Certainty equivalent
Also known as: Cash equivalent
The guaranteed amount that feels exactly as good to you as taking a given gamble.
What it means
The certainty equivalent of a risky prospect is the sure amount of money that a person values equally to the gamble, so they are indifferent between taking the certain sum and playing the lottery. It is the workhorse measure of risk attitude: a certainty equivalent below the gamble's expected value signals risk aversion, above it signals risk seeking, and the gap between them is the risk premium. Because prospect theory predicts that risk attitudes vary by domain and probability, certainty equivalents trace out the fourfold pattern — falling below expected value for high-probability gains but rising above it for long-shot gains. Eliciting certainty equivalents is a standard experimental method for estimating the value and weighting functions, though the figures can diverge from choice-based measures, as preference reversals show. The concept is the operational bridge between abstract utility curves and the prices people actually attach to risk.
From the utility curve to a single number
Under expected utility the certainty equivalent is not a free-standing preference but a consequence of the shape of the utility function: it is the sure amount whose utility equals the gamble's expected utility, so u(CE) equals the probability-weighted average of the utilities of the outcomes. When utility is concave, each extra pound worth a little less than the last, the average of the utilities sits below the utility of the average, and the certainty equivalent lands below the expected value. Pratt (1964) made this precise for small stakes: the risk premium is roughly half the gamble's variance times the local curvature of utility, the coefficient of absolute risk aversion. That is why the same person can look almost risk-neutral over pocket change yet sharply risk-averse over a month's salary: the premium scales with the stakes.
How it is elicited
Two families of procedure dominate. Matching asks directly for the number: state the sure amount that would leave you indifferent to the gamble. Choice-based methods never ask for a value at all; they present a sequence of binary choices, a sure 300 or the gamble, then a sure 350 or the gamble, and let the switching point bracket the certainty equivalent, narrowing it by bisection. Tversky and Kahneman's 1992 estimation of the prospect-theory value and weighting functions worked this way, reconstructing each subject's curves from certainty equivalents for dozens of two-outcome gambles. Gonzalez and Wu (1999) used the same raw material to pin down the inverse-S shape of probability weighting. The attraction is that a certainty equivalent compresses a whole distribution into one comparable figure, a price.
Why the number moves with the method
The certainty equivalent is less stable than its definition suggests. Ask for it by matching and by a probability-equivalence judgment on the same gamble and the two disagree systematically; Hershey and Schoemaker (1985) traced the gap and argued the numbers cannot both come from one underlying utility curve. Response mode matters more starkly in preference reversals: people routinely price a long-shot, large-prize bet above a safer one, yet choose the safer bet when the two are set side by side, so the pricing-based and choice-based certainty equivalents rank the same gambles in opposite orders. Bostic, Herrnstein and Luce (1990) showed the reversal shrinks when the indifference point is itself elicited through choices rather than stated as a price. The measure, in short, partly reflects how you asked for it.
Where it does the work
For all its instability the certainty equivalent remains the practical unit of risk. An insurer's premium is essentially the distance between a policyholder's wealth and the certainty equivalent of the loss they face: the more risk-averse the buyer, the lower that value and the more they will pay to be rid of the gamble. Corporate capital budgeting uses certainty-equivalent cash flows to strip risk out of a project before discounting, replacing a volatile future payoff with the smaller sure sum management would accept for it. Wherever a decision forces a swap of a distribution for a number, an out-of-court settlement, a buyout offer, a guaranteed-versus-variable pay package, the figure under negotiation is a certainty equivalent, whether or not anyone names it as one.
Examples
If you would just barely trade a 50% chance at $1,000 for a guaranteed $400, then $400 is your certainty equivalent for that gamble — and the $100 shortfall from its $500 expected value is your risk premium.
Offered a guaranteed £8,000 bonus or a coin-flip at £20,000, an employee takes the sure money. Her certainty equivalent sits below the £10,000 average, which is just to say she is risk averse.
A claimant with a strong case that might win £100,000 settles for £35,000 to be done with it. That settlement is his certainty equivalent, and the other side is trying to guess it.
A farmer signs a forward contract fixing next season's crop price, giving up the higher average from selling at market. That contract price is her certainty equivalent for the uncertain harvest revenue.
A retiree converts a volatile investment pot into a fixed lifetime annuity paying less than the portfolio's expected return. That guaranteed income is the certainty equivalent she accepts for surrendering the upside.
First described in Expected utility theory (von Neumann & Morgenstern, 1944); used throughout prospect theory.
Key references
- Gonzalez, R., & Wu, G. (1999). On the shape of the probability weighting function. Cognitive Psychology, 38(1), 129-166. doi.org/10.1006/cogp.1998.0710
- Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297-323. doi.org/10.1007/BF00122574
- Bostic, R., Herrnstein, R. J., & Luce, R. D. (1990). The effect on the preference-reversal phenomenon of using choice indifferences. Journal of Economic Behavior & Organization, 13(2), 193-212. doi.org/10.1016/0167-2681(90)90086-S
- Hershey, J. C., & Schoemaker, P. J. H. (1985). Probability versus certainty equivalence methods in utility measurement: Are they equivalent? Management Science, 31(10), 1213-1231. doi.org/10.1287/mnsc.31.10.1213
- Pratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica, 32(1/2), 122-136. doi.org/10.2307/1913738