Present values with dynamic pricing and dynamic uptake

Calculate present value for a payoff with dynamic (lifecycle) pricing and dynamic uptake (stacked cohorts).

Usage

dynpv(
  uptakes = 1,
  payoffs,
  horizon = length(payoffs),
  tzero = 0,
  prices = rep(1, length(payoffs) + tzero),
  discrate = 0
)

Arguments

uptakes: Vector of patient uptake over time
payoffs: Vector of payoffs of interest (numeric vector)
horizon: Time horizon for the calculation (length must be less than or equal to the length of payoffs)
tzero: Time at the date of calculation, to be used in lookup in prices vector
prices: Vector of price indices through the time horizon of interest
discrate: Discount rate per timestep, corresponding to price index

Value

A tibble of class "dynpv" with the following columns:

j: Time at which patients began treatment
k: Time since patients began treatment
l: Time offset for the price index (from tzero)
t: Equals $j+k-1$
uj: Uptake of patients beginning treatment at time $j$ (from uptakes)
pk: Cashflow amount in today's money in respect of patients at time $k$ since starting treatment (from payoffs)
R: Index of prices over time $l+t$ (from prices)
v: Discounting factors, $(1+i)^{1-t}$, where i is the discount rate per timestep
pv: Present value, $PV(j,k,l)$

Details

Suppose payoffs in relation to patients receiving treatment (such as costs or health outcomes) occur over timesteps $t=1, ..., T$. Let us partition time as follows.

Suppose $j=1,...,T$ indexes the time at which the patient begins treatment.
Suppose $k=1,...,T$ indexes time since initiating treatment.

In general, $t=j+k-1$, and we are interested in the set of $(j,k)$ such that $1 \leq t \leq T$.

For example, $t=3$ comprises:

patients who are in the third timestep of treatment that began in timestep 1: (j,k)=(1,3);
patients who are in the second timestep of treatment that began in timestep 2, (j,k)=(2,2); and
patients who are in the first timestep of treatment that began in timestep 3, (j,k)=(3,1)

The Present Value of a cashflow $p_k$ for the $u_j$ patients who began treatment at time $j$ and who are in their $k$th timestep of treatment is as follows $$PV(j,k,l) = u_j \cdot p_k \cdot R_{j+k+l-1} \cdot (1+i)^{2-j-k}$$ where $i$ is the risk-free discount rate per timestep, $p_k$ is the cashflow amount in today’s money, and $p_k \cdot R_{j+k+l-1}$ is the nominal amount of the cashflow at the time it is incurred, allowing for an offset of $l = tzero$.

The total present value, $TPV(l)$, is therefore the sum over all $j$ and $k$ within the time horizon $T$, namely: $$TPV(l) = \sum_{j=1}^{T} \sum_{k=1}^{T-j+1} PV(j,k, l) \\ \; = \sum_{j=1}^{T} \sum_{k=1}^{T-j+1} u_j \cdot p_k \cdot R_{l+j+k-1} \cdot (1+i)^{2-j-k}$$

This function calculates $PV(j,k,l)$ for all values of $j$, $k$ and $l$, and returns this in a tibble.

Examples

# Simple example
pv1 <- dynpv(
   uptakes = (1:10), # 1 patient uptakes in timestep 1, 2 patients in timestep 2, etc
   tzero = c(0,5), # Calculations are performed with prices at times 0 and 5
   payoffs = 90 + 10*(1:10), # Payoff vector of length 10 = (100, 110, ..., 190)
   prices = 1.02^((1:15)-1), # Prices increase at 2\% per timestep in future
   discrate = 0.05 # The nominal discount rate is 5\% per timestep;
                   # the real discount rate per timestep is 3\% (=5\% - 3\%)
)
pv1
#> # A tibble: 110 × 9
#>        j     k     l     t    uj    pk     R     v    pv
#>    <int> <int> <dbl> <dbl> <int> <dbl> <dbl> <dbl> <dbl>
#>  1     1     1     0     1     1   100  1    1      100 
#>  2     1     1     5     1     1   100  1.10 1      110.
#>  3     1     2     0     2     1   110  1.02 0.952  107.
#>  4     1     2     5     2     1   110  1.13 0.952  118.
#>  5     1     3     0     3     1   120  1.04 0.907  113.
#>  6     1     3     5     3     1   120  1.15 0.907  125.
#>  7     1     4     0     4     1   130  1.06 0.864  119.
#>  8     1     4     5     4     1   130  1.17 0.864  132.
#>  9     1     5     0     5     1   140  1.08 0.823  125.
#> 10     1     5     5     5     1   140  1.20 0.823  138.
#> # ℹ 100 more rows
summary(pv1)
#> Summary of Dynamic Pricing and Uptake
#>      Number of cohorts:             10 
#>      Number of times:               2 
#>      Total uptake:                  55 
#> 
#>  Total and mean present values by timepoint: 
#> # A tibble: 2 × 3
#>   tzero  total  mean
#>   <dbl>  <dbl> <dbl>
#> 1     0 22123.  402.
#> 2     5 24426.  444.

# More complex example, using cashflow output from a heemod model

# Obtain dataset
democe <- get_dynfields(
   heemodel = oncpsm,
   payoffs = c("cost_daq_new", "cost_total", "qaly"),
   discount = "disc"
   )

# Obtain short payoff vector of interest
payoffs <- democe |>
   dplyr::filter(int=="new", model_time<11) |>
   dplyr::mutate(cost_oth = cost_total - cost_daq_new)

# Example calculation
pv2 <- dynpv(
   uptakes = rep(1, nrow(payoffs)),
   payoffs = payoffs$cost_oth,
   prices = 1.05^(0:(nrow(payoffs)-1)),
   discrate = 0.08
)
summary(pv2)
#> Summary of Dynamic Pricing and Uptake
#>      Number of cohorts:             10 
#>      Number of times:               1 
#>      Total uptake:                  10 
#>      Total present value:           16404.32 
#>      Mean present value:            1640.432