Minimum-variance versus tangent portfolios – A note

Abstract

In this study, we analyze three research items found in the literature on the comparative performance of the global minimum-variance to the tangent portfolio. First, some authors assume that the global minimum-variance portfolio does not include average returns, but we show this is not the case. Average returns are used in the calculation of variance-covariances and the global minimum-variance portfolio and the tangent portfolio share the ‘optimal descent’ (return-to-variance) ratio. Second, contrary to what some authors assume, using cross-returns instead of the usual variance-covariance matrix keeps the tangent portfolio unchanged, but yields an inefficient ‘minimum-variance’ portfolio. In general, with the noted exception of the tangent portfolio, using cross-returns instead of variance-covariance matrices deforms the risk matrix in very complex ways. Third, again, contrary to what is stated in the literature, the global minimum-variance portfolio is not only likely to be the most complicated portfolio along the efficient frontier, but it has properties that tend to underestimate the risk taken by the investor. Two important sources of underestimated risk are the lower quality of the securities being held and the changes in the structural stability of the risk-matrix. Both these sources of risk would make the optimal portfolio weights of the ‘global minimum-variance’ portfolios less reliable than those of the tangent portfolio. This possible underestimation of risk, especially in the forgiving climate of a growing stock market, may explain results favoring the global minimum-variance over the tangent portfolio. Another possibility, however, is that alternative conceptualizations of risk that rely less on averaging of returns may actually provide better portfolios.