Minimum variance hedge with more than one asset

Question

My portfolio comprises of 3 assets A,B,C that are correlated and the variance-covariance structure is known. At any given point in time, my position in Asset A say is given to me.

I need to construct a variance minimizing hedge using both B and C, given this position in A.

Basically I am approaching the problem by directly constructing the Variance, as a function of say x and y, the positions in B and C, and minimizing the function by setting the partials w.r.t x and y to 0. This gives me 2 equations in x and y and I can solve them for x and y.

My question is that, is this approach sound? What notion of risk is this really minimizing? I did not explicitly construct a risk model here?

Answer 1

If the variances are known to be $\sigma_0$, $\sigma_1$ and $\sigma_2$ and the correlations are $\rho_{01}$, $\rho_{02}$ and $\rho_{12}$ then you can do exactly as you suggest - write down the variance of the total portfolio as a function of your holdings $x_0$, $x_1$ and $x_2$ and set the partial derivatives with respect to $x_1$ and $x_2$ to zero. You end up with the following matrix equation

$$ \left[ \begin{matrix} \sigma_1^2 && \rho_{12}\sigma_1\sigma_2 \\ \rho_{12}\sigma_1\sigma_2 && \sigma_2^2 \end{matrix} \right] \left[ \begin{matrix} x_1\\x_2 \end{matrix} \right] = \left[ \begin{matrix} \rho_{01}\sigma_1 \\ \rho_{o2}\sigma_2 \end{matrix} \right] \sigma_0 x_0$$

which you can solve by inverting the matrix, as long as $\rho_{12}\neq \pm 1$ (ie your hedging assets aren't perfectly correlated or anticorrelated).

As to whether this is sound - well, it depends what you mean by "sound". You are minimizing the total variance of your portfolio, conditional on you knowing the covariance matrix. That's about all you can say. Your "risk model" is a single dimension - you are saying that the only notion of risk that you care about is the total variance.

A full risk evaluation would need a procedure for determining the covariance matrix, and some level of backtesting to determine if your forecast risk after the hedge is a reflection of the true risk you would see if you were to hold this portfolio.