In this fifth volume of the series ‘Mastering Mathematical Finance’ we present a self-contained rigorous account of mean-variance portfolio theory, as well as a simple introduction to utility functions and modern risk measures.

Portfolio theory, exploring the optimal allocation of wealth among different assets in an investment portfolio, based on the twin objectives of maximising return while minimising risk, owes its mathematical formulation to the work of Harry Markowitz in 1952; for which he was awarded the Nobel Prize in Economics in 1990. Mean-variance analysis has held sway for more than half a century, and forms part of the core curriculum in financial economics and business studies. In these settings mathematical rigour may suffer at times, and our aim is to provide a carefully motivated treatment of the mathematical background and content of the theory, assuming only basic calculus and linear algebra as prerequisites.

Chapter 1 provides a brief review of the key concepts of return and risk, while noting some defects of variance as a risk measure. Considering a portfolio with only two risky assets, we show in Chapter 2 how the minimum variance portfolio, minimum variance line, market portfolio and capital market line may be found by elementary calculus methods. Chapter 3 contains a careful account of the method of Lagrange multipliers, including a discussion of sufficient conditions for extrema in the special case of quadratic forms. These techniques are applied in Chapter 4 to generalise the formulae obtained for two-asset portfolios to the general case.