Financial Risk Manager Part 1

Explanation

This is a Poisson distribution probability calculation problem with a rate conversion from monthly to yearly.

Key Steps:

1. Monthly Rate: The firm receives clients at a rate of λ = 4 clients per month.

2. Yearly Rate: Since there are 12 months in a year, the yearly rate is: \lambda_{year} = 4 \times 12 = 48$`$3`. **Poisson Probability Formula:** For a Poisson distribution with rate λ, the probability of exactly x events is: P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}$`$4. **Calculation:** We need to find P(X = 44) with λ = 48: $$P(X = 44) = \frac{e^{-48} \times 48^{44}}{44!}$$5`. Result: This calculation yields approximately 0.0506.

Why other options are incorrect:

• A (0.025): This might be an approximation error or using incorrect parameters.
• C (0.24): This is too high and doesn't match the Poisson distribution calculation.
• D (0.00363): This could result from using the monthly rate (λ = 4) instead of converting to yearly rate.

Conceptual Understanding:

• The Poisson distribution is appropriate for modeling the number of events occurring in a fixed interval of time when events occur independently at a constant average rate.
• When converting time periods, the rate parameter scales linearly with time (monthly rate × 12 = yearly rate).
• The probability mass function accounts for both the expected number of events (λ) and the specific count we're interested in (x).