Risk in Perspective

In this chapter we provide a non-mathematical discussion of various issues that form the background to the rest of the book. In Section 1.1 we begin with the nature of risk itself and discuss how risk relates to randomness; in the financial context (which includes insurance) we summarize the main kinds of risks encountered and explain what it means to measure and manage such risks.

A brief history of financial risk management and the development of financial regulation is given in Section 1.2, while Section 1.3 contains a summary of the regulatory framework in the financial and insurance industries.

In Section 1.4 we take a step back and attempt to address the fundamental question of why we might want to measure and manage risk at all. Finally, in Section 1.5 we turn to quantitative risk management (QRM) explicitly and set out our own views concerning the nature of this discipline and the challenge it poses. This section in particular should give more insight into our choice of methodological topics in the rest of the book.

1.1 Risk

The Concise Oxford English Dictionary defines risk as "hazard, a chance of bad consequences, loss or exposure to mischance". In a discussion with students taking a course on financial risk management, ingredients that are typically discussed are events, decisions, consequences and uncertainty. It is mostly only the downside of risk that is mentioned, rarely a possible upside, i.e. the potential for a gain. While for many people risk has largely negative connotations, it may also represent an opportunity. Much of the financial industry would not exist were it not for the presence of financial risk and the opportunities afforded to companies that are able to create products and services that offer more financial certainty to their clients.

For financial risks no single one-sentence definition of risk is entirely satisfactory. Depending on context, one might arrive at notions such as "any event or action that may adversely affect an organization's ability to achieve its objectives and execute its strategies" or, alternatively, "the quantifiable likelihood of loss or less-than-expected returns".

1.1.1 Risk and Randomness

Regardless of context, risk strongly relates to uncertainty, and hence to the notion of randomness. Randomness has eluded a clear, workable definition for many centuries; it was not until 1933 that the Russian mathematician A. N. Kolmogorov gave an axiomatic definition of randomness and probability (see Kolmogorov 1933). This definition and its accompanying theory provide the language for the majority of the literature on risk, including this book.

Our reliance on probability may seem unsatisfactorily narrow to some. It bypasses several of the current debates on risk and uncertainty (Frank Knight), the writings on probabilistic thinking within economics (John Maynard Keynes), the unpredictability of unprecedented financial shocks, often referred to as Black Swans (Nassim Taleb), or even the more political expression of the known, the unknown and the unknowable (Donald Rumsfeld); see the Notes and Comments section for more explanation. Although these debates are interesting and important, at some point clear definitions and arguments are called for and this is where mathematics as a language enters. The formalism of Kolmogorov, while not the only possible approach, is a tried-and-tested framework for mathematical reasoning about risk.

In Kolmogorov's language a probabilistic model is described by a triplet (Ω, F, P). An element ω of Ω represents a realization of an experiment, in economics often referred to as a state of nature. The statement "the probability that an event A occurs" is denoted (and in Kolmogorov's axiomatic system defined) as P(A), where A is an element of F, the set of all events. P denotes the probability measure. For the less mathematically trained reader it suffices to accept that Kolmogorov's system translates our intuition about randomness into a concise, axiomatic language and clear rules.

Consider the following examples: an investor who holds stock in a particular company; an insurance company that has sold an insurance policy; an individual who decides to convert a fixed-rate mortgage into a variable one. All of these situations have something important in common: the investor holds today an asset with an uncertain future value. This is very clear in the case of the stock. For the insurance company, the policy sold may or may not be triggered by the underlying event covered. In the case of a mortgage, our decision today to enter into this refinancing agreement will change (for better or for worse) the future repayments. So randomness plays a crucial role in the valuation of current products held by the investor, the insurance company and the home owner.

To model these situations a mathematician would now define the value of a risky position X to be a function on the probability space Ω, F, P; this function is called a random variable. We leave for the moment the range of X (i.e. its possible values) unspecified. Most of the modelling of a risky position X concerns its distribution function FX(x) = P(X ≤ x): the probability that by the end of the period under consideration the value of the risk X is less than or equal to a given number x. Several risky positions would then be denoted by a random vector (X1, ..., Xd), also written in bold face as X; time can be introduced, leading to the notion of random (or so-called stochastic) processes, usually written (Xt). Throughout this book we will encounter many such processes, which serve as essential building blocks in the mathematical description of risk.

1.1. Risk

We therefore expect the reader to be at ease with basic notation, terminology and results from elementary probability and statistics, the branch of mathematics dealing with stochastic models and their application to the real world. The word "stochastic" is derived from the Greek "stochazesthai", the art of guessing, or "stochastikos", meaning skilled at aiming ("stochos" being a target). In discussing stochastic methods for risk management we hope to emphasize the skill aspect rather...