Two-Fund Separation under Model Mis-Specification

The two-fund separation theorem tells us that an investor with quadratic utility can separate her asset allocation decision into two steps: First, find the tangency portfolio (TP), i.e., the portfolio of risky assets that maximizes the Sharpe ratio (SR); and then, decide on the mix of the TP and the risk-free asset, depending on the investor’s attitude toward risk. In this paper, we describe an extension of the two-fund separation theorem that takes into account uncertainty in the model parameters (i.e., the expected return vector and covariance of asset returns) and uncertainty aversion of investors. The extension tells us that when the uncertainty model is convex, an investor with quadratic utility and uncertainty aversion can separate her investment problem into two steps: First, find the portfolio of risky assets that maximizes the worst-case SR (over all possible asset return statistics); and then, decide on the mix of this risky portfolio and the risk-free asset, depending on the investor’s attitude toward risk. The risky portfolio is the TP corresponding to the least favorable asset return statistics, with portfolio weights chosen optimally. We will show that the least favorable statistics (and the associated TP) can be found efficiently by solving a convex optimization problem.