The optimal hedge ratio in unbiased futures markets

Abstract

n extensive body of recent research in futures markets has dealt with the A determination of the minimum variance hedging ratio.' The strength of these results is mitigated, however, by two factors: First, the researchers assume (implicitly or explicitly) that the hedger has a quadratic utility function. This is well-known to be a problematic assumption, since quadratic utility functions have many undesirable properties.2 Second, the hedge ratio which is determined is not an optimal hedge ratio, but rather one which minimizes the variance of the producer's income. Even if the hedger is assumed to have a quadratic utility function, there is no reason to believe that the utility will be maximized when the variance of the spot cum futures position is minimized?In this article we show that-when futures markets are assumed to be unbiasedthe minimum variance hedge position is also on optimal hedge ratio. Optimality holds irrespective of the hedger's utility function. Since existing research on futures prices appears to indicate that unbiasedness is a property of many futures markets, our results are considerably stronger than those of previous authors.In Section I of the article we set out our model and indicate the intuition behind 'The seminal papers are Johnson (1960) and Stein (1961). Interest in the problem was re-inspired by Ederington (1979), and much of the recent research has its roots in his paper. For papers which deal with the JohnsonlEderington model in various future markets, see Franckle (1980) and Hill and Schneeweis (1982).'For a discussion, see Baron (1977). 3The potential nonoptimality of the minimum variance hedge portfolio has been recognized by all of the authors who have dealt with the theoretical aspects of the problem, but this distinction has tended to become blurred in the empirical papers which have focussed on the determination of the hedge ratio.

Hedging price risk in the presence of crop yield and revenue insurance

“…It is equal to 1.29 for the selected farm in Centre and to 0.68 for the selected farm in Ile de France. 12 If there was no production risk, the optimal unbiased futures hedge ratio would thus reduce to the beta coefficient, a well known result in the optimal hedging literature (see, for example, Benninga, Eldor and Zilcha, 1984). Table 3 shows the optimal hedge ratios under alternative insurance policies when futures markets are perceived as unbiased, i.e., f E F= .…”

Hedging with Stock Index Options: A Mean-Extended Gini Approach

“…When investors maximize expected utility of portfolio returns, the optimal hedge ratio can be obtained by using the mean-variance (MV) model since, as shown by Levy and Markowitz [9], MV approximates expected utility, regardless of the utility and the probability distribution. Furthermore, Benninga, Eldor and Zilcha [10,11] showed that the optimal hedge ratio is equal to the minimum-variance hedge ratio which is:…”

Securitizing Area Insurance: A Risk Management Approach

“…Expression (10), on its part, states that the insurance net payoff should itself equal its risk premium. Indicating with the sign * the vectors and the submatrices corresponding to non zero activities, and applying the variance and covariance definitions, we can write: of total revenue per ha of the area i olved i the scheme and the corresponding mean according to the so called "regressability assumption" (Benninga et al, 1984): Going ba to the maximizing conditions, expression (11) states that prices of all non zero activities should equal marginal costs at the optimum, while expression (12) indicates that the quantity of insurance acquir ck ed by the i-th firm equals the net value of the insurance, including its utility from risk diversification. This result can be seen more clearly, dropping the asterisks for simplicity, by writing it as follows:…”