Annuity Factor

Annuity Factor in Health Economic Evaluation

An annuity factor is a present-value multiplier built by adding one discount factor for each period in which an equal amount falls, so multiplying a level yearly cost, QALY gain or monetised benefit by the factor gives the present value of the whole stream in one step. Dividing a present value by the same factor runs the calculation the other way and gives the equal yearly amount that a lump sum is worth over a set number of years. Health economists meet the factor whenever a model counts a sustained treatment effect, annualises the price of equipment, sets instalments for a one-off therapy or discounts the years of life lost in a DALY. The factor has no inputs other than the rate and the length of the stream, together with a convention for when each amount falls, yet the rate fed into it carries the welfare judgements about time that lie behind discounting.

Summing discount factors for a level stream

Each future amount is weighted by one divided by one plus the rate, raised to the number of periods of delay, and the present value of a stream is the sum of the weighted amounts. When every amount is the same, the amount comes outside the sum and what remains is a geometric series with a closed form. The OpenStax finance text by Dahlquist and Knight gives the present value of an ordinary annuity as the payment times this expression, which it calls the annuity factor.

$$A(n,r) = \sum_{t=1}^{n} \frac{1}{(1+r)^{t}} = \frac{1 - (1+r)^{-n}}{r}$$

where $A(n,r)$ is the annuity factor, $r$ is the discount rate per period written as a decimal, $n$ is the number of equal amounts and $t$ counts the periods, with the first amount one period after the reference date. A level amount $X$ per period then has a present value of $X \times A(n,r)$.

The ordinary form assumes each amount falls at the end of its period. In an annuity due the first amount falls on the reference date, so every amount is discounted for one period less. Dahlquist and Knight convert an ordinary annuity into an annuity due by multiplying by one plus the rate, and their five-payment example at 8% gives an annuity-due value exactly 8% above the ordinary value.

$$\ddot{A}(n,r) = (1+r) \times A(n,r)$$

where $\ddot{A}(n,r)$ is the annuity-due factor and the other symbols are as defined above.

Two limits help in checking a spreadsheet. As $n$ grows without bound the factor approaches $r^{-1}$, the value of a perpetuity, given in the same text as the constant payment divided by the rate. At 3.5% no level stream, however long, is worth more than 28.57 times its yearly amount. As $r$ approaches zero the factor approaches $n$, the plain count of periods.

A continuous version applies where an outcome is treated as accruing evenly through time, as in the Fox-Rushby and Hanson DALY formulae that Larson works from. Larson shows that, without age weighting, discounted years of life lost reduce to the expression below.

$$YLL(r) = \int_{0}^{L} e^{-rt}\, dt = \frac{1 - e^{-rL}}{r}$$

where $YLL(r)$ is the discounted years of life lost for one death, $L$ is the remaining life expectancy at the age of death, $r$ is the continuous discount rate and $t$ is time since the death.

Where the factor appears in health economic evaluation

The most frequent use is the discounted health gain from a sustained effect. If a treatment adds a constant number of QALYs a year for $n$ years, the discounted gain is the yearly gain multiplied by $A(n,r)$. Section 4.5.1 of the NICE manual (PMG36) states that cost-effectiveness results should reflect the present value of the stream of costs and benefits over the time horizon, with costs and health effects discounted at 3.5% a year in the reference case.

Burden of disease measures use the same arithmetic. In Larson's example a man dies on his 55th birthday with a remaining life expectancy of 10 years. At 3%, with each lost year counted at its start, the spreadsheet gives 8.786 discounted years of life lost, which is the annuity-due factor $A(10, 0.03) \times 1.03$, while the continuous formula gives 8.6394. Larson judges the 1.7% gap minor and unlikely to change any ranking in a cost-effectiveness analysis that uses the DALY.

Costing uses the reciprocal. Turner and colleagues describe the economic cost of a capital item as its replacement cost divided by an annualisation factor based on the expected lifespan and an assumed discount or interest rate. Because the factor is smaller than the lifespan, 4.58 against 5 in their example, the annualised economic cost exceeds the financial cost from straight-line depreciation, and 4.58 matches $A(5, 0.03) = 4.580$. The method is set out on the annualisation and capital cost pages. Instalment schemes for one-off therapies divide a price by an annuity factor at a negotiated financing rate, in its annuity-due form when the first instalment is paid at treatment, as described under annuity payment.

