Signature
rho = log(1 + r); Q_c = q * (1 - exp(-rho * T)) / rho
| Inputs | Definition | Unit |
|---|---|---|
r | Annual discount rate in the discrete form 1/(1 + r)^t, written as a decimal | a proportion per year, above zero |
q | Amount arising over each year, spread evenly through the year | QALYs, life years or currency per year |
T | Length of the flow from time 0 | years, and it need not be a whole number |
rho | Continuous discount rate equal to the natural log of (1 + r) | per year |
|---|---|---|
Q_c | Present value of the flow from time 0 to time T under continuous discounting | 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
Continuously discounted even flow from named cells
The first formula returns the continuous rate. In the second, EXP(minus rho times T) equals (1 + r) to the power minus T, so the total needs only the annual rate.
=LN(1+DiscRate); =YearlyAmount*(1-(1+DiscRate)^-Years)/LN(1+DiscRate)
Assumptions
Even flow within each year of a continuously discounted stream
The amount accrues at a constant pace through time, as a QALY gain does while a person lives in a health state. A cost paid at one moment, such as a treatment given at the start, is discounted with the single-value factor instead.
Continuous rate taken from the annual rate in force
rho is derived from the annual rate r set by the governing guidance, so the continuous and discrete factors agree at each whole year. Larson notes that a continuous rate of 0.03 corresponds to an annual rate of exp(0.03) minus 1, about 0.0305.
Worked examples
Larson's ten years discounted continuously at a continuous rate of 0.03
Ten years of 1 discounted continuously at 0.03 are worth 8.6394 present years, Larson's continuous-time value. The annual rate that gives the same factors is about 0.0305.
r = 0.0304545; q = 1; T = 10; rho = 0.03; Q_c = 8.6394
Article's 0.8 QALYs a year as an even flow discounted at 3.5 per cent
Treating the article's gain as an even flow over 30 years gives about 14.97 discounted QALYs, between 14.71 for end-of-year and 15.23 for start-of-year counting (computed here for illustration).
r = 0.035; q = 0.8; T = 30; rho = 0.0344; Q_c = 14.9696
Common errors
Annual discount rate used as the continuous rate for an even flow
Putting 0.035 in place of rho for the article's gain gives about 14.86 discounted QALYs instead of 14.97 (computed here for illustration). The annual and continuous rates differ, and the continuous rate is the natural log of (1 + r).
Continuous QALY totals set against end-of-year cost totals
Discounting QALYs continuously while costs or the comparator use end-of-year counting mixes conventions. In the article's example the cost per QALY at 3.5% falls from about GBP 20,389 to GBP 20,041, about 1.7% lower (computed here for illustration), only because of the convention.
Sources
Continuous discounting of a flow and its discrete-time equivalent
Larson BA. Calculating disability-adjusted-life-years lost (DALYs) in discrete-time. Cost Effectiveness and Resource Allocation. 2013;11 article 18. Equation 1, the integral of exp(minus rt) from 0 to L, equal to (1/r)(1 minus exp(minus rL)), with 8.6394 for 10 years at 3%, the statement that an annual rate of exp(0.03) minus 1 gives the identical factor, and Table 3 with the continuous-time adjustment factor.
Canonical Identity
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