Question

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Neighborhood Insurance sells fire insurance policies to local homeowners. The premium is $200, 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 $190,000 a. Make a table of the two possible payouts on each policy with the probability of each. Outcome A: No Fire $ o Outcome B: Fire! 190,000 Payout 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 Variance Standard Deviation

Answer 1

a)

	Outcome A; No Fire	Outcome B : No Fire
Payout	$ -200	$1,90,000
Probability	99.9%	0.1%

The probability of a fire is 0.1% is given

Hence the probability of No Fire = 1- Probability of a Fire (Because these are the only two possible outcomes in this scenario; either there is a fire or there is no fire.

In case there is no fire the Insurance company makes no payout to the customer rather he earns $200 on the policy. Since there is a net earning on the policy hence there is a negative payout for the Insurance firm of $200.

b)

Expected Value of profit	Variance of profit	Standard Deviation of profit
$ 9.8	76020.04	$275.7173

Expected value of the profit = Probability of Outcome A* Payout in Outcome A + Probability of Outcome B* Payout in Outcome B

Therefore Expected Value of Profit = 0.999 X (200) + 0.001 X (-190000) = 199.8 -190 = 9.8

The expected value of profit can also be called the mean value of the profit.

Variance is the Sum of the squared differences of each data point from the mean divided by the number of data points minus 1.

Here there are two outcomes and hence two data points.  

These data points are $199.8 and $190 and the mean profit is $9.8.

Therefore the variance here is (199.8-9.8)^2 + (-190-9.8)^2 = 36100 + 39920.04 = 76020.04

We need to divide this by the no of outcome minus 1. Since here the nos of outcomes are just 2 we have to divide by 2-1 i.e 1

Variance is a sum of squares and hence a difficult parameter for practical purpose. The variance mentioned above is Profit squared.

Hence standard deviation is a much more useful term for all practical purposes.

Thus standard deviation is square root of Variance divided by no of outcomes minus 1 (again 1 in this case)

Therefore Standard deviation of profit is $ 275.7173

c)

	Outcome: No Fire	Outcome: One Fire	Outcome Two Fires
Payout	-$400	$189800	$380000
Probability	99.8001%	0.1998%	0.0001%

Since the two policies are independent of each other i.e the probability incidence of Fire on the first policy is not anyway related to the incidence of fire on the second policy, probability of No fire is the product of the probabilities of No fire on the first policy and No fire on the second policy.

In this Instance the probability of No fire is 0.999 X 0.999 = 0.998001

In this case the payout would be negative $200 from the premiums of each of the policies i.e -$400 and earning for the firm.

The Second case of one fire would be probability of Fire on one while no fire on the Other and there could be two sch instances 0.001 X 0.999 x 2 = 0.001998 and the payout in this instance is -200+190000= $189800

The Third case would be the one where there is a fire incident on both the policies the probability of which would be 0.001 x 0.001 = 0.000001 or 0.0001% and the payout in this instance would be $ 190000 x 2 = $380000

d)

Expected Value of Profit	Variance	Standard Deviation
$19.6	151776.688

Expected Value of the profit is calculated as by multiplying the profits with the associated probabilities. Note that the profits to the form would be negative of the payouts to the customer.

Thus the Expected profits =0.998001 x $400 - 0.001998 x $189800 - 0.000001 x $380000 = $399.2004 - $379.2204 - $ 0.38 = $19.6

Variance is again calculated as in earlier instance only that we have three outcomes (data points)

Hence Variance = [(399.2004-19.6)^2 + (-379.2204 - 19.6)^2 + (-0.38-19.6)^2]/(3-1) = 144096.464 + 159057.711 + 399.2004 = 303553.375/2 = 151776.688

Standard Deviation is just the square root of Variance = Sqrt (151776.688) = 389.585

Hence Standard deviation = 389.585

e) As we can see combining these two similar risk profiles increased the Variance