Multiperiod hedging using futures: Mean reversion and the optimal hedging path

Rao, Vadhindran K. Article Multiperiod hedging using futures: Mean reversion and the optimal hedging path Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Rao, Vadhindran K. (2011) : Multiperiod hedging using futures: Mean reversion and the optimal hedging path, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 4, Iss. 1, pp. 133-161, https://doi.org/10.3390/jrfm4010133 This Version is available at: https://hdl.handle.net/10419/178532 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-sa/3.0/ Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 133 Multiperiod Hedging using Futures: Mean Reversion and the Optimal Hedging Path Vadhindran K. Raoa a College of Management, Metropolitan State University, Tel: 001 612 659 7291, Email: [email protected] ABSTRACT This paper considers the multiperiod hedging decision in a framework of mean-reverting spot prices and unbiased futures markets. The task is to determine the optimal hedging path, i.e., the sequence of positions in futures contracts with the objective of minimizing the variance of an uncertain future cash flow. The model is used to illustrate both hedging using a matchedmaturity futures contract and hedging by rolling over a series of nearby futures contracts. In each case, the paper derives the conditions under which a single period (myopic) strategy would be optimal as opposed to a dynamic multiperiod strategy. The results suggest that greater the market power of the hedging entity, closer the optimal strategy is to a myopic hedge. The paper also highlights the difference in the optimal hedging path when hedging is based on matched-maturity as opposed to nearby contracts. KEYWORDS: Multiperiod Hedging, Futures, Mean Reversion JEL Classification: G32, G13 ACKNOWLEDGEMENTS: I would like to thank Mike Sher and an anonymous referee for their comments on an earlier draft of this paper. Errors, of course, remain my own responsibility. Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 134 1. INTRODUCTION Consider the hedging problem of a firm facing an uncertain future cash flow at a certain future time, T. The uncertain future cash flow (referred to as the hedged item) may arise from a fixed cash or spot position or, more generally, from revenues/costs/profits in a certain future period referred to as the terminal or target period. The hedging horizon (i.e. the time between now and the future terminal date T) is broken up into a series of discrete intervals, and the decisionmaker's task is to choose futures positions for each period so as to minimize the variance of the future cash flow. For example, the model may be applied to a commodity trader hedging a forward commitment or a firm hedging future input costs. In a classic, widely-cited study, Howard and D’Antonio (1991), henceforth referred to as HD, derived the optimal hedging strategy in a framework of mean-reverting spot returns and unbiased futures markets. Their main result is that the optimal hedging strategy depends crucially on the rate of mean reversion of the spot process (i.e., the price process of the hedged item, also referred to as the “hedged process”). The current study extends the analysis by explicitly allowing for mean reversion in the price process of the underlying of the futures contract as well (referred to as the “hedging process”). In this extended framework, the HD model may be viewed as a special case in which hedging is carried out by rolling over a series of nearby contracts (“stack and roll” hedging, or just “stack” hedging). The current study considers hedging based on matched-maturity futures contracts (i.e., contracts maturing at the same time as the occurrence of the cash flow being hedged) as well as hedging based on nearby futures contracts. It is seen that if hedging is based on matched-maturity contracts, the optimal hedging strategy depends not so much on the absolute mean reversion rate of the hedged process (as in the case of hedging using nearby contracts), but rather on the relative mean reversion rates of the hedged and hedging processes. The main contributions of the current paper are as follows. A simple, but empirically relevant setting of mean-reverting prices is used to illustrate matched-maturity hedging and compare it to stack hedging. However, the focus is not on determining if one is better than the other. Rather, the objective is to determine the optimal hedging path in each case. Possible Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 135 multiperiod