FRM Part I · FRM Exam Part I · Random Variables
A credit analyst models the number of defaults in a portfolio of 20 independent loans as a binomial random variable, with each loan having a default probability of 0.05. What is the expected number of defaults and the variance of the number of defaults?
The expected number of defaults is 1.00 and the variance is 0.95. A binomial variable has mean np = 20 × 0.05 and variance np(1−p) = 20 × 0.05 × 0.95, so the variance is slightly smaller than the mean.
- AMean 1.00; variance 0.95Correct
- BMean 1.00; variance 1.00
- CMean 0.95; variance 1.00
- DMean 1.00; variance 0.0475
Explanation
For a binomial, mean = np = 20 × 0.05 = 1.00 and variance = np(1−p) = 20 × 0.05 × 0.95 = 0.95. Option B uses the Poisson property that variance equals the mean, which ignores the (1−p) factor. Option D reports the per-trial variance, not the portfolio variance.
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