The Risk Premium

The certainty equivalent ($CE$) is the guaranteed amount that gives the same utility as the risky gamble: $u(CE)=E(u(w))$. The risk premium is what a risk-averse person would pay to eliminate risk — the difference between expected wealth and the certainty equivalent: Risk Premium $=E(w)-CE$. It measures the dollar cost of risk to this person.

On the graph of a concave utility function, the risk premium has a clean geometric interpretation. The expected utility $E(u)$ lies on the chord connecting the two outcomes. The certainty equivalent is the wealth level where the curve reaches that same utility. Since the curve is above the chord (concavity), the $CE$ is to the left of $E(w)$. The horizontal gap between $CE$ and $E(w)$ is the risk premium. More concavity (more curvature) means a bigger gap — the more risk-averse you are, the more you’d pay to eliminate risk.