Signature
A = P * i / ((1 - (1+i)^(-n)) * (1+i))
| Inputs | Definition | Unit |
|---|---|---|
P | Upfront price of the one-off therapy that the instalment schedule replaces | currency, for example pounds |
i | Annual financing rate agreed between the payer and the manufacturer, entered as a decimal (0.05 for 5%). Above zero; at zero the instalment is P divided by n | rate per year |
n | Number of instalments, one a year, including the payment at treatment | count |
A | Instalment paid at the start of each year, the first at the point of treatment | currency per instalment, for example pounds |
|---|
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.
Computational function
Computational function: therapy price to annuity instalment, nominal total and present value
Takes the terms that a payer and a manufacturer negotiate, the upfront price, the number of yearly instalments and the financing rate, together with the discount rate of the evaluation, and returns the instalment, the nominal total and the present value that enters the economic evaluation. It chains HE-FM-AP-001, which sets the instalment at the financing rate, and HE-FM-AP-002, which values the instalments at the discount rate, so the second step uses the result of the first and the two rates play different parts. A financing rate of zero, as in an interest-free split, is handled as a separate case because the general instalment formula then divides zero by zero.
Inputs and outputs:
P: Upfront price of the one-off therapy; required, above zero. Unit: currency.;n: Number of yearly instalments, the first at treatment; required, a whole number of 1 or more. Unit: count.;i: Annual financing rate agreed in the contract, as a decimal; required, zero or above. Unit: rate per year.;r: Annual discount rate for costs in the evaluation, as a decimal, for example 0.035 in the NICE reference case; required, zero or above. Unit: rate per year.;A: Instalment paid at the start of each year. Unit: currency.;T_nom: Nominal total of the instalments, n times A. Unit: currency.;PV: Present value of the instalments at the evaluation discount rate, at the date of treatment. Unit: currency.Assumption: Instalments are equal, fixed and paid at the start of each year, the first at treatment, and every instalment is paid once the patient is treated. For an outcomes-based schedule the present value is replaced by the expected present value HE-FM-AP-003.
Worked example (£1 million therapy, five instalments, 5% financing, 3.5% discounting): The instalment is about £219,976, the nominal total about £1,099,880 and the present value about £1,027,965, about 2.8% above the upfront price because financing costs more than the evaluation discounts. The article's £1,027,948 uses rounded discount factors.
P = 1000000; n = 5; i = 0.05; r = 0.035; A = 219976; T_nom = 1099880; PV = 1027965Worked example (Financing rate equal to the discount rate): With both rates at 3.5% the instalment is about £213,992 and the nominal total about £1,069,958, but the present value equals the £1 million price, so the schedule leaves the incremental cost-effectiveness ratio unchanged.
P = 1000000; n = 5; i = 0.035; r = 0.035; A = 213992; T_nom = 1069958; PV = 1000000Worked example (Same schedule valued at a 1.5% discount rate): Where the committee considers a non-reference-case rate of 1.5%, which PMG36 section 4.5.3 allows only when all three criteria are met (people who would otherwise die or have a very severely impaired life, likely restoration to full or near-full health, and benefits sustained over a very long period) and which applies to health effects as well as costs, the same 5% schedule has a present value of about £1,067,848, about 6.8% above the price, because the gap between the two rates is wider.
