Conditional Distribution in Portfolio Theory

We present a new method to obtain a conditional mean vector and a conditional covariance matrix when given an investor's view about return profiles of certain assets. The method extends earlier results that were limited to the conditional mean. The new method allows an investor to express views on return means, volatilities, and correlations. An application of our results illustrates how a single anticipated volatility shock spreads to other assets and increases the correlation coefficients among assets. Another application shows how a flight-to-quality event affects volatilities and correlations. Based on the conditional mean and covariance matrix, we then derive analytically an optimal mean–variance portfolio and discuss its implications for asset allocation. Asset allocation managers in search of diversification have a difficult time constructing portfolios based on mean–variance optimization. The reason is that the outcome of a mean–variance optimization is very sensitive to its inputs. Given a covariance matrix and a risk-aversion factor, a set of seemingly reasonable forecasts often gives rise to a portfolio that is overwhelmingly concentrated in a few assets. This conflict between diversification and mean–variance optimization needs to be resolved.Current methodology that resolves the conflict first chooses implied equilibrium expected returns (extracted inversely from observed market value weights) as a reference point. The investor must express views on a few assets relative to this reference. Conditional distribution theory is then used to adjust the entire mean vector to reflect the active views. This conditional adjustment is crucial; without it, small differences between the equilibrium expected returns and the investor's forecasts again result in nondiversified portfolios. By adjusting the mean vector, this method uses the mean–variance optimization to obtain portfolios that not only reflect investor expectations but also provide a diversified mix of assets.The limitation of this method is that it deals only with active views on the expected returns. In practice, some investors also have opinions on the volatilities of certain assets and correlations among assets. So, conditional adjustment to the covariance matrix is as important as conditional adjustment to the mean vector.We present a theoretical framework that unifies conditional adjustments for the mean vector and the covariance matrix that is markedly different from the current Bayesian approach. We first derive the multivariate regression relationship implied by conditional distributions; we incorporate the fact that an investor has active views on certain assets. We assume that this regression relationship remains valid when the returns of certain assets change from the equilibrium to the distribution dictated by the investor's view. On the basis of this assumption, we deduce the expected returns and covariance matrix for all assets that are consistent with the investor's view.Our result for the conditional mean vector is equivalent to that of the current method, but the result for the conditional covariance matrix is new. And it is found to be consistent with observed market dynamics. When a volatility shock is anticipated in one asset, our result indicates increases in the volatilities of all assets and increases in correlations between the one asset and other assets.