Question

Neighborhood Insurance sells fire insurance policies to local homeowners. The premium is $270, 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 $260,000.

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

Answer A)

There are two possible payout scenarios - 1) There is a fire 2) There is no fire. So payout table will look like below:

	Scenario 1	Scenario 2
	Fire	No-Fire
Payout	260,000	0
Probability	0.10%	99.90%

Answer B)

Expected Value of profit = Profit from scenario 1 * probability of scenario 1 + Profit from scenario 2 * probability of scenario 2

Profit from scenario 1 = Premium Collected - Insurance Payout = 270 - 260,000 = -259730

Profit from scenario 2 = Premium Collected

	Scenario 1	Scenario 2
	Fire	No-Fire
Profit	-259730	270	given
Probability	0.10%	99.90%	given
Profit*Probability	-259.73	269.73	10	=Expected Value

Variance

= (Profit from scenario 1)^2 * probability of scenario 1 + (Profit from scenario 2)^2 * probability of scenario 2 - (Expected Value)^2

Standard Deviation = Square root of variance

	Scenario 1	Scenario 2
Variance	Fire	No-Fire
Profit^2	67,459,672,900.00	72,900.00
Probability	0.10%	99.90%
Profit^2 * Probability	67,459,672.90	72,827.10	67,532,400.00	=Variance
			8,217.81	=Standard Deviation

So, the expected income is $10, with a variance of $8,217.81

Answer C)

Joint Probability of two independent events = probability of event 1*probability of event 2

Scenario	Probability
Single Case - No fire	99.90%
Single Case - fire	0.10%
Scenario	Join Probability	Payouts
No Fire	99.8001%	0
One Fire	0.0999%	260,000
Two Fire	0.0001%	520,000

Answer D)

Scenario	Join Probability (p)	Payout (b)	Premium (a)	Profit (x) = (a-b)	xp	x2p
No Fire	99.8001%	0	540	540	538.92	291,017.09
One Fire	0.0999%	260000	540	(259,460)	(259.20)	67,252,172.11
Two Fire	0.0001%	520000	540	(519,460)	(0.52)	269,838.69
				Summation	279.20	67,813,027.89
Expected Value (E)= Sum(XP)=	279.20
Variance (Var)= x2p-E2=	67,735,074.95
Standard Deviation = Sqrt(Var)	8,230

Answer E) From b & d, we can see that expected profits have increased drastically with increase in number of policies, though variance and S.D. of this expected profits remained similar.