Tag Archives: Insurance and risk management

Exam P Practice Problem 96 – Expected Insurance Payment

Problem 96-A

An insurance policy is purchased to cover a random loss subject to a deductible of 1. The cumulative distribution function of the loss amount X is:

    \displaystyle  F(x) = \left\{ \begin{array}{ll}           \displaystyle  0 &\ \ \ \ \ \ x<0 \\            \text{ } & \text{ } \\          \displaystyle  \frac{3}{25} \ x^2 - \frac{2}{125} \ x^3 &\ \ \ \ \ \ 0 \le x<5 \\           \text{ } & \text{ } \\           1 &\ \ \ \ \ \ 5<x           \end{array} \right.

Given a random loss X, determine the expected payment made under this insurance policy?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 1.54

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 1.72

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 4.63

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 6.26

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Problem 96-B

An insurance policy is purchased to cover a random loss subject to a deductible of 2. The density function of the loss amount X is:

    \displaystyle  f(x) = \left\{ \begin{array}{ll}                     \displaystyle  \frac{3}{8} \biggl(1- \frac{1}{4} \ x + \frac{1}{64} \ x^2 \biggr) &\ \ \ \ \ \ 0<x<8 \\           \text{ } & \text{ } \\           0 &\ \ \ \ \ \ \text{otherwise}           \end{array} \right.

Given a random loss X, what is the expected benefit paid by this insurance policy?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.51

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.57

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.63

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 1.60

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 2.00

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\copyright \ 2016 - \text{Dan Ma}

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Exam P Practice Problem 93 – Determining Average Claim Frequency

Problem 93-A

An actuary performs a claim frequency study on a group of auto insurance policies. She finds that the probability function of the number of claims per week arising from this set of policies is P(N=n) where n=1,2,3,\cdots. Furthermore, she finds that P(N=n) is proportional to the following function:

    \displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=1,2,3,\cdots

What is the weekly average number of claims arising from this group of insurance policies?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   2.900

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   3.015

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   3.036

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   3.069

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   3.195

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Problem 93-B

Let N be the number of taxis arriving at an airport terminal per minute. It is observed that there are at least 2 arrivals of taxis in each minute. Based on a study performed by a traffic engineer, the probability P(N=n) is proportional to the following function:

    \displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=2,3,4,\cdots

What is the average number of taxis arriving at this airport terminal per minute?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   2.740

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   2.900

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   3.339

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   3.489

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   3.692

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\copyright \ 2015 \ \ \text{ Dan Ma}

Exam P Practice Problem 92 – Expected Claim Payment

Problem 92-A

The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function:

    \displaystyle f(x)=\frac{1}{5000} \ (100-x) \ \ \ \ \ \ \ 0<x<100

By subjecting the insured to a deductible of 12 per claim, what is the expected reduction in claim payment?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   9.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   10.6

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   11.1

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   11.8

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   12.0

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Problem 92-B

The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function:

    \displaystyle f(x)=\frac{1}{3200} \ (80-x) \ \ \ \ \ \ \ 0<x<80

By subjecting the insured to a deductible of 10 per claim, by what percent is the expected claim payment reduced?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   10 \%

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   15 \%

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   22 \%

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   25 \%

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   33 \%

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\copyright \ 2015 \ \ \text{ Dan Ma}

Exam P Practice Problem 90 – Insurance Benefits

Problem 90-A

A random loss follows an exponential distribution with mean 20. An insurance reimburses this random loss up to a benefit limit of 30.

When a loss occurs, what is the expected value of the benefit not paid by this insurance policy?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   4.5

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   5.1

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   6.3

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   8.5

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   11.2

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Problem 90-B

A random loss follows an exponential distribution with mean 100. An insurance reimburses this random loss up to a benefit limit of 200.

When a loss occurs, what is the expected value of the benefit not paid by this insurance policy?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   12.6

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   13.5

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   24.6

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   40.6

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   40.7

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\copyright \ 2014 \ \ \text{ Dan Ma}

Exam P Practice Problem 88 – Expected Value of Insurance Payments

Problem 88-A

A random loss X has a uniform distribution over the interval 0<x<20. An insurance policy is purchased to reimburse the loss up to a maximum limit of m where 0<m<20.

The expected value of the benefit payment under this policy is 8.4. Calculate the value of m.

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   8.7

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   9.0

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   12.0

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   13.6

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   18.3

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Problem 88-B

An individual purchases an insurance policy to cover a loss X whose density function is:

    \displaystyle f(x)=\frac{2}{25} \ (5-x) \ \ \ \ \ \ \ \ 0<x<5

The insurance policy reimburses the policy owner up to a benefit limit of 4 for each loss. What is the expected value of insurance payment made to the policy owner?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 1.35

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 1.41

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 1.49

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 1.65

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 1.67

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\copyright \ 2014 \ \ \text{ Dan Ma}

Exam P Practice Problem 87 – Modeling Insurance Payments

Problem 87-A

A business owner is facing a risk whose economic loss is modeled by the random variable X. The following is the density function of X.

