Risk Premium

Risk Premium in Health Economics

A risk premium is a form of willingness to pay that measures in money, or by extension in health units, how much a person dislikes bearing a risk. It is the gap between the expected value of an uncertain prospect, such as a year of income exposed to a hospital bill or an uncertain gain in quality of life, and the certain amount the person would accept in its place. In health economics the idea explains why households pay insurers more than their expected medical costs, and a related systematic risk premium is built into the discount rate in HM Treasury's Green Book. Standard cost-effectiveness analysis sets the risk premium over health to zero by valuing treatments at their expected QALYs, an assumption that recent work on risk aversion in health relaxes.

Pratt's definition and the certainty equivalent

John Pratt's 1964 paper in Econometrica, on utility functions for money, discussed the risk premium or insurance premium for an arbitrary risk and related it to a measure of risk aversion in the small. The definition rests on the certainty equivalent. In Yildiz's MIT lecture notes, the certainty equivalent of a lottery is the sure wealth level that yields the same expected utility as the lottery, so it is found by inverting the utility function at the lottery's expected utility.

$$\pi = E[X] - CE(X), \qquad CE(X) = u^{-1}\big(E[u(X)]\big)$$

where $X$ is the uncertain outcome, $E[X]$ is its expected value, $u$ is an increasing von Neumann-Morgenstern utility function, $CE(X)$ is the certainty equivalent and $\pi$ is the risk premium. Yildiz shows that a decision maker is risk averse if and only if the certainty equivalent never exceeds the expected value, which is the same as a risk premium that is never negative. A risk-neutral person has a premium of zero, and a risk seeker has a negative one.

Osborne and Rubinstein give the behavioural content. A strictly risk-averse individual is willing to pay a positive amount of money to replace a lottery with its expected value, so buying insurance suggests strictly risk-averse preferences. They prove that under expected utility risk aversion holds exactly when the Bernoulli function is concave, so the risk premium is the money value of the curvature that risk aversion and diminishing marginal utility of income describe.

How large the premium is and the Arrow-Pratt approximation

For small risks Pratt approximated the premium with a second-order expansion. Gollier, Hammitt and Treich describe this as the origin of his index of risk aversion in the small and note that the risk premium for a small lottery is proportional to the Arrow-Pratt index, which Arrow developed independently.

$$\pi \approx \frac{1}{2}\,\sigma^{2}\, r_A(\mu), \qquad r_A(x) = -\frac{u''(x)}{u'(x)}$$

where $\mu$ and $\sigma^{2}$ are the mean and variance of the outcome, $u'$ and $u''$ are the first and second derivatives of the utility function, and $r_A$ is the Arrow-Pratt coefficient of absolute risk aversion. The premium grows with the variance of the risk and with the curvature of utility at the mean. Dividing by $u'$ normalises the curvature, which Yildiz explains by the uniqueness of utility only up to affine transformations. Yildiz shows that the relation is exact under constant absolute risk aversion with a normally distributed outcome.

Two refinements matter in health applications. Under decreasing absolute risk aversion, which Yildiz describes as richer people taking higher risks, the same medical bill commands a smaller premium from a richer household. Relative risk aversion, $r_R(x) = x\,r_A(x)$, scales the measure to the level of wealth or health. Both refinements tie a risk premium to a person at a given baseline, so a figure estimated for one income group or one severity of illness does not transfer unchanged to another.

Risk premium and the demand for health insurance

The best-known health application is insurance against medical costs. Arrow's 1963 paper on the welfare economics of medical care assumed that individuals are normally risk-averters and argued that a person offered cover on an actuarially fair basis, with a premium equal to the mean of medical costs, will certainly prefer to take out a policy. For a household with wealth $w$ facing a loss $L$ with probability $p$, full cover at premium $P$ leaves a certain $w - P$, and the household is indifferent when that sum equals the certainty equivalent of staying uninsured.

$$P_{\max} = w - CE = pL + \pi$$

where $P_{\max}$ is the highest premium the household would pay for full cover, $CE$ is the certainty equivalent of the uninsured position, $pL$ is the expected loss and $\pi$ is the risk premium. The insurer's loading, its charge above expected claims, has to fit inside $\pi$ for the household to buy. Arrow noted that in 1958 expenses made up 51.6 per cent of total premium income for individual health insurance policies. A household example with loadings and deductibles is worked through under health insurance.

