Risk Aversion

Risk Aversion in Health Economics

Risk aversion is a property of a person's preferences over uncertain prospects, in which receiving the average outcome for sure is ranked at least as highly as facing the gamble that produces it. In health economics the idea explains why households pay health insurance premiums above their expected medical bills, and why a patient may turn down an operation that offers the larger expected gain in quality of life. It also bears on how health state values are elicited and on whether a health system that ranks treatments by expected QALYs should ignore the spread of outcomes around it.

Lotteries, expected values and concave utility

Osborne and Rubinstein state the idea for lotteries over money. A person is risk-neutral when only the expectation of a lottery matters, risk-averse when the prize equal to that expectation is always at least as good as the lottery, and strictly risk-averse when it is strictly better for every lottery with more than one prize. They note that the property applies to any preference relation, whether or not it fits expected utility theory. When it does fit, they prove that preferences are risk-averse if and only if the function that scores each prize is concave.

$$u\left( E[X] \right) \geq E\left[ u(X) \right]$$

where $X$ is an uncertain outcome such as income after medical costs or a quality of life weight, $E[X]$ is its probability-weighted mean, $u$ is the person's utility function over that outcome, and $E[u(X)]$ is the expected utility of the gamble. A linear $u$ turns the inequality into an equality, which is risk neutrality.

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 added that gambling makes its full application difficult, and Osborne and Rubinstein likewise read buying insurance as a sign of strict risk aversion and paying to gamble as a sign against it. The certainty equivalent and the risk premium turn the inequality into amounts of money or health.

Measuring how strongly a person avoids risk

Concavity shows only that a person is risk-averse, so its strength is measured from the curvature of the utility function. Pratt's 1964 paper, as its abstract describes, relates a measure of risk aversion in the small to the risk premium and also treats risks as a proportion of total assets. The two standard measures are these.

$$r_A(x) = -\frac{u''(x)}{u'(x)}, \qquad r_R(x) = -\frac{x \, u''(x)}{u'(x)}$$

where $x$ is the level of the outcome, $u'(x)$ and $u''(x)$ are the first and second derivatives of the utility function, $r_A$ is the coefficient of absolute risk aversion and $r_R$ is the coefficient of relative risk aversion. Both are positive where the utility function is strictly concave, with $u''(x) < 0$ and $x > 0$, and zero where it is linear.

Cutler and Zeckhauser use the absolute coefficient to put a money value on fair health insurance, and show that the value of spreading risk rises with risk aversion and with the variability of medical spending.

Lakdawalla and Phelps define a coefficient of relative risk aversion over quality of life in the same way, with the quality weight in place of income. A common functional form keeps that coefficient constant.

$$W(h) = \frac{h^{1-r}}{1-r} \quad (r \neq 1), \qquad W(h) = \ln h \quad (r = 1)$$

where $h$ is a health-related quality of life weight above 0 and at most 1, $W(h)$ is the utility of living at that weight, and $r$ is the constant coefficient of relative risk aversion. Setting $r = 0$ gives $W(h) = h$, the linear scoring of quality that standard QALY arithmetic uses.

Insurance, pooling and the payer's attitude to risk

The demand for health insurance is the classic health application. Arrow argued that a risk-averse person offered actuarially fair cover will take it and gain in welfare, that pooling independent risks shrinks the insurer's own risk to small proportions, and that insurance is more valuable the greater the uncertainty in the risk insured. The risk pooling and moral hazard in health insurance pages take the argument further.

Pooling also explains why a large payer can act as if risk-neutral. Lakdawalla and Phelps note that cost-effectiveness analysis typically assumes risk-averse consumers are insured against spending and that third-party payers are typically thought of as risk-neutral. They treat risk aversion over costs as a more specialised matter than risk aversion over quality of life.

Risk attitudes in standard gamble and time trade-off values

Health state valuation brings risk aversion into the QALY itself. Lipman, Brouwer and Attema report that standard gamble weights are systematically higher than time trade-off weights, so evaluation results can depend on the valuation method. They write that both weights are commonly derived under expected utility and the linear QALY model, which assumes linear utility of life duration, while many studies find diminishing marginal utility of life years. By the concavity result above, linear utility of duration under expected utility is risk neutrality over life years.

The same authors summarise Bleichrodt's 2002 proposal that the departures described by prospect theory bias the two methods differently. Loss aversion pushes both weights upwards, utility curvature affects only the time trade-off, and probability weighting affects only the standard gamble. In their own sample the standard gamble weights exceeded the time trade-off weights, and the gap disappeared only after nonparametric correction. Bleichrodt's 1995 paper adds that reference levels strongly influence risk attitudes, so the design of a valuation task shapes the number that enters a cost-utility analysis.

Risk aversion over quality of life and the severity of illness

Lakdawalla and Phelps argue that standard cost-effectiveness analysis lives in a risk-neutral world in which a QALY is a QALY however it is gained. They introduce risk aversion in quality of life and show that it makes willingness to pay thresholds rise with the severity of illness. They report that ignoring it overvalues treatments for minor illnesses and undervalues treatments for very severe illnesses, perhaps by an order of magnitude, and they keep the assumption of risk neutrality over survival.

Mulligan and colleagues, in a 2023 NBER working paper that had not been peer reviewed, elicited risk preferences over quality of life from a nationally representative US sample. They found risk-seeking preferences at low levels of health and a switch to risk-averse preferences at a quality weight of 0.485. Their estimates imply that each unit of health is worth three times more to patients at a weight of 0.5 than to people in perfect health, and that traditional cost-effectiveness analysis overvalues treatments for the mildest diseases by more than a factor of two.

