Mean-variance portfolio optimization is the core of modern portfolio theory, which was introduced by Harry Markowitz in 1952. The main idea is to construct an optimal portfolio that maximizes the expected return for a given level of risk or minimizes the risk for a given level of expected return. And using variance as a measure of risk.

The mean-variance optimization framework is based on the following assumptions: Investors are rational and risk-averse, meaning they prefer higher returns for a given level of risk. Investors have a one-period investment horizon. No transaction costs or taxes are involved in buying and selling assets. Only the mean and variance of asset returns are relevant for decision-making. (Explained in the first section below)

We will explore the following topics in this section:

  1. From Optimized Utility to Mean-Variance Analysis
  2. Solution to the Mean-Variance Optimization Problem
  3. Efficient Frontier without Risk-Free Asset
  4. Solution to the Mean-Variance Optimization Problem with Risk-Free Asset
  5. Market Portfolio and Security Market Line

This topic will provide a comprehensive understanding of mean-variance portfolio optimization and its applications in finance. It further relates to the CAPM (Capital Asset Pricing Model) and APT (Arbitrage Pricing Theory), which are essential concepts in modern finance.

Note: this section are math focused, since this is a quant blog. We will delve into the math details of the mean-variance analysis.

For more contentes optimization methods, please refer to Optimization Models in Finance.