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?
20% chance of default. why do we use 0.80(chance of not default)??
below is the answer.
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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.
The probability of default for a single given bond is 0.2, so the probability that it will pay off is 0.8. Zero defaults is equivalent to all 5 paying off, which has a probability of 0.8^5. The probability that ALL the bonds default is 0.2^5.
We can find the probability of event xyz happening in two ways. First is the direct way, you calculate the probabilty of xyz happening directly and second is you find the probability of “xyz not happening” and subtract it from 1. The second method uses the complement rule.
Probabilities of all mutually and exhaustive outcomes must sum up to 1. Since happening and not happening are mutually exclusive events (both can’t happen at the same time) and exhaustive event (there’s no third outcome: either the event will happen or it will not), there probabilites must sum up to 1.
The binomial distribution applies to Bernoulli trials. The characteristics the binomial distribution and Bernoulli trials are:
Each trial has two possible outcomes: success or failure
The probability of success on each trial is a constant (thus, the trials are statistically independent)
The outcome for which we are computing the distribution is the number of successes in n trials
In this problem, we’re told that:
The bonds have two outcomes: default, or no default
The probability of any one bond defaulting is the same as the probability of any other bond defaulting: the probability of failure (or success) is a constant
The outcome for which we are computing the distribution is the number of bonds that default in a sample of n bonds: the number of successes in n bonds.
Thus, this problem has all of the characteristics of Bernoulli trials, and the binomial distribution applies.