In this problem, we were given the covariance, but not the correlation. We can calculate the correlation using the following formula:

$$\rho = \frac{\text{Cov}(S,F)}{\sigma_S \sigma_F} = \frac{0.4132}{(0.75)(0.68)} = 0.81$$

With the correlation calculated, we can determine a few key values. First, the coefficient of determination ($R^2$) between the change in futures prices and the change in spot prices is the square of the correlation coefficient ($\rho^2$) between spot and futures prices, or $0.81^2 = 0.656$. This tells us that the oil futures are 65.6% effective at hedging the variance in spot prices.

We can also calculate the optimal hedge ratio as:

$$\text{HR} = \rho_{S,F} \frac{\sigma_S}{\sigma_F} = 0.81 \times \frac{0.75}{0.68} = 0.89$$

This hedge ratio suggests that the ratio of the size of the futures to the spot should be 0.89.

(Book 3, Module 34.1, LO 34.d)