Portfolio Optimization with Correlation Matrices: How, Why, and Why Not

Correlation is used frequently both in the classroom and in professional environments to illustrate and summarize investment know-how, especially with regard to diversification. Pedagogically, the initial build-up on correlation, which reaches its climax while describing a hypothetical two-variable optimization case, abruptly disappears when the discussion reaches optimizations of several securities, thereby stopping short of running a full-fledged, correlation-based optimization. Why is that so? We offer some explanations. First, correlations initially seem to provide clarification of the workings of the optimization, specifically with respect to how security risk-relations affect optimal weights. However, the variable transformation required changes coordinates, thus making correlation-based optimal weights and the desired information hard to understand. Second, correlation-based optimizations may be counterproductive. Nobody with a minimum of financial sophistication would try to make up covariance estimates; correlations, however, are easy to make up, which may make one overstate their practical value. Third, while mean-variance optimal weights can be easily constructed from correlation-based optimal numbers, not transforming the optimal numbers back to the mean-variance values deforms the information processed. We do not expect correlation-based optimizations to replace mean-variance ones except in specialized cases (e.g., small portfolios where investors may have extra-knowledge of security relationships).