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Portfolio 5: Maximum Risk-Adjusted Return

We solved this portfolio previously but we did not examine its properties. We shall do that now. The objective is to find the portfolio that maximizes risk-adjusted return, that is,

The first term, w′r, is the portfolio return, while w′Vw is the variance of the returns on the portfolio w. This is particularly straightforward to solve by differentiation with respect to w′:

Solving this first order condition yields the portfolio:

Let's examine the intuition here. If we revert to the two-asset case that we have been studying, then we would write the objective function as follows:

This has two first order conditions:

Simplifying, and rewriting in matrix format, we get the following two-equation system:

The covariance matrix is diagonal in our preceding ...