Signature
rho = delta + L; W_e = mu * g; STPR = rho + W_e; r_HD = STPR - W_e
| Inputs | Definition | Unit |
|---|---|---|
delta | Pure time preference, society's impatience, its preference for social benefits now rather than later | a proportion per year, 0.005 in the Green Book |
L | Catastrophic or systemic risk, an allowance for unpredictable risks that are not normally included in appraisal | a proportion per year, 0.01 in the Green Book |
mu | Elasticity of marginal utility of consumption, how wellbeing changes as consumption changes | none, 1.0 in the Green Book |
g | Expected annual growth rate of future real per capita consumption | a proportion per year, 0.02 in the Green Book |
rho | Time preference, the sum of pure time preference and catastrophic risk | a proportion per year |
|---|---|---|
W_e | Wealth effect, the elasticity of marginal utility times consumption growth | a proportion per year |
STPR | Social time preference rate, the real annual rate applied to social costs and benefits in years 1 to 30 | a proportion per year |
r_HD | Health discount rate for effects on health or life, the social time preference rate without its wealth effect | a proportion per year |
Function
Social time preference rate, stepped long-term discount factors and implied individual rates for health
Maps the components of social time preference to the HM Treasury social time preference rate and its health discount rate, maps a stepped long-term schedule of rates to the discount factor for any year of an appraisal, and maps one answer to a stated choice between ill health at two dates to the constant annual rate it implies. The rates it returns feed the single-amount factor of the Discount Rate page (HE-FM-DR-001) and the stream totals of the Discounting page (HE-FN-DISC-001, HE-FM-DISC-001 to HE-FM-DISC-004 and HE-CF-DISC-001). A health rate derived from growth in the consumption value of health is HE-FM-DDIS-002 under HE-FN-DDIS-001, and the quasi-hyperbolic weight D(k) = beta lambda^k of the article is HE-FM-HYPD-004 under HE-FN-HYPD-001, so neither is repeated here. Notation follows the Time Preference article.
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Implementations
Excel
Green Book social time preference rate and health rate from named component cells
With pure time preference in PureTimePref, catastrophic risk in CatastropheRisk, the elasticity in ElasticityMU and consumption growth in ConsGrowth, all entered as decimals, the first formula returns the STPR and the second the health discount rate. The tests below read the two results from cells named STPROut and HealthRateOut.
=PureTimePref+CatastropheRisk+ElasticityMU*ConsGrowth; =PureTimePref+CatastropheRisk
Assumptions
Ramsey components added as real annual rates
The formula sums the components, as paragraph 2.2 of the guidance states, and every rate is real, so the result applies only to values with general inflation removed (paragraph 2.19). Compounding the parts instead, (1 + delta)(1 + L)(1 + mu g) minus 1, gives about 3.535% with the Table 2.A values and is not the Treasury's form (computed here for illustration).
Ramsey component values fixed for years 1 to 30 only
Table 2.A gives the parameter values for years 1 to 30. For later years the guidance states the declining rates directly in Table 3.A and attributes the decline to uncertainty about the future values of the parameters (paragraph 3.1), without new component values, so the formula is not used to rebuild the 3.0% or 2.5% rates. Their discount factors are in HE-FM-TPR-003.
Health discount rate as a Green Book rule and not the NICE reference case
Paragraphs 2.16 and 2.17 discount health and life effects at the health discount rate in every year of the appraisal, 1.5% for years 1 to 30 and the lower rates of Table 3.A after that, with all other social costs and benefits at the STPR. Section 4.5.1 of the NICE manual PMG36 discounts costs and health effects at the same 3.5% a year in the reference case, and section 4.5.3 allows 1.5% for both only when all of its criteria are met, so r_HD belongs to Treasury cost-benefit appraisal.
Normative status of the Ramsey components of time preference
Attema and colleagues note that building a social rate from the Ramsey components assumes that every component, impatience included, is normatively relevant. The guidance cites Freeman, Groom and Spackman for a plausible range of 0.0% to 1.0% for pure time preference (paragraph 2.10), so the 0.5% value is a choice within that range.