NICE treats severity by another route. Its manual PMG36, NICE technology appraisal and highly specialised technologies guidance, has the committee consider absolute and proportional QALY shortfall when it assesses the severity of a condition, a measure of future health lost rather than of attitudes to gambles.

Worked example with a risky operation and a certain treatment

All numbers in this example are illustrative. A patient chooses between a medicine that gives a quality of life weight of 0.70 for sure and an operation that gives 0.95 with probability 0.6 and 0.40 with probability 0.4. Both outcomes last one year, so each weight is also a number of QALYs.

1. Expected QALYs. The operation gives 0.6 × 0.95 + 0.4 × 0.40 = 0.73 QALYs, made up of 0.57 and 0.16, against 0.70 for the medicine. A QALY maximiser, who is risk-neutral over quality, chooses the operation.

2. Spread of outcomes. The variance of the operation's outcome is 0.6 × 0.4 × (0.95 − 0.40)² = 0.24 × 0.3025 = 0.0726, a standard deviation of about 0.269.

3. Mild risk aversion with r of 0.5. With $W(h) = h^{0.5}$ up to a constant factor, expected utility is 0.6 × 0.9747 + 0.4 × 0.6325 = 0.8378, against 0.8367 for the medicine. The operation is still preferred, and its certainty equivalent is 0.8378² = 0.7019, just above 0.70.

4. Stronger risk aversion with r of 1. With $W(h) = \ln h$, expected utility is 0.6 × (−0.0513) + 0.4 × (−0.9163) = −0.0308 − 0.3665 = −0.3973, against ln 0.70 = −0.3567 for the medicine. The certainty equivalent is $e^{-0.3973}$ = 0.6721, so the medicine is preferred, and the patient would give up 0.73 − 0.6721 = 0.0579 QALYs to avoid the gamble.

5. Switching point. The certainty equivalent equals 0.70 at $r$ of about 0.53.

6. Interpretation. The options differ by only 0.03 expected QALYs, so modest risk aversion reverses the ranking a cost-effectiveness model would report. Risk-seeking below a weight of about 0.485, as Mulligan and colleagues found, would change the shape of $W$ near the poor outcome, so a constant coefficient is a simplification.

Expected values in appraisal and the welfarist question

Appraisal guidance mostly works with expected values. Claxton argued in 1999 that decisions should rest only on mean net benefit, the basis of expected value analysis, leaving the distribution of net benefit to value of information analysis. PMG36 asks for the expected value of each cost component and the expected QALYs for each option to be presented. HM Treasury's Green Book 2026 allows an expected value as a central estimate but tells decision makers to consider their risk appetite, since a high-reward but high-risk option may not offer the best value for money if they are particularly risk-averse.

Whether individual risk attitudes should count is part of the wider choice between the two traditions of welfare economics in health. Brouwer and colleagues quote Hurley's summary that the welfarist tenets require any policy to be judged solely by the resulting utilities of individuals, as assessed by those individuals. On that view a patient's risk premium is part of the benefit, and willingness to pay measured in a cost-benefit analysis should capture it. Extra-welfarism, which Brouwer and colleagues describe as supplementing individual utility with other characteristics of people, often takes health itself as the maximand, and a decision maker who maximises expected QALYs is then risk-neutral over health by construction.

Misreadings of risk aversion

Risk aversion is often confused with loss aversion. Lipman, Brouwer and Attema describe loss aversion as a utility function that is steeper for losses than for gains around a reference point, and they correct for it separately from utility curvature.

A second error is to treat one coefficient as a fixed trait. The link between concavity and risk aversion is proved for expected utility preferences, curvature measured over money need not carry over to quality of life or life years, and Mulligan and colleagues found the attitude changing sign with the level of health.

A third error is to read the risk neutrality of a payer as a finding about patients. Pooling makes the payer's risk small, but a patient facing a risky operation still bears the full spread of outcomes, and decision uncertainty about which treatment is better is a separate matter.

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
• 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
• Claxton K. The irrelevance of inference: a decision-making approach to the stochastic evaluation of health care technologies. Journal of Health Economics. 1999;18(3):341-364. https://doi.org/10.1016/S0167-6296(98)00039-3
• Cutler DM, Zeckhauser RJ. The anatomy of health insurance. NBER Working Paper 7176. National Bureau of Economic Research; 1999. https://www.nber.org/papers/w7176
• HM Treasury. The Green Book: UK government guidance on appraisal. 2026. https://www.gov.uk/government/publications/the-green-book--2
• Lakdawalla DN, Phelps CE. Evaluation of medical technologies with uncertain benefits. NBER Working Paper 26058. National Bureau of Economic Research; 2019, revised 2020. https://www.nber.org/papers/w26058
• Lipman SA, Brouwer WBF, Attema AE. QALYs without bias? Nonparametric correction of time trade-off and standard gamble weights based on prospect theory. Health Economics. 2019;28(7):843-854. https://doi.org/10.1002/hec.3895
• Mulligan K, Baid D, Doctor JN, Phelps CE, Lakdawalla DN. Risk preferences over health: empirical estimates and implications for healthcare decision-making. NBER Working Paper 31524. National Bureau of Economic Research; 2023. https://www.nber.org/papers/w31524
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Published 31 January 2022, last updated 13 October 2026. https://www.nice.org.uk/process/pmg36
• Osborne MJ, Rubinstein A. Models in Microeconomic Theory. Expanded 2nd ed. Open Book Publishers; 2023. Chapter 3. https://doi.org/10.11647/OBP.0362
• Pratt JW. Risk aversion in the small and in the large. Econometrica. 1964;32(1/2):122-136. https://www.jstor.org/stable/1913738