Certainty Equivalent

Certainty Equivalent in Health Economics

A certainty equivalent is a guaranteed outcome, such as a fixed sum of money or a fixed number of life years, that a person rates just as highly as a gamble over uncertain outcomes. Health economists use it to turn a risky prospect, such as an uninsured hospital bill or a risky operation, into one sure figure. The gap between that figure and the expected value of the gamble shows how much the person dislikes the risk, which is why the idea sits behind the demand for health insurance. In health state valuation the same idea gives a lesser known relative of the standard gamble, in which the sure outcome is varied while the probability is held fixed.

Indifference between a gamble and a sure amount

Bonanno defines the certainty equivalent of a money lottery as the sum of money that the individual considers just as good as the lottery. He notes that it will typically differ between individuals. For a risk-neutral person it coincides with the expected value of the lottery, for a risk-averse person it is lower, and for a risk-loving person it is higher.

Under expected utility the certainty equivalent can be computed from a utility function.

$$U(CE) = E\big[U(X)\big] = \sum_{i} p_i \, U(x_i), \qquad CE = U^{-1}\Big(E\big[U(X)\big]\Big)$$

where $X$ is the uncertain outcome, $x_i$ is its $i$th possible value, $p_i$ is the probability of that value, $U$ is the person's utility function, $U^{-1}$ is its inverse and $CE$ is the certainty equivalent. Osborne and Rubinstein show that a person whose preferences follow expected utility is risk-averse if and only if the Bernoulli function is concave, so a concave $U$ pulls the certainty equivalent below the expected value.

Bonanno derives the risk premium from the certainty equivalent. It is the amount by which the expected value of the lottery can be reduced while leaving the person indifferent between the lottery and the reduced amount for certain.

$$RP = E[X] - CE$$

where $RP$ is the risk premium and $E[X]$ is the expected value of the lottery. The risk premium is zero for a risk-neutral person and positive for a risk-averse one. Risk aversion describes the person and the risk premium prices the risk, while the certainty equivalent values the whole prospect.

Two properties make the measure useful. It is stated in pounds or years, so it can be observed and compared in a way that utility numbers cannot. Osborne and Rubinstein note that Bernoulli functions representing the same preferences differ only by a positive affine transformation, and such a transformation leaves the certainty equivalent unchanged. Bonanno defines the certainty equivalent from a ranking of lotteries alone, without expected utility, so the measure still applies where expected utility fails, although it can then no longer be read as the curvature of $U$.

Pricing health insurance with the certainty equivalent

Arrow's 1963 paper on the welfare economics of medical care assumed that individuals are normally risk-averters, which in utility terms means a diminishing marginal utility of income. He argued that a person with a strict preference for an actuarially fair policy over bearing the risk alone would still prefer an actuarially unfair policy, provided it is not too unfair. The certainty equivalent sets how unfair the policy can be. Bonanno shows that a customer offered full cover at the highest acceptable premium, the expected loss plus the risk premium, is guaranteed the certainty equivalent of the no-insurance lottery.

For a household with illustrative wealth of GBP 10,000, a 10% chance of a GBP 6,400 treatment bill and $U(w) = \sqrt{w}$, the certainty equivalent is GBP 9,216, so it would pay up to GBP 784 for full cover, the fair premium of GBP 640 plus a risk premium of GBP 144.

The certainty equivalent method in health state valuation

Wakker and Stiggelbout explain that the standard gamble in medical decision making conventionally adopts the variable probability equivalent method, in which the probability in the gamble is varied until the respondent is indifferent between the gamble and the certain outcome. In the variable certainty equivalent method the certain outcome is varied instead, and in medical decision making it has been used to elicit the utility of life years. They report that it has been used to adjust time trade-off results to agree with expected utility, a method introduced by Pliskin and colleagues.

The two methods approach the same indifference from opposite ends. A probability equivalent question fixes the sure outcome, such as ten years in good health, and asks for the chance that makes a gamble just as attractive, while a certainty equivalent question fixes the gamble and asks for the number of sure years. Wakker and Stiggelbout state that under expected utility the two should yield the same utility values, and that utility curves derived from the certainty equivalent method should not depend on the probability used.

The evidence points the other way. Wakker and Stiggelbout report that the standard gamble usually gives higher utility values than the certainty equivalent method, and that certainty equivalent curves are higher the closer the probability used is to 1. They explain the gap with rank-dependent utility, in which people transform probabilities into decision weights, so part of the dislike of gambles cannot be captured by the curvature of utility. Later evidence is mixed, since in an experiment comparing the two Ruggeri and Coretti found that on average the technique did not change risk attitude significantly, while respondents were risk averse in gambles over life years and risk seeking in gambles over quality of life.

Worked example with a risky operation valued in life years

All numbers in this example are illustrative. A patient with 20 years of remaining life in good health is offered an operation with a 50% chance of keeping those 20 years and a 50% chance of immediate death. The patient's utility of life years is $U(T) = (T/20)^{0.8}$, which sets $U(20) = 1$ and $U(0) = 0$ and is concave because the exponent is below 1.

1. Expected life years from the operation are 0.5 × 20 + 0.5 × 0 = 10.
2. Expected utility of the operation is 0.5 × 1 + 0.5 × 0 = 0.5.
3. The certainty equivalent solves $(T/20)^{0.8} = 0.5$, so $T = 20 \times 0.5^{1.25} = 20 \times 0.4204 = 8.41$ years.
4. The risk premium is 10 − 8.41 = 1.59 years, so the patient would accept about 8.4 sure years in place of the gamble.
5. A standard gamble question for 10 sure years asks for the probability that gives indifference, which under expected utility equals $U(10) = 0.5^{0.8} = 0.574$.
6. A certainty equivalent question at a probability of 0.5 returns 8.41 years, since $U(8.41) = 0.5$. Both answers lie on the same utility curve, as expected utility requires.

