Introduction

In the realm of financial risk management, understanding the relationship between risk and return is paramount. The mean-variance framework provides a foundational approach to analyzing the trade-off between these two critical factors. This chapter delves into the mean-variance framework and its application through the concept of the efficient frontier. By comprehending these principles, you’ll be equipped with a powerful tool for optimizing investment portfolios while considering risk and return.

Mean-Variance Framework

The mean-variance framework is a cornerstone concept in modern portfolio theory (MPT), introduced by Harry Markowitz. It establishes a mathematical approach to constructing efficient portfolios that maximize expected returns for a given level of risk or minimize risk for a given level of expected returns. Central to this framework is the understanding that investors seek to strike a balance between achieving higher returns and managing risk effectively.

Expected Return and Variance

The mean-variance framework involves two essential statistical measures: expected return and variance. The expected return of an asset or portfolio represents the weighted average of potential outcomes, each multiplied by its respective probability. Mathematically, it is expressed as:

$$E(R)=\sum_{i=1}^n P_i \times R_i$$

Where $E(R)$ is the expected return, $P_i$ is the probability of outcome $i$, and $R_i$ is the return associated with outcome $i$.

The variance measures the dispersion of returns around the expected return. A lower variance indicates less volatility and, consequently, lower risk. Mathematically, it is computed as:

$$\operatorname{Var}(R)=\sum_{i=1}^n P_i \times\left(R_i-E(R)\right)^2$$

Where $\operatorname{Var}(R)$ is the variance, $P_i$ is the probability of outcome $i, R_i$ is the return associated with outcome $i$, and $E(R)$ is the expected return.

Efficient Frontier

Building upon the mean-variance framework, the efficient frontier illustrates the optimal set of portfolios that offer the highest expected return for a given level of risk or the lowest risk for a given level of expected return. By diversifying assets with varying degrees of risk and return, investors can position themselves on the efficient frontier to maximize their returns while minimizing risk.