5. Binomial Distribution & Discrete Random Variables
Poisson Distribution
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A financial analyst is assessing the risk of credit defaults in a large bond portfolio. The portfolio contains 2,000 corporate bonds, & the probability of any one bond defaulting in a year is 0.002.
(C) The analyst considers any probabilities less than 5% to be significant. If more than 5 bonds default in a year, should the analyst be concerned?
A
Should be concerned
B
Should not be concerned
Verified step by step guidance1
Step 1: Identify the type of probability distribution applicable to the problem. Since we are dealing with a large number of trials (2,000 bonds) and a small probability of success (defaulting, p = 0.002), this is a Binomial distribution. However, because the number of trials is large and p is small, we can approximate it using a Poisson distribution with λ = n * p, where n is the number of trials and p is the probability of success.
Step 2: Calculate the mean (λ) of the Poisson distribution. Using the formula λ = n * p, substitute n = 2000 and p = 0.002. This will give you the expected number of defaults in a year.
Step 3: Determine the probability of more than 5 defaults. In a Poisson distribution, the probability of observing more than 5 defaults is calculated as P(X > 5) = 1 - P(X ≤ 5). To find P(X ≤ 5), sum the probabilities for X = 0, 1, 2, 3, 4, and 5 using the Poisson probability mass function: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of defaults.
Step 4: Compare the calculated probability P(X > 5) to the analyst's threshold of 5% (0.05). If P(X > 5) is less than 0.05, the event of more than 5 defaults is not significant, and the analyst should not be concerned. Otherwise, the event is significant, and the analyst should be concerned.
Step 5: Conclude based on the comparison. If the probability of more than 5 defaults is less than 5%, the analyst should not be concerned. Otherwise, the analyst should be concerned. Ensure to interpret the result in the context of the problem.
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