Optimal Hedge Ratio - Colorado School of Mines

1 Optimal Hedge RatioIn hedging, you can Hedge your whole portfolio or some portion of it. The Hedge Ratio isthe size of the futures contract relative to the cash transaction. In a previous example involving atrader with oil en route from the Gulf, the Hedge Ratio was one, since she sold a futures contractrepresenting each barrel of oil. When using turnips to Hedge , the Ratio was still 1, since the valueof the crude was hedged with an equal value of turnips. If the hedger had sold _ barrel for eachbarrel in transit, the Hedge Ratio would have been _. It turns out that it is not always Optimal tohedge your whole product and Hull (2000) works out what the Optimal Hedge Ratio is to minimizerisk.

2 In his exposition, he defines S = ST - St = the change in the spot price over the life of the contract, F = FT - Ft = the change in the futures price over the life of the contract, S = the standard deviation of S, F = the standard deviation of F, = SF/( S F) = the correlation coefficient between S and F, which is the covarianceof S and F divided by the standard deviations of S and F, andh = the Hedge the hedger owns the products and sells the future, his portfolio value is (S - hF). The change invalue of the portfolio is S - h , if the hedger buys the future and is short the product, his portfolio value is hF - change in value of the portfolio ish F - variance ( 2) for the above two portfolios is 2 = S2 + h2 F2 - 2h SF = S2 + h2 F2 - 2h S find the Optimal Hedge Ratio , which minimizes risk or variance, minimize the above expressionwith respect to h.

3 2/ h = 2h F2 - 2 S F = second order conditions 2 2/ h2 = 2 F2 > h is a minimum. Solving for h, we geth = S/ if S and F are for the same products it is likely that S and F are close to the same value and is close to 1. Then the Optimal Hedge Ratio is near 1. Where this becomes more interesting iswhere you are hedging one product with a different products future contract. For example, youmight use Henry Hub gas to Hedge for gas at Waha or some other hub. Then S and F may notbe close to the same and may not be close to 1. The closer is to one, and the larger is thevariance of the product you are hedging, the more you Hedge .

4 The larger is the variance of theproduct used to Hedge the lower the Hedge Ratio . It is even possible that h would be greater than1.