Refresher Reading

Common Probability Distributions

2021 Curriculum CFA Program Level I Quantitative Methods

Introduction

In nearly all investment decisions we work with random variables. The return on a stock and its earnings per share are familiar examples of random variables. To make probability statements about a random variable, we need to understand its probability distribution. A probability distribution specifies the probabilities of the possible outcomes of a random variable.

In this reading, we present important facts about four probability distributions and their investment uses. These four distributions—the uniform, binomial, normal, and lognormal—are used extensively in investment analysis. They are used in such basic valuation models as the Black–Scholes–Merton option pricing model, the binomial option pricing model, and the capital asset pricing model. With the working knowledge of probability distributions provided in this reading, you will also be better prepared to study and use other quantitative methods such as hypothesis testing, regression analysis, and time-series analysis.

After discussing probability distributions, we end the reading with a brief introduction to Monte Carlo simulation, a computer-based tool for obtaining information on complex problems. For example, an investment analyst may want to experiment with an investment idea without actually implementing it. Or she may need to price a complex option for which no simple pricing formula exists. In these cases and many others, Monte Carlo simulation is an important resource. To conduct a Monte Carlo simulation, the analyst must identify risk factors associated with the problem and specify probability distributions for them. Hence, Monte Carlo simulation is a tool that requires an understanding of probability distributions.

Before we discuss specific probability distributions, we define basic concepts and terms. We then illustrate the operation of these concepts through the simplest distribution, the uniform distribution. That done, we address probability distributions that have more applications in investment work but also greater complexity.

Learning Outcomes

The member should be able to:

define a probability distribution and distinguish between discrete and continuous random variables and their probability functions;
describe the set of possible outcomes of a specified discrete random variable;
interpret a cumulative distribution function;
calculate and interpret probabilities for a random variable, given its cumulative distribution function;
define a discrete uniform random variable, a Bernoulli random variable, and a binomial random variable;
calculate and interpret probabilities given the discrete uniform and the binomial distribution functions;
construct a binomial tree to describe stock price movement;
define the continuous uniform distribution and calculate and interpret probabilities, given a continuous uniform distribution;
explain the key properties of the normal distribution;
distinguish between a univariate and a multivariate distribution and explain the role of correlation in the multivariate normal distribution;
determine the probability that a normally distributed random variable lies inside a given interval;
define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution;
define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy’s safety-first criterion;
explain the relationship between normal and lognormal distributions and why the lognormal distribution is used to model asset prices;
distinguish between discretely and continuously compounded rates of return and calculate and interpret a continuously compounded rate of return, given a specific holding period return;
describe Monte Carlo simulation.

Summary

In this reading, we have presented the most frequently used probability distributions in investment analysis and the Monte Carlo simulation.

A probability distribution specifies the probabilities of the possible outcomes of a random variable.
The two basic types of random variables are discrete random variables and continuous random variables. Discrete random variables take on at most a countable number of possible outcomes that we can list as x ₁, x ₂, … In contrast, we cannot describe the possible outcomes of a continuous random variable Z with a list z ₁, z ₂, … because the outcome (z ₁ + z ₂)/2, not in the list, would always be possible.
The probability function specifies the probability that the random variable will take on a specific value. The probability function is denoted p(x) for a discrete random variable and f(x) for a continuous random variable. For any probability function p(x), 0 ≤ p(x) ≤ 1, and the sum of p(x) over all values of X equals 1.
The cumulative distribution function, denoted F(x) for both continuous and discrete random variables, gives the probability that the random variable is less than or equal to x.
The discrete uniform and the continuous uniform distributions are the distributions of equally likely outcomes.
The binomial random variable is defined as the number of successes in n Bernoulli trials, where the probability of success, p, is constant for all trials and the trials are independent. A Bernoulli trial is an experiment with two outcomes, which can represent success or failure, an up move or a down move, or another binary (two-fold) outcome.
A binomial random variable has an expected value or mean equal to np and variance equal to np(1 − p).
A binomial tree is the graphical representation of a model of asset price dynamics in which, at each period, the asset moves up with probability p or down with probability (1 − p). The binomial tree is a flexible method for modeling asset price movement and is widely used in pricing options.
The normal distribution is a continuous symmetric probability distribution that is completely described by two parameters: its mean, μ, and its variance, σ².
A univariate distribution specifies the probabilities for a single random variable. A multivariate distribution specifies the probabilities for a group of related random variables.
To specify the normal distribution for a portfolio when its component securities are normally distributed, we need the means, standard deviations, and all the distinct pairwise correlations of the securities. When we have those statistics, we have also specified a multivariate normal distribution for the securities.
For a normal random variable, approximately 68 percent of all possible outcomes are within a one standard deviation interval about the mean, approximately 95 percent are within a two standard deviation interval about the mean, and approximately 99 percent are within a three standard deviation interval about the mean.
A normal random variable, X, is standardized using the expression Z = (X − μ)/σ, where μ and σ are the mean and standard deviation of X. Generally, we use the sample mean X ¯ as an estimate of μ and the sample standard deviation s as an estimate of σ in this expression.
The standard normal random variable, denoted Z, has a mean equal to 0 and variance equal to 1. All questions about any normal random variable can be answered by referring to the cumulative distribution function of a standard normal random variable, denoted N(x) or N(z).
Shortfall risk is the risk that portfolio value will fall below some minimum acceptable level over some time horizon.
Roy’s safety-first criterion, addressing shortfall risk, asserts that the optimal portfolio is the one that minimizes the probability that portfolio return falls below a threshold level. According to Roy’s safety-first criterion, if returns are normally distributed, the safety-first optimal portfolio P is the one that maximizes the quantity [E(R_P ) − R_L ]/σ _P , where R_L is the minimum acceptable level of return.
A random variable follows a lognormal distribution if the natural logarithm of the random variable is normally distributed. The lognormal distribution is defined in terms of the mean and variance of its associated normal distribution. The lognormal distribution is bounded below by 0 and skewed to the right (it has a long right tail).
The lognormal distribution is frequently used to model the probability distribution of asset prices because it is bounded below by zero.
Continuous compounding views time as essentially continuous or unbroken; discrete compounding views time as advancing in discrete finite intervals.
The continuously compounded return associated with a holding period is the natural log of 1 plus the holding period return, or equivalently, the natural log of ending price over beginning price.
If continuously compounded returns are normally distributed, asset prices are lognormally distributed. This relationship is used to move back and forth between the distributions for return and price. Because of the central limit theorem, continuously compounded returns need not be normally distributed for asset prices to be reasonably well described by a lognormal distribution.
Monte Carlo simulation involves the use of a computer to represent the operation of a complex financial system. A characteristic feature of Monte Carlo simulation is the generation of a large number of random samples from specified probability distribution(s) to represent the operation of risk in the system. Monte Carlo simulation is used in planning, in financial risk management, and in valuing complex securities. Monte Carlo simulation is a complement to analytical methods but provides only statistical estimates, not exact results.