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Preference-free optimal hedging using futures
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Vadhindran K. Rao
Business Administration Department, Embry-Riddle Aeronautical University, 600 S. Clyde Morris Blvd., Daytona Beach, FL 32114-3900, USA
Received 19 October 1998; accepted 29 July 1999
Abstract
This paper provides an alternative formulation and proof of the conditions under which optimal hedge ratios are independent of risk preference and points out that the conditions are satisfied by a wide class of models of spot and futures returns. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Hedging; Futures; Hedge ratio; Separation JEL classification: G13
1. Introduction
A set of sufficient conditions under which the optimal hedge ratio (OHR) for an investor with a fixed spot position is independent of the investor’s utility function was first discovered by Benninga et al. (1983), and then independently by Baillie and Myers (1989). This line of research was extended by Lence (1995) who derived both sufficient and necessary conditions for the existence of a preference-free OHR.
The main objectives of this paper are: (a) to present the conditions in an alternative form that yields some additional insights; (b) to provide an alternative proof based on the concept of stochastic dominance, and, in the process, draw attention to the connection between the optimal hedge ratio problem and the well-known mutual fund separation theorems of Ross (1978).
One advantage of this alternative approach is that, unlike as in Lence (1995), the approach does not assume the existence of variances and covariances of spot and futures prices. Relaxing this assumption is important owing to evidence that futures prices may follow stable distributions with
*Tel.:11-904-226-6246; fax: 11-904-226-6696. E-mail address: raov@cts.db.erau.edu (V.K. Rao)
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infinite variance. Further, Lence (1995) relies on Holthausen (1979) for a proof of the necessity of unbiased futures markets for the existence of a preference-free OHR. In contrast, this paper shows that the necessity of unbiased futures markets follows directly from the necessary and sufficient conditions for preference-free optimal hedging provided in this paper.
A noteworthy implication of our results is that, assuming unbiased futures markets, a sufficient condition for the existence of preference-free OHRs is that spot and futures returns follow a
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distribution of the elliptical class. This implies that popular GARCH models used in studies such as Baillie and Myers (1989, 1991) and Park and Switzer (1995a,b) are consistent with the existence of preference-free optimal hedge ratios.
2. Optimal hedging: one-fund separation
Consider an investor with a fixed long spot position in one or more risky assets. Suppose that there exist n futures contracts that can be used to hedge the spot portfolio, or, more generally, to enhance its risk–return characteristics. Let x be the random return over the hedging period on the spot portfolio; let z be an a(n31) vector of random returns on the futures contracts, with the return on the jth contract denoted by z (the term ‘return’ is just a convenient way of referring to the percentage pricej change on the futures contract over the hedging period); let p be an a(n31) vector representing a portfolio of futures contracts. The jth element of p, p , is the value of the position in the jth futuresj contract divided by the total value of the spot position (the value of the position in the jth futures contract is the number of futures contracts held times the futures price). Thus, a futures portfolio is essentially a vector of hedge ratios (note that, despite the use of the term ‘portfolio’ for the vector p, its components do not have to sum to unity). A negative value for any of the components of p indicates a short position in the corresponding futures contract.
Throughout, subscripts denote individual elements of a vector, and the superscript ‘t’ denotes transposition of a vector or matrix.
The return on the spot-cum-futures portfolio (or the ‘joint portfolio’) is then given by
t
x1p z. (1)
Let U denote the class of all monotone increasing, concave utility functions. Assume that the investor has a utility function, u, from this class and that the objective is to maximize
t
Ehu(x1p z)j, (2)
3 the expected utility of return, where E denotes the expectations operator.
An important concept used in the paper is that of stochastic dominance. A random return y is said to stochastically dominate in the second degree another return w if and only if for every u in U
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See Cornew et al. (1984), So (1987) and Liu and Brorsen (1995) for evidence in favor of, and Gribbin et al. (1992) for evidence against, the hypothesis that futures prices follow stable distributions.
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See Muirhead (1982, Chapter 1) and Owen and Rabinovitch (1983) for a good introduction to elliptical distributions.
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Ehu( y)j$Ehu(w)j. (3)
As noted in Ross (1978), this is equivalent to asserting the existence of two random variablesj and´
such that
w|y1j 1´, withj#0 and Eh´uy1jj50. (4)
Above, the symbol ‘|’ is read ‘‘has the same distribution as.’’ Observe that a necessary condition for
y to stochastically dominate w is that Ehyj$Ehwj.
Proposition. A vector of hedge ratios p is universally optimal if and only if it satisfies t
Ehz ux1p zj50, for every j. (5)
j
Proof. The proposition is analogous to the theorem on one-fund separation in Ross (1978). For a
proof of sufficiency, suppose that (5) holds for some p. Define the return on the corresponding spot-cum-futures portfolio by
t
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r ;x1p z. (6)
Given any other vector of hedge ratios h, let the return on the corresponding spot-cum-futures portfolio be given by
by hypothesis. This implies that r stochastically dominates r , and therefore that p is universallyh optimal. The intuition underlying the result is straightforward. For any other futures portfolio h, the
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joint portfolio return r has the same expectation as rh but has some additional noise.
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For a proof of necessity, suppose that p is universally optimal, and that r denotes the return on the corresponding joint portfolio. For any other futures portfolio h, there will exist random variables j(h) and ´(h) such that
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rh|r 1j(h)1´(h), withj(h)#0 and Eh´(h)ur 1j(h)j50. (10)
It can be shown that j(h) has to be zero almost surely. Suppose not. Then, using (8), it is seen that
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which would contradict the hypothesis that r stochastically dominates the return on all other joint portfolios.
