Signature
rho = log(1 + d); DF_t = exp(-rho * t)
| Inputs | Definition | Unit |
|---|---|---|
d | Discount rate per year in the discrete form, such as 0.035 | proportion per year |
t | Time at which the cost or health outcome occurs | years |
rho | Instantaneous discount rate equivalent to the annual rate | per year |
|---|---|---|
DF_t | Present value of one unit of cost or health received at time t | none |
Function
Continuous-time transition probabilities and expected state occupancy from transition intensities
Maps a matrix of constant transition rates (intensities) between health states to the probability of being in each state after any interval, by the matrix exponential, and to the expected, optionally discounted, time spent in each state. A fixed-cycle Markov model approximates this continuous process; the exact interval probabilities show what the approximation should reproduce. The notation follows the Continuous Model article and its illness-death example.
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Implementations
Excel
Continuous discount rate and factor from named cells
With AnnualRate and Years named, the first formula returns the continuous rate, held in ContRate, and the second the discount factor, held in DiscFactor.
=LN(1+AnnualRate); =EXP(-ContRate*Years)
Assumptions
Annual discount rate applied as compound annual discounting
d is the rate in the discrete formula 1 / (1 plus d) to the power t, as used in reference-case analyses; t is in years.
Same discount rate for costs and health unless stated
One rho is used for both costs and QALYs when the annual rates are equal; separate rates need separate rho values.
Worked examples
Continuous equivalent of a 3.5 per cent annual rate at 10 years
rho is log 1.035, about 0.0344, and the factor at 10 years is about 0.7089, the same as 1.035 to the power minus 10 (computed here for illustration).
d = 0.035; t = 10; rho = 0.0344; DF_t = 0.7089
Continuous equivalent of a 1.5 per cent annual rate at 10 years
rho is about 0.0149 and the factor at 10 years about 0.8617 (computed here for illustration).
d = 0.015; t = 10; rho = 0.0149; DF_t = 0.8617
Common errors
Using the annual rate as the continuous rate
exp(minus 0.035 times 10) gives 0.7047 instead of 0.7089, so every discounted total is slightly too low; in a continuous discount function the rate is not the same as in the discrete form.
Assuming a reference-case rate other than the one stated
The NICE reference case discounts costs and health effects at the same rate of 3.5 per cent a year, which corresponds to a continuous rate of about 0.0344; alternative analyses at 1.5 per cent need their own rho.
Sources
Continuous discount rate of ln(1 + r) equivalent to the discrete rate r
Murray CJL. Quantifying the burden of disease: the technical basis for disability-adjusted life years. Bulletin of the World Health Organization. 1994;72(3):429-445. Note h, p. 441: in a continuous discount function r is not precisely the same as r in the discrete form, whose formula is 1/(1 + r) to the power t; if the discount rate in the discrete formula is r, the equivalent result is achieved with a continuous discount rate of ln(1 + r).
Reference-case discount rate of 3.5 per cent for costs and health effects
National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 31 March 2026. Section 4.5.1: for the reference case, costs and health effects should be discounted at the same rate of 3.5 per cent per year; section 4.5.2: alternative analyses using rates of 1.5 per cent for both costs and health effects may be presented alongside the reference-case analysis in specific circumstances.
Canonical Identity
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