Pooling shrinks the premium a risk commands. Arrow argued that, with medical risks on different individuals basically independent, risk pooling reduces the risk to the insurer to relatively small proportions and yields a net social gain that may be of quite substantial magnitude.

Systematic risk premium in the Green Book discount rate

Public payers are usually treated as close to risk neutral for the same pooling reason. Lakdawalla and Phelps note that cost-effectiveness analysis typically assumes risk-averse consumers are insured against health-care spending and that third-party payers are typically thought of as risk-neutral. HM Treasury's Green Book supplementary guidance on discounting, last updated on 28 October 2026, takes a similar line for UK public spending, with one exception built into the discount rate.

Its Social Time Preference Rate adds a systematic risk premium to a risk-free discount rate. The guidance defines it as the premium that society demands in compensation for the riskiness of government proposals, arising because the social benefits of government proposals generally correlate with the wider economy. Such risk moves with the economy as a whole, unlike the independent risks in Arrow's argument.

$$r = r_f + \pi_s = 2.00\% + 1.00\% = 3.00\%$$

where $r$ is the Social Time Preference Rate, used as the real discount rate for years 1 to 30 of a proposal, $r_f$ is the risk-free discount rate and $\pi_s$ is the systematic risk premium. HM Treasury reports that the STPR approach implies a premium of around 0.50% and that triangulation with other evidence implies 1.00%. The guidance accepts that proposals may in theory have different systematic risk premiums, yet states that practitioners should not deviate from 1.00%. The possible outcomes of a proposal should generally be reflected in expected social costs and benefits, rather than by varying the discount rate, and the premium does not cover obsolescence risk.

For an illustrative health project with a benefit of GBP 10 million in year 11, the Annex A discount factor of 0.7441 gives a present value of GBP 7.441 million. At the risk-free rate of 2.00% alone the factor would be $(1.02)^{-10} = 0.8203$, giving GBP 8.203 million, so the systematic risk premium reduces the present value by about GBP 0.762 million. Uncertain uptake of the project belongs in its expected benefits, not in a higher rate.

Risk premiums over health outcomes in cost-effectiveness analysis

Risk also attaches to health, and a risk-averse patient may value a treatment with widely spread outcomes at less than its mean gain. Lakdawalla and Phelps state that standard CEA models assume that consumers are risk-neutral in health, which sets the risk premium over quality of life to zero. In that world they show that the cost per QALY gained, regardless of how the QALY was gained, is a sufficient index of welfare.

Their own model introduces risk aversion in quality of life. It leads to willingness-to-pay thresholds that rise with illness severity, potentially by an order of magnitude, and treatments that reduce variance gain value beyond their average improvement. Their risk-adjusted QALY uses the certainty-equivalent gain in quality of life in place of the average gain, which amounts to deducting a risk premium measured in health units. Shafrin and colleagues, in a 2024 user guide to generalised cost-effectiveness analysis, name this model Generalized and Risk-Adjusted Cost Effectiveness (GRACE). They call the ex-ante value that healthy individuals obtain from a health technology its insurance value, value it with certainty-equivalent quality of life, and map it to the ISPOR value flower. An expected-QALY model leaves this value out.

Worked example with an uncertain quality-of-life gain

All numbers in this example are illustrative. Two treatments for a chronic condition each last one year. Treatment A gives a certain health-related quality of life of 0.70, while Treatment B gives 0.95 or 0.45 with equal probability. The patient values quality of life $h$ with $W(h) = \sqrt{h}$, which up to a scale factor is the constant relative risk aversion form with a coefficient of one half.

1. Expected outcome. 0.5 × 0.95 + 0.5 × 0.45 = 0.70, so an expected-QALY analysis scores the two treatments as equal.

2. Expected utility of B. With $\sqrt{0.95} = 0.97468$ and $\sqrt{0.45} = 0.67082$, expected utility is 0.5 × 0.97468 + 0.5 × 0.67082 = 0.82275.

