Financial Economics: A Concise Introduction to Classical and Behavioral Finance, by T. Hens and M. O. Rieger

Financial Economics: A Concise Introduction to Classical and Behavioral Finance, by T. Hens and M. O. Rieger, Springer, Heidelberg (2010), 373 pp., $99.00, 64.99, 74.95, CHF 107.50. ISBN 978-3-540-36148-0.

I found this book to be an interesting and thought-provoking introduction to decision theory and behavioural finance, which is an increasingly prominent part of financial economics. Decision theory studies the properties of preferences for one risky set of outcomes over another, which is obviously a central concern in economic decision-making! If we define 'lottery' to mean a specified set of outcomes and their probabilities, then decision theory asks how should we (and, not quite the same, how do we) decide to prefer one lottery to another?

We need some notation to describe a preference relation: for lotteries A and B, we write A ≻ B (or B ≺ A) if we prefer lottery A to lottery B, A ∼ B if we are indifferent, and A ≽ B if either is true. If A and B are lotteries and 0 ≤ p ≤ 1, then pA + (1 −p)B denotes the combined lottery where with probability p lottery A is played, and with probability 1 − p lottery B is played.

A simple example of a preference relation is given by an expected utility functional U: for a real-valued utility function u, defined on lottery outcomes, define U(A) = E[u(x)], where E denotes the expectation with respect to the probabilities of the lottery A having outcomes x. The choice of utility function determines a familiar kind of preference relation by the rule A ≺ B if and only if U(A) < U(B).

This is a more important example than appears at first glance because it turns out every 'rational' preference relation is given by an expected utility functional. More precisely, Hens and Rieger state the following version of the classic von Neumann–Morgenstern theorem: a preference relation is given by an expected utility functional if and only if it satisfies the following four axioms.

• i. Completeness. For any two lotteries A and B, either A ≺ B, A ∼ B or A ≻ B.
• ii. Transitivity. For every A, B, C, if A ⪯ B and B ⪯ C, then A ⪯ C.
• iii. Independence. If A ≻ B and 0 < p ≤ 1, then for any lottery C,

•

Graph

• iv. Continuity. If A ≽ B ≽ C then there exists p such that B ∼ pA + (1 − p)C.

If we accept these axioms as necessary for a 'rational' preference relation, then Expected Utility Theory (EUT) becomes a complete description of rational preferences. Behavioural finance is born from the observation that EUT is nevertheless not a complete description of the preferences people actually express.

Here is an example from Chapter 2: the 'Allais paradox'. There are four lotteries, A, B, C, D, with payoffs shown in table 1. In each lottery, a random number (the 'state') is drawn from the set {1, 2, 3, ... , 100}, each equally likely, and, depending on the draw, the lottery pays either 0, 2400 or 2500.

Table 1. The Allais paradox lotteries

If you were granted the opportunity to choose to receive the payoff of either A or B, which would you choose? How about the same question for C and D? Hens and Rieger report that it turns out for most people B ≻ A, and also C ≻ D.

We could argue that such preferences are not 'rational' for the following reason. In states 34–99, A and B have the same outcomes, so those states can be safely ignored when trying to distinguish between A and B. The same is true of C and D. However, when states 34–99 are removed, the pair A, B is identical to the pair C, D, and so preferring B to A should imply a corresponding preference of D to C, contrary to what is observed in experiments. To put it more concisely, the preferences exhibited in the Allais paradox do not satisfy the independence axiom above. As a result, they cannot be explained by an expected utility functional.

There are many similar phenomena where we appear to be sometimes risk averse and other times risk seeking, as for example when we buy both insurance and lottery tickets. Behavioural finance seeks a quantitative way to explain such risk preferences.

