Hedging ?nancial risk is a very important practical issue in economics. In this exercise, you will derive your optimal hedge ratio, assuming that you are an expected utility maximizer with quadratic tastes over rates of return who has a spot position in a single risky asset.
Here's the notation. The random return on a portfolio that consists of a spot position in a single risky asset is
If you hedge your risk by selling a fraction h of your asset forward then your return becomes
Rp = Rs - h ⋅ Rf
where Rf is the payoff on the forward contract. Your utility function, where γ is a risk preference parameter, is
u (Rp) = E ( Rp) - γ var (Rp)
Here's the story. Say that all of your wealth is invested in a single asset whose uncertain return is Rs over t. Now suppose that you want to reduce the riskiness of your spot position (is risk aversion reason enough?) as measured by its variance. One way to hedge the risk is to sell the asset forward in a forward or futures market. For example, you might be a manufacturer of electric guitars who exports to the United States. Chances are, you will be paid in US dollars, say, a month later. Your spot position then is the one-month rate of return on manufacturing guitars. As an exporter, you face a number of risks: one is default risk, the risk of not being paid; another is unexpected changes in the rate of in?ation; and still another is foreign exchange risk. Let's ignore default risk by assuming that you're dealing with a longtime and ?nancially stable customer. Let's also ignore in?ation risk because, after all, this is Canada - eh? - and it's only one month. That leaves foreign exchange risk. A naive currency hedge would be one-for-one or dollar-for dollar (h =1); so, if you're owed US $1,000 at the end of the month, you'd sell US $1,000 forward one month. If spot and futures prices on the dollar are highly correlated, then any change in the spot price at month's end will be largely offset by changes in the futures price. Since we're talking in terms of rates of return rather than dollars, that simple hedge ratio would be 1, which is the same as saying that 100% of your spot position is hedged. But is a hedge ratio of 1 optimal?