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Neighborhood Insurance sells fire insurance policies to local homeowners
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?
Expert Solution
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 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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