Hedging Basis Risk
Let's study the case of the hedger who takes a short position in futures at time t = 1 and knows that he will close out this position at time t = 2. The basis at t = 1 is therefore b1 = S1 – F1, while the basis at t = 2 is b2 = S2 – F2. In the case of our cattle rancher, the market price received for selling his herd would be S2 and the profit on his futures position when he closes it out would be F1 – F2. The effective price he receives is therefore S2 + (F1 – F2), which by definition equals F1 + b2.
The value of F1 is known at t = 1, but the basis b2 at t = 2 is not. Were it so, then we would have constructed the perfect hedge. In fact, for the case in which the futures position is closed out on the delivery date and the hedging asset is the underlying, then the basis would be zero at t = 2. Since this is not always the case, then the final profit or loss includes b2 as a risk. To see this more clearly, let S2′ be the spot price of the asset used for hedging. Adding and subtracting this from the effective price the hedger receives at t = 2, which from before, is S2 + (F1 – F2), we get:
The two terms in parentheses are the components of the basis risk. S2 – S2′ is the part of the basis represented by the imperfect hedging choice, and S2′ – F2 represents the nonconvergence in the spot to the futures price. If the asset hedged were identical to the hedging asset, then S2 = S2′ and the two basis components disappear, leaving F1.
Let's return to our dollar/euro trade as an example. Suppose on January 1 the exporter knows she will receive 500,000 euros on May 1. Suppose euros futures contracts traded on the CME are for 100,000 euros with delivery dates at the close of every month. The exporter therefore decides to short five contracts with delivery date April 30. Suppose that the futures price on January 1 is $1.30 per euro and the spot and futures prices for euros turn out to be $1.275 and $1.27, respectively, on April 30.
The gain on the futures contract is F1 – F2 = $1.30 – $1.27 = $0.03. The basis b2 = $0.005. The effective price obtained is therefore:
Looked at differently, it is also equal to:
The total amount received by the exporter is therefore $1.305∗500,000 = $652,500. That is, the exporter's hedged position consisted of the spot sale of her euros on April 30 at S2 along with the (in this case) positive impact of the hedge (F1 – F2).
Cross-Hedging
The hedging asset is oftentimes not the same as the asset being hedged. In these instances, the investor must select a hedging asset that is a close substitute (for example, orange juice futures to hedge grapefruit exposure). More specifically, the investor chooses a hedge position that is a function of the correlation between the asset being hedged (the underlying) and the hedging instrument. It should be obvious that if these are the same assets, then the correlation is one and the optimal hedge is to take an equal and opposite position in the futures market for the underlying. If they are not the same asset, then the investor desires to minimize the risk of an imperfect hedge.
Intuitively, the problem lies in the fact that changes in the spot price for the asset being hedged (S) do not vary one for one with changes in the futures price of the hedging instrument (denoted by F). That is, their correlation coefficient (ρ) is not 1. If the investor knew the correlation between S and F, then she would be able to adjust the size of her hedge to minimize the misfit caused by using an imperfect hedging asset.
If the investor had at her disposal a history of futures prices on the hedge instrument as well as the spot prices for the asset being hedged, then regressing S on F would give her the information she'd need to size the hedge. To see this, suppose she estimates the parameters of the following regression:
In standard statistics texts it is demonstrated that (also see the appendix to Chapter 5):
If the asset hedged and the hedging asset were the same (no basis risk), then it should be clear that β would equal 1 and we'd hedge using an equal and opposite position in the futures market. Consider then, the case in which β is estimated to be equal to 2. In that case, the covariation between the two assets would be twice the variation in the hedging asset alone. This would imply that a hedge would need to consist of twice the normal position in the futures market because the variation in the underlying futures is half the covariation between it and the underlying asset. Thus, β is the optimal hedge ratio.
Let's prove this. In our livestock futures case, the rancher will go to market and sell his Q pounds of cattle at S2 per pound. His hedge position has value equal to (F1 – F2) h, where h is the amount of the hedge (assume this is unknown for the moment). The total cash flow, C, at the time of sale is therefore equal to:
Note that (F1 – F2) h represents the profit on the futures position. Absent futures, the cash flow at t = 2 will simply be QS2. We seek the value of h that minimizes the variance of this cash flow. The variance is equal to the sum of the variance of QS2 denoted as Q2 var(S), the variance of (F1 – F2) h, which we'll denote by h2 var(F), and twice their covariance denoted as 2 cov(S,F). This can be written as:
Taking the derivative with respect to h and solving yields h = –Qβ = –Qcov(S,F)/var(F). Thus, the optimal hedge is the asset's beta to the hedging asset scaled by the size of the position Q.