8,004 8,183 8,342 8,537 8,723 8,943 9,164 9,393 9,666 10,028 10,237 10,237 10,237

6,817 7,099 7,292 7,446 7,586 7,707 7,819 7,921 8,029 8,120 8,226 8,336 8,438 8,560 8,679 8,835 9,014 9,267 9,634 10,078


The effective rates are ex post measures based on actual outcomes. For example, if (from Table 5) the actual repayment is $10,792 for the 250-day call with X = $2.158 (at 45% probability), the effective rate falls from the maximum of 34.93% (at 100%) to 7.92%. Likewise, for the 250-day put with X = $3.237 in Table 6, and an actual repayment of $10,941 (at 50% probability), the effective rate would be 9.41% which compares with the maximum possible effective rate (at 100%) of 25.63%. Similar calculations can be made for the discount loans in Tables 7 and 8 except the denominator must be reduced by the option premium from Table 3, i.e., Repayment/(10,000 ! Premium ). For example, the minimum effective rates (at 100%) for the 250-day at-the-money ($2.698) call in Table 7 is (1.0 ! [10,672/ (10,000 ! 691.20)]) = 0.146 (14.6%), and the maximum effective rate for the put (Table 8) would be (1.0 ! [10,672/ (10,000 ! 664.45)]) = 0.143 (14.3% ). Put this way, lenders may prefer to offer loans with maximum effective interest rates rather than stated dollar-value premiums per se. For example, the lender may offer a loan at 6.5% interest with no downside risk protection at all, or offer a loan with a maximum effective rate of 13.81% with downside risk protection if future prices fall below $2.698/bu. The basis point differential (maximum effective rate minus base rate) = (13.81% ! 6.5% ) = 7.31% would then be interpreted as a risk premium.

Commodity-Linked Loans, Lines of Credit, and Farm Mortgages The ability to hedge agricultural credit through commodity-linked loans is a new concept, at least in the context of agricultural finance, and lenders willing to pursue commodity-linked loans as a new product offering will need to be astute in its application to specific products and services. The simplest application would

Jin and Turvey 55

be for a single period loan with fixed terms. An application to fixed-term loans with variable rates will require a more sophisticated approach that includes randomness in variable interest rates as well as randomness in commodity prices (see Carr). However, if default risk is low, the lender can assume a constant interest rate and the option can be based on principal repayment alone without any loss of generality. At the other end of the spectrum are revolving and non-revolving lines of credit in which both commodity prices and interest rates might vary, but so will the principal amount. The problem with lines of credit is that interest is not charged until funds are released and principal can be repaid at any time. From a lender’s perspective there are three random variables involved. These are principal balance, interest rate, and commodity prices. For a commodity-linked loan of the type discussed in this study to become operational, a product manager could consider writing a contract on an average loan balance at a fixed interest rate, with settlement occurring at a specified date in the future (e.g., annually for a revolving line of credit and at contract termination for non-revolving lines). The model can also be used to hedge the current portion of mortgage principal plus interest. As discussed throughout this article, a commodity rider, based upon the farmer’s primary source of risk, would create the option value. For example, a corn producer could hedge total fixed mortgage payments by purchasing an option rider derived from the corn futures contract. If a $20,000 principal and interest payment is required at the end of a period, then F = $20,000. If the future option value is $1,000, then $21,000 would be repaid. If the corn option expires in-the-money with an intrinsic value of $3,000, then only $18,000 of the loan and interest would be paid back. If the price of corn rises, then the full $21,000 face value of the mortgage payment plus option value will be repaid.


If a loan or mortgage contract is comprised of n periodic payments of principal plus interest, then an option can be established for each period. The expected value of the loan is designated by: N

(19) A(P, 0) ' j Fe &rnt & W(P, Xn , nt) , n'1

where the term Xn is added to the definition of W(·) to indicate the strike price does not necessarily have to remain constant, and F represents the periodic loan repayment. If repayment is semiannual, then t = 0.5; if it is annual, t = 1, and so on. The first part of (19) calculates the notional value of the loan or mortgage without the option, and states that the value of a loan today must equal the present value of future principal and interest payments. Deducting from this the periodic option component gives the present value of the loan. The value of the option component will increase in time due to the variance of the underlying commodity, which, by definition, increases linearly with time. An option of this type is called a compound option since it is made up of N separate options, each with a fixed start and expiry date. The specification in (19) is a simple version of a compound option since the payoff in one period is uncorrelated with the expected payoff in the next period. Alternatively, a compound option can have built-in flexibility. For example, it may include a strike price that is fixed for the duration of the mortgage or one that changes as market conditions change. In addition, it may provide the mortgagee with the choice of accepting or rejecting the option rider for any period given changing market conditions. The present value of the mortgage will ultimately depend on the nature of the compound option. The calculation of the compound option value, which is beyond the scope of this analysis, will become more complex as the complexity of the option increases.

