Certainty Equivalent Return

Offer someone a coin flip paying $200 on heads and $0 on tails — the expected value is $100, but almost nobody treats it as equivalent to a guaranteed $100 in hand, because the flip carries risk the sure thing doesn't. The certainty equivalent is the guaranteed amount that person would find exactly as attractive as taking the gamble: if they'd be indifferent between the flip and a sure $85, then $85 is their certainty equivalent for that bet.

The gap between the expected value and the certainty equivalent is the risk premium — here, $100 − $85 = $15 — and it's a direct measure of how risk-averse that investor is. A more risk-averse investor discounts the same gamble further, giving a lower certainty equivalent and a bigger risk premium; a risk-neutral investor's certainty equivalent equals the expected value exactly. In portfolio choice, the certainty equivalent return lets you rank uncertain strategies on a common, risk-adjusted scale: a strategy with a higher expected return but wilder swings can have a lower certainty equivalent than a steadier one with a smaller expected return, and a risk-averse investor should prefer the latter.

Mean-variance utility functions compute a certainty equivalent directly as expected return minus a penalty proportional to variance, scaled by a risk-aversion coefficient — the higher that coefficient, the more variance erodes the certainty-equivalent number even when expected return is unchanged.