# Negative certainty equivalent

Let us consider an agent of initial wealth $$w_0$$ whose utility function is $$u(x)=\sqrt{x}$$. This individual faces a risk of loss $$Z$$ which occurs with probability $$p$$.

It is assumed that $$w_0=60000$$, $$Z=10000$$ and $$p=0.1$$. What is the certainty equivalent for the risk incurred by the individual? What is his risk premium? Interpret.

For the certainty equivalent, I have found:

$$\mathbb{E}(u(w_0+L))=u(w_0+c)\iff c=(0.1\sqrt{50000}+0.9\sqrt{60000})^2-60000\approx-1040.99$$

How can this result be interpreted?

$$\mathbb{E}(u(w_0+L))=u(w_0+\mathbb{E}(L)-\pi)\iff \pi\approx60040.99$$

Thanks.

• "How can this result be interpreted?" I am not sure what you are asking. A negative certainty equivalent is what you would expect for a possible loss. Or are you asking what a certainty equivalent generally is? Does your course give an explanation? Nov 20, 2021 at 14:22
• Sorry, I'm French so it's a little bit hard to explain and translate in English. In my course is written that a certainty equivalent is "the monetary amount for which the economic agent is indifferent between the lottery and this amount," thus the minimum price for which he would accept to free himself from the lottery. Actually, I don't understand how this definition can make sense with a negative certainty equivalent. Thanks. Nov 20, 2021 at 14:29

The certainty equivalent in your example is $$w_0+c$$, this certain payoff's utility is equivalent with the lottery's.
The amount $$c$$ is not the certainty equivalent, but the amount the consumer is willing to forego in expected value in exchange for the certainty. Perhaps $$-c$$ is what you call the risk premium.
• Thanks for the answer. Sorry for the delay, I didn't have access to my computer. Thus the certainty equivalent is $w_0+c=(0.1\sqrt{50000}+0.9\sqrt{60000})^2\approx58959$ I can't understand the logic here. Does this mean the agent is willing to pay $58959 to get rid of the risk? Nov 25, 2021 at 15:17 • No. This certain payoff's utility is equivalent with the lottery's. This is the smallest certain payoff the consumer would be willing to trade his uncertain lottery based cashflow for. Nov 25, 2021 at 17:36 • Since his initial wealth is 60,000 dollars, I assume the consumer is willing to lose 1,041 dollars (the difference between his initial wealth and the certainty equivalent) to trade his uncertain lottery. Then how would you interpret the risk premium (which is defined as the difference between the mathematical expectation of the lottery and its certainty equivalent)?$\pi=\mathbb{E}(w_0+L)-(w_0+L)\approx59000-58949\approx41\$ I am totally lost with all these terms, and my course has no examples. Thanks! Nov 27, 2021 at 11:55