# Certainty Equivalent

The certainty equivalent is a guaranteed return that someone would accept now, rather than taking a chance on a higher, but uncertain, return in the future. Otherwise, some definitions say that the certainty equivalentis the mean income of a gamble. If we see the vast literature of microeconomic and game theory papers, then we shall see that they assume CARA Gaussian normal preferences (or negative expnential utility, i.e. $$\mathbb{U}(\widetilde{W})=-e^{\rho\widetilde{W}}$$). Specifically the certainty equivalent in such case is $$CE(\widetilde{W})=\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$$ How can someone claim that this return is guaranteed, since we need to use $$\mathbb{V}ar(\widetilde{W})$$ to measure the utility gain of an individual?

• While $\widetilde W$ is a random variable, $\mathbb Var (\widetilde W )$ is not random. Jun 7 '20 at 18:01

How can someone claim that this return is guaranteed, since we need to use $$\mathbb{V}ar(\widetilde{W})$$ to measure the utility gain of an individual?
Maybe I misunderstand your point, but though $$\widetilde{W}$$ is a random variable, $$\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$$ and hence $$CE(\widetilde{W})$$ is a real number. So $$CE(\widetilde{W})$$ is a measure of a certain income which to the consumer has the same utility as $$\widetilde{W}$$.
• Why $CE(\widetilde{W})$ is a real number? What is the intuition behind this? Jun 6 '20 at 22:52
• The "intuition" is that $\mathbb{E}(\widetilde{W})$ is a number, $\frac{\rho}{2}$ is a number and $\mathbb{V}ar(\widetilde{W})$ is a number, so performing the mathematical operations $$\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$$ gives you a number, which you defined as $CE(\widetilde{W})$. Jun 7 '20 at 9:25