For a certain class of junk bonds, the probability of default in a given year is 0.2. Whether one bond defaults is independent of whether another bond defaults. For a portfolio of five of these junk bonds, what is the probability that zero or one bond of the five defaults in the year ahead?
A) 0.4096. B) 0.0819. C) 0.7373. The outcome follows a binomial distribution where n = 5 and p = 0.2. In this case p(0) = 0.85 = 0.3277 and p(1) = 5 × 0.84 × 0.2 = 0.4096, so P(X=0 or X=1) = 0.3277 + 0.4096. I answered A because I did not multiply the second part of the equation by 5. Why must you do this? Thanks!
You are multiplying by 5 because there are 5 different ways that one out of the 5 bonds defaults (either the 1st one, or the 2nd one, or the 3rd one…). As the solution pointed out, you are dealing with a binomial distribution where two things can happen (default or no default) and each individual event is independent from the other events.
The formula for the binomial distribution is: n!/r!(n-r)! * p^® (1-p)^(n-r)
For the first case, there is only one way that no bond defaults (you pick 5 out of 5), for the second case, there are 5 ways.
Binomial Formula can be found in the curriculum in:
Reading 8, Chapter 4.8