Hedge ratio and residual risk with fresh numbers

Question

You are long an asset S and hedge by shorting a related instrument F. Their return volatilities are \sigmaS = 30\% and \sigmaF = 20\%, with correlation \rho = 0.6. How many units of F do you short per unit of S, and what is your residual volatility? Follow-up: what correlation would you need to bring the residual down to 15\%?

Accepted Answer

The variance-minimizing hedge ratio is the beta of S on F: h^ = ho\,\frac{\sigmaS}{\sigmaF} = 0.6 \cdot \frac{0.30}{0.20} = 0.6 \cdot 1.5 = 0.9. Short 0.9 units of F per unit of S. Here S is the more volatile leg, so the ratio exceeds the correlation: you need almost a full unit of the quieter hedge to match the asset's moves. Residual risk. The leftover volatility is \sigma{	ext{residual}} = \sigmaS\sqrt{1 - ho^2} = 0.30\sqrt{1 - 0.36} = 0.30\sqrt{0.64} = 0.30 \cdot 0.8 = 24\%. The hedge removes the fraction ho^2 = 36\% of the variance, cutting volatility from 30\% to 24\%. The rest i…

Hedge ratio and residual risk with fresh numbers

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