The rate inside the factor and its welfare basis

The factor itself is mechanical, while the rate inside it is a value judgement. HM Treasury's Green Book (2026) bases discounting on social time preference, the principle that people generally prefer benefits sooner and costs later, and sets the social time preference rate (STPR) at 3.00% in real terms for years 1 to 30, 2.50% for years 31 to 75 and 2.25% from year 76. It treats the STPR as a real rate to be applied only to values already in real terms, and it warns that adding the inflation rate to the discount rate gives a mathematically incorrect result.

A stepped schedule breaks the single closed form, so the factor is built in bands. Assuming the 2.50% rate applies to years 31 to 40 on top of the discounting accumulated by year 30, a 40-year factor is calculated as follows.

$$A_{40} = A(30, 0.03) + (1.03)^{-30} \times A(10, 0.025) = 19.600 + 0.41199 \times 8.752 = 23.206$$

where $A_{40}$ is the stepped 40-year factor, 19.600 is the 30-year factor at 3%, 0.41199 is the discount factor at year 30 and 8.752 is the 10-year factor at 2.5%. A flat 3% would give 23.115, so the step adds less than half a per cent over this horizon.

The Green Book gives a willingness-to-pay value for a QALY of GBP 70,000 in 2020-21 prices. It adds that the social welfare from additional years of life rises with income, so the value of future real health effects should be uprated by 2% a year. A money value that grows at rate $g$ and is discounted at rate $r$ has the ordinary annuity factor at a net rate.

$$r^{*} = \frac{1+r}{1+g} - 1 = \frac{1.03}{1.02} - 1 = 0.0098$$

where $r^{*}$ is the net rate, $r$ is the STPR and $g$ is the uprating rate, here 3% and 2%.

Welfarist and extra-welfarist evaluation meet at this point, and both draw on welfare economics. Brouwer and colleagues describe cost-utility analysis as using something like need where cost-benefit analysis uses effective demand and willingness to pay, and they read QALYs as a measure of health rather than of utility. The annuity factor works identically under either view. What changes is the stream and its rate, with QALYs and costs both at 3.5% in the NICE reference case, against a money value of health growing at 2% under a 3% STPR in a Green Book appraisal. Where a rate is meant to reflect what capital would earn elsewhere, as with the social opportunity cost of capital, the same factor applies with that rate.

Worked example of a one-off therapy with a 30-year QALY gain (illustrative)

All figures in this example are illustrative. A one-off therapy for a severe inherited condition adds 0.5 QALY a year over standard care for 30 years, counted at the end of each year. It costs GBP 300,000 at treatment and GBP 2,000 a year more in monitoring, also paid at year ends.

1. The factors. At 3.5%, $(1.035)^{-30} = 0.35628$ and the factor is 18.392, and at 1.5%, $(1.015)^{-30} = 0.63976$ and the factor is 24.016. The discounting page uses these factors for a one-off treatment without running costs, and this example adds a yearly monitoring cost.

2. Discounted QALYs. At 3.5% the gain is 0.5 × 18.392 = 9.196 QALYs, and at 1.5% it is 0.5 × 24.016 = 12.008 QALYs, against 15 undiscounted.

3. Discounted costs. At 3.5% the cost is 300,000 + 2,000 × 18.392 = GBP 336,784, and at 1.5% it is 300,000 + 2,000 × 24.016 = GBP 348,032.

4. The ICERs. At 3.5% the ICER is 336,784 / 9.196 = GBP 36,623 per QALY. At 1.5% it is 348,032 / 12.008 = GBP 28,983 per QALY. Undiscounted, 360,000 / 15 = GBP 24,000.

5. Why the rate matters so much here. Most of the cost falls at treatment and is not discounted, while all of the health gain is spread over time. Lowering the rate raises the QALY factor by 24.016 / 18.392 = 1.306 but the discounted cost by only 348,032 / 336,784 = 1.033.

6. Reading the result against PMG36. Section 6.3.4 says that below GBP 25,000 per QALY the decision is normally based on the cost-effectiveness estimate, and section 6.3.7 says that as the ICER rises within GBP 25,000 to GBP 35,000 the committee refers explicitly to the factors in section 6.3.5. At 3.5% the example lies above that range and at 1.5% inside it. Section 4.5.3 allows the 1.5% rate only for a technology for people who would otherwise die or have a very severely impaired life, which is likely to restore full or near-full health with benefits sustained over a very long period, and section 4.5.4 asks for a highly plausible case that benefits are maintained. Section 4.5.5 adds that the committee must be satisfied that irrecoverable costs, such as the acquisition cost, are captured in the model or mitigated through commercial arrangements.

