Build an annual commutation table (discrete ages) — commutation

The function builds annual commutation columns from x, lx, and an interest-rate specification. The interest rate is converted internally to an original log accumulation rate before forming discounted products.

Usage

commutation_table(lt, i, i_type = "effective", m = 1L, check = TRUE)

Arguments

lt: A life table object as produced by lifetable. It must contain columns x and lx. If available, qx or px may also be present, but dx is computed robustly from successive values of lx.
i: Numeric scalar. Annual interest-rate input.
i_type: Character string indicating the interest-rate type. Allowed values are "effective", "nominal_interest", "nominal_discount", and "force".
m: Positive integer. Conversion frequency for nominal rates. Ignored for i_type = "effective" and i_type = "force".
check: Logical. If TRUE, performs basic input checks.

Value

A tibble with columns x, lx, dx, v, Dx, Nx, Sx, Cx, Mx, and Rx.

Details

Constructs classical annual commutation functions $D_x$, $N_x$, $S_x$, $C_x$, $M_x$, and $R_x$ from a life table defined at integer ages, using compact actuarial notation.

This function follows the compact actuarial notation used throughout tidyactuarial: lt is the life table, i is the interest-rate input, i_type is the interest-rate type, and m is the conversion frequency for nominal rates.

The rate specification is validated through standardize_interest. If $i_e$ denotes the annual effective rate, then $$v = \frac{1}{1+i_e}.$$

The annual deaths are computed as $$d_x = l_x - l_{x+1},$$ closing the table with $l_{\omega+1}=0$. The main commutation functions are then computed as $$D_x = v^x l_x, \qquad C_x = v^{x+1} d_x,$$ Discounted products retain the original log accumulation rate to avoid intermediate factor overflow or underflow. This constructor always closes the terminal count, including when the input carries open-table metadata. The horizon is therefore different from the truncated integral in e_x. Reverse cumulative sums are used to obtain $N_x$, $S_x$, $M_x$, and $R_x$.

Examples

lt <- data.frame(
  x = 60:65,
  lx = c(100000, 99000, 97500, 95500, 93000, 90000)
)

commutation_table(
  lt = lt,
  i = 0.05
)
#> # A tibble: 6 × 10
#>       x     lx    dx     v    Dx     Nx     Sx     Cx    Mx     Rx
#>   <int>  <dbl> <dbl> <dbl> <dbl>  <dbl>  <dbl>  <dbl> <dbl>  <dbl>
#> 1    60 100000  1000 0.952 5354. 27423. 90450.   51.0 4048. 23116.
#> 2    61  99000  1500 0.952 5048. 22070. 63026.   72.8 3997. 19069.
#> 3    62  97500  2000 0.952 4734. 17022. 40956.   92.5 3924. 15072.
#> 4    63  95500  2500 0.952 4416. 12288. 23934.  110.  3831. 11148.
#> 5    64  93000  3000 0.952 4096.  7871. 11646.  126.  3721.  7317.
#> 6    65  90000 90000 0.952 3775.  3775.  3775. 3595.  3595.  3595.

commutation_table(
  lt = lt,
  i = 0.06,
  i_type = "nominal_interest",
  m = 12
)
#> # A tibble: 6 × 10
#>       x     lx    dx     v    Dx     Nx     Sx     Cx    Mx     Rx
#>   <int>  <dbl> <dbl> <dbl> <dbl>  <dbl>  <dbl>  <dbl> <dbl>  <dbl>
#> 1    60 100000  1000 0.942 2757. 13771. 44982.   26.0 1957. 11157.
#> 2    61  99000  1500 0.942 2571. 11014. 31211.   36.7 1931.  9200.
#> 3    62  97500  2000 0.942 2385.  8443. 20198.   46.1 1894.  7269.
#> 4    63  95500  2500 0.942 2200.  6058. 11755.   54.3 1848.  5375.
#> 5    64  93000  3000 0.942 2018.  3858.  5697.   61.3 1794.  3527.
#> 6    65  90000 90000 0.942 1840.  1840.  1840. 1733.  1733.  1733.