Since (10) has to hold with j(h)50 for every h, by Theorem A2 in Ross (1978) (reproduced in Appendix A)
t
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Eh(h2p) zur j50, (13)
which is equivalent to (5). h
3. Discussion
t
Observe that Ehzjux1p zj50 implies that Ehzjj50. Thus, unbiased futures markets are a necessary condition for the existence of preference-free OHRs. Further, the conditions have an intuitively appealing interpretation: a set of hedge ratios is universally optimal only if the return on the spot-cum-futures portfolio contains no information about returns on any of the futures contracts.
A noteworthy point is that nowhere in the proof do we require the existence of a covariance matrix for spot and futures returns. However, if it exists, then for any vector of hedge ratios h, it is seen from Eq. (8) that
t t t
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Varhx1h zj5Varhr j1Varh(h2p) zj12Covhr ,(h2p) zj, (14)
where Var denotes variance and Cov covariance. But (5) implies that the covariance term is zero. Thus, the universally optimal futures portfolio p (provided it exists) is the global minimum variance portfolio and is accordingly given by
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2Vz Vzx, (15)
where Vz is the covariance matrix of z, andVzx is the column vector of covariances between each of the elements of z and x.
It is easy to show that (5) is equivalent to the conditions in Lence (1995), who considers the problem of an investor hedging the sale of a fixed quantity Q of a certain good at a future time T. As stated in Lence (1995), necessary and sufficient conditions to obtain a preference-free hedge ratio are that the futures market should be unbiased and the relationship between the spot and futures price should have the following form:
PT5bFT1u ,T (16)
where P is the random spot price at the end of the period, F is the random futures price at the end ofT T
the period, b is some constant, and u a random variable with F being conditionally independent ofT T
u . Under these conditions, the optimal futures quantity is given byT
X5bQ. (17)
To show that these conditions are equivalent to (5), observe that (5) is equivalent to
where F is the (known) futures price at the beginning of the period. Given (18), define u by0 T
uT5PT2FT b. (19)
Then, F is conditionally independent of u because, from (18),T T
EhFT2F0uu QT j50. (20)
Conversely, given the conditions in Lence (1995)
F05EhFTj5EhFTuu QT j5EhFTuP QT 1 (F02F ) XT j. (21)
The form in which the conditions are stated in Lence (1995) has its advantages. For example, as pointed out by Lence (1995), the formulation makes it easy to see that the log–log model in spot and
4 futures prices does not satisfy the conditions required for preference-free OHRs.
However, the alternative formulation in (5) may be more convenient in other contexts. For one, it is more general in its scope in that it allows the investor to have any kind of a spot position and hedge the position using multiple futures contracts. For example, the model can be applied in the context of a corporation hedging its stock returns against multiple factor risks.
Consider the result in Baillie and Myers (1989), who show that, given unbiased futures markets, a sufficient condition for the existence of preference-free OHRs is that the distribution of spot and futures returns is jointly normal. The conditions as stated in (5) suggest that the class of joint distributions of spot and futures returns for which OHRs are independent of risk preferences is much wider. It is not difficult to show that the class includes not just the multivariate normal but all
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elliptical distributions. Therefore (assuming that the unbiased futures markets condition is imposed), not only are popular GARCH models that postulate a multivariate normal distribution for conditional spot and future returns consistent with the conditions required for preference-free OHRs, but also more general GARCH models that postulate non-normal elliptical distributions such as the
6 multivariate t-distribution or symmetric stable distributions.
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Lence (1995) mistakenly supposes that Baillie and Myers (1991) use a log–log model in price levels, and thereby incorrectly concludes that their model does not permit preference-free optimal hedge ratios (see p. 388 of Lence, 1995). However, as can be inferred from the first sentence in the last paragraph on page 110 of the Baillie and Myers paper (namely, that ‘‘As with most price data, it is reasonable to base inference on the change in the logarithm of price,’’), and from Eq. (8) on p. 117 of their paper, the optimal hedge ratio in Baillie and Myers is based on the covariance matrix of spot and futures returns and is the same as Eq. (15) of this present paper. Therefore, as discussed by Baillie and Myers in the two paragraphs following Eq. (8) of their paper, and as mentioned in the last paragraph of this paper, the Baillie and Myers model is fully consistent with preference-free optimal hedge ratios.
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A proof of the proposition is available from the author on request.
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Appendix 1
Theorem A2 from Ross (1978). Given random variables y and z, if for every l, there exists a
random variable ´(l) such that
y1lz|y1´(l), where Eh´(l)uyj50,
then
Ehzuyj50.
References
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Baillie, R.T., Myers, R.J., 1991. Bivariate GARCH estimation of the optimal commodity futures hedge. Journal of Applied Econometrics 6, 109–124.
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Gribbin, D.W., Harris, R.W., Lau, H.S., 1992. Futures prices are not stable-Paretian distributed. The Journal of Futures Markets 12, 475–487.
Holthausen, D., 1979. Hedging and the competitive firm under uncertainty. American Economic Review 69, 989–995. Lence, S.H., 1995. On the optimal hedge under unbiased futures prices. Economics Letters 47, 385–388.
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Park, T.H., Switzer, L.N., 1995a. Time-varying distributions and the optimal hedge ratios for stock index futures. Applied Financial Economics 5, 131–137.
Park, T.H., Switzer, L.N., 1995b. Bivariate GARCH estimation of the optimal hedge ratios for stock index futures: a note. The Journal of Futures Markets 15, 61–67.
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