Chapter 2

Fundamentals of Probability*

The preceding chapter has shown how a risk manager can characterize the risk of a portfolio using a frequency distribution. This process uses the tools of probability, a mathematical abstraction that constructs the distribution of random variables. These random variables are financial risk factors, such as movements in stock prices, in bond prices, in exchange rates, and in commodity prices. These risk factors are then transformed into profits and losses on the portfolio, which can be described by a probability distribution function.

This chapter reviews the fundamental tools of probability theory for risk managers. Section 2.1 lays out the foundations, characterizing random variables by their probability density and distribution functions. These functions can be described by their principal moments, mean, variance, skewness, and kurtosis. Distributions with multiple variables are described in Section 2.2. Section 2.3 then turns to functions of random variables. Section 2.4 presents some examples of important distribution functions for risk management, including the uniform, normal, lognormal, Student's t, binomial, and Poisson distributions. Finally, Section 2.5 discusses limit distributions, which can be used to characterize the average of independent random variables.

2.1 CHARACTERIZING RANDOM VARIABLES

The classical approach to probability is based on the concept of the random variable (RV). This can be viewed as the outcome from throwing ...