Signature
EPV = sum_(t=0)^T [S_t * A / (1+r)^t]
| Inputs | Definition | Unit |
|---|---|---|
S_t | Probability, from the date of treatment, that the patient is alive and still meets the response criterion when the instalment for year t falls due, so that it is payable. It equals 1 for the payment at treatment | probability from 0 to 1 |
A | Instalment payable at the start of each year while the response criterion is met | currency per instalment |
r | Annual discount rate for costs in the economic evaluation, as a decimal | rate per year |
t | Number of whole years since treatment at which an instalment falls due, from 0 for the payment at treatment to T | years |
EPV | Expected present value at the date of treatment of the payments under the outcomes-based schedule | currency |
|---|
TYears since treatment at which the last instalment falls due, equal to the number of instalments minus 1, so 4 for a five-instalment schedule (years)
Function
Annuity payment schedule valuation function
Turns the upfront price of a one-off therapy into equal yearly instalments at an agreed financing rate, and values the resulting schedule, fixed or conditional on continued response, at the discount rate of the economic evaluation. The instalment is set so that the schedule matches the upfront price at the financing rate. The cost that enters a cost-effectiveness model is the present value of the expected payments at the evaluation's own rate, not the list price or the nominal sum of the instalments.
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Implementations
Excel
Expected present value of outcomes-based instalments with SUMPRODUCT
With the probabilities for years 0 to T in a range named PayProb and the matching years in a range named YearIndex, SUMPRODUCT weights each discounted instalment by the probability that it is payable.
=SUMPRODUCT(PayProb,Instalment/(1+DiscRate)^YearIndex)
Assumptions
Outcomes-based payments stop once response is lost
Each later instalment is due only if the patient meets the response criterion on that date, and payments stop when the criterion is not met, so S_t does not rise from one year to the next. Contract rules decide what happens for patients who die, move to another payer or are lost to follow-up.
Outcomes-based instalment fixed at the start
The instalment A is agreed at the start and is not recalculated when payments stop, so the manufacturer bears the cost of the instalments that are not paid.
Worked examples
Outcomes-based £1 million schedule with falling response
With instalments of £219,976 discounted at 3.5%, and probabilities of 0.90, 0.85, 0.80 and 0.75 that the patient is alive and still responding at years 1 to 4, the expected present value is about £888,304, about 11% below the upfront price. The article's £888,285 uses rounded discount factors. The reduction reflects payments avoided for patients whose benefit ends, and those patients also contribute fewer QALYs, so it is not a price discount.
T = 4; A = 219976; r = 0.035; t = [0,1,2,3,4]; S_t = [1,0.90,0.85,0.80,0.75]; EPV = 888303.82
Common errors
Payment probabilities that differ from those in the health model
Weighting payments by response probabilities from a different source from those that drive the QALYs breaks the link between paying and benefiting. The model can then credit benefit for which nothing is paid, or charge for benefit that it does not count.
Year-on-year continuation probabilities entered as payment probabilities
S_t is the probability from treatment that instalment t is payable. If a source reports the share of responders at one payment date who still respond at the next, those conditional values have to be multiplied together first. Entering conditional values of 0.90, 0.85, 0.80 and 0.75 directly gives about £888,304 in the worked example, against about £777,765 with the cumulative values 0.90, 0.765, 0.612 and 0.459.
Sources
Outcomes-based annuity payments discontinued when response is not sustained
Jørgensen J, Kefalas P. Annuity payments can increase patient access to innovative cell and gene therapies under England's net budget impact test. Journal of Market Access and Health Policy. 2017;5(1):1355203. Introduction, section on outcomes-based managed entry agreements, which describes payment over time for a one-off therapy in which payments can be discontinued if the patient does not sustain the desired response.
Outcome-based reduction of annuity payments and social discounting
Van Dyck W, Michelsen S, Veredas D, Huys I, Luyten J, Simoens S. When do annuity-based payments help to address the affordability challenge of funding advanced therapies? Insights from a budget impact simulation. Journal of Market Access and Health Policy. 2026;14(2):23. Section 3.1.2, equation 2, on discounting annuity-based payments at a social discount rate, and the discussion of models in which the annuity is reduced when the therapy works for less time than expected or the patient dies early.
Canonical Identity
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