Question

Emina has an amount of wealth $w_0$ and she faces a random loss $L$. This loss will happen with probability $q$ (assume $0 < q < 1$). If it happens, then the amount of the loss has a Uniform(0,1) distribution (otherwise, $L = 0$). Emina's utility function is $v(w) = -e^{-w}$. She has the possibility to insure the totality of the loss $L$ by paying a premium equal to: $c cdot E[L]$, where $c > 0$ is a constant set by the insurer. 1. (1pt) What can you say about the risk profile of Emina? Justify your answer. 2. (1pt) Show that $E[L] = q/2$. 3. (2pt) If she does not buy insurance, show that her Expected Utility has the following expression: $-e^{-w_0}[1 + q(e - 2)]$. 4. (2pt) Denote by $c^$ the value of $c$ which makes the premium the maximum premium Emina is willing to pay for insurance. Show that $c^ = 2log[1 + q(e - 2)]/q$. 5. (2pt) Let $c^$ be as before. Explain, conceptually, why it is the case that: For any $q$, we have that $c^$ is bigger than 1. $c^*$ decreases as $q$ increases. 6. (2pt) State one potential benefit for Emina and one potential benefit for the insurer to include a deductible on this insurance policy. [No calculations are required to answer this question.]

          Emina has an amount of wealth $w_0$ and she faces a random loss $L$. This loss will happen with probability $q$ (assume $0 < q < 1$). If it happens, then the amount of the loss has a Uniform(0,1) distribution (otherwise, $L = 0$). Emina's utility function is $v(w) = -e^{-w}$. She has the possibility to insure the totality of the loss $L$ by paying a premium equal to: $c cdot E[L]$, where $c > 0$ is a constant set by the insurer.

1. (1pt) What can you say about the risk profile of Emina? Justify your answer.
2. (1pt) Show that $E[L] = q/2$.
3. (2pt) If she does not buy insurance, show that her Expected Utility has the following expression: $-e^{-w_0}[1 + q(e - 2)]$.
4. (2pt) Denote by $c^*$ the value of $c$ which makes the premium the maximum premium Emina is willing to pay for insurance. Show that $c^* = 2log[1 + q(e - 2)]/q$.
5. (2pt) Let $c^*$ be as before. Explain, conceptually, why it is the case that:
For any $q$, we have that $c^*$ is bigger than 1.
$c^*$ decreases as $q$ increases.
6. (2pt) State one potential benefit for Emina and one potential benefit for the insurer to include a deductible on this insurance policy.
[No calculations are required to answer this question.]

Added by Jeffrey A.

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