Financial Risk Manager Part 1
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In the context of an insurance company's projections, it is anticipated that 40% of policyholders who have only an auto policy will renew it in the upcoming year. Additionally, 70% of those with solely a homeowner policy are expected to renew. Furthermore, 80% of individuals holding both types of policies are estimated to renew at least one policy. Based on the company’s records, 70% of policyholders have an auto policy, 50% have a homeowner policy, and 20% possess both policies. Given these statistics, what proportion of the entire pool of policyholders is projected to renew at least one policy in the coming year?
In the context of an insurance company's projections, it is anticipated that 40% of policyholders who have only an auto policy will renew it in the upcoming year. Additionally, 70% of those with solely a homeowner policy are expected to renew. Furthermore, 80% of individuals holding both types of policies are estimated to renew at least one policy. Based on the company’s records, 70% of policyholders have an auto policy, 50% have a homeowner policy, and 20% possess both policies. Given these statistics, what proportion of the entire pool of policyholders is projected to renew at least one policy in the coming year?
Explanation:
The correct answer to the question is D, which is 57%. The explanation for this can be derived from the given probabilities and the relationships between the events A (having an auto policy) and H (having a homeowner policy). Here's a comprehensive breakdown:
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Understanding the Events:
- A: Policyholder has an auto policy.
- H: Policyholder has a homeowner policy.
- A ∩ H: Policyholder has both an auto and a homeowner policy.
- A°: Policyholder has only an auto policy (A without H).
- H°: Policyholder has only a homeowner policy (H without A).
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Given Probabilities:
- P(A ∩ H) = 0.20 (20% have both policies).
- P(A) = 0.70 (70% have an auto policy).
- P(H) = 0.50 (50% have a homeowner policy).
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Calculating Complementary Probabilities:
- P(A°) = P(A) - P(A ∩ H) = 0.70 - 0.20 = 0.50 (50% have only an auto policy).
- P(H°) = P(H) - P(A ∩ H) = 0.50 - 0.20 = 0.30 (30% have only a homeowner policy).
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Renewal Probabilities:
- 40% of those with only an auto policy will renew (P(A°) * 0.40).
- 70% of those with only a homeowner policy will renew (P(H°) * 0.70).
- 80% of those with both policies will renew at least one (P(A ∩ H) * 0.80).
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Calculating the Total Renewal Rate:
- Total renewal rate = (0.50 * 0.40) + (0.30 * 0.70) + (0.20 * 0.80) = 0.20 + 0.21 + 0.16 = 0.57 or 57%.
This calculation shows that 57% of policyholders will renew at least one policy next year, which is the correct answer provided in the file content.