Signature
Q_s = q * (1 + r) * (1 - (1 + r)^(-T)) / r
| Inputs | Definition | Unit |
|---|---|---|
q | Amount arising in each year, the same every year | QALYs, life years or currency per year |
r | Annual discount rate written as a decimal | a proportion per year, above zero in this closed form |
T | Number of yearly amounts, falling in years 0 to T minus 1 | years, a whole number |
Q_s | Present value of the T yearly amounts counted at the start of each year | same as q |
|---|
Function
Discounting of yearly cost and QALY streams to present-value totals
Maps a stream of yearly costs or QALYs, an annual discount rate, a time horizon and a timing convention to one discounted total, by multiplying each year's amount by its discount factor and adding the results. Totals for costs and for QALYs are formed separately and then compared across options, for example as an incremental cost per QALY. The present value of a single amount, the general sum over a stream and separate rates for costs and health are on the Discount Rate page (HE-FM-DR-001 to HE-FM-DR-003), and unequal rates on the Differential Discounting page (HE-FN-DDIS-001). This package adds the totals of a level yearly stream counted at the end of each year, at the start of each year and continuously, and the discounted cost per QALY of a one-off treatment with a lasting gain, as in the article's worked example.
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Implementations
Excel
Start-of-year discounted level stream with Excel PV
The fifth argument of PV set to 1 places each amount at the start of its period. The second formula is the same closed form written out.
=PV(DiscRate,Years,-YearlyAmount,0,1); =YearlyAmount*(1+DiscRate)*(1-(1+DiscRate)^-Years)/DiscRate
Assumptions
First yearly amount falls at the start of the analysis
The first amount arises in year 0 and carries a factor of 1, and the last arises in year T minus 1. A model that starts the stream later shifts every factor and needs the general stream sum of HE-FM-DR-002.
Constant annual rate for a start-of-year discounted stream
One rate r applies to every year of the stream, with amounts in constant prices when they are costs.
Worked examples
Green Book benefit of GBP 1,000 in real terms from year 0 to year 5
The Green Book's Table 8 applies its factors of 1, 0.9662, 0.9335, 0.9019, 0.8714 and 0.8420 to social benefits constant at GBP 1,000 in real terms in years 0 to 5. The six amounts are worth about GBP 5,515 in present value terms at 3.5% (computed here for illustration from the formula).
q = 1000; r = 0.035; T = 6; Q_s = 5515.05
Larson's ten years counted at the start of each year at 3 per cent
Ten years of 1 counted at the start of each year and discounted at 3% are worth about 8.786 present years, Larson's discrete-time value for 10 years of life lost.
q = 1; r = 0.03; T = 10; Q_s = 8.7861
Article's 0.8 QALYs a year counted at the start of each year at 3.5 per cent
Counting the article's 0.8 QALYs a year for 30 years at the start of each year gives about 15.23 discounted QALYs instead of 14.71 (computed here for illustration).
q = 0.8; r = 0.035; T = 30; Q_s = 15.2286
Common errors
Timing convention changed between costs and QALYs
Counting QALYs at the start of each year while the analysis states end-of-year counting raises the article's discounted gain from 14.71 to 15.23 QALYs and lowers the cost per QALY at 3.5% from about GBP 20,389 to GBP 19,700 (computed here for illustration). The convention is stated and applied the same way to every option.
Sources
Discrete-time discounting with each year counted at the beginning of the year
Larson BA. Calculating disability-adjusted-life-years lost (DALYs) in discrete-time. Cost Effectiveness and Resource Allocation. 2013;11 article 18. Table 1 and the text before it, where time runs from 0 to 9 so that each of 10 years is counted at the beginning of the year, the same as a stream of payments each made at the beginning of the year, giving 8.786 at 3%.
Green Book table applying discount factors from year 0
HM Treasury. The Green Book (2026). London, HM Treasury, updated 5 February 2026. Table 8, Discounting and present values, which applies the factors 1.000 for year 0 to 0.8420 for year 5 to social benefits constant in real terms.
Canonical Identity
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