All investment decisions are made in an environment of risk. The tools that allow us to make decisions with consistency and logic in this setting come under the heading of probability. This reading presents the essential probability tools needed to frame and address many real- world problems involving risk. We illustrate how these tools apply to such issues as predicting investment manager performance, forecasting financial variables, and pricing bonds so that they fairly compensate bondholders for default
PROBABILITY, EXPECTED VALUE, AND VARIANCE
The probability concepts and tools necessary for most of an analyst’s work are relatively few and simple but require thought to apply. This section presents the essentials for working with probability, expectation, and variance, drawing on examples from equity and fixed income analysis.
An investor’s concerns center on returns. The return on a risky asset is an example of a random variable, a quantity whose outcomes (possible values) are uncertain. For example, a portfolio may have a return objective of 10 percent a year. The portfolio manager’s focus at the moment may be on the likelihood of earning a return that is less than 10 percent over the next year. Ten percent is a particular value or outcome of the random variable “portfolio return.” Although we may be concerned about a single outcome, frequently our interest may be in a set of outcomes: The concept of “event” covers both.
We may specify an event to be a single outcome—for example, the portfolio earns a return of 10 percent. (We use italics to highlight statements that define events.) We can capture the portfolio manager’s concerns by defining the event as the portfolio earns a return below 10 percent. This second event, referring as it does to all possible
returns greater than or equal to −100 percent (the worst possible return) but less than 10 percent, contains an infinite number of outcomes. To save words, it is common to use a capital letter in italics to represent a defined event. We could define A = the portfolio earns a return of 10 percent and B = the portfolio earns a return below 10 percent.
To return to the portfolio manager’s concern, how likely is it that the portfolio will earn a return below 10 percent?
The answer to this question is a probability: a number between 0 and 1 that measures the chance that a stated event will occur. If the probability is 0.40 that the portfolio earns a return below 10 percent, there is a 40 percent chance of that event happening. If an event is impossible, it has a probability of 0. If an event is certain to happen, it has a probability of 1. If an event is impossible or a sure thing, it is not random at all. So, 0 and 1 bracket all the possible values of a probability.