Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Annuity payment schedule valuation function

f(P, n, i, r, S_t) = (A, PV, EPV)

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.

  • Annuity-due instalment for a one-off therapy price

    A = P * i / ((1 - (1+i)^(-n)) * (1+i))

    Gives the equal instalment paid at the start of each year, the first at the point of treatment, so that the n payments have the same present value at the financing rate i as the upfront price P. It is the present value of an annuity due rearranged for the payment. When payments fall at the end of each year instead, the final division by (1 + i) is dropped, which gives the ordinary annuity used to annualise capital costs.

  • Present value of a fixed annuity payment schedule at the evaluation discount rate

    PV = A * (1 - (1+r)^(-n)) * (1+r) / r

    Values n equal instalments A, the first at treatment, at the discount rate r used in the economic evaluation. It equals the sum of A divided by (1 + r) to the power t for t from 0 to n minus 1, written here in closed form. This present value, not the list price or the nominal total, is the cost of the therapy in the model. When r equals the financing rate, the result equals the upfront price.

  • Expected present value of an outcomes-based annuity payment schedule

    EPV = sum_(t=0)^T [S_t * A / (1+r)^t]

    Values a schedule in which each later instalment is due only if the patient still meets a pre-agreed response criterion on that date. The sum runs over the years t from 0, the payment at treatment, to T, the year of the last instalment, which is the number of instalments minus 1. Each instalment A is weighted by the probability S_t that it falls due and discounted at the evaluation rate r. The probabilities should be the same ones that drive the modelled health benefit, so that a patient who stops responding stops generating both QALYs and payments.