is the present value of a newly issued 20-year bond, with a notional value of $100, and a 5 percent annual coupon? Assume a constant 4 percent discount rate, and no risk of default.
Answer:
This question utilizes discount factors and finite geometric series.
The bond will pay 20 coupons of $5, starting in a year's time. In addition, the notional value of the bond will be returned with the final coupon payment in 20 years. The present value, V, is then:
We start by evaluating the summation, using a discount factor of δ =
Inserting this result into the initial equation we obtain our final result:
Note that the present value of the bond, $113.59, is greater than the notional value of the bond, $100. In general, if there is no risk of default, and the coupon rate on the bond is higher than the discount rate, then the present value of the bond will be greater than the notional value of the bond.
When the price of a bond is less than the notional value of the bond, we say that the bond is selling at a discount. When the price of the bond is greater than the notional value, as in this example, we say that it is selling at a premium. When the price is exactly the same as the notional value, we say that it is selling at par.
Part II Probabilities
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, xi. Further assume that the probability of any given xi occurring is pi. 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