hedging paths can be grouped into three categories: (i) a static multiperiod hedge, wherein the hedge position is taken at the start of the hedging horizon and then kept unchanged until the terminal period, (ii) a myopic or single-period hedge, wherein the hedge position is initiated only at the start of the terminal period, and (iii) a dynamic hedge, wherein hedging is initiated at the start of the hedging horizon and updated each period. The focus is on the conditions under which each of these categories is optimal. It is shown that, in general, the optimal hedging path or pattern can be very different depending on whether hedging is carried out using matched-maturity or nearby futures contracts. The exception to this is when futures prices follow a random walk. In addition to HD, another study to which the current paper has a strong relation is Myers and Hanson (1996), hereafter referred to as MH. Their paper deals with the same problem of hedging a fixed cash position in the presence of basis risk. They show that provided futures prices evolve as a martingale, it is possible to derive an optimal dynamic hedging strategy that is independent of risk preferences under fairly general assumptions about the relationship between spot and futures prices. As will be elaborated upon later, the current paper may be viewed as a special case of the MH model. As a result, the variance-minimizing hedge derived in the current study is also the expected-utility maximizing hedge. The next section contains a review of the literature, the following section develops the model and discusses implications and the final section concludes with a brief summary of the main results. 2, LITERATURE REVIEW Several studies have examined the hedging behavior of firms facing price or exchange rate uncertainty in input/output markets. A seminal study by Johnson (1960) analyzed the hedging decision as an optimal portfolio problem and derived the minimum variance hedge ratio. While Sandmo (1971) showed that a firm’s output decision would be affected by price uncertainty, Danthine (1978) and Holthausen (1979) showed that in the presence of a forward market to hedge away this uncertainty, the output decision would be independent of this risk and the firm’s risk preferences. Extensions and refinements were made by Losq (1982), Fishelson (1984) and Zilcha and Broll (1992) among many others. De Meza and von Ungern Sternberg (1980) derived Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 136 the result that a monopolistic firm is more likely to hedge input risk than a competitive firm as volatile input prices will have a bigger impact on the profits of the former. Koppenhaver and Swidler (1996) investigated how the degree of market power enjoyed by a firm in the output market affects the extent to which the firm will hedge its input price risk. They concluded that lower the market power of the firm, more the importance of hedging. These studies are set in a single-period framework and focus on the relationship between hedging strategies and output decisions in the context of specific market structures. The decision-maker’s objective is typically assumed to be maximizing expected utility or minimizing the variance of end-of-period wealth. Examples of discrete-time, multi-period models investigating the dependency between the production and the hedging decision include Zilcha and Eldor (1991) and Donoso (1995). Among other things, these studies show that the firm will overhedge when current shocks can adversely affect future cash flows. Another set of studies focuses on the optimal hedging strategy for a trader or producer who will liquidate a non-tradable cash position at a certain future time, T. The hedging horizon (i.e. the time between now and the future terminal date T) is broken up into a series of discrete intervals, and the decision-maker's task is to choose futures positions for each period so as to maximize expected utility of end-of-period wealth or minimize the variance of end-of-horizon wealth/cash flow. Some of these models ignore basis risk (Anderson and Danthine, 1983) and others assume negative exponential utility with a joint normal distribution for cash and futures price changes (Karp, 1987, Martinez and Zering, 1992, and Vukina and Anderson, 1993). A few studies have tackled the optimal dynamic hedging problem in a discrete-time setting without making restrictive assumptions either about risk preferences or the distribution of prices. Notable studies of this kind include HD and MH, mentioned earlier in the