# Is the expected utility the inverse of the utility function?

Can somebody explain to me if that it's true and also graphically explain it?

• No. It is not the inverse of the utility function. Why would you believe this? – Bayesian Dec 3 '20 at 9:45

This is trivially not true. Consider simple example of utility:

$$u(x) = x^{1/2}$$

Expected utility $$E(u(x)) = E[x^{1/2}]$$

Inverse utility is $$u^{-1} \implies x = u^2$$

clearly generally $$E(u) \neq u^{-1}$$.

• Why do you substitute the EU in the inverse of the utility funcciton to obtain the risk premium? – Teko JR Dec 3 '20 at 10:09
• @TekoJR I don’t know what you are talking about. In economic literature risk premium is commonly defined as the difference between expected payoff from gamble and its certainty equivalence of the same gamble or in more finance oriented literature as a difference between expected return on risky and riskless asset of the same kind. – 1muflon1 Dec 3 '20 at 10:16