Quantitative Risk Management: A Practical Guide to Financial Risk, + Website

2.2 Risk Measures

One important consequence of viewing risk as the distribution of future P&L is that risk is multifaceted and cannot be defined as a single number; we need to consider the full distribution of possible outcomes. In practice, however, we will rarely know or use the full P&L distribution. We will usually use summary measures that tell us things about the distribution because the full distribution is too difficult to measure or too complicated to easily grasp or because we simply want a convenient way to summarize the distribution.

These summary measures can be called risk measures: numbers that summarize important characteristics of the distribution (risk). The first or most important characteristic to summarize is the dispersion, or spread, of the distribution. The standard deviation is the best-known summary measure for the spread of a distribution, and it is an incredibly valuable risk measure. (Although it sometimes does not get the recognition it deserves from theorists, it is widely used in practice.) But plenty of other measures tell us about the spread, the shape, or other specific characteristics of the distribution.

Summary measures for distribution and density functions are common in statistics. For any distribution, the first two features that are of interest are location, on the one hand, and scale (or dispersion), on the other. Location quantifies the central tendency of some typical value, and scale or dispersion quantifies the spread of possible values ...