Question

Neighborhood Insurance sells fire insurance policies to local homeowners. The premium is $150, the probability of a fire is 0.1%, and in the event of a fire, the insured damages (the payout on the policy) will be $140,000.

a. Make a table of the two possible payouts on each policy with the probability of each.

b. Suppose you own the entire firm, and the company issues only one policy. What are the expected value, variance and standard deviation of your profit?

c. Now suppose your company issues two policies. The risk of fire is independent across the two policies. Make a table of the three possible payouts along with their associated probabilities. (Round your "Probability" answers to 4 decimal places.)

d. What are the expected value, variance and standard deviation of your profit?

e. Compare your answers to (b) and (d). Did risk pooling increase or decrease the variance of your profit?

f. Continue to assume the company has issued two policies, but now assume you take on a partner, so that you each own one-half of the firm. Make a table of your share of the possible payouts the company may have to make on the two policies, along with their associated probabilities. (Round your "Probability" answers to 4 decimal places.)

g. What are the expected value and variance of your profit?

Answer 1

a). Expected payout of $150 (in case of no fire) and expected payout of 140,000 - 150 = 139,850(in case of fire)

b). Expected return = sum of probability*payout = (0.10%*-139850) + (99.9%*150) = 10

Variance = sum of probability*(return for a given probability - expected return)^2 =

0.10%*(-139850-10)^2 + 99.9%*(150-10)^2 = 19,580,400

Standard deviation = variance^0.5 = 19,480,400^0.5 = 4414

c). Expected payout (in case of no fire) = 150+ 150 = 300 (Probability = 100% -0.1%*2 = 99.8%)

Expected payout (in case of one fire) = 150 + (-140,000 + 150) = -139700 (Probability = 100% - prob. of no fire - prob. of tow fires = 100% -99.8% -0.0001% = 0.1999%)

Expected payout (in case of two fires) = (-140,000 + 150) + (-140,000 + 150) = -279,700 (Probability = 0.1%*0.1% = 0.0001%)

d). Expected return = (99.8%*300) + (0.1999%*-139700) + (0.0001%*-279700) = 19.86 (or 20)

Variance = 99.8%*(300-19.86)^2 + 0.1999%*(-139700-19.86)^2 + 0.0001%*(-279700-19.86) = 39,102,078

Standard deviation = 39,102,078^0.5 = 6253

f). When a partner is taken on, then payouts will be divided by half but probabilities of occurrence will remain the same as before.

Expected payout (in case of no fire) = 300/2 = 150 (Probability = 99.8%)

Expected payout (in case of one fire) = -139700/2 = -69850 (Probability = 0.1999%)

Expected payout (in case of two fires) = -279700/2 = -139,850 (Probability = 0.0001%)

g). Expected return = (99.8%*150) + (0.1999%*-69850) + (0.0001%*-139850) = 9.93 (or 10)

Variance = 99.8%*(150-9.93)^2 + 0.1999%*(-69850-9.93)^2 + 0.0001%*(-139850 - 9.93)^2 = 9,795,080

Standard deviation = 9,795,080^0.5 = 3130