# Chapter 3

# Mean-Variance Analysis

Virtue or excellence is a characteristic involving choice, and it consists in observing the mean relative to us, a mean which is defined by a rational principle. It is the mean by reference to two vices: the one of excess and the other of deficiency.

Aristotle, *Nichomachean Ethics, Book II*

## 3.1 Preview

Mean-variance analysis studies the performance of portfolios in the space of reward versus risk in a single-period context. Reward is measured by the portfolio expected (mean) return, and risk is measured by the portfolio variance. In this chapter we develop the classic quadratic programming model to support mean-variance analysis. We discuss variants of the model, addressing limits on trading size, borrowing, transaction costs, and liabilities. We then develop factor models for generating the data required in mean-variance analysis, and show how these models can be used to simplify the quadratic programs. Finally we discuss the sensitivity of mean-variance analysis to noisy or erroneous input data, and introduce ways to alleviate the potential problems.

## 3.2 Mean-Variance Optimization

Balancing rewards against risks is at the heart of financial engineering, and the foundations for this balancing act were laid with the work of Markowitz (1952) on mean-variance analysis. Mean-variance analysis studies the tradeoffs between portfolio reward, as measured by the portfolio expected (mean) return, and portfolio risk, as measured by its variance. Mean-variance ...