# A Probability Metrics Approach to Financial Risk Measures.

A Probability Metrics Approach to Financial Risk Measures, by Svetlozar T. Rachev, Stoyan V. Stoyanov, and Frank J. Fabozzi, 2011, Oxford, UK: Wiley Blackwell, 392 pp. ISBN: 978-1-4051-8369-7

In its essence, financial economics is a science of random elements--daily returns of stock prices are modeled as random variables, the price evolution of an exchange rate is modeled as a stochastic process, and the daily movement of the shape of a yield curve is modeled as a more complex random object. While each individual random object has been extensively studied as a mathematical entity, I am unaware of a comprehensive work focusing on the mathematical quantification of the relationship between such random financial elements.

To motivate the idea of "relationship quantification," consider the following problems arising from seemingly disjoint theories of financial economics:

(P1) An investor seeks to find a portfolio that is closest to a given benchmark, where the idea of "closeness" is not necessarily determined by the typical tracking error, the standard deviation, but by some other desired measure of dispersion.

(P2) An investor prefers portfolios X and Y over a third portfolio Z, where preference is with respect to a stochastic dominance relation, and seeks to determine whether X is more strongly preferred to Z than Y is.

(P3) A risk manager is looking for the difference between the probabilities that two stocks X and Y lose more than a prespecified loss level L.

The first is a problem arising in the field of error tracking, the second in economic choice theory, and the third in risk analysis. The common denominator is an underlying notion of distance between entities of a stochastic nature. The solution to each of these problems utilizes the notion of probability metric, a probabilistic object measuring distances between random quantities.

With these problems in mind, the aim of A Probability Metrics Approach to Financial Risk Measures is twofold. On the one hand, it provides an accessible, technically rigorous yet intuitive, introduction to the theory of probability metrics for the nonspecialist. On the other hand, it describes applications in finance, with particular focus on risk, uncertainty, and modern portfolio theory (Markowitz, 1952). In doing so, the authors have given us an in-depth, self-contained treatment of the basics of risk and uncertainty in mathematical finance and economics.

THE THEORY OF PROBABILITY METRICS

A probability metric is an abstract tool enabling the measurement of relationships and differences between random elements irrespective of their precise nature. Generally speaking, a function that measures the distance between random quantities is called a probability metric. More formally, a probability metric calculates the distance between two probability measures. It is defined axiomatically as follows.

Definition 1 (Probability Metric)

Denote by X: = X (R) the set of all random variables on a given probability space ([OMEGA], F, P) taking values in (R, Bt), where B, denotes the Borel algebra of Borel subsets of R, and by L[X.sub.2] the space of all joint distributions [Pr.sub.X,Y] generated by the pairs X, Y [epsilon] X. A mapping

[mu] : L[X.sub.2] [right arrow] [0; [infinity]],

where [mu] (X, Y): = [greater than or equal to] 0, ([Pr.sub.X,Y]), is a probability metric if it satisfies the following axioms:

1. Identity: [mu] (X, Y) [greater than or equal to] 0, and [mu] (X, Y) = 0 iff X ~ Y. (1)

2. Symmetry: [mu] (X, Y) = [mu] (Y, X).

3. Triangle Inequality: [mu] (X, Y) [less than or equal to] [mu] (X, Z) + [mu] (Z, Y) for all X, Y, Z [epsilon] X.

The first two chapters of the book are dedicated to the technical development of the theory of probability metrics and their applicability in finance. As the authors point out, "there are no limitations in the theory of probability metrics concerning the nature of the random quantities. This makes its methods fundamental and appealing" (p. 2). To illustrate their viewpoint, they provide various examples of metrics in probability theory and interpretations from a financial economics perspective. Following this formal development of the theory, the remainder of the book is committed to two particular applications, namely, within the theories of stochastic dominance and financial risk measures.

STOCHASTIC DOMINANCE AND PROBABILITY METRICS

Stochastic dominance, a form of stochastic ordering, arises in decision theory in situations where one gamble, also known as a prospect and formalized as a probability distribution over possible outcomes, can be ranked as superior to another gamble for a broad class of decision makers. The theory of stochastic orders is closely related to expected utility theory, pioneered by the influential work of von Neumann and Morgenstern (1944). It describes how choices under uncertainty are made and represents investor's preferences via a utility function. Essentially, stochastic dominance induces an ordering by comparing probability distributions of random variables. The simplest example of stochastic dominance is state-by-state dominance, where a random variable X on the set is said to dominate the random variable Y, written X [??] Y where [??] is a preference relation (2), if X ([omega]) [greater than or equal to] Y ([omega]) for [omega] [epsilon] [omega].

This means that prospect X gives at least as good a result in every possible set of outcomes. More generally, stochastic dominance is defined by comparing the probability distribution functions of prospects X and Y, formalized as follows.

Definition 2 (Stochastic Dominance)

Denote by [F.sub.X] (x) and [F.sub.Y] (y), the cumulative distribution functions (c.d.f.) of two uncertain prospects X and Y. X is said to dominate Y in the sense of the n-th order stochastic dominance, written X [??] Y, if and only if

[F.sup.(n)] X (X) > [F.sup.(n)] Y (y),

where [F.sup.(n)] x (x) stands for the n-th integral of the c.d.f. of X.