3. Certainty equivalent. Squaring 0.82275 gives 0.67692, the sure quality of life that this patient values as highly as Treatment B.

4. Risk premium. 0.70 − 0.67692 = 0.02308 QALYs, about 3.3% of the mean outcome.

5. Certainty-equivalence ratio. Untreated, the patient would live at 0.40, so the mean gain from B is 0.30 and its certainty-equivalent gain is 0.67692 − 0.40 = 0.27692, a ratio of 0.923. This ratio is the $\varepsilon$ of Lakdawalla and Phelps, and 0.27692 is the risk-adjusted QALY gain from B.

6. Interpretation. For this patient Treatment B is worth 0.02308 QALYs less than Treatment A, so B would need a lower cost or a higher mean to match it. A risk-neutral analysis misses that difference, and the risk-adjusted QALY records it.

Welfarist and extra-welfarist readings of the risk premium

Whether a patient's risk premium should count depends partly on the tradition an evaluation follows. In welfarist cost-benefit analysis, grounded in welfare economics, aversion to risk enters through ex ante willingness to pay for cover or for risk reduction. Brouwer and colleagues describe extra-welfarist cost-utility analysis as using something like need, instead of effective demand and willingness to pay as in cost-benefit analysis.

An extra-welfarist decision maker may therefore value expected health and treat aversion to uncertain outcomes as a separate, explicit judgement. GRACE moves cost-effectiveness analysis towards the welfarist reading by building patients' risk preferences into the outcome measure, so a report using it should state whose risk attitudes it reflects.

Common misreadings of the risk premium

Most errors come from mixing the risk premium with other premiums or with other kinds of uncertainty. Five confusions recur.

• Risk premium and insurance premium. The insurance premium is the price of cover and includes the expected claim. The risk premium is only the most a buyer would pay above that expected claim.
• A fixed number for a given risk. The premium varies with baseline wealth or health and with the person's curvature, and the quadratic approximation can fall well short when a loss is a large share of income.
• More risk aversion, more spending on any risk reduction. Gollier, Hammitt and Treich note that a more risk-averse decision maker has a larger premium for eliminating a risk, but need not pay more for a partial reduction.
• The Green Book premium as a project contingency. The 1.00% covers correlation with the wider economy. Obsolescence and other project-specific risks belong in expected costs and benefits.
• Risk premium and decision uncertainty. A risk premium describes the preferences of those who bear an outcome. Decision uncertainty is the chance that the option preferred on current evidence is not the best.

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
• 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
• Gollier C, Hammitt JK, Treich N. Risk and choice: a research saga. Journal of Risk and Uncertainty. 2013;47(2):129-145. https://doi.org/10.1007/s11166-013-9175-7
• HM Treasury. Green Book supplementary guidance: discounting. London: HM Treasury; October 2026, page last updated 28 October 2026. https://www.gov.uk/government/publications/green-book-supplementary-guidance-discounting
• Lakdawalla DN, Phelps CE. Health technology assessment with risk aversion in health. Journal of Health Economics. 2020;72:102346. https://doi.org/10.1016/j.jhealeco.2020.102346
• 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
• Pratt JW. Risk aversion in the small and in the large. Econometrica. 1964;32(1):122-136. https://www.econometricsociety.org/publications/econometrica/1964/01/01/risk-aversion-small-and-large
• Shafrin J, Kim J, Cohen JT, Garrison LP, Goldman DA, Doshi JA, et al. Valuing the societal impact of medicines and other health technologies: a user guide to current best practices. Forum for Health Economics and Policy. 2024;27(1):29. https://doi.org/10.1515/fhep-2024-0014
• Yildiz M. Attitudes towards risk. Chapter 3 of lecture notes for 14.123 Microeconomic Theory III. MIT OpenCourseWare; Spring 2015. https://ocw.mit.edu/courses/14-123-microeconomic-theory-iii-spring-2015/resources/mit14_123s15_chap3/