FXP Stock Trading

EXAMPLE 56 The Expected Number of Defaults in a Bond Portfolio

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Suppose as a bond analyst you are asked to estimate the number of bond issues expected to default over the next year in an unmanaged high-yield bond portfolio with 25 U.S. issues from distinct issuers. The credit ratings of the bonds in the portfolio are tightly clustered around Moody's B2/Standard & Poor's B, meaning that the bonds are speculative with respect to the capacity to pay interest and repay principal. The estimated annual default rate for B2/B rated bonds is 10.7 percent.

1. Over the next year, what is the expected number of defaults in the portfolio, assuming a binomial model for defaults?

2. Estimate the standard deviation of the number of defaults over the coming year.

3. Critique the use of the binomial probability model in this context.

Solution to 1. For each bond, we can define a Bernoulli random variable equal to 1 if the bond defaults during the year and zero otherwise. With 25 bonds, the expected number of defaults over the year is np = 25(0.107) = 2.675 or approximately 3.

Solution to 2. The variance is np( 1 - p) = 25(0.107)(0.893) = 2.388775. The standard deviation is (2.388775)1'2 = 1.55. Thus a two standard deviation confidence interval about the expected number of defaults would run from approximately 0 to approximately 6, for example.

Solution to 3. An assumption of the binomial model is that the trials are independent. In this context, a trial relates to whether an individual bond issue will default over the next year. Because the issuing companies probably share exposure to common economic factors, the trials may not be independent. Nevertheless, for a quick estimate of the expected number of defaults, the binomial model may be adequate.

10 We can show that p( 1 - p) is the variance of a Bernoulli random variable as follows, noting that a Bernoulli random variable can take on only one of two values, 1 or 0: <t2(Y) = E[(Y - EY)2] = E[{Y - pf ] = (1 - pfp + (0 - pf{ 1 - p) = (1 - p)[(l ~ p)p + p2] = p(l - p).

Earlier, we looked at a simple one-period model for stock price movement. Now we extend the model to describe stock price movement on three consecutive days. Each day is an independent trial. The stock moves up with constant probability p (the up transition probability); if it moves up, u is 1 plus the rate of return for an up move. The stock moves down with constant probability 1 - p (the down transition probability); if it moves down, d is 1 plus the rate of return for a down move. We graph stock price movement in Figure 5-2, where we now associate each of the n = 3 stock price moves with time indexed by t. The shape of the graph suggests why it is a called a binomial tree. Each boxed value from which successive moves or outcomes branch in the tree is called a node; in this example, a node is potential value for the stock price at a specified time.