## The Minimumvariance Hedge

It is not always possible to ionn a perfect hedge with futures contracts. There may be no contract involving the exact asset whose value must be hedged, the delivery dates of the available contracts may not match the asset obligation date, the amount of the asset obligated may not be an integral multiple of the contract size, there may be a lack of liquidity in the futures market, or the delivery terms may not coincide with the those of the obligation In these situations, the original risk cannot be eliminated completely with a futures contract, but usually the risk can be reduced

One measure of the lack of hedging perfection is the basis, defined as the mismatch between the spot and futures prices Specifically, basis — spot price of asset to be hedged — futures price of contract used

If the asset to be hedged is identical to that of the futures contract, then the basis will be zero at the delivery date However, in general, for the reasons mentioned, the final basis may not be zero as anticipated. Usually the final basis is a random quantity, and this precludes the possibility of a perfect hedge The basis risk calls for alternative hedging techniques

One common method of hedging in the presence of basis risk is the minimum-variance hedge The general formula for this hedge can be deduced quite readily. Suppose that at time zero the situation to be hedged is described by a cash flow v to occur at time I For example, if the obligation is to purchase IV units of an asset at time / , we have „v = VV5, where S is the spot price of the asset at 7 Let F denote the futures price of the contract that is used as a hedge, and let h denote the futures position taken. We neglect interest payments on margin accounts by assuming that all profits (or losses) in the futures account are settled at f The cash flow at time 7 is therefore equal to the original obligation plus the profit in the futures account Hence, cash flow = y = v -f ( Fj - F\$)h .

We find the variance of the cash flow as var(y) = Efv — 74- {F'j — F / )h]2 = var(.v) -}- 2cov(.v, Fy )h + var(Fy)/r

This is minimized by setting the derivative with respect to It equal to zero. This leads to the following result:

Minimum-variance hedging formula The minimum-variance hedge and the resulting ~7 variance are covfv, F t )

When the obligation has the form of a fixed amount W of an asset whose spot price is Sj, (10 7) becomes

This, of course, reminds us of the general mean-variance formulas of Chapter 7; and indeed it is closely related to them.

Example 10.11 (The perfect hedge) As a special case, suppose that the futures commodity is identical to the spot commodity being hedged, In that case Fj — Sj . Suppose that the obligation is W units of the commodity, so that .v = IV Sj In that case cov(x, Fj) ™ cov(-Sr, Fj)W ~ variF/)^. Therefore, according to (10 7) we have h = — W, and according to (10..8) we find var(y) = 0. In other words, the minimum-variance hedge reduces to the perfect hedge when the futures price is perfectly correlated with the spot price of the commodity being hedged.

Example 10.12 (Hedging foreign currency with alternative futures) The BIG

H Corporation (a U.S. corporation) has obtained a large order from a Danish firm Payment will be in 60 days in the amount of 1 million Danish kroner. BIG H would like to hedge the exchange risk, but there is no futures contract for Danish kroner. The vice president for finance of BIG H decides that the company can hedge with German marks, although DM and kroner do not follow each other exactly

He notes that the current exchange rates are K ~ .164 dollar/kroner and M = .625 doliar/DM. Hence the exchange rate between marks and kroner is K/M ~ . 164/ 625 — .262 DM/kroner. Therefore receipt of 1 million Danish kroner is equivalent to the receipt of 262,000 DM at the current exchange rate He deduces that an equal and opposite hedge would be to short .262,000 DM.

An intern working at BIG H suggests that a minimum-variance hedge be considered as an alternative. The intern is given a few days to work out the details. He does some quick historical studies and estimates that the monthly fluctuations in the U.S. exchange rates K and M are correlated with a correlation coefficient of about 8. The standard deviation of these fluctuations is found to be about .3% of its value per month for marks and slightly less, 2 5%, for kroner In this problem the v of (10.7) denotes die dollar value of 1 million Danish kroner in 60 days, and F-] is the dollar value of a German mark at that time We may put .v = K x 1 million The intern therefore estimates beta as h = -pW

10.11 OPTIMAL HEDGING* 285

Hence the minimum-variance hedge is

The minimum-variance hedge is smaller than that implied by a lull hedge based on the exchange ratios; it is reduced by the correlation coefficient and by the ratio of standard deviations

We can go a bit further and find out how effective this hedge really is, compared to doing nothing. We have ,.v = K x 1 million. Hence cov(.,v, M) — 1 million x aKl\t and crv = 1 million x aK. Combining these two, we have cov(a\ m) — ok mox/a^ < Using the minimum-variance hedging formula, we find

Hence the minimum-variance hedge reduces risk by a factor of .6. A hedge with lower risk would be obtained if a hedging instrument could be found that was more highly correlated with Danish kroner

Example 10.13 (Changing portfolio beta with stock index futures) Mrs Smith owns a large portfolio that is heavily weighted toward high technology stocks She believes that these securities will perform exceedingly well compared to the market as a whole over the next several months However, Mrs, Smith realizes that her portfolio, which has a beta (with respect to the market) of 14, is exposed to a significant degree of market risk. If the general market declines, her portfolio will also decline, even if her securities do achieve significant excess return above that predicted by, say, CAPM, as she believes they will.

Mrs Smith decides to hedge against this market risk She can change the beta of her portfolio by selling some stock index futures. She might decide to construct a minimum-variance hedge of her \$2 million portfolio by shorting \$2 million x 1.4 = \$2.8 million of S&P 500 stock index futures with maturity in 120 days Since the normal beta of her portfolio is based on the S&P 500, this beta is the same beta as that in the general equation, (10,9) The overall new beta of her hedged portfolio, after taking the short position in the stock index futures, is zero.

Hence, stdev(v) = 1 — .B- ) stdev(.,v) = .6 x stdev(.v)