introduction and discussed in more detail in the next section. The interested reader is referred to Low, Muthuswamy, Sakar and Terry (2002) for a more extensive review. While much of the discrete-time literature is based on expected utility maximization or variance minimization, some studies have explored alternative hedging objectives such as regret minimization and approaches based on the mean-extended Gini coefficient and the semivariance. Chen, Lee and Shrestha (2003) provide a fairly comprehensive review of these alternative theoretical approaches to hedging. Yet another branch is concerned with empirical testing of Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 137 alternative estimation models and techniques, such as Ordinary Least Squares Regression, Cointegration and Error Correction models, Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models and stochastic volatility models. The interested reader is referred to Lien and Tse (2002) for an excellent review of these various econometric approaches and estimation methods. Some noteworthy recent empirical studies include Hung and Lee (2007), who focus on minimizing down-side risk in the presence of volatility clustering and price jumps, Power and Vedenov (2008), who estimate a copula-GARCH model, Huang, Pan and Lo (2010), who compare a duration-dependent Markov-switching vector autoregression approach with GARCH models, Chan (2010), who employs a GARCH model that is modified to allow for jumps, Cao, Harris and Shen (2010), who use a non-parametric approach to estimate hedge ratios designed to minimize Value-at-Risk, and Chen and Tsay (2011), who use a novel approach to estimate a Markov regime-switching, Autoregressive Moving Average model. There is also a vast literature that investigates optimal hedging in continuous-time. Early classics include Breeden (1984), Adler and Detemple (1988), Duffie and Jackson (1990) and Duffie and Richardson (1991). Breeden (1984) provides the dynamic optimal hedging strategy in terms of the value function for an agent maximizing expected utility of intertemporal consumption, while Adler and Detemple (1988) consider the problem of an agent hedging a nontraded spot position and maximizing logarithmic utility of terminal wealth, and derive an explicit, analytical solution for the case in which markets are complete. Duffie and Jackson (1990) and Duffie and Richardson (1991) consider the optimal hedging problem in a setting in which prices follow Geometric Brownian Motion (GBM) and derive explicit solutions for various mean-variance hedging objectives. Among recent studies, Basak and Chabakauri (2008) provide a good summary of this literature and also provide explicit, tractable solutions to the dynamic hedging problem of a non-tradable asset. Ankirchner and Heyne (2012) solve the problem for the case of stochastic correlation between the process being hedged and the hedging instrument, and Ankirchner, Dimitroff, Heyne and Pigorsch (2012), the case in which the (log) spread between the hedged and hedging processes is stationary (which is not the case when the two processes follow GBM). Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 138 As is apparent from the above, there is an extensive literature on dynamic hedging using futures. Below are described some recent studies (other than HD and MH) to which the current study is closely related and a summary of how the current study complements these prior studies. Hilliard (1999) uses a continuous time setting in which prices follow Geometric Brownian motion to derive optimal hedging results using a “stack and roll” strategy. Neuberger (1999) studies a similar problem, but with a focus on “strip” hedges (i.e., hedging using futures contracts of multiple maturities). The model is based on assumptions about the cross-sectional character of futures prices of different maturities rather than assumptions about their time-series dynamics. In both these studies, the future commitment that is being hedged is far enough in the future so that hedging using a matched-maturity futures contract is either not possible or not economical. In contrast, the current study explicitly considers hedging based on matchedmaturity contracts and is set in a different setting of discrete-time, autoregressive prices. Another strand of research extends from a noteworthy study by Lien and Luo (1993) in which they derive the optimal multiperiod hedging strategy and apply it in a setting of cointegrated spot and futures prices. Their empirical results clearly show the superiority of