P = 1000000; n = 5; i = 0.05; r = 0.015; A = 219976; T_nom = 1099880; PV = 1067848Excel:
=PMT(FinRate,NInstal,-Price,0,1); =NInstal*Instalment; =PV(DiscRate,NInstal,-Instalment,0,1)Three cells that return A, T_nom and PV, with the instalment cell named Instalment. With a type argument of 1, PMT and PV place payments at the start of each year, and they return P divided by n and n times A when a rate is zero.R:
annuity_schedule <- function(P, n, i, r) { A <- if (i == 0) P / n else P * i / ((1-(1+i)^(-n)) * (1+i)); pv <- if (r == 0) n * A else A * (1-(1+r)^(-n)) * (1+r) / r; c(instalment = A, nominal_total = n * A, present_value = pv) }Returns a named vector, with separate branches for zero rates.Python:
def annuity_schedule(P, n, i, r): A = P / n if i == 0 else P * i / ((1-(1+i)**(-n)) * (1+i)); pv = n * A if r == 0 else A * (1-(1+r)**(-n)) * (1+r) / r; return A, n * A, pvReturns the instalment, the nominal total and the present value as a tuple, with no imports needed.Test (Zero financing rate returns equal shares of the price): With a financing rate of zero the instalment equals the price divided by the number of instalments. Expected result: TRUE. Excel check:
=ABS(PMT(0,NInstal,-Price,0,1)-Price/NInstal)<1E-9Test (Present value above the price only when financing exceeds discounting): With the instalment set at the financing rate, the present value at the discount rate exceeds the price exactly when the financing rate is the higher of the two. Expected result: TRUE whenever the rates differ. Excel check:
=(PV(DiscRate,NInstal,-PMT(FinRate,NInstal,-Price,0,1),0,1)>Price)=(FinRate>DiscRate)Common error (Applying the general formula to an interest-free split): Entering a financing rate of zero in the explicit instalment formula divides zero by zero, which gives #DIV/0! in Excel and NaN in R or Python. An interest-free split takes the P divided by n branch, which in the worked example gives £200,000 a year and a present value at 3.5% of about £934,616.
Source: 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 3.1.3, equation 4 (annuity due as a function of the principal), and section 3.1.2, equation 2, which discounts annuity payments at a social discount rate r to compare them with upfront payment.
A = P * i / ((1 - (1+i)^(-n)) * (1+i)); T_nom = n * A; PV = A * (1 - (1+r)^(-n)) * (1+r) / r
Try this function
Implementations
Excel
Annuity-due instalment with Excel PMT
Excel PMT with a type argument of 1 places each payment at the start of the period. The price is entered as a negative present value so that the instalment is returned as a positive amount. The explicit form =PriceFinRate/((1-(1+FinRate)^(-NInstal))(1+FinRate)) gives the same result for any financing rate above zero.
=PMT(FinRate,NInstal,-Price,0,1)
Assumptions
First annuity instalment paid at treatment
Each instalment falls at the start of a year and the first is paid at the point of treatment, as in the annuity-due set-up of Van Dyck et al. With payments at the end of each year the ordinary annuity form applies instead, and it gives a larger instalment for the same price and rate.
Fixed financing rate and equal annuity instalments
The financing rate is fixed for the whole schedule and every instalment is the same size. The rate is a commercial term set in negotiation rather than by HTA methods guidance, and reflects the cost to whoever finances the deferral of waiting for the money and of the risk that later payments are not made.
Worked examples
£1 million therapy paid in five instalments at 5% financing
A one-off therapy priced at £1,000,000 is paid in five yearly instalments, the first at treatment, at a financing rate of 5%. The instalment is about £219,976 and the nominal total is about £1,099,880, about 10% more than the upfront price, as in the article's worked example.
P = 1000000; i = 0.05; n = 5; A = 219976
€1 million advanced therapy over five years at a 3% bond rate
Van Dyck et al. price an advanced therapy at €1 million paid with five annuities. At a 3% bond rate the instalment due at the start of each year is about €211,995, and at a 10% corporate bond rate it rises to about €239,816.
P = 1000000; i = 0.03; n = 5; A = 211995
Common errors
End-of-year annuity used when the first instalment is paid at treatment
Applying the ordinary annuity formula, without the division by (1 + i), to a schedule whose first payment falls at treatment overstates every instalment by a factor of (1 + i). In the £1 million example it gives about £230,975 instead of £219,976, and the five payments then have a present value at 5% of £1.05 million rather than the price.
Sources
Annuity due for advanced therapy payments in a budget impact simulation
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 3.1.3, equation 4, which gives the annuity due at the beginning of each payment period as a function of the principal, and the €1 million, five-year example at 3% and 10%.
Canonical Identity
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