Basis Risk A final issue (caveat) regarding commoditylinked loans is the issue of basis risk. In Table 2 and Figure 1, we included the cash price of Kansas City wheat as a comparison to the CBOT futures price. A simple regression of the cash price against the futures price yielded a hedge ratio (slope coefficient) of 1.08, which, with a standard error of 0.005, was statistically different from 1.0 over the long run. However, calculating the cash pricefutures price basis revealed the average basis was approximately $0.23/bu. with a standard deviation of $0.24/bu. and a range from !0.515 to 1.443. Since the payoff for a commodity-linked loan is contingent on the futures price rather than the cash price, there exists the possibility that cash prices can be higher or lower than the futures price. When this occurs it is possible for the option component of the loan to expire out-of-themoney (P # X for a call option, or P $ X for a put option) while the cash price adversely moves against the borrower (cash price increases for a buyer, cash price decreases for a seller). Given the strong correlation between the Kansas City cash price for wheat and the nearby wheat futures contract over the long run, the likelihood of adverse basis changes is quite remote. However, for cash markets not so strongly correlated with the futures market—for example, a cross-hedge—the impact of basis risk on the effectiveness of the commodity-linked loan should be considered.

Conclusions and Extensions In this study, we analyzed the firm’s risks in the context of commodity-linked loans. Commodity-linked loans are contingent claim securities with a payoff structure contingent on the derived value of an underlying commodity. We provided some illustrative examples of linked loans, developed a theoretical framework for


calculating the value of the loans, and then illustrated the value and repayment probabilities of the loans with an example using historic wheat prices and Monte Carlo simulation. Finally, we discussed the role of commodity-linked loans within the context of the bond problem. We suggest that new financial instruments which are linked to the value of an underlying commodity can be used for primary production in agriculture, and agribusiness in general. The purpose of these loans is to reduce the financial risk arising when business risk is unfavorable. We argue that commoditylinked loans can be extended to amortized loans and mortgages by converting the current portion of debt into the value of a bond-clone. Then the procedures developed here may be used directly.

References Anderson, R. W., C. L. Gilbert, and A. Powell. “Securitization and Commodity Contingency in International Lending.” Amer. J. Agr. Econ. 71(1989): 523S30. Black, F. “The Pricing of Commodity Contracts.” J. Finan. Econ. 3(1976): 167S79. Black, F., and M. Scholes. “The Pricing of Options and Corporate Liabilities.” J. Polit. Econ. 81(1973):637S59. Carr, P. “A Note on the Pricing of Commodity-Linked Bonds.” J. Finance 42(1987):1071S76. Collins, R. A. “Expected Utility, DebtEquity Structure, and Risk Balancing.” Amer. J. Agr. Econ. 67(1985):627S29. Cromwell, J. B., W. C. Labys, and E. Kouassi. “What Color Are Commodity Prices? A Fractal Analysis.” Empirical Econ. 25(2000):563S80.

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Featherstone, A. M., C. B. Moss, T. G. Baker, and P. V. Preckel. “The Theoretical Effects of Farm Policies on Optimal Leverage and the Probability of Equity Losses.” Amer. J. Agr. Econ. 70(1988):572S79. Merton, R. C. Continuous-Time Finance. Oxford, England: Blackwell Publishers, 1990. Myers, B., and S. Thompson. “Optimal Portfolios of External Debt in Developing Countries: The Potential Role of Commodity-Linked Bonds.” Amer. J. Agr. Econ. 71(1989):517S22. Priovolos, T., and R. C. Duncan. “Commodity Risk Management and Finance.” The World Bank, Washington, DC, 1992. Schwartz, E. S. “The Pricing of CommodityLinked Bonds.” J. Finance 37(1982): 525S41. Stokes, J. R., W. I. Nayda, and B. C. English. “The Pricing of Revenue Assurance.” Amer. J. Agr. Econ. 79(1997): 439S51. Turvey, C. G. “Random Walks and Fractal Structure in Futures Markets.” Selected paper presented at the annual meetings of the AAEA/CAES, Chicago, IL, August 2001. Turvey, C. G., and T. G. Baker. “Optimal Hedging Under Alternative Capital Structures and Risk Aversion.” Can. J. Agr. Econ. 37(1989):135S43. Winston, W. Simulation Modelling Using @ Risk. Belmont, CA: Duxbury Press, 1996. Yin, S., and C. G. Turvey. “The Pricing of Options on Non-Traded Assets: Implications for Market-Driven Revenue Assurance.” Selected paper presented at the annual meetings of the AAEA, Tampa, FL, August 2000.