multiperiod hedging over a myopic (single-period) strategy. Building on this work, Low et al (2002) study multiperiod hedging in a setting in which the futures price follows a cost of carry model. This allows them to capture the effect of the maturity of the futures contract on the spotfutures basis. Their empirical tests show that their “dynamic cost-of-carry hedge” outperforms other strategies such as the conventional (myopic) hedge, the cointegrated price hedge of Lien and Luo (1993), and the GARCH hedge of Kroner and Sultan (1993), although a static, cost-ofcarry hedge is found to perform slightly better. Lien and Shaffer (2002) use a three period setting to compare the effectiveness of “strip” hedging with “stack” hedging in managing the risk of a forward commitment. They use a Vector Autoregressive framework to model spot and futures prices and show that “strip” hedging outperforms “stack” hedging especially when forward prices are subject to multiple risk factors. The current study is similar to these studies in that it investigates the same multiperiod hedging problem. However, it differs from these studies in some important respects. Firstly, the current study is set in a framework of mean-reverting rather than non-stationary prices. It is, of course, true that price processes of many assets – especially financial assets - have been determined to be non-stationary, as noted in the above studies. Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 139 However, as concluded by Bessembinder, Coughenour, Smoller, and Seguin (1995) and noted in Schwartz (1997), a mean reverting process is a good description of the price dynamics of many commodities. Secondly, the main focus of the current study is different. Unlike these prior studies, the current paper does not attempt to assess the empirical performance of alternative hedging approaches. Instead, using a simple, but empirically useful framework, the paper makes clear (i) the connection between the relative rates of mean reversion of hedged and hedging processes on the one hand and the optimal multiperiod hedging strategy on the other, (ii) the conditions under which a single period (myopic) strategy, a static multiperiod strategy and a dynamic multiperiod strategy would each be optimal in this framework, and (iii) the difference in the optimal hedging path when hedging is based on matched-maturity as opposed to nearby contracts. 3. MODEL Suppose that the hedger has a fixed cash position, x. The hedger will liquidate this position at terminal time T and faces the problem that the price of the hedged item as of time T is uncertain. Suppose that the price process of the hedged item (the “hedged process”) is given by: tttt uppp      ))(1( 11   (1) Above, pt is the value of the process in period t. If φ = 1, then pt follows a random walk. If φ is strictly between 0 and 1, µ is the long run mean of pt and (1-φ) is the speed of adjustment of pt to the long run mean. If φ = 0, then pt is independently and identically distributed (i.i.d.) each period. Thus, higher the value of φ, lower is the degree of mean reversion. Similar to the model in HD, it is assumed that ut is i.i.d. with variance 2 u  . The process for pt may be equivalently expressed as ttt upp    1 )1(    (2) Similarly, let yt denote the price process of the underlying of the futures contract (the “hedging process”). tttt yyy         ))(1( 11 (3) Above, yt is the value of the process in period t, κ is the long run mean, and (1-  ) is the rate of mean-reversion. ξt is assumed to be i.i.d. with variance 2   . Further, ut and ξt are assumed to be Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 140 contemporaneously correlated with covariance   u, but all noncontemporaneous covariances are assumed to be zero. Let T kT f denote the futures price in period T-k of the contract that matures in period T. Similar to HD and MH, the current study assumes unbiased futures prices.1 This assumption can be stated as follows: )( TkT T kT yEf   for k ≥ 0 (4) where Et is the expectations operator conditional on information available in period t. Take the hedger’s objective to be to minimize the variance of the cash flow at time T. (As discussed subsequently, the variance-minimizing hedge is also the strategy that maximizes the expected utility for any increasing, concave utility function.) 