A linear QALY model values the operation at its 10 expected life years and so ignores the illustrative gap of 1.59 years.

Why public appraisal works with expected values

Health technology assessment works with expected values. NICE's PMG36 manual for technology appraisal states in section 4.2.14 that cost-utility analysis is used to determine whether differences in expected costs can be justified by changes in expected health effects. Section 4.2.16 adds that reported ICERs must be the ratio of expected additional total cost to expected additional QALYs. The decision rule therefore values a treatment at its mean QALY gain even when each patient's certainty equivalent lies below that mean. Risk attitude can still enter through health state utilities, since Wakker and Stiggelbout argue that the standard gamble overestimates risk aversion and the utilities of impaired health states.

HM Treasury's Green Book (2026), the UK government's guidance on appraisal, takes the same line for wider public spending. It says practitioners may use an expected value as a central estimate of a social cost or social benefit, found by weighting each possible value by its probability. The Green Book supplementary guidance on discounting, published in October 2026, states that the future outcomes of any proposal are uncertain and should generally be reflected in expected social costs and social benefits, rather than by varying the discount rate. Its standard rate of 3.00% does contain a systematic risk premium of 1.00%, for the riskiness of government proposals whose benefits correlate with the wider economy. The Green Book leaves remaining risk attitude to the final judgement, noting that a high-reward but high-risk option may not offer the best value for money if decision makers are particularly risk-averse.

The certainty equivalent also marks a line in the welfarist versus extra-welfarist debate. In the welfarist tradition of welfare economics, Brouwer and colleagues write, net utility is expressed through willingness and ability to pay. A person's willingness to pay to remove a risk then embeds their certainty equivalent, as the GBP 784 maximum premium shows. Brouwer and colleagues describe extra-welfarism as rejecting individual utility as the only outcome of interest, with health usually measured in QALYs because they measure health. A cost-benefit analysis built on willingness to pay before the outcome is known therefore counts the value of reduced risk, while a cost-utility analysis built on expected QALYs leaves out risk attitude over the spread of outcomes, apart from what standard gamble utilities already carry.

Boundaries with neighbouring equivalence measures

Several measures in health economics use a sure quantity to stand in for something harder to value. They differ in what is being replaced, and mixing them up leads to the wrong comparison.

• Equivalent variation is an income change that leaves a person as well off as a price, health or policy change would. The certainty equivalent replaces a gamble, so it describes risk rather than a change in circumstances.
• A healthy year equivalent is a number of years in full health valued as highly as a lifetime health profile. Bleichrodt treats it as an argument in the utility function rather than a utility, while a certainty equivalent is read off a utility function for one gamble.

Limits and common misreadings of certainty equivalents

A certainty equivalent is personal and depends on context. Four cautions follow for its use in economic evaluation.

• The value belongs to one person at one level of wealth or health, so applying a sample average to a whole population is a further judgement.
• A certainty equivalent below the expected value need not come from concave utility. Wakker and Stiggelbout show that probability weighting can produce the same pattern, so reading utility curvature from certainty equivalent answers assumes expected utility holds.
• Elicited values depend on the gamble, as both the probability used and the kind of outcome change the answers reported above.
• Adding a separate risk premium on top of expected QALYs built from standard gamble utilities can count risk attitude twice, because those utilities already carry part of the respondents' aversion to risk.

Sources

• Arrow KJ. Uncertainty and the welfare economics of medical care. American Economic Review. 1963;53(5):941-973. https://www.aeaweb.org/aer/top20/53.5.941-973.pdf
• Bleichrodt H. QALYs and HYEs: under what conditions are they equivalent? Journal of Health Economics. 1995;14(1):17-37. https://doi.org/10.1016/0167-6296(94)00033-Z
• Bonanno G. Uncertainty, Risk and Information. Davis: University of California, Davis; 2019. Sections 2.3 and 2.6. https://faculty.econ.ucdavis.edu/faculty/bonanno/URI_Book.html
• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. https://doi.org/10.1016/j.jhealeco.2007.07.003
• HM Treasury. The Green Book (2026). London: HM Treasury; updated 28 October 2026. https://www.gov.uk/government/publications/the-green-book--2/the-green-book-2026
• HM Treasury. Green Book supplementary guidance: discounting. London: HM Treasury; October 2026. https://www.gov.uk/government/publications/green-book-supplementary-guidance-discounting
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Sections 4.2.14 and 4.2.16. https://www.nice.org.uk/process/pmg36
• Osborne MJ, Rubinstein A. Models in Microeconomic Theory. Expanded second edition. Cambridge: Open Book Publishers; 2023. Chapter 3. https://doi.org/10.11647/OBP.0362
• Ruggeri M, Coretti S. Do probability and certainty equivalent techniques lead to inconsistent results? Evidence from gambles involving life-years and quality of life. Value in Health. 2015;18(4):413-424. https://doi.org/10.1016/j.jval.2014.12.019
• Wakker P, Stiggelbout A. Explaining distortions in utility elicitation through the rank-dependent model for risky choices. Medical Decision Making. 1995;15(2):180-186. https://doi.org/10.1177/0272989X9501500212