This book is part of the Mastering Mathematical Finance Series, which is described as a series of short books covering the core topics of a Masters programme in mathematical or quantitative finance. The series starts financially from the start, and assumes only undergraduate calculus, linear algebra and elementary probability theory. The book is fifth in the series and, in certain sections, relies on results found in previous books in the series.
The approach in the book is a formal mathematical approach with propositions and theorems. There are a number of exercises and examples, which the student would need to complete, as many form part of the proofs. The examples are mathematical in nature and add to the detailed understanding of the subject.
The book is based on financial risk and starts with the mathematical formulation of the subject following Markowitz. This is one of the two cases where the authors make a reference to an outside work, as the course is intended to be self-contained. Chapter 1 introduces in mathematical terms the key concepts of risk and return, and indicates some shortfalls in the use of variance as a risk measure. Chapter 2 looks at a portfolio with two assets, bringing in the correlation function and attainable sets. The authors look at short selling, adding a risk-free security and indifference curves. The next chapter is entitled “Lagrange multipliers”. The purpose of this is to extend the scenarios set in Chapter 2 to portfolios of more than two assets. This extension is found in Chapter 4. Lagrangians are extremely useful tools and the chapter is a little short, giving many of the basic results without practical uses, but this may be because of the need to concentrate on Modern Portfolio Theory.
Chapter 4 deals with extension to multiple assets, with the market portfolio, the capital market line and the definition of risk premium. At this stage, it is worth noting that the portfolios may not be “investments” but could be portfolios of insurance liabilities. This often gives an insight into the risk/reward of an insurance operation with constraints as to minimal portfolios met by the Lagrangian, and selling represented by reinsurance.
Chapter 5 deals with the Capital Asset Pricing Model (CAPM), which is derived mathematically by a theorem. Again this chapter is short.
The books gives a good basic background in the subject. The authors then proceed to three important developments, namely utility theory, value at risk (VaR) and coherent risk measures. Owing to the limited number of references, the reader is not actively encouraged to explore beyond the basics, yet these are the areas where a significant volume of new thoughts are being developed.
Utility theory is set out in axiomatic format. An important conclusion is that the CAPM theorem is an approximation for the optimal portfolio choice for arbitrary utility functions. The section closes with review of risk aversion. Owing to the mathematical nature of the book, few real-life examples are given, for example, why should an investor have exponential or logarithmic utility functions.
In the VaR chapter, we are given, in the introduction, a reason why VaR was first used, namely following the collapse of Barings Bank and Orange County, as symmetric risk measures such as variance did not necessarily focus on the downside. Real-life examples such as this would have helped the reader understand the earlier sections. The majority of this section concentrates on VaR, as it relates to the Black–Scholes model. The results of this section are of value both to investments, and could be utilised to assess the VaR of complex insurance-based investments. Again, the limited practical examples in the book are a deficiency.
The final chapter is on coherent risk measures. In the early 1990s, Professor Artzner put forward a series of four axioms, which he felt a risk measure should satisfy, and labelled those that did coherent risk measures. Unfortunately, VaR did not satisfy these axioms, though other measures did, including the spectral risk measure, which is also analysed here.
In general, the book is a strong mathematical introduction to the subject and is the basis for a good foundation course. However, its strong mathematical basis may put many readers off, the results cover development in the theory to 1999 (or thereabouts). As the results are mathematical, there are very few examples that relate directly to the real world, such as how the recent crashes affected the theory. There are no references for further reading, which again suggests that the book should be viewed as a self-contained academic course. The models are all treated in a static (or smooth) environment when portfolio management and risk are dynamic, and often involve radical changes in perception and a Bayesian approach with entropic risk measures. Superfast computers and “Flash Boy” trading has meant that in the real world some of the techniques described in the book have been overtaken. Having said this, the book is a good starting point to learn and understand the underlying theory, which is essential before one can progress further and understand more recent developments. In addition, portfolios need not be restricted to investments, as the theory was being applied to insurance portfolios to help management understand their business mix better.
The book is also available in ebook format, which makes it easier to use as a reference book if the reader wishes to concentrate on certain sections, and to quickly back reference for definitions and theorems.
