3. Consider an insurance contract which pays 100000 euro's within 20 years if the insured (x) is still alive at this moment. In the other situation, there is no payment. The probability that (x) survives for 20 years is p= . The insurer has the possibility to invest the earned premiums in a bank account which pays a compound interest of 3%. (a) Assume the insurer is a utility maximizer with utility function given by U1(x) = x and zero initial wealth. Calculate the minimal premium P the insurer wants to receive when selling this contract. This premium is paid at the beginning of the contract, whereas the benefit is paid after 20 years. (Remark: The payment of the premium and the payment of the loss happen in different periods in time. It is only meaningful to compare utilities at the same period in time.) (b) Assume the utility function of the insurer is given by U2 (2) = -ae-ax, where a = 0.001. Determine the minimal premium P2 the insurer wants to receive in this case. (c) Determine the Arrow-Pratt measure of absolute risk-aversion for the utility function U2. 7 3. Consider an insurance contract which pays 100000 euro's within 20 years if the insured (x) is still alive at this moment. In the other situation, there is no payment. The probability that (x) survives for 20 years is p= . The insurer has the possibility to invest the earned premiums in a bank account which pays a compound interest of 3%. (a) Assume the insurer is a utility maximizer with utility function given by U1(x) = x and zero initial wealth. Calculate the minimal premium P the insurer wants to receive when selling this contract. This premium is paid at the beginning of the contract, whereas the benefit is paid after 20 years. (Remark: The payment of the premium and the payment of the loss happen in different periods in time. It is only meaningful to compare utilities at the same period in time.) (b) Assume the utility function of the insurer is given by U2 (2) = -ae-ax, where a = 0.001. Determine the minimal premium P2 the insurer wants to receive in this case. (c) Determine the Arrow-Pratt measure of absolute risk-aversion for the utility function U2. 7