Discount Factor

Discount Factor Formula and Tables for Health Economic Evaluation

A discount factor is a weight attached to one year of an appraisal, equal to one divided by one plus the discount rate, raised to the number of years after the reference date. At the 3.5% a year of the NICE reference case, a QALY gained 10 years from now carries a factor of 0.709 and one gained 30 years from now a factor of 0.356, so a cost-effectiveness analysis counts the later gain at about half the weight of the earlier one. Every discounted total in an evaluation is a sum of amounts multiplied by these factors, so a short table of them explains most of what discounting does to an ICER. In welfare economics the same number has a second reading, as the relative weight a society places on wellbeing or consumption in a later year compared with now.

A weight on later wellbeing in welfare economics

The factor began as a parameter of the discounted utility model. Frederick, Loewenstein and O'Donoghue trace that model to a five-page article by Paul Samuelson in 1937, in which a person's intertemporal utility is the sum of each period's utility multiplied by a discount function.

$$U_0 = \sum_{k=0}^{T} D(k)\, u(c_k), \qquad D(k) = \left( \frac{1}{1+\rho} \right)^{k}$$

where $U_0$ is intertemporal utility judged now, $u(c_k)$ is the utility from consumption $c_k$ in period $k$, $T$ is the last period, $D(k)$ is the discount factor for period $k$ and $\rho$ is the pure rate of time preference. Frederick and colleagues describe $D(k)$ as the relative weight a person attaches, now, to wellbeing in a later period. They also report that Samuelson disavowed any connection between this utility and a welfare concept, so using the model to weigh social outcomes is a normative choice, not something the model itself supplies.

Social appraisal borrows the form and changes the interpretation. A social welfare function that adds wellbeing across years needs a weight for each year, and HM Treasury's supplementary guidance on discounting derives its rate mainly by the social time preference approach, in which the rate reflects how far society is willing to trade off consumption today for consumption tomorrow. Read that way, the discount factor for a year is the amount of consumption today that society treats as equal to one unit of consumption in that year, which makes it a relative price between periods.

A factor applied to consumption mixes two separate reasons for giving the future less weight.

$$d_t = (1+\delta)^{-t} \left( \frac{c_t}{c_0} \right)^{-\mu}$$

where $d_t$ is the discount factor for consumption in year $t$, $\delta$ is pure time preference, $c_t$ and $c_0$ are consumption per head in year $t$ and now, and $\mu$ is the elasticity of marginal utility of consumption. The first term is impatience and the second is diminishing marginal utility, since a richer future population gains less from an extra pound. With consumption growing by $g$ a year the second term becomes $(1+g)^{-\mu t}$, which gives the Ramsey rate of roughly $\delta + \mu g$. The time preference page sets out how HM Treasury builds its 3.00% rate from these terms.

Health raises a further question. Attema, Brouwer and Claxton note that if the consumption value of health grows over time, a decision maker seeking to maximise welfare must account for that growth, and that this holds whether the underlying social welfare function is welfarist or extra-welfarist. A factor built for consumption is therefore not automatically the right weight for a future QALY, a point the differential discounting page develops.

Computing factors from a constant rate or a schedule

Most evaluations use one constant rate, and the factor then follows from a single formula. Attema and colleagues give the factor as one divided by one plus the rate, raised to the power of the period, and note that because of compounding the farther in the future an outcome occurs, the less weight it gets.

$$d_t = \frac{1}{(1+r)^t} = \left( \frac{1}{1+r} \right)^{t}$$

where $d_t$ is the discount factor for year $t$, $r$ is the annual discount rate as a decimal and $t$ is the number of years after the reference date. The bracketed term is the one-period factor, 0.9662 at 3.5%, and the factor for year $t$ is that number raised to the power $t$. Because factors multiply, the factor between two future years is the ratio of their factors, so at 3.5% the year-10 factor divided by the year-5 factor equals 0.8420, the same as the factor for 5 years. This lets an analyst move a model's reference date without recomputing every weight.

A schedule of rates is handled one year at a time. The HM Treasury Green Book supplementary guidance states that its declining rates are forward rates, not spot rates, so each year's factor is the previous factor divided by one plus that year's rate.

$$d_t = \frac{d_{t-1}}{1+r_t}, \qquad d_0 = 1$$

where $r_t$ is the rate that applies in year $t$ alone. At 3.00% for years 1 to 30 the year-30 factor is 0.4120, and dividing by 1.025 for year 31 gives 0.4019, the value printed in the guidance's own table. Inverting the constant-rate formula gives the average rate implied by any factor, $d_t^{-1/t}$ minus 1, which for the Green Book's year-40 factor of 0.3218 is about 2.87% a year, neither of the two published rates.

Continuous-time models use $e^{-\rho t}$ in place of $(1+r)^{-t}$, with $\rho = \ln(1+r)$, and the present value page converts annual rates to per-cycle and continuous equivalents.

Discount factors at the NICE and Green Book rates

The rate a body sets fixes every factor in an analysis, so factors give a direct comparison of guidelines. The NICE manual (PMG36, section 4.5.1) requires costs and health effects to be discounted at the same rate of 3.5% a year in the reference case, and section 4.5.3 allows the committee to consider 1.5% for both only when all its criteria are met, including that the technology is likely to restore people who would otherwise die or have a very severely impaired life to full or near-full health. The Green Book sets 3.00% for years 1 to 30, 2.50% for years 31 to 75 and 2.25% for years 76 to 125, and tells practitioners to multiply social costs and benefits in each future year by the factor for that year.

