Classic Mean-Variance Optimization

Abstract This chapter describes in relatively simple terms some of the essential technical issues that characterize MV optimization and portfolio efficiency. For the sake of compact discussion, the introduction of some basic assumptions and mathematical notation will be useful. An example of an asset allocation optimization illustrates the techniques presented here and throughout the text. Suppose estimates of expected returns, variances or standard deviations, and correlations for a universe of assets. The expected return, µ(mu), of a portfolio of assets P, µ is the portfolio-weighted expected return for each asset. The variance σ2 (sigma squared) of a portfolio of assets P, σP2, depends on the portfolio weights, the variance of the assets in the portfolio, and the correlation, Ρ (rho), between pairs of assets.3 The standard deviation σ is the square root of the variance and is a useful alternative for describing asset risk. One reason for preferring the standard deviation to the variance is that it is in the same units of return as the mean. Exhibit 2.1 shows the mean and standard deviation for a portfolio consisting of two assets. It illustrates some essential properties of portfolio expected return and risk. Asset 1 has an expected return of 5% and risk of 10%, and asset 2 has an expected return of 10% and risk of 20%. Five curves connect the two assets and display the risk and expected return of portfolios, ranging from 100% of capital in asset 1 to 100% in asset 2. The asset correlations associated with the five curves (from right to left) are 1.0, 0.5, 0, –0.5, and –1.0. The five curves illustrate how correlations and portfolio weights affect portfolio risk and expected return. When the correlation is 1, as in the extreme right-hand curve in the exhibit, portfolio risk and expected return is a weighted average of the risk and return of the two assets.