Financial Risk Manager Part 1
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An insurance company writes business in three territories: A, B and C. They have 150 policyholders in territory A, 250 in territory B and 300 in territory C. A person is twice as likely to have a claim in territory B than territory A and 3 times as likely to have a claim in territory C than territory A. On average, 50 people have a claim every policy period. Given a claim occurs, what is the probability it was a policyholder in territory C?
Explanation:
This is a conditional probability problem using Bayes' theorem. Let's solve it step by step:
Given:
- Policyholders: A=150, B=250, C=300 (Total=700)
- Claim probabilities: P(Claim|B) = 2 × P(Claim|A), P(Claim|C) = 3 × P(Claim|A)
- Average claims per period = 50
Step 1: Define variables Let P(Claim|A) = x Then P(Claim|B) = 2x And P(Claim|C) = 3x
Step 2: Calculate overall claim probability P(Claim) = P(Claim|A)×P(A) + P(Claim|B)×P(B) + P(Claim|C)×P(C) = x × (150/700) + 2x × (250/700) + 3x × (300/700) = x × (150 + 500 + 900)/700 = x × (1550/700)
We know from the data that P(Claim) = 50/700 So: x × (1550/700) = 50/700 x = 50/1550 = 0.03226
Step 3: Apply Bayes' theorem P(C|Claim) = [P(Claim|C) × P(C)] / P(Claim) = [3x × (300/700)] / (50/700) = [3 × 0.03226 × 300] / 50 = [29.034] / 50 = 0.58068 ≈ 58%
Therefore, the probability that a claim came from territory C is approximately 58%.