3.1 Hedging with Matched-Maturity contracts The following assumes that the hedger uses matched-maturity futures contracts (i.e., contracts maturing at time T). The case of hedging with nearby futures contracts is considered later. Suppose that the earliest the hedger may initiate hedging is N periods prior to T. (This may be due to non-availability or lack of liquidity of longer-term futures contracts.) For 1 ≤ k ≤ N, let bTk denote the hedge ratio (i.e., futures position per unit of the spot position). The post-hedge cash flow in period T is given by xprffxbC N k T kT kT T kTkTT      1 1 1)1)(( (5) Above, it is assumed that futures positions are marked-to-market each period, and r is the constant, periodical interest rate used to compound intermediate, mark-to-market cash flows. As in HD, the optimal hedge ratios can now be derived by working backwards from the terminal period. As derived in Appendix A, the optimal hedge ratio at time T-k is given by 2 1 1 * )1( 1       u k k kT r b            for 1 ≤ k ≤ N (6) Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 147 APPENDIX A Hedging with matched-maturity futures contracts As in the main body of the paper, N stands for the number of periods from today to terminal time T. For 1 ≤ k ≤ N, bT-k denotes the hedge ratio (i.e., futures position per unit of the spot position taken at time T-k in order to hedge time T cash flows). By repeated substitution, for n ≥ 1, the price process for the hedged item, equation (2), can be expressed as: jT n j j nT nn Tupp      1 0 )1(  (A1) Similarly, the price process of the futures contract’s underlying, the hedging process in equation (3) can be expressed as: jT n j j nT nn Tyy       1 0 )1( (A2) Using the unbiased futures price assumption, it is seen that, for 1 ≤ k ≤ N      1 0 k j jT jT kTT fy  (A3) kT kT kT T kT ff    1 (A4) Note that futures prices are unbiased, but do not follow a random walk. As mentioned in the main body of the paper, the post-hedge cash flow in period T is given by      N k T kT kT T kTkTT prffbC 1 1 1)1)(( (A5) Above, the non-stochastic spot position has been normalized to unity for convenience. As in HD, the optimal hedge ratios can now be derived by working backwards from the terminal period. With one period to go (that is, as of time T-1), T T TTTT pfybC   )( 11 (A6) Using (A3) and (2), this can be expanded as follows: TTTTT upbC      11 )1(     (A7) The variance of CT (focusing only on the error terms) is given by Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 148   uTTu bb 1 22 1 22  (A8) Differentiating with respect to bT-1, and setting the derivative equal to zero yields 2 * 1     u T b  (A9) The second derivative is 2σξ2, which is obviously positive, and it is thus clear that the second order condition for a minimum is satisfied. Next, with two periods to go, T T T T TT T TTTT prffbfybC   )1)(()( 21211 (A10) Using (A1) and (A4), TTTTTTTT uuprbbC   12 22 121 )1()1)((  (A11) The variance of CT is given by ))1((2))1(()1( 12 2 1 222 2 222   TTuTTu brbbrb   (A12) Noting that optimal bT-1 is a known constant and using the first order condition gives the result 2 * 21 1       u Tr b         . (A13) The second derivative is 222 )1(2   r (A14) This is clearly positive, and thus the second order condition for a minimum is satisfied. In general, k periods prior to T, the variance of the cash flow in period T is given by                      1 0 1 1 0 222 1 2 1 0 22 )1(2)1( k j jjj jTu k j jj jT k j j urbrb   (A15) In any prior period, the optimal futures positions for subsequent periods are known constants. Using the first order condition for the optimal hedge ratio at time T-k, we arrive at the optimal hedge ratio, which is equation (6) in the main body of the paper: 2 1 1 * )1( 1       u k k kT r b            for 1 ≤ k ≤ N (A16) The second derivative is 2)1(2)1(2 )1(2    kk r (A17) This is clearly positive and thus the second order condition for a minimum is satisfied. Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 149 B Hedging with matched-maturity forward contracts The derivation of the optimal hedge ratio is very similar to that above, and is being provided mainly for completeness. Let T kT f denote the forward price in period T-k of the forward contract that matures in period T. As per the assumption of unbiased forward prices: )( TkT T kT yEf   for 1 ≤ k ≤ N (B1) where Et is the expectation operator conditional on information available in period t. The post-hedge cash flow in period T is given by xpfyxbC N k T T kTTkTT     1 )( (B2) Normalizing the cash position x to unity for convenience, using equations (A1) and (A3), and collecting terms,       1 01 1 0 )()1( N j N jk jT j kTjT N j j NT NN TbupC  (B3) For 1 ≤ n ≤ N, the variance of the cash flow in period T is given by (focusing on the error terms)                      1 01 1 01 222 1 0 22 )(2)( n j n jk jj kTu n j n jk j kT n j j ubb   (B4) Above, 2 u  is the variance of the error term, u; 2   is the variance of the error term, ξ; and   uis the covariance of these error terms. The optimal hedge ratios can be derived by working backwards. Let hT-k denote the cumulative hedge ratio (cumulative futures position per unit of spot position) as of time T-k.    