# Solving the "Corner Solution Problem" of Portfolio Optimization

Many portfolio optimization methods (e.g., Markowitz/Modern Portfolio Theory in 1952) face the well-known predicament called the “corner portfolio problem”. When short selling is allowed, they usually give efficient allocation weighting that is highly concentrated in only a few assets in the portfolio. This means that the portfolio is not as diversified as we would like, which makes the optimized portfolio less practically useful.

In [Corvalan, 2005], the author suggests to look for instead an “almost efficient” but “more diversified” portfolio within the close neighborhood of the Mean-Variance (MV) optimal solution. The paper shows that there are many eligible portfolios around the MV optimal solution on the efficient frontier. Specificially, given the MV optimal solution, those “more diversified” portfolios can be computed by relaxing the requirements for the portfolio return $R$ and risk $\sigma$ in an additional optimization problem:

max_{w} D(w) \ \ \textup{s.t.,} \\ \begin{aligned} \sqrt{w' \Sigma w} & \le \sigma^* + \Delta \sigma \\ R^* - \Delta R & \le w'r \\ w' 1 & = 1 \\ w_i & \ge 0 \end{aligned}

where $( \sigma^* , R^* ) = ( \sqrt{w^{*'} \Sigma w^*} , w^{*'} r )$, $w^*$ is the Markowitz MV optimal weights, $\Delta \sigma, \Delta R$ are the relaxation tolerance parameters, and $D(w)$ is a diversification measure for the portfolio (for example, $\sum_i w_i, ln(w_i)$, $\prod_i w_i$). In other words, the new optimization problem looks for a portfolio with the maximal diversification around the optimal solution.

Corvalan’s approach can be extended to create an approximate, sufficiently optimal and well diversified portfolio from the optimal portfolio. The approximate portfolio keeps the constraints from the original optimization problem.

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