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Discrete Random Variables

The concept of probability is central to risk management. Many concepts associated with probability are deceptively simple. The basics are easy, but there are many potential pitfalls.

In this chapter, we will be working with both discrete and continuous random variables. Discrete random variables can take on only a countable number of values – for example, a coin, which can only be heads or tails, or a bond, which can only have one of several letter ratings (AAA, AA, A, BBB, etc.). Assume we have a discrete random variable X, which can take various values, x_i. Further assume that the probability of any given x_i occurring is p_i. We write:

(3.16)

where P[ ·] is our probability operator.

An important property of a random variable is that the sum of all the probabilities must equal one. In other words, the probability of any event occurring must equal one. Something has to happen. Using our current notation, we have:

(3.17)

Continuous Random Variables

In contrast to a discrete random variable, a continuous random variable can take on any value within a given range. A good example of a continuous random variable is the return of a stock index. If the level of the index can be any real number between zero and infinity, then the return of the index can be any real number greater than –1.

Even if the range that the continuous variable occupies is finite, the number of values that it can take is infinite. For this reason, for a continuous variable, the probability of any specific value occurring is zero.

Even though we cannot talk about the probability of a specific value occurring, we can talk about the probability of a variable being within a certain range. Take, for example, the return on a stock market index over the next year. We can talk about the probability of the index return being between 6 percent and 7 percent, but talking about the probability of the return being exactly 6.001 percent or exactly 6.002 percent is meaningless. Even between 6.001 percent and 6.002 percent there are literally an infinite number of possible values. The probability of any one of those infinite values occurring is zero.

For a continuous random variable X, then, we can write:

(3.18)

which states that the probability of our random variable, X, being between r₁ and r₂ is equal to p.

Mutually Exclusive Events

For a given random variable, the probability of any of two mutually exclusive events occurring is just the sum of their individual probabilities. In statistics notation, we can write:

(3.19)

where A∪B is the union of A and B. This is the probability of either A or B occurring. This is true only of mutually exclusive events.

This is a very simple rule, but, as mentioned previously, probability can be deceptively simple, and this property is easy to confuse. The confusion stems from the fact that and is synonymous with addition. If you say it this way, then the probability that A or B occurs is equal to the probability of A and the probability of B. It is not terribly difficult, but you can see where this could lead to a mistake.

This property of mutually exclusive events can be extended to any number of events. The probability that any of n mutually exclusive events occurs is simply the sum of the probabilities of those n events.

Sample Problem

Question:

Calculate the probability that a stock return is either below –10 percent or above 10 percent, given:

Answer:

Note that the two events are mutually exclusive; the return cannot be below –10 percent and above 10 percent at the same time. The answer is: 14 percent + 17 percent = 31 percent.

Independent Events

In the preceding example, we were talking about one random variable and two mutually exclusive events, but what happens when we have more than one random variable? What is the probability that it rains tomorrow and the return on stock XYZ is greater than 5 percent? The answer depends crucially on whether the two random variables influence each other or not. If the outcome of one random variable is not influenced by the outcome of the other random variable, then we say those variables are independent. If stock market returns are independent of the weather, then the stock market should be just as likely to be up on rainy days as it is on sunny days.

Assuming that the stock market and the weather are independent random variables, then the probability of the market being up and rain is just the product of the probabilities of the two events occurring individually. We can write this as follows:

(3.20)

We often refer to the probability of two events occurring together as their joint probability.

Sample Problem

Question:

According to the most recent weather forecast, there is a 20 percent chance of rain tomorrow. The probability that stock XYZ returns more than 5 percent on any given day is 40 percent. The two events are independent. What is the probability that it rains and stock XYZ returns more than 5 percent tomorrow?

Answer:

Since the two events are independent, the probability that it rains and stock XYZ returns more than 5 percent is just the product of the two probabilities. The answer is: 20 percent × 40 percent = 8 percent.

Probability Matrices

When dealing with the joint probabilities of two variables, it is often convenient to summarize the various probabilities in a probability matrix or probability table. For example, pretend we are investigating a company that has issued both bonds and stock. The bonds can either be downgraded, be upgraded, or have no change in rating. The stock can either outperform the market or underperform the market.

In Table 3.1, the probability of both the company's stock outperforming the market and the bonds being upgraded is 15 percent. Similarly, the probability of the stock underperforming the market and the bonds having no change in rating is 25 percent. We can also see the unconditional probabilities, by adding across a row or down a column. The probability of the bonds being upgraded, irrespective of the stock's performance, is: 15 percent + 5 percent = 20 percent. Similarly, the probability of the equity outperforming the market is: 15 percent + 30 percent + 5 percent = 50 percent. Importantly, all of the joint probabilities add to 100 percent. Given all the possible events, one of them must happen.

TABLE 3.1

Sample Problem

Question:

You are investigating a second company. As with our previous example, the company has issued both bonds and stock. The bonds can either be downgraded, be upgraded, or have no change in rating. The stock can either outperform the market or underperform the market. You are given the following probability matrix, which is missing three probabilities: X, Y, and Z. Calculate values for the missing probabilities.

Answer:

All of the values in the first column must add to 50 percent, the probability of the equity outperforming the market; therefore, we have:

We can check our answer for X by summing across the third row: 5 percent + 30 percent = 35 percent.

Looking down the second column, we see that Y is equal to 20 percent:

Finally, knowing that Y = 20 percent, we can sum across the second row to get Z: