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20 | 20 | #' |
21 | 21 | #' Calculate present value for a payoff with dynamic (lifecycle) pricing and dynamic uptake (stacked cohorts). |
22 | 22 | #' |
23 | | -#' Let us partition time as follows. |
| 23 | +#' Suppose payoffs in relation to patients receiving treatment (such as costs or health outcomes) occur over timesteps \eqn{t=1, ..., T}. Let us partition time as follows. |
24 | 24 | #' |
25 | | -#' - Suppose \eqn{j=1,...,T} indexes the time at which the patient begins treatment, where \eqn{T} is the time horizon of the decision-maker. |
| 25 | +#' - Suppose \eqn{j=1,...,T} indexes the time at which the patient begins treatment. |
26 | 26 | #' - Suppose \eqn{k=1,...,T} indexes time since initiating treatment. |
27 | 27 | #' |
28 | | -#' In general, \eqn{t=j+k-1}, and we are interested in \eqn{t=1,...,T}. |
| 28 | +#' In general, \eqn{t=j+k-1}, and we are interested in the set of \eqn{(j,k)} such that \eqn{1 \leq t \leq T}. |
| 29 | +#' |
| 30 | +#' For example, \eqn{t=3} comprises: |
| 31 | +#' |
| 32 | +#' - patients who are in the third timestep of treatment that began in timestep 1: (j,k)=(1,3); |
| 33 | +#' - patients who are in the second timestep of treatment that began in timestep 2, (j,k)=(2,2); and |
| 34 | +#' - patients who are in the first timestep of treatment that began in timestep 3, (j,k)=(3,1) |
29 | 35 | #' |
30 | 36 | #' The [Present Value](https://en.wikipedia.org/wiki/Present_value) of a cashflow \eqn{p_k} for the \eqn{u_j} patients who began treatment at time \eqn{j} and who are in their \eqn{k}th timestep of treatment is as follows |
31 | 37 | #' \deqn{PV(j,k,l) = u_j \cdot p_k \cdot R_{j+k+l-1} \cdot (1+i)^{2-j-k}} |
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