Financial Risk Manager Part 1

Explanation

This is a classic Bayes' Theorem application. We need to find the probability that a policyholder came from the ultra preferred tier given that they have a claim: $P(U|C)$.

Given Information:

• Distribution: Standard (25%), Preferred (50%), Ultra Preferred (25%)
• Claim probabilities:
  • Standard: 10% (0.10)
  • Preferred: 5% (0.05)
  • Ultra Preferred: 2% (0.02)

Using Bayes' Theorem:

$P(U|C) = \frac{P(C|U)P(U)}{P(C)}$

Step 1: Calculate $P(C)$ - Overall probability of a claim

$P(C) = P(C|S)P(S) + P(C|P)P(P) + P(C|U)P(U)$ $P(C) = (0.10 \times 0.25) + (0.05 \times 0.50) + (0.02 \times 0.25)$ $P(C) = 0.025 + 0.025 + 0.005 = 0.055$

Step 2: Calculate $P(U|C)$

$P(U|C) = \frac{P(C|U)P(U)}{P(C)} = \frac{0.02 \times 0.25}{0.055} = \frac{0.005}{0.055} = 0.0909 \approx 9\%$

Verification:

The result makes intuitive sense - although ultra preferred policyholders have the lowest claim rate (2%), they represent a significant portion of the population (25%), and their low claim rate combined with their population proportion gives them about 9% of all claims.