Question

Neighborhood Insurance sells fire insurance policies to local homeowners. The premium is $300, 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 $290,000.

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	Probability of no fire=1-0.001=0.999
			NoFire	Fire
		Payout	$0	$290,000
		Probability	0.999	0.001
	b	Profit in case of Fire=300-290000=			($289,700)
			A	B	C=A*B	D=B-10	E=D^2	F=E*C
			Probability	Profit	Profit*Probability	Deviation fromexpected	Deviation Squared	Deviation Squared* Probability
		Fire	0.001	($289,700)	($289.70)	($289,710)	$83,931,884,100	$83,931,884.10
		No Fire	0.999	$300	$299.70	$290	$84,100	$84,015.90
				Total	$10.00		TOTAL	$84,015,900
		Expected Profit	$10.00
		Variance	$84,015,900
		Standard Deviation =Square Root of Variance
		Standard Deviation	$9,166	(SQRT(84015900)
	c	Probability of 0 Fire =0.999*0.999=		0.998001
		Probability of 1 Fire =2*0.001*0.999=		0.001998		1
		Probability of 2 Fire =0.001*0.001=		0.000001
			Total	1
			No Fire	One Fire	TwoFire
		Payout	$0	$290,000	$580,000
		Probability	0.998001	0.001998	0.000001
		Profit for No fire	$600
		Profit for One fire	($289,400)	(600-290000)
		Profit for two fire	($579,400)
			A	B	C=A*B	D=B-20	E=D^2	F=E*C
			Probability	Profit	Profit*Probability	Deviation fromexpected	Deviation Squared	Deviation Squared* Probability
		No Fire	0.998001	$600	$598.80	$580	$336,400	$335,727.54
		One Fire	0.001998	($289,400)	($578.22)	($289,420)	$83,763,936,400	$167,360,344.93
		Two Fire	0.000001	($579,400)	($0.58)	($579,420)	$335,727,536,400	$335,727.54
				TOTAL	$20.00		TOTAL	$168,031,800
		Expected Profit	$20.00
		Variance	$168,031,800
		Standard Deviation =Square Root of Variance
		Standard Deviation	$12,963	(SQRT(168031800)