Financial Risk Manager Part 1

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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:

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).
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).
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).
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).
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.

Explanation:

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).
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).
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).
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).
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.

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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?

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29%

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12.1%

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57%