Minimum Variance Hedge Ratio and Risk capital

Question

I understand that the minimum variance hedge ratio minimizes the second moment of the portfolios. My question is how is it related to the size of the risk capital (which is calculated as the Value at Risk - Expected Value). Is the Risk capital also minimized at the optimal hedge ratio?

Answer 1

No. Say we have a very simple portfolio $$ Z = (1-\lambda)X + \lambda Y $$ with $E(X) = 1, E(Y) = 2, \text{Var}(X) = 1$, $\text{Var}(Y) = 2$. Then $$ \begin{align} \text{Var}(Z) &= (1-\lambda)^2 \cdot \text{Var}(X) + \lambda^2 \cdot \text{Var}(Y) \\ &= (1-\lambda)^2 \cdot 1 + \lambda^2 \cdot 2 \\ &= 1 -2\lambda + 3\lambda^2 \end{align} $$ which we minimize for $\lambda = \frac{1}{3}$. But this gives the expectation $$ E(Z) = \frac{2}{3} \cdot 1 + \frac{1}{3} \cdot 2 = 1 $$ which is clearly lower than what we would get if we choose $\lambda = 1$. In this latter case $$ E(Z) = 1 \cdot 2 = 2 $$ So if we minimize the variance, this is sometimes going to come at the cost of reducing the expectation. This is called the mean-variance tradeoff.