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standard deviation of the wealth

Suppose Mary can have good health with probability 0.95 and bad health with probability 0.05. If Mary has a good health her wealth will be $90,000, if she has bad health her wealth will be $10,000 (after paying the health bill). Suppose that the utility of wealth come from the following utility function: U(W)=W^0.5

Note: When the question is about utilities, report the answer with one decimal. When the question is about money, report integers (without comas for thousands). In both cases, round appropriately. 

1.    Find the reduction in wealth if Mary has bad health.

2.    Find the expected wealth of Mary if she has no insurance.

3.    Find her utility if she has bad health and she has no insurance.

4.    Find her utility if she has good health and she has no insurance.

5.    Find the expected utility of Mary if she has no insurance.

6.    Find the certain equivalent wealth of this lottery.

7.    If she has full insurance, find the payment the insurance company made to her if she has bad health.

8.    Find the maximum premium she is willing to pay for full insurance.

9.    Find the fair premium if she is fully insured.

10. Find her expected utility if she paid the fair premium and has full insurance.

11. Calculate the standard deviation of the wealth of Mary if she has no insurance.

12. Calculate the standard deviation of the wealth of Mary if she has full insurance.

13. Calculate the profits made with Mary's policy for the insurance company if the premium is fixed at $5,000 with full insurance.

14. Calculate Mary's expected utility if she pays a premium of $5,000 and she is full insured.

15. Suppose the case of fair premium and full insurance, and the insurance company has only one client (Mary). Calculate the standard deviation of the profits.

16. Suppose the case of fair premium and full insurance, and the insurance company has two identical clients like Mary. Under the assumption that the covariance between both clients' risk is zero, what would be the standard deviation of the profits of the insurance company. 

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