Question

Homeowner's Policy . An insurance company wants to design a homeowner's policy for mid priced homes. From data compiled by the company. it is known that the annual claim amount, X in thousands of dollars, per homeowner is a random variable with the following probability distribution.

Part (a) Determine the expected annual claim amount per homeowner.

Part (b) How much should the insurance company charge for the annual premium if it wants to average a net profit of $50 per policy?

Short answer

Part (a) $760

Part (b) $810

Step-by-step solution

Part (a) Step 1. Given information. 

Consider the following data table:

x	0	10	50	100	200
P(X = x)	0.95	0.045	0.004	0.0009	0.0001

Part (a) Step 2. Annual claim amount per homeowner is estimated.

Let X be the annual claim amount per owner event.

$$\begin{gathered} \mu =\sum _{}x·P(X=x) \\ \mu =0+\left(10\times 0.045\right)+\left(50\times 0.004\right)+\left(100\times 0.0009\right)+\left(200\times 0.0001\right) \\ \mu =0+0.45+0.20+0.09+0.02 \\ \mu =0.76 \end{gathered}$$

As a result, the predicted value is $760.

Part (b) Step 1. Given information. 

Consider the following data table:

x	0	10	50	100	200
P(X = x)	0.95	0.045	0.004	0.0009	0.0001

Average net profit per policy is $50.

Part (b) Step 2. The insurance firm should charge the amount as an annual premium.

Part a shows that the average annual claim amount per homeowner is $760.

Premium.

$$\begin{gathered} =50+E\left(X\right) \\ =50+760 \\ =810\$ \end{gathered}$$

As a result, the insurance company should be penalised $810.