N kj jTkT bh for 1 ≤ k ≤ N (B5) With one period to go (that is, as of time T-1), T T TTTT pfyhC   )( 11 (B6a) This can be expanded as follows: TTTTT uphC      11 )1(     (B6b) and the variance of CT is given by (taking n to be 1 in equation B4) 1 2 1 22 2  TuTu hh   (B7) Differentiating with respect to hT-1, and setting the derivative equal to zero gives the result that Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 150 2 * 1     u T h  (B8) The second derivative is 2σξ2, and it is thus clear that the second order condition is satisfied. Next, with two periods to go, T T TTT T TTTT pfyhfybC   )()( 2211 (B9) This can be rewritten as TTTTTTTTT uuphbC   12 22 121 )1()(  (B10) And the variance of CT is given by )(2)2()1( 22112 22 2 2 2 2 1 222   TTTuTTTTTu hhbbhhhb   . (B11) The first order condition for optimal hT-2 is 0)1()( 1 2 22 2    uTTT bhh (B12) Note that bT-1 is a known constant given by 2 2 211   T u TTT hhhb     (B13) Using the first order condition gives the result 2 * 2       u T h       . (B14) The second derivative is seen to be )1(2 22   (B15) This is clearly positive, and thus the second order condition for a minimum is satisfied. In any prior period, the optimal forward positions for subsequent periods are known constants. Proceeding along the same lines as above, we arrive at equation (7) in the paper: 2 1 *       u k kT h         for 1 ≤ k ≤ N (B16) C The Myers and Hanson (1996) framework As in the current paper, the Myers and Hanson (MH) model considers a hedger with a nonstochastic cash position, x, which will be liquidated on a certain future terminal date, T. The hedger’s problem relates to the stochastic cash price at that time, pT. Each period, the hedger chooses a hedge ratio, bt, and enters into a futures position, bt x, at the prevailing futures price, ft. Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 151 Futures positions are marked to market at the end of each period. The hedger’s wealth at the end of each period is given by: xbffxpxcwrw TTTTTTT 1111 )()]()[1(        (C1) xbffxcwrw tttttt 1111 )()]()[1(       for 1 ≤ t < T (C2) Above, r is the constant interest rate per period, and ct(x) represents non-stochastic costs. The hedger’s problem is to design a strategy to choose the futures position, bt, each period so as to maximize the expected utility of terminal wealth subject to the wealth constraints above:  T t t T b wUE 1 1 0)]([max   or  N k kT TNT b wUE 1 )]([max    (C3) Above Et is the expectations operator conditional on information available at time t, and U is an increasing and strictly concave utility function. N is the number of periods prior to the target period that the hedging activity can be initiated, and may be dictated either by internal corporate policy or external constraints such as availability of hedging contracts. MH make three assumptions regarding the behavior of spot and futures prices in order to derive their preference-free optimal hedging strategy: Assumption 1: Futures prices, ft, follow a martingale with a zero-mean random shock, et: ttt eff  1 (C4) Assumption 2: The expected value of the terminal price, Et(pT), follows a martingale with a zeromean random shock, vt: tTtTt vpEpE   )()( 1 (C5) Assumption 3: The two error terms, et and vt are linearly related in the following manner: tttt ev     (C6) Above, εt is an unpredictable, zero-mean error term, and is independent of et at all lags. δt is the (possibly time-varying) slope coefficient of the linear regression of vt on et, and, as such, is the ratio of the covariance between et and vt to the variance of et Based on these three assumptions, they derive an optimal hedging strategy that is valid for all increasing and strictly concave utility functions: tT t tr b   )1( * 1  for 1 ≤ t ≤ T Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 152 or (C7) 1 1 * )1(     k kT kT r b  for 1 ≤ k ≤ N Thus, the optimal hedge ratio essentially depends on the slope coefficient of the linear relationship between the innovations driving the expected spot price of the hedged item and the innovations driving the futures price. Note that the framework allows for basis