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Neighborhood Insurance sells fire insurance policies to local homeowners

Finance

Neighborhood Insurance sells fire insurance policies to local homeowners. The premium is $350, 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 $340,000.
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?

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ANSWER -

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

Payout of No Fire: ANS. 350

Payout of Fire: ANS. (339,650); 350-340,000=339650

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?
Expected return: ANS. $10; E(r) = (99.9% *350) + (.1%* - 339650) = 10

Variance: 1158400; (99.9% *(350-10)^2 +.10%(-339650-10)^2 = 115484400

Standard deviation: 10746; Square root or Sqrt(115484400)

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

  outcome no fire outcome one fire outcome two fires

payout

700; 350*2

-339300; 350-339650

-67300; (340*2)-700

probability

99.8%

.1999%

.0001%
       

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

expected return

variance

standard deviation

$20; (99.8%(700)+.1999%(-339300)+.0001%(-679300))=19.66 rounds to 20

231083938; (99.8%(700-19.66)^2 + .1999%(-339300-19.66)^2 + .0001%(-67300-19.66)^2)

15201; sqrt(231083938)
     

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

 

outcome no fire
outcome one fire outcome two fires

payout

$350; was half of part c

$-69650

$-339650

probability

99.8%

.1999%

.0001%
       
       
       

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

Expected Return

Variance

Standard deviation

$10

57770984

7601
     

take previous expected return from part d and divide by 2… Variance divide by 4 and that’s it.

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