The Green Book schedule column matches Annex A of the supplementary guidance. A factor halves after about 20.1 years at 3.5%, 23.4 years at 3.00% and 46.6 years at 1.5%, so the choice between the two NICE rates moves the weight on long-term benefits a long way.

The last column reflects the Green Book's treatment of health. It uprates future real health effects by 2% a year, as the present value page explains.

Worked example with a one-off preventive treatment

All numbers here are illustrative. A one-off preventive treatment costs GBP 3,000,000 now, at year 0, and prevents a late complication, adding 50 QALYs in year 10 and another 50 QALYs in year 30 compared with no treatment.

1. Cost. The cost falls at the reference date, so its factor is 1 and its present value stays GBP 3,000,000.

2. Factors at 3.5%. The year-10 factor is 1 divided by 1.035 to the power 10, or 0.70892, and the year-30 factor is 1 divided by 1.035 to the power 30, or 0.35628.

3. Discounted QALYs at 3.5%. The year-10 gain is worth 50 × 0.70892 = 35.45 QALYs and the year-30 gain 50 × 0.35628 = 17.81 QALYs, a total of 53.26 against 100 undiscounted.

4. Cost per QALY at 3.5%. The ratio is 3,000,000 / 53.26 = GBP 56,327 per QALY gained, against 3,000,000 / 100 = GBP 30,000 undiscounted.

5. Factors at 1.5%. The factors become 0.86167 and 0.63976, the gains 43.08 and 31.99 QALYs, the total 75.07 and the ratio 3,000,000 / 75.07 = GBP 39,963 per QALY.

6. Green Book weights. With the Green Book factors of 0.74409 and 0.41199 and uprating factors of 1.21899 and 1.81136 (1.02 to the power 10 and 30), the net weights are 0.90705 and 0.74626. The gains are then worth 45.35 and 37.31, or 82.66 QALYs at today's value.

The undiscounted ratio falls inside the range of GBP 25,000 to GBP 35,000 per QALY that NICE's PMG36 (sections 6.3.4 to 6.3.8) uses when judging cost-effectiveness, while the reference-case ratio lies above GBP 35,000, where PMG36 says the committee needs an increasingly stronger case. The factors alone make that difference, since the cost has a weight of 1 and every QALY a weight below 1. A preventive technology like this one would not obviously meet the section 4.5.3 criteria, so the committee would be unlikely to consider the 1.5% result, a non-reference-case analysis in PMG36. The same health gain carries far more weight under the Green Book weights than in a NICE appraisal.

Reading discount factors in models and reports

In a Markov model each cycle's costs and QALYs are multiplied by that cycle's factor before they are summed. The convention for when amounts fall within a cycle changes every factor, so it has to match the half-cycle correction and cycle length.

PMG36 (section 4.10.4) asks for the disaggregated costs, QALYs and life years to be presented with and without discounting, which lets a reader see how much of the result comes from the factors and not from the clinical inputs. A sensitivity analysis over the rate changes the whole set of factors at once.

The reference date is also a choice. Attema and colleagues report that under differential discounting the starting year can change the ICER, and that a fixed rule such as the year of treatment initiation avoids strategic choice of it.

Discount factor and neighbouring terms

Several terms describe parts of the same calculation and are easily run together. Each answers a different question.

• Discount rate. The discount rate is the percentage chosen by a body or an analyst, and the factor is the weight that rate implies for one particular year.
• Present value. A present value is the sum of amounts multiplied by their factors, and the net present value subtracts the present value of costs from that of benefits.
• Annuity factor. The annuity factor is the sum of the factors over a run of years.
• Hyperbolic discount function. Hyperbolic discounting uses a discount function whose implied rate falls with delay.
• Price discounts. A confidential discount on a medicine's list price is a commercial term with no link to timing.

Limitations and common misreadings

The factor carries the value judgements of the rate behind it, and compounding magnifies them over long horizons. At 3.5% the factor for year 100 is 0.0321, against 0.0778 under the Green Book schedule. Attema and colleagues observe that constant discounting over a very long horizon leaves close to zero weight for future generations, which has been argued to be unfair, and that this matters in health economics for programmes against infectious diseases whose herd protection spans many cohorts. They note that the French and UK guidelines recommend declining rates for projects longer than 30 years, unlike the NICE reference case.

Several errors come from the arithmetic and not from the choice of rate.

• Shifting the year count. The Green Book table gives year 0 a factor of 1, and applying the year-1 factor to amounts that fall at the start of the analysis lowers every weight by one year's discount.
• Reading the factor as a probability. A survival probability also lies between 0 and 1 and multiplies the same cells, but it reflects the risk of death, not time.

Sources

• Attema AE, Brouwer WBF, Claxton K. Discounting in economic evaluations. PharmacoEconomics. 2018;36(7):745-758. doi:10.1007/s40273-018-0672-z
• Frederick S, Loewenstein G, O'Donoghue T. Time discounting and time preference: a critical review. Journal of Economic Literature. 2002;40(2):351-401. doi:10.1257/002205102320161311
• HM Treasury. The Green Book (2026). London: HM Treasury; updated 28 October 2026. Chapter 6, Discounting, and chapter 8, Life and health. https://www.gov.uk/government/publications/the-green-book-appraisal-and-evaluation-in-central-government/the-green-book-2026
• HM Treasury. Green Book supplementary guidance: discounting. London: HM Treasury; October 2026. Chapters 2 and 3 and Annex A. https://assets.publishing.service.gov.uk/media/6ae0b34016428df537606216/Green_Book_supplementary_guidance_-_discounting.pdf
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Sections 4.5, 4.10.4 and 6.3.4 to 6.3.8. Published 31 January 2022, last updated 13 October 2026. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation