The most exciting financial innovation of the 1980s might be the introduction of stock index futures contracts. These contracts, written on the value of various stock index portfolios as defined in the text of this chapter, provide important benefits to stock portfolio managers.
The first stock index futures contract was introduced in February 1982 by the Kansas City Board of Trade. This Value Line futures contract is written on the Value Line Composite Index, a stock index that consists of approximately 1,700 stocks from the New York, American, and OTC stock markets. The Chicago Mercantile Exchange quickly followed suit in April 1982 with a futures contract on the S&P 500 stock index, and then the Chicago Board of Trade in July 1984 followed with a futures contract on the Major Market Index.
Most recently, the Chicago Board of Trade introduce Dow Jones Index Futures in October 1997.
Stock portfolio manager can use index futures to hedge his (or her) portfolio. Now, we discuss how the minimum variance type of hedge ratio can be derived in accordance with method used to derive the optimal weights of a portfolio which has been discussed in
Appendix 1 of
Chap. 13. We first define
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ΔS: Change in spot price, S, during a period of time equal to the life of the hedge
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ΔF: Change in futures price, F, during a period of time equal to the life of the hedge
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σ S : Standard deviation of ΔS
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σ F : Standard deviation of ΔF
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ρ: Coefficient of correlation between ΔS and ΔF
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h: Hedge ratio
When the hedger is long for the asset and short futures, the change in the value of the hedger’s position during the life of the hedge is
In either case the variance,
σ, of the change in value of the hedged position is given by
$$ \sigma =\sigma_S^2+{h^2}\sigma_F^2-2h\rho {\sigma_S}{\sigma_F} $$
so that
$$ \frac{{\partial \sigma }}{{\partial h}}=2h\sigma_F^2-2\rho {\sigma_S}{\sigma_F} $$
Setting this equal to zero and noting that ∂
2 σ/∂
h 2 is positive, we see that the value of
h which minimizes the variance is
$$ h=\rho \frac{{{\sigma_S}}}{{{\sigma_F}}} $$
(19.20)
The optimal hedge ratio is therefore the product of the coefficient of correlation between ΔS and ΔF and the ratio of the standard deviation of ΔS to the standard deviation of DF.
If ρ = 1 and σ F = σ S , the optimal hedge ratio, h, is 1.0. This is to be expected since in this case the futures price mirrors the spot price perfectly. If ρ = 1 and σ F = 2σ S , h is 0.5. This result is also expected since in this case the futures price always changes by twice as much as the spot price.
The optimal hedge ratio,
h, defined in Eq.
19.20 can be estimated by using the following regression
$$ \Delta {S_t}={a_0}+{a_1}\Delta {F_t}+{e_t} $$
(19.21)
where ΔS
t = change in spot price in period
t ΔF t = change in futures price in period t
a 0 and a 1 are the intercept and slope of a regression, respectively. The estimated a 1 is the estimated hedge ratio, h. Application of Eq. 19.21 has been shown in Problem 60 of Chap. 14.