risk. This is captured by assumption 3 which allows for imperfect correlation between the innovations to the expected value of the cash price on the terminal date and futures price changes. Further, there is obviously no requirement that the futures price should be the expected terminal date spot price of the cash position. As pointed out by MH, their framework can accommodate a wide range of behavior of spot and futures prices. Consider the following special case. As in the main body of the current paper, suppose that the spot price of the hedged item (cash position), pt, and the spot price of the underlying of the futures contract follow mean-reverting processes given by equations (1) and (3) in the main body of the current paper. Given the assumption of unbiased futures prices, the futures price as of time t of the contract maturing at T will evolve as follows: t tTT t T tff    1 (C8) The expected future spot price of the hedged item will evolve as follows: t tT TtTt upEpE    )()( 1 (C9) Further, given that ut and ξt are contemporaneously correlated, their relationship can be represented as follows: tttt u      (C10) Above, λt is the slope coefficient of the regression of ut on ξt, and is therefore the ratio of the covariance of the two terms to the variance of ξt. ηt is a zero-mean random shock and unrelated to ξt at all lags. Comparing the above three equations to the three assumptions of the MH framework, it is apparent that the model in the current study fits into the MH framework by letting t tT tuv    (C11) t tT t e    (C12) Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 153 tT tT tt       (C13) t tT t    (C14) The optimal hedging strategy is accordingly given by: Nkfor r b or Ttfor r b kT k k kT t tT tT t                            1 )1( 1 1 )1( 1 1 1 1 * * 1       (C15) The above is, of course, essentially the same as equation (6) in the main body of the current paper (except that in this framework, λ is allowed to be time-varying, whereas in the main paper, it is assumed to be a constant for ease of exposition). Thus, the variance-minimizing strategy also maximizes expected-utility. D Hedging with nearby forwards or futures The derivation is very similar to the one in Appendix A and is being provided mainly for completeness. The objective remains to hedge a cash flow that will occur in a future period T. The difference is that hedging is carried out by rolling over a series of nearby forwards or futures contracts. Specifically, the forward or futures contract used in period t is the one maturing in period t+1. As the contract matures in the very next period, there is no need to distinguish between forwards and futures. The following exposition is in terms of forward contracts, but obviously applies to futures as well. Using equation (A3), 1 11 11       kT kT kT kT kTkT ffy  , for k ≥ 1 (D1) Note that forward prices are not only unbiased, but follow a random walk. Assuming that hedging is carried out over a horizon of N periods, and normalizing the spot position to unity for convenience, the post-hedge cash flow in period T is given by        N k T kkT kT kT kTkTT prffbC 1 111 1)1)(( (D2) Vadhindran K. Rao / Journal of Risk and Financial Management 5(2011) 133-161 154 Now, we derive the optimal hedge ratios, the bT-k, using dynamic programming. With one period to go (that is, as of time T-1), T T TTTT pfybC   )( 11 (D3) This can be expanded as below: TTTTT upbC      11 )1(     (D4) The variance of CT is given by 1 2 1 22 2  TuTu bb   (D5) Above, bT-1 is the forward position as at time T-1, 2 u  is the variance of the error term, u, 2   is the variance of the error term, ξ, and   uis the covariance between these error terms. Differentiating with respect to bT-1, and setting the derivative equal to zero gives the result that 2 * 1     u T b  (D6) This is of course the same as the standard single-period optimal hedge ratio. Next, with two periods to go, T T T T TT T TTTT prffbfybC      )1)(()( 1 2 1 1211 (D7) This can be rewritten as TTTTTTTT uuprbbC   12 22 121 )1()1(  (D8) The variance of CT is given by ))1((2))1(()1( 12 2 1 22 2 222   TTuTTu bbrbrb   . (D9) Noting that optimal bT-1 is a known constant and using the first order condition gives the result 2 * 21      u Tr b         (D10) In any prior period, the optimal forward positions for subsequent periods are known constants. Therefore, proceeding in the same manner as above, we arrive at equation (8) in the paper: 2 1 * 1      u k kT r b          for 1 ≤ k ≤ N (D11) Vadhindran K. 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