One notably crucial issue is identified by the authors, namely the fact that the stochastic ordering [??] is only qualitative in nature. The statement X [??] n Y gives no indication over whether Y would be categorically disregarded because it is far less preferred to X, or whether the "differences" between X and Y are very small, hence making prospects X and Y "similar" in some sense. Similarly, consider Problem (P2) we introduced at the beginning, where an investor is unable to deduce the magnitude of his preference from a stochastic ordering. The means for describing this lacking quantitative element to preference orders is through probability metrics, which calculate the distance between two prospects with respect to the underlying stochastic order. The task of finding such a suitable metric is far from elementary. In particular, one cannot expect that one probability metric will be suitable for all stochastic orders. The authors discuss these technical issues by giving examples of suitable and unsuitable distances and imposing conditions on probability metrics for them to be practically meaningful.

Five chapters later, the discussion of the relationship between probability metrics and decision theoretic preference relations is revisited. Even though to the reader this choice of chapter ordering appears incoherent, the authors seem to have saved the best for last. We are introduced to the notion of quasi-semidistance, which allows measuring by how much one prospect dominates another, or, in case they are incomparable, allows measuring the degree of violation of the stochastic dominance rule. The authors then make an elegant unification of the three recurring themes of their work: probability metrics, preference orders, and risk measures. In particular, new stochastic dominance rules based on a specific risk measure are explored and a utility-type representation of quasi-semidistances is given. Finally, the rather theoretical stochastic dominance theory is compared to the more practical mean-risk approach: even though stochastic dominance is more general, the authors make the compelling remark that it does not provide a method for portfolio construction from a collection of assets. Once again, a probability metrics framework is outlined as a step toward resolving this issue.

RISK, UNCERTAINTY, AND PROBABILITY METRICS

The second part of the book (Chapters 5 through 7) discusses the topic advertised in its title: applications of probability metrics in the context of financial risk and uncertainty. For the less acquainted, the subject matter is introduced with a conceptual discussion a-la-Knight on the subtle distinction between risk and uncertainty, two concepts that are closely linked but not synonymous. (3)

Mathematically, the rather subjective notion of risk is quantified by means of a risk measure, which is a functional [rho] associating a real number to a portfolio's loss or return distribution. On the other hand, the notion of uncertainty, which in the context of investment risk relates to the probable deviations from expectation, is quantified via a deviation measure, the most classic example being the variance.

Following a list of concrete examples of such risk and deviation measures used in practice, the relationship between dispersion measures and probability metrics is introduced, albeit rather superficially. Even though references to the technical appendix and to the authors' academic work are provided, the exposition appears abrupt, particularly as this is one of the main examples of the link between probability metrics and risk. For example, the stunning result that all deviation measures arise from probability-quasi-metrics is only mentioned in passing. However, the authors make up for this lack in their following exposition, as they proceed to thoroughly describe a neat mathematical formalization of the link between risk and uncertainty in terms of risk and deviation measures. Tying the discussion to the first theme of stochastic orders, the authors conclude this segment by pointing to the importance of the consistency of risk measures with the second-order stochastic dominance order, in particular by addressing the following thought-provoking question: "If all risk-averse investors prefer X to Y, then does it follow that [rho] (X) [less than or equal to] [rho] (Y)?"

The title may indeed imply a confined set of potential readers. It may also indicate that the authors attack a narrow range of problems. By nature of my personal research focus, which essentially spans choice theoretic preference relations and the theory of financial risk measures, coupled with an enthusiastic interest in probability theory that one may deem obsessive, my volunteering to review this work could be construed as a pure self-indulgent pleasure. To the extent that I can be objective, however, I strongly recommend A Probability Metrics Approach to Financial Risk Measures to anyone who (1) seeks a self-contained introduction to the theory of risk and uncertainty in financial economics, (2) seeks an accessible account of the theory of probability metrics, or (3) appreciates the omnipresence of elegant abstract mathematical entities in financial economics. In the spirit of the latter audience, this work may serve as an inspiration to unify other seemingly impractical areas of probability theory with financial economics.

Economists have long realized that economic agents inhabit a universe of randomness and uncertainty. Using an elegant probabilistic tool, the authors help illuminate relationships within this universe. They seamlessly bridge the fields of probability metrics, risk measures, and stochastic dominance. Serious technical learners will be thrilled by the rigorous conciseness of the exposition, while the less technically inclined will welcome the approachable introduction to a highly technical field. Dense with information on every page and presented in formal yet intuitive manner, the authors effectively usher the reader into a realm of an abstract theoretical concept with a surprising number of links to the less abstract and highly applied field of financial economics.

DOI: 10.1111/jori.12157

REFERENCES

Knight, F. H., 1921, Risk, Uncertainty, and Profit (New York: Houghton Mifflin). Markowitz, H. M., 1952, Portfolio Selection, Journal of Finance, 7(1): 77-91.

von Neumann, J., and O. Morgenstern, 1944, Theory of Games and Economic Behavior (Princeton, NJ: Princeton University Press).

Reviewer: Ola Mahmoud, University of St. Gallen and University of California, Berkeley; ola.mahmoud@unisg.ch, olamahmoud@berkeley.edu

(1) The notation X_Y denotes that X is equivalent to Y, where the meaning of equivalence depends on the type of metrics, for example, in almost sure sense, in the sense of equality of distribution, in the sense of equality of some characteristics of X and Y.

(2) A preference relation is a binary relation on a choice set X that is and negative transitive.

(3) Knight (1921) initiated the debate about risk and uncertainty in his seminal work Risk, Uncertainty, and Profit: "Uncertainty must be taken in a sense radically distinct from the familiar notion of risk, from which it has never been properly separated. The essential fact is that risk means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomena depending on which of the two is really present and operating. It will appear that a measurable uncertainty, or risk proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all." (p. 19).
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