Baylor University Hankamer School of Business Department of Finance, Insurance & Real Estate Risk Management Dr. Garven Problem Set 5 Name: SOLUTIONS Suppose you have initial wealth (W 0 ) of $1,000 that is fully invested in an asset that is worth $1,000. However, there is a 20% probability that this asset will be completely destroyed by a fire, and an 80% probability that a fire won’t occur. Your utility function is U (W ) = ln W, and the price of an insurance policy which fully insures this asset against risk of loss is $240. A. In dollar and percentage terms, what is the premium loading for a full coverage insurance policy which costs $240? Solution: The actuarially fair premium for a full coverage insurance policy is E (L) = .20($1,000) - $200. Therefore, the premium loading for a full coverage insurance policy in dollar terms is $240 - $200 = $40. In percentage terms, it is 240/200 – 1 = 20%. B. Determine the “optimal” level of insurance coverage. Specifically, what coinsurance rate maximizes your expected utility? Solution: To find the “optimal” coinsurance rate, we must first write down the equation for expected utility. Let α be the coinsurance rate (note that α cannot be less than 0 or greater than 1; α = 0 implies that you self-insure, whereas α = 1 implies that you purchase full coverage). Since there is a 20% chance of a $100 loss and a 80% chance that there is no loss, this implies that state-contingent wealth (W s ) is calculated as follows: • 20% of the time, W s = W 0 – αP i - (1-α)L, where P i is the price of full insurance coverage = $240. Therefore, W s = 1,000 – α240 - (1-α)1,000 = 760α. • 80% of the time, W s = W 0 – αP i = 1,000 – 240α. Therefore, expected utility is E (U (W )) = .20 ln(760α) + .80 ln(1,000 – 240α). The optimal value for α maximizes expected utility; therefore, dE(U (W )) = .2(760/α760) − .8(240/(1, 000 − α240)) = 0; dα ∴ .2(760/α760) = .8(240/(1, 000 − α240)) ⇒ .2α−1 = 192(1, 000 − 240α)−1 ; 200 − 48α ∴ = 192 ⇒ 200 − 48α = 192α ⇒ α = 5/6. α C. Suppose you make an “optimal” insurance purchase. What will be the expected value and standard deviation of your wealth? Solution: Since the “optimal” insurance purchase involves a coinsurance rate (α) of 5/6, this means that E (W q )= .20((5/6)760) + .80(1,000 - 240(5)/6) = $766.67. The standard deviation is σW = .2((5/6)760 − 766.67)2 + .8((1, 000 − 240(5)/6) − 766.67)2 = 66.67. Please also answer the following two additional questions based upon the above information: D. Suppose you have a friend who is identical in all respects to you, except her utility is U (W ) = -W −1 . What coinsurance rate maximizes her expected utility? Who is more risk averse – you or your friend? Solution: Since U (W ) = -W −1 , this implies that E (U (W )) = – .2/(760α) – .8/(1,000 – 240α). Therefore, dE(U (W )) dα ∴ 152/(α760)2 ∴ 12.33(1, 000 − α240) ∴α = .2(760/(α760)2 ) − .8(240/(1, 000 − α240)2 ) = 0; = 192/(1, 000 − α240)2 ⇒ 152(1, 000 − α240)2 = 192(α760)2 ; = 13.86(α760) ⇒ 12, 330 − 2959.2α = 10533.6α; = .9139. Since my friend’s optimal coinsurance rate is higher than mine, this implies that she is more risk averse than me (note also that if you compute the Arrow-Pratt absolute risk aversion coefficients for these utility functions, your Arrow-Pratt coefficient is 1/W, whereas your friend’s Arrow-Pratt coefficient is 2/W; therefore, it is not surprising that she prefers a higher coinsurance level than me). D. Now suppose that you can fully insure this fire risk for $200. What is your optimal level of coinsurance at this price? What is your friend’s optimal level of coinsurance? Solution: We know from the Bernoulli hypothesis that if insurance is actuarially fair, then all risk averse consumers fully insure. Therefore, the optimal level of coinsurance (α) must be 1 (note that this can also be show by solving for α from the first order condition equations, but if done correctly both of these equations would produce this result). 2