Question

Problem Set I

1) Rick’s Toy Store faces the following probability distribution of fire losses in its store over the next year:

Probability	0.50	0.40	0.10
Loss	$0	$20,000	$70,000

Calculate the expected value and standard deviation of Rick’s losses for the year.

Assume that Rick pools his losses with Mark’s store, which has an identical loss distribution. Mark’s losses are independent of Rick’s. Rick and Mark agree to split the total losses in the pool equally. Show the revised probability distribution for the mean loss from the pool.

Calculate the expected value and standard deviation of the pooled mean losses

2) Maria is analyzing the workers’ compensation (WC) losses of the employees in the firm that occurred over a one-year period, based on the following data:

Number of WC Claims Filled/Worker	Number of Workers	Total Number of Claims
0	500	0
1	270	270
2	50	100

Use the information in the table to find the average frequency of losses per worker.

Use the information in the table to estimate a probability distribution for the frequency distribution of losses per worker in a year.

3) You are given the following table:

Range of Loss Amount	Midpoint Dollar Amount of Loss	Number of Losses	Total $ Amt. of Losses
$1-2,000	$1,000	300	$300,000
$2,000-10,000	$6,000	15	$90,000
Greater than $12,000	NA	0	0

Use the information in the table to find the average severity per claim

Use the information in the table to estimate a probability distribution for the loss severity per claim.

Using your answers from question 3, part (a) and question 2, part( b), use convolution to find the average loss.

Answer 1

Rick’s Toy Store:

Expected Loss = 0.50 x 0 + 0.40 x 20,000 + 0.10 x 70,000 = $ 15,000

Variance of the loss = 0.50 x (0-15,000)² + 0.40 x (20,000-15,000)² + 0.10 x (70,000-15,000)² = $ 425 x 10⁶

Standard deviation of the loss = sqrt(Variance) = $ 20,615.53

For the pooled loss, the mean loss is the average of the losses of both Rick's store and Mark's store. The probability distribution of the mean loss of the pooled account is as per the below table:

Rick's store	Mark's store	Mean Loss	Probability
0	0	0	0.25
0	20000	10000	0.20
0	70000	35000	0.05
20000	0	10000	0.20
20000	20000	20000	0.16
20000	70000	45000	0.04
70000	0	35000	0.05
70000	20000	45000	0.04
70000	70000	70000	0.01

Expected value of the mean loss = 0.25 x 0 + 0.2 x 10,000 + 0.05 x 35,000 + 0.2 x 10,000 + 0.16 x 20,000 + 0.04 x 45,000 + 0.05 x 35,000 + 0.04 x 45,000 + 0.01 x 70,000 = 15,000

Variance = 212.5 x 10⁶

Standard deviation = $ 14,577.38