Hens and Rieger describe a way forward: prospect theory and its variants, pioneered in the Nobel-prize-winning work of Kahneman and Tversky. Prospect theory proposes a utility functional similar to EUT, except that probabilities are adjusted by a weighting function that overweights small probabilities and underweights large ones. This can be worked out in such a way as to explain the Allais paradox. Roughly speaking, the small probability of a zero payoff in lottery A is magnified in significance, increasing the extent to which B is more desirable than A. For lotteries C and D, the probability of zero payoff is of moderate size and so C and D are closer together in preference value. With a well-chosen utility function and probability weightings, this effect is enough to reverse the preferences when moving from the pair A, B to C, D.

Hens and Rieger give a good discussion of all this and aim in their book to introduce prospect theory alongside EUT as they develop the standard equilibrium topics of financial economics that traditionally rest on expected utility. EUT and prospect theory – and its preferred variant Cumulative Prospect Theory (CPT) – are then positioned as the foundations of classical finance and behavioural finance, respectively, the two topics in this book's subtitle. Each of these is also nicely compared to a third topic: Mean Variance Theory (MVT), because of its ubiquity in applications. EUT, CPT and MVT are the three themes of this book.

I learned a lot from the book about how behavioural finance fits into the classical theory, and recommend the book to readers looking for an intuitive and inviting overview of this topic.

Readers like me may appreciate some comments about what this book is not. Mainly what it is not is a reliable place to learn the details of the proofs. In this it is by no means alone. An anecdote: my colleague over in the economics department now teaches beginning graduate financial economics using this book and loves it. Yet he reports that the math students in his class tend to dislike the book. This got me thinking more broadly about the cultural divide between financial mathematicians (like me) and financial economists (like my colleague).

From my mathematician's perspective, a lot of writing on financial economics is frustrating to read. Mathematics is the subject's language, but so often that language is used in a way that works from a distance but is fuzzy up close. The mathematician accustomed to understanding a topic from the bottom up can find the exposition punishes careful scrutiny. At worst, proofs are promised and then not delivered, definitions are omitted or overlooked, notation defined late or not at all. At the opposite extreme, presentations can be overly abstract and untethered from application. Service to a reader who wants to understand the mathematical details as simply as possible (but no simpler!) is often just not a priority, as much as I wish it were.

Some of these comments apply to parts of the book under review. Examples: in the discussion above of the von Neumann–Morgenstern theorem, missing is a statement of the conditions the utility function u must satisfy. I couldn't find this in the book at hand, even in the 'complete proof' promised in Appendix A.6. The proof given there is only a sketch, referring to other papers and invoking 'one can prove that ...' here and there.

Several other times proofs are promised that don't fully materialize. Another favorite topic is the Fundamental Theorem of Asset Prices in Chapter 4. The central term 'arbitrage' is described but not clearly defined. Two proofs are promised: a geometric proof for an easy case, and a general proof of the full result. Yet the geometric proof relies on a figure that is not fully defined, and the general proof covers only an easy case and concludes with 'a proof for the general case can be found in the book by Magill and Quinzii'. (Proofs are rarely wrong, but one apparent exception is the second proof of Lemma 3.1, which seems to rely on the erroneous assertion that a continuous image of a closed set is closed.)

I freely admit that I am not in the target audience for this textbook, and that the authors' priorities are not the same as mine. A standard joke is that reviewers and referees so often complain because a work under review is not the one that the reviewer would have written!

I do think there remains an unmet need to address the many mathematics students motivated to learn financial economics, especially those at the beginning graduate level, who desire an introductory treatment of the definitions and proofs in detail. The cultural divide mentioned above could probably be described as a difference of opinion about the importance of this desire. For these readers, the role of mathematics is to clarify understanding more than simply express or validate. For my part, I prefer to start at the bottom of things (which means internalizing the proofs), and my economist friends freely admit that is not their usual approach. I see this as no cause for pessimism. On the contrary, it signals a tremendous opportunity for further fruitful collaboration across the cultural divide.

Alec N. Kercheval

Department of Mathematics

Florida State University

© 2012, Alec N. Kercheval