Investment Risk and Uncertainty: Advanced Risk Awareness Techniques for the Intelligent Investor

CHAPTER 14

Portfolio Optimization

Sebastian Ceria, PhD, and Kartik Sivaramakrishnan, PhD

Every portfolio manager faces the challenge of building portfolios that achieve an optimal trade-off between risk and return. Harry Markowitz1 was the first to develop a mathematical framework to solve this problem in the 1950s. The Markowitz model, as it is known in the literature, considers the first two moments of the asset returns, namely the mean and the variance, to measure the return and the risk of the portfolio, respectively. The model is known as Markowitz’s MVO (mean-variance optimization) in the financial world.

Let the weight of an asset in a portfolio be the proportion of the total funds invested in this asset. The portfolio return is modeled as a linear function of the weights, representing the portfolio’s expected return; and the portfolio risk is modeled as a quadratic function in the weights, representing the variance of the portfolio. The trade-off between risk and return is obtained by solving a simple quadratic program (QP) of the form:

(14.1)

where α is the vector of expected returns, Q is the covariance matrix of returns, and λ > 0 is the portfolio manager’s risk aversion parameter that represents the investor’s preference as how to trade-off risk and return. The solution to the QP determines the asset weights in an efficient portfolio—the portfolio with the minimum risk ...