Quantitative Financial Risk Management: Theory and Practice

CHAPTER 7

Portfolio Optimization: Theory and Practice

William T. Ziemba

Alumni Professor of Financial Modeling andStochastic Optimization (Emeritus),University of British Columbia

Distinguished Visiting Research AssociateSystemic Risk Center, London School of Economics

STATIC PORTFOLIO THEORY

In the static portfolio theory case, suppose there are n assets, i = 1, …, n, with random returns ξ₁, …, ξ_n. The return on asset i, namely ξ_i, is the capital appreciation plus dividends in the next investment period such as monthly, quarterly, or yearly or some other time period. The n assets have the distribution F(ξ₁, …, ξ_n) with known mean vector ξ = (ξ₁, …, ξ_n) and known n × n variance-covariance matrix Σ with typical covariance σ_ij for i ≠ j and variance for i = j. A basic assumption (relaxed in section 6) is that the return distributions are independent of the asset weight choices, so F ≠ φ(x).

A mean-variance frontier is

where e is a vector of ones, x = (x₁, …, x_n) are the asset weights, K represents other constraints on the x, and w₀ is the investor's initial wealth.

When variance is parameterized with δ > 0, it yields a concave curve, as in Figure 7.1(a). This is a Markowitz (1952, 1987, 2006) mean-variance efficient frontier and optimally trades off mean, which is desirable, with variance, ...