7. Timing. If the QALY gain fell at the start of each year, the annuity-due factor at 3.5% would be 1.035 × 18.392 = 19.036, the gain 0.5 × 19.036 = 9.518 QALYs and, with costs unchanged, the ICER 336,784 / 9.518 = GBP 35,384.

8. The same therapy in a Green Book appraisal. With the QALY valued at GBP 70,000 and, as an assumption, uprated by 2% a year from the appraisal year, the health stream uses the net rate of 0.98%, so $A(30, 0.0098) = 25.882$. The benefit is 0.5 × 70,000 × 25.882 = GBP 905,870. Costs at the 3% STPR are 300,000 + 2,000 × 19.600 = GBP 339,200, giving a net present value of 905,870 − 339,200 = GBP 566,670. The GBP 70,000 figure is a societal valuation of health and not a NICE threshold, so this result answers a different question from the ICER.

Reading discounted totals through the factor

A factor can be read as the number of present-value years that a level stream is worth. The table gives the factor at the two NICE rates and the Green Book's first-band STPR, with the perpetuity value as the limit.

The gap between rates widens with the horizon. At 10 years the 1.5% factor is 9.222 / 8.317 = 1.109 times the 3.5% factor, while at 60 years it is 39.380 / 24.945 = 1.579 times as large, so the rate weighs most in appraisals of vaccines and one-off therapies with long-lasting benefits. The factor also gives a quick check on a model, since a Markov model with a constant yearly gain cannot produce more discounted QALYs than the gain multiplied by the factor for its horizon, and mortality and waning will pull the modelled total below that ceiling.

Boundaries with neighbouring concepts

A discount factor weights a single period, and the annuity factor sums those weights over a run of equal amounts, so the two coincide only for a one-period stream. Present value is the general calculation for any stream, and the factor is its shortcut for the level case.

Annualisation and annuity payment both divide by the factor, the first at the evaluation's discount rate and the second at a financing rate agreed between payer and manufacturer, and all of them rest on the time value of money.

Limitations and common errors

Most errors with the factor arise when a stream breaks one of its assumptions.

1. Using it for a stream that is not level. Survival and waning change the yearly amount, so the analysis needs a year-by-year sum. For a stream that can only decline, the first-year amount multiplied by the factor gives an upper bound.
2. Mixing timing conventions. Ordinary and annuity-due factors differ by one plus the rate, and a model that counts costs at the start of each year and QALYs at the end builds in a bias between them.
3. Pairing a nominal rate with real values. The Green Book's warning against adding inflation to the discount rate applies to the factor as much as to single amounts.
4. Using one closed form across a stepped schedule. Beyond year 30 under the Green Book STPR the factor has to be built in bands.
5. Confusing the factor with its reciprocal. Multiplying by the factor gives a present value, while dividing by the factor gives an equal yearly amount.
6. Treating continuous and discrete rates as the same. Larson shows that a discrete rate of $e^{0.03} - 1 = 0.03045$ reproduces the continuous 3% discount factor, which closes part of the gap between his discrete and continuous DALY figures.

Sources

• 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 (author copy https://eprints.whiterose.ac.uk/id/eprint/3789/)
• Dahlquist J, Knight R. Principles of Finance. Houston: OpenStax; 2022. Sections 8.1 Perpetuities and 8.2 Annuities. https://openstax.org/books/principles-finance/pages/8-2-annuities
• HM Treasury. The Green Book (2026). London: HM Treasury; updated 28 October 2026. Sections on discounting, assets and life and health. https://www.gov.uk/government/publications/the-green-book-appraisal-and-evaluation-in-central-government/the-green-book-2026
• Larson BA. Calculating disability-adjusted-life-years lost (DALYs) in discrete-time. Cost Effectiveness and Resource Allocation. 2013;11:18. https://doi.org/10.1186/1478-7547-11-18
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; published 31 January 2022, last updated 13 October 2026. Sections 4.5.1 to 4.5.5, 6.3.4 and 6.3.7. https://www.nice.org.uk/process/pmg36
• Turner HC, Sandmann FG, Downey LE, Orangi S, Teerawattananon Y, Vassall A, Jit M. What are economic costs and when should they be used in health economic studies? Cost Effectiveness and Resource Allocation. 2023;21:31. https://doi.org/10.1186/s12962-023-00436-w