Social time preference rate, stepped long-term discount factors and implied individual rates for health
tp(delta, L, mu, g, t, r_1, r_2, r_3, y, s) = (STPR, r_HD, DF_t, r)
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.
Green Book social time preference rate from its Ramsey components with the health discount rate
rho = delta + L; W_e = mu * g; STPR = rho + W_e; r_HD = STPR - W_e
Adds pure time preference delta and catastrophic risk L to give time preference rho, and adds the wealth effect, the elasticity of marginal utility mu times the growth rate g of real per capita consumption, to give the social time preference rate, as in paragraphs 2.2 to 2.8 of the Treasury's supplementary guidance on discounting (February 2026). With the Table 2.A values for years 1 to 30 the two parts are 1.5% and 2.0% and the rate is 3.5%. The health discount rate r_HD leaves out the wealth effect (paragraph 2.16), so it equals rho, 1.5%, which is broader than pure time preference because it keeps catastrophic risk. Freeman, Groom and Spackman, in their report for HM Treasury, and Attema and colleagues write the same sum as the Ramsey rule.
Implied annual discount rate for health from a TEMPUS open-ended stated choice
r = (y / 20)^(1 / (s - 2)) - 1
Converts one answer to a TEMPUS open-ended question into the constant annual rate it implies. The respondent states the largest number of days y of ill health in year s at which a one-off minor treatment, delaying 20 days of ill health expected in 2 years' time, would still be worthwhile. With b = 1/(1 + r) the answer implies 20 b^2 = y b^s, which gives equation (1) of Cairns and van der Pol (2000). An answer above 20 days gives a positive rate, exactly 20 days a zero rate and fewer than 20 a negative rate. The rate is the r of the factor in HE-FM-DR-001, and rates that fall as the delay grows point to the non-constant forms of HE-FN-HYPD-001.
Green Book discount factor for a year under the stepped long-term schedule
DF_t = (1 + r_1)^(-min(t, 30)) * (1 + r_2)^(-max(min(t, 75) - 30, 0)) * (1 + r_3)^(-max(t - 75, 0))
Gives the discount factor for year t when the rate steps down after year 30 and again after year 75, the declining regime of Table 3.A of the supplementary guidance. Each band's rate compounds only over the years that fall in that band, so the factor for a year after 30 is the year-30 factor times (1 + r_2) to the power minus the years past 30. With 3.5%, 3.0% and 2.5% it reproduces the standard factors of Annex A Table A.1, 0.3459 for year 31 and 0.1468 for year 60, and with the health rates 1.5%, 1.286% and 1.071% it gives 0.4955 for year 50 as in Table A.2. For years up to 30 it is the single-amount factor of HE-FM-DR-001, and totals over a stream are in HE-CF-TPR-001.