Financial Risk Manager Part 1

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Explanation:

The question is a probability problem that involves calculating the percentage of policyholders who will renew at least one policy next year based on the given probabilities and the distribution of policy types among policyholders.

Let's define the events:

$A$ = event that a policyholder has an auto policy
$H$ = event that a policyholder has a homeowner policy

From the company records:

$P(A \cap H) = 0.20$ (20% of policyholders have both an auto and a homeowner policy)
$P(A) = 0.70$ (70% of policyholders have an auto policy)
$P(H) = 0.50$ (50% of policyholders have a homeowner policy)

To find the probabilities of the complementary events:

$P(A^c \cap H^c) = P(A) - P(A \cap H) = 0.70 - 0.20 = 0.50$ (50% have only an auto policy)
$P(A^c \cap H) = P(H) - P(A \cap H) = 0.50 - 0.20 = 0.30$ (30% have only a homeowner policy)

The company estimates the renewal rates:

40% of those with only an auto policy will renew
70% of those with only a homeowner policy will renew
80% of those with both policies will renew at least one

The calculation for the proportion of policyholders that will renew at least one policy is: $0.40 \times P(A^c \cap H^c) + 0.70 \times P(A^c \cap H) + 0.80 \times P(A \cap H)$ $= 0.40 \times 0.50 + 0.70 \times 0.30 + 0.80 \times 0.20$ $= 0.20 + 0.21 + 0.16$ $= 0.57$

Thus, the percentage of policyholders that will renew at least one policy next year is 57%, which corresponds to option D.

Explanation:

Let's define the events:

$A$ = event that a policyholder has an auto policy
$H$ = event that a policyholder has a homeowner policy

From the company records:

$P(A \cap H) = 0.20$ (20% of policyholders have both an auto and a homeowner policy)
$P(A) = 0.70$ (70% of policyholders have an auto policy)
$P(H) = 0.50$ (50% of policyholders have a homeowner policy)

To find the probabilities of the complementary events:

$P(A^c \cap H^c) = P(A) - P(A \cap H) = 0.70 - 0.20 = 0.50$ (50% have only an auto policy)
$P(A^c \cap H) = P(H) - P(A \cap H) = 0.50 - 0.20 = 0.30$ (30% have only a homeowner policy)

The company estimates the renewal rates:

40% of those with only an auto policy will renew
70% of those with only a homeowner policy will renew
80% of those with both policies will renew at least one

Thus, the percentage of policyholders that will renew at least one policy next year is 57%, which corresponds to option D.

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Based on the projections of an insurance company, the likelihood of policy renewals for the upcoming year varies depending on the type of insurance held by the policyholders. Here are the specific probabilities for policy renewals:

40% of individuals who have only an auto insurance policy are expected to renew.

70% of individuals who have only a homeowner's insurance policy are anticipated to renew.

80% of individuals who hold both auto and homeowner's insurance policies are estimated to renew at least one of these policies.

Furthermore, the distribution of policyholders in terms of their insurance coverage is as follows:

70% of the policyholders have auto insurance.

50% of the policyholders have homeowner's insurance.

20% of the policyholders have both auto and homeowner's insurance.

Using this information, calculate the proportion of policyholders that is projected to renew at least one policy in the next year.

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29%

8.8%

41%

17.5%

53%

18.2%

57%

55.5%