Worked examples
Green Book 3.5% rate and 1.5% health rate from the Table 2.A values
With pure time preference of 0.5%, catastrophic risk of 1.0%, an elasticity of 1.0 and growth of 2.0%, time preference is 1.5%, the wealth effect 2.0%, the STPR 3.5% and the health rate 1.5%, as in paragraphs 2.6 to 2.8 and 2.16 of the guidance.
delta = 0.005; L = 0.01; mu = 1; g = 0.02; rho = 0.015; W_e = 0.02; STPR = 0.035; r_HD = 0.015
Green Book rate with pure time preference set to zero
Removing pure time preference leaves 3.0%, the reduced rate for years 1 to 30 in Table 3.B used for sensitivity analysis beyond 50 years. The health output of 1.0% is computed here for illustration only, since Table 3.B gives no reduced health rate.
delta = 0; L = 0.01; mu = 1; g = 0.02; rho = 0.01; W_e = 0.02; STPR = 0.03; r_HD = 0.01
Green Book rate with the pooled elasticity of 1.5 reported by Groom and Maddison
Paragraph 2.12 reports a pooled elasticity of 1.5 from Groom and Maddison, while the Green Book assumes 1.0. With 1.5 and the other Table 2.A values the wealth effect rises to 3.0% and the STPR to 4.5%, while the health rate stays at 1.5% because it has no wealth effect (illustrative sensitivity, computed here for illustration).
delta = 0.005; L = 0.01; mu = 1.5; g = 0.02; rho = 0.015; W_e = 0.03; STPR = 0.045; r_HD = 0.015
Common errors
Treasury 3.5% read as the public's impatience
Only 0.5 of the 3.5 points is pure time preference. Setting it to zero lowers the rate to 3.0% (Table 3.B), so a value in year 20 keeps a weight of 0.5537 instead of 0.5026 (Annex A Tables A.3 and A.1), far from the full weight of 1 that a rate of zero would give.
Green Book health rate used in a NICE reference case
Discounting QALYs at 1.5% in place of the 3.5% of PMG36 section 4.5.1 puts a weight of 0.7425 on a QALY in year 20 against 0.5026, about 48% more discounted health (computed here for illustration), which lowers the cost per QALY of programmes with distant gains.
Green Book health rate taken as pure time preference alone
Discounting health at 0.5% drops catastrophic risk from the time preference term. A QALY in year 20 then keeps a weight of 0.9051 against 0.7425 at the 1.5% health rate (computed here for illustration), as in step 3 of the article's worked example.
Sources
Treasury supplementary guidance deriving the social time preference rate and health rate
HM Treasury. Discounting, Green Book supplementary guidance. London, HM Treasury, February 2026 (ISBN 978-1-918417-18-0), page last updated 5 February 2026. Chapter 2, paragraphs 2.1 to 2.8 and Table 2.A, which give STPR = rho + mu g with rho = delta + L, the values 0.5%, 1.0%, 1.0 and 2.0% for years 1 to 30 and the sums 1.5%, 2.0% and 3.5%; paragraphs 2.10 and 2.12 for the ranges of pure time preference and the elasticity; paragraphs 2.16 and 2.17, which set the health discount rate at 1.5% because the wealth effect is excluded and apply it in every year of the appraisal; paragraph 2.19 on real terms.
Ramsey rule for the social rate of time preference in the report for HM Treasury
Freeman M, Groom B, Spackman M. Social discount rates for cost-benefit analysis, a report for HM Treasury. London, HM Treasury, February 2018. Part 1, page 3, section on the derivation of a social rate of time preference, which writes the Ramsey rule as SDR = STP = rho + eta g, with rho made up of pure time preference delta and, in HM Treasury practice, a component L for certain types of risk.
Ramsey equation with a catastrophic risk term for economic evaluation
Attema AE, Brouwer WBF, Claxton K. Discounting in economic evaluations. PharmacoEconomics. 2018;36(7) 745-758. Box 1, discounting in equations, which gives the Ramsey equation D = delta + epsilon g and its extension with a factor gamma for catastrophic risk, and the section on the normative or positive approach, which states that the Ramsey route assumes all components, pure time preference included, are normatively relevant.
NICE reference-case rate of 3.5% for costs and health effects alike
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. Chapter 4, economic evaluation, section 4.5.1, which states that for the reference case costs and health effects should be discounted at the same rate of 3.5% per year, and section 4.5.3, which allows 1.5% a year for both when all of its criteria are met.
Canonical Identity
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