Question 25 You are a famous musician and have 1,500,000 of wealth. You own a Stradivarius violin valued at 600, 000. During the year you travel a lot and estimate that there is a 5% chance that your violin is lost, stolen or damaged. However, you are able to purchase insurance for a premium of 5 per 100 of insured violin value for the year. In the event that your violin is lost, stolen or damaged, the insurance company makes a payment equal to its insured value. You are a risk averse agent with utility of end-of- year wealth given by U(W) = In(W). Assume that the interest rate for the year is 0% and that the value of the violin, if intact, remains unchanged at the end of the year. a) Let u be the amount of insurance that you purchase. In terms of x, what is wealth at the end of the period if your violin survives intact? b) Let x be the amount of insurance that you purchase. In terms of x, what is wealth at the end of the year if your violin is lost, stolen or damaged? c) What is expected utility in terms of the quantities given in parts a) and b)? d) How much insurance should you purchase? e) What percentage of the violin's value did you choose to insure? f) Given the relationship between the premium and the insured value, is the expected profit for the insurance company positive? Is the insurance contract actuarially fair? g) How would the decision change if the risk-free rate was 5% rather than 0%? Ex- plain. (No numerical calculations are required.) Question 25 You are a famous musician and have 1,500,000 of wealth. You own a Stradivarius violin valued at 600, 000. During the year you travel a lot and estimate that there is a 5% chance that your violin is lost, stolen or damaged. However, you are able to purchase insurance for a premium of 5 per 100 of insured violin value for the year. In the event that your violin is lost, stolen or damaged, the insurance company makes a payment equal to its insured value. You are a risk averse agent with utility of end-of- year wealth given by U(W) = In(W). Assume that the interest rate for the year is 0% and that the value of the violin, if intact, remains unchanged at the end of the year. a) Let u be the amount of insurance that you purchase. In terms of x, what is wealth at the end of the period if your violin survives intact? b) Let x be the amount of insurance that you purchase. In terms of x, what is wealth at the end of the year if your violin is lost, stolen or damaged? c) What is expected utility in terms of the quantities given in parts a) and b)? d) How much insurance should you purchase? e) What percentage of the violin's value did you choose to insure? f) Given the relationship between the premium and the insured value, is the expected profit for the insurance company positive? Is the insurance contract actuarially fair? g) How would the decision change if the risk-free rate was 5% rather than 0%? Ex- plain. (No numerical calculations are required.)
