# Certainty equivalent

Please see attached document.

If an individual with initial wealth w that is facing a random risk X that

takes values Ã¨ 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(-Ã£y)

1. How can I show that the certainty equivalent of not taking insurance is

= 1 - exp(-Ã£w)[p exp(Ã£Ã¨) + 1 - p].

and that the certainty equivalent of taking insurance is?

= 1 - exp(-Ã£w) exp(Ã£Ã¨).

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 Preview

1. How can I show that the certainty equivalent of not taking insurance is

= 1 - exp(-Ã Â£w)[p exp(Ã Â£Ã¨) + 1 - p].

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

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(-Ã Â£w)

The expected utility of not taking insurance is

u(NI) = p* u(w -Ã¨) +(1-p)* u(w) = p*[1 - exp(-Ã Â£(w -Ã¨))] +(1-p)* [1 - ...

#### Solution Summary

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