Simulate future lifetimes from a life table

Simulates curtate and, optionally, complete future lifetimes from a life table containing one-year death probabilities.

Usage

simulate_lifetime(
  lt,
  x,
  n_sim = 10000L,
  x_col = "x",
  qx_col = "qx",
  method = c("inverse", "multinomial", "antithetic"),
  frac = c("udd", "cml", "balducci", "none", "constant_force"),
  seed = NULL,
  include_distribution = FALSE,
  truncation = c("conditional", "error"),
  tol = 1e-10
)

Arguments

lt

A data frame or tibble containing the life table.

x

Nonnegative integer initial actuarial age. The age must appear explicitly in the column selected by x_col.

n_sim

Positive integer number of simulations.

x_col

Character scalar naming the age column.

qx_col

Character scalar naming the one-year death-probability column.

method

Simulation method for the curtate future lifetime: "inverse", "multinomial", or "antithetic".

frac

Fractional-age assumption used inside the year of death: "udd", "cml", "balducci", or "none". The legacy value "constant_force" is accepted as an alias for "cml".

seed

Optional nonnegative integer seed. When supplied, the caller's random-number state is restored after simulation.

include_distribution

Logical scalar. If TRUE, the probability distribution used for simulation is attached as a list-column.

truncation

Treatment of a life table whose available death probabilities do not exhaust the lifetime distribution. With "conditional", the historical behavior is retained and the probabilities are normalized conditional on death within the available ages, with a warning. With "error", the simulation stops.

tol

Nonnegative numeric tolerance used for age-grid and probability checks.

Value

A tibble with one row per simulation and standardized columns including sim_id, Kx, curtate_lifetime, Tx, complete_lifetime, death_age, method, and frac. The logical column distribution_conditioned indicates whether a truncated table was normalized conditionally.

Details

For a life aged \(x\), the curtate future lifetime has probability mass function $$ P(K_x=k)={}_kp_xq_{x+k}. $$ The selected life-table ages must therefore start at \(x\) and proceed in consecutive one-year steps. Missing or duplicated ages are rejected because the survival recursion would otherwise no longer represent \({}_kp_x\).

Under UDD, the fractional part of \(T_x\) is uniform conditional on the year of death. Under CML, it follows the conditional constant-force distribution. Under Balducci, it follows the corresponding conditional Balducci distribution. Positive annual survival is retained even below tol; only exact zero survival gives the established zero fractional lifetime under CML and Balducci. Tolerances do not replace a positive probability by zero.

References

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., and Nesbitt, C. J. (1997). Actuarial Mathematics. Second Edition. Society of Actuaries.

See also

simulate_lifetimes, mc_insurance, mc_annuity, mc_premium, mc_loss, mc_reserve

Other monte-carlo: mc_annuity(), mc_insurance(), mc_loss(), mc_multilife_status(), mc_premium(), mc_reserve(), simulate_lifetimes(), summary_mc()

Examples

lt <- tibble::tibble(
  x = 40:43,
  qx = c(0.10, 0.20, 0.30, 1)
)

lt |>
  simulate_lifetime(
    x = 40,
    n_sim = 25,
    method = "inverse",
    frac = "udd",
    seed = 123
  )
#> # A tibble: 25 × 14
#>    sim_id simulation_id     x   age method  frac  fractional    Kx
#>     <int>         <int> <int> <int> <chr>   <chr> <chr>      <int>
#>  1      1             1    40    40 inverse udd   udd            2
#>  2      2             2    40    40 inverse udd   udd            3
#>  3      3             3    40    40 inverse udd   udd            2
#>  4      4             4    40    40 inverse udd   udd            3
#>  5      5             5    40    40 inverse udd   udd            3
#>  6      6             6    40    40 inverse udd   udd            0
#>  7      7             7    40    40 inverse udd   udd            3
#>  8      8             8    40    40 inverse udd   udd            3
#>  9      9             9    40    40 inverse udd   udd            3
#> 10     10            10    40    40 inverse udd   udd            2
#> # ℹ 15 more rows
#> # ℹ 6 more variables: curtate_lifetime <int>, Tx <dbl>,
#> #   complete_lifetime <dbl>, death_age <int>, qx_at_death_year <dbl>,
#> #   distribution_conditioned <lgl>

lt |>
  simulate_lifetime(
    x = 40,
    n_sim = 25,
    method = "antithetic",
    frac = "cml",
    seed = 123
  )
#> # A tibble: 25 × 14
#>    sim_id simulation_id     x   age method     frac  fractional    Kx
#>     <int>         <int> <int> <int> <chr>      <chr> <chr>      <int>
#>  1      1             1    40    40 antithetic cml   cml            2
#>  2      2             2    40    40 antithetic cml   cml            3
#>  3      3             3    40    40 antithetic cml   cml            3
#>  4      4             4    40    40 antithetic cml   cml            1
#>  5      5             5    40    40 antithetic cml   cml            2
#>  6      6             6    40    40 antithetic cml   cml            3
#>  7      7             7    40    40 antithetic cml   cml            3
#>  8      8             8    40    40 antithetic cml   cml            1
#>  9      9             9    40    40 antithetic cml   cml            3
#> 10     10            10    40    40 antithetic cml   cml            0
#> # ℹ 15 more rows
#> # ℹ 6 more variables: curtate_lifetime <int>, Tx <dbl>,
#> #   complete_lifetime <dbl>, death_age <int>, qx_at_death_year <dbl>,
#> #   distribution_conditioned <lgl>