    \displaystyle f(x)=\frac{1}{8} \ (4-x) \ \ \ \ \ \ \ \ 0<x<4

The business owner purchases an insurance policy to cover this potential loss. The insurance policy pays the business owner 80% of the amount of each loss.

Given that a loss has occurred, what is the probability that the amount of the insurance payment to the business owner is less than 2?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   0.25

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   0.36

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   0.64

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   0.75

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   0.86

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Problem 87-B

An individual purchases an insurance policy to cover a loss X whose density function is:

    \displaystyle f(x)=\frac{1}{1800} \ x \ \ \ \ \ \ \ \ 0<x<60

The insurance policy reimburses the policy owner 50% of each loss. Given that a loss has occurred, what is the median amount of the insurance payment made to the policy owner?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 15.00

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 18.65

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 21.21

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 23.63

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 42.43

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\copyright \ 2014 \ \ \text{ Dan Ma}

Exam P Practice Problem 84 – When Random Loss is Doubled

Problem 84-A

A business owner faces a risk whose economic loss amount X follows a uniform distribution over the interval 0<x<1. In the next year, the loss amount is expected to be doubled and is expected to be modeled by the random variable Y=2X.

Suppose that the business owner purchases an insurance policy effective at the beginning of next year with the provision that any loss amount less than or equal to 0.5 is the responsibility of the business owner and any loss amount in excess of 0.5 is the responsibility of the insurer. When a loss occurs next year, what is the expected payment made by the insurer to the business owner?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{16}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \frac{6}{16}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \frac{7}{16}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \frac{8}{16}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \frac{9}{16}

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Problem 84-B

A business owner faces a risk whose economic loss amount X whose density function is:

    \displaystyle f(x)=\frac{x}{2} \ \ \ \ \ \ 0<x<2

In the next year, the loss amount is expected to be doubled and is expected to be modeled by the random variable Y=2X.

Suppose that the business owner purchases an insurance policy effective at the beginning of next year with the provision that any loss amount less than or equal to 1 is the responsibility of the business owner and any loss amount in excess of 1 is the responsibility of the insurer. When a loss occurs next year, what is the expected payment made by the insurer to the business owner?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{12}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \frac{2}{3}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \frac{19}{12}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \frac{27}{16}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \frac{21}{12}

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\copyright \ 2014 \ \ \text{ Dan Ma}

Exam P Practice Problem 83 – Claim Size of Auto Insurance Policies

Problem 83-A

An insurance company has a block of auto insurance policies. The claim size (in thousands) for a policy in this block of auto insurance policies is modeled by the random variable Y=X^2 where X has a normal distribution with mean 0 and variance 1.5.

What is the expected claim size for such an auto insurance policy?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 1250

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 1500

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 1750

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 2250

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 2500

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Problem 83-B

An insurance company has a block of auto insurance policies. The claim size (in thousands) for a policy in this block of auto insurance policies is modeled by the random variable Y=X^2 where X has a normal distribution with mean 0 and variance 3.

What is the standard deviation of the claim size for such an auto insurance policy?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 1732

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 3000

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 4243

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 4987

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 5732

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\copyright \ 2014 \ \ \text{Dan Ma}

Exam P Practice Problem 80 – Total Insurance Payment

Problem 80-A

An individual purchases an insurance policy to cover a random loss. If a random loss occurs during the year, the amount of loss is at least 1. Once a random loss occurs, the insurance payment to the insured is modeled by the random variable X with the following density function

    \displaystyle f(x)=\frac{1}{x^2} \ \ \ \ \ 1<x<\infty

If there is a loss, there is only one loss in each year. In each year, the probability of a loss is 0.25. What is the probability that the annual amount paid to the policyholder under this policy is less than 2?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.250

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.500

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.750

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.875

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.925

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Problem 80-B

An individual purchases an insurance policy to cover a random loss. If a random loss occurs during the year, the loss amount is at least 1. Once a loss occurs, the insurance payment to the insured is modeled by the random variable X with the following density function

    \displaystyle f(x)=\frac{1}{30} \ x(1+3x) \ \ \ \ \ 1<x<3

If there is a loss, there is only one loss in each year. In each year, the probability of a loss is 0.15. What is the probability that the annual amount paid to the policyholder under this policy is less than 2?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.1500

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.2833

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.8500

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8735

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.8925

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Exam P Practice Problem 77 – Estimating Random Claim Sizes

Problem 77-A

The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.

    \displaystyle f(x)=\frac{3}{1000} \ (50-5x+\frac{1}{8} \ x^2), \ \ \ \ \ \ \ \ \ \ 0<x<20

What is the probability that a randomly selected claim from this insurance policy is within 120% of the mean claim size?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.85

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.88

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.91

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.95

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Problem 77-B

The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.

    \displaystyle f(x)=\frac{3}{2500} \ (100x-20x^2+ x^3), \ \ \ \ \ \ \ \ \ \ 0<x<10

What is the probability that a randomly selected claim from this insurance policy is within one-half of a standard deviation of the mean claim size?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.37

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.60

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.62

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.64

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\copyright \ 2013 \ \ \text{Dan Ma}