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Certainty equivalent

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If an individual with initial wealth w that is facing a random risk X that
takes values è with probability p and value zero with probability 1 - p. If the individual does
not take insurance, his wealth will be w - X. If he takes insurance, his wealth will be w - a,
where a is the insurance premium. Suppose that the investor has utility function

u(y) = 1 - exp(-ãy)

1. How can I show that the certainty equivalent of not taking insurance is
= 1 - exp(-ãw)[p exp(ãè) + 1 - p].

and that the certainty equivalent of taking insurance is?
= 1 - exp(-ãw) exp(ãè).

2. What is the largest premium that the investor will be willing to pay?

3. How does the premium change as the parameter ã increases?

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Solution Summary

These questions discuss the certainty equivalent of taking or not taking insurance in a given situation and the premiums for the insurance.

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1. How can I show that the certainty equivalent of not taking insurance is
= 1 - exp(-à£w)[p exp(à£è) + 1 - p].

if he doesn't take insurance, when there's a loss, his wealth will be w -è, and the probability is p. Then the according utility is u(w -è) = 1 - exp(-à£(w -è))

When there's no loss, his wealth will be w-0 = w and the probability is 1-p.
Then the according utility is u(w) = 1 - exp(-à£w)
The expected utility of not taking insurance is
u(NI) = p* u(w -è) +(1-p)* u(w) = p*[1 - exp(-à£(w -è))] +(1-p)* [1 - ...

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