Compute Monte Carlo net premiums for life contingencies

Estimates net premiums from simulated present values of benefits and premium annuities, while distinguishing the annualized premium from the amount paid at each premium date.

Usage

mc_premium(
  .data = NULL,
  col_Z = "pv_benefit",
  col_Y = "pv_annuity",
  col_P = "P",
  by = NULL,
  na_rm = TRUE,
  k = NULL,
  annuity_payment = NULL,
  tol = 1e-10,
  ...
)

Arguments

.data: A data frame or tibble containing simulated present values. Usually obtained after applying mc_insurance and mc_annuity to the same simulated lifetime sample.
col_Z: Character scalar naming the simulated present value of benefits. Default is "pv_benefit".
col_Y: Character scalar naming the simulated present value of the premium annuity. Default is "pv_annuity".
col_P: Character scalar naming the compatibility output containing the coefficient that directly multiplies col_Y. Default is "P". For a normalized $k$-thly premium annuity constructed with payment = 1 / k, this coefficient is the annualized premium.
by: Optional character vector of grouping columns. If supplied, the premium is estimated separately within each group. If by = NULL and .data is already grouped, the current grouping is used.
na_rm: Logical scalar. If TRUE, simulations with a missing benefit or annuity present value are removed as complete pairs. The same simulation rows are therefore used in both means.
k: Optional positive integer number of premium payments per year. By default it is inferred from payments_per_year or k in .data; if neither exists, k = 1 is used.
annuity_payment: Optional positive numeric scalar giving the amount paid at each date when col_Y was constructed. It is normally inferred from payment_per_payment, payment, or payment_annualized in the output of mc_annuity.
tol: Nonnegative numeric tolerance for consistency checks.
...: Transitional compatibility for older calls using data, benefit_col, annuity_col, and premium_col.

Value

A data frame or tibble containing the original simulations and standardized premium columns:

premium_annualized: Estimated annualized premium $\widehat P^{(k)}$.
premium_per_payment: Amount collected at each premium date.
payments_per_year: Premium payment frequency k.
premium_annuity_scale: Coefficient $\widehat q$ multiplying col_Y.
mc_equivalence_residual: Difference $\overline Z-\widehat q\overline{Y_c}$ within the estimation group.

The column selected by col_P stores premium_annuity_scale for backward compatibility. Unless col_P = "premium", a compatibility column premium is also created with the same scale coefficient. Existing grouping is used for estimation and removed from the result.

Details

Let $Z$ denote the simulated present value of benefits. Suppose the simulated premium annuity is $$ Y_c = c\sum_j v^{t_j}, $$ where $c$ is the amount used at each of the $k$ premium dates per year. The Monte Carlo coefficient that directly multiplies this annuity is $$ \widehat q = \frac{\overline Z}{\overline{Y_c}}. $$

The corresponding premium amount at each payment date is $$ \widehat P_{\mathrm{per\ payment}} = \widehat q c, $$ and the annualized premium is $$ \widehat P^{(k)} = k\widehat q c. $$

The recommended premium-annuity normalization uses payment = 1 / k in mc_annuity. In that case, $$ \widehat q = \widehat P^{(k)}, \qquad \widehat P_{\mathrm{per\ payment}} = \frac{\widehat P^{(k)}}{k}. $$

If instead payment = 1, then $\widehat q$ is the amount paid at each premium date, while the annualized premium is $k\widehat q$.

When na_rm = TRUE, missing values are removed jointly: a simulation contributes only if both col_Z and col_Y are observed. This prevents the numerator and denominator from being estimated from different simulated samples. Monetary premium amounts retain the ratio before intermediate overflow or underflow. A displayed annuity-scale coefficient may still be zero or infinite when the coefficient itself is outside double precision, while the per-payment and annualized amounts remain representable. The equivalence residual uses the ratio before that loss of range. Positive subnormal observations are retained using scaled means when an ordinary mean rounds to zero. A genuinely zero annuity exposure is rejected. A positive infinite payment_annualized metadata value is accepted only when it agrees with the product of a finite payment basis and finite frequency. The payment basis cannot be inferred from that infinite value alone; retain payment_per_payment or supply annuity_payment.

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.

Examples

simulated <- tibble::tibble(
  pv_benefit = c(800, 1200),
  pv_annuity = c(8, 12),
  payment_per_payment = 1 / 12,
  payment_annualized = 1,
  payments_per_year = 12
)

simulated |>
  mc_premium()
#> # A tibble: 2 × 11
#>   pv_benefit pv_annuity payment_per_payment payment_annualized payments_per_year
#>        <dbl>      <dbl>               <dbl>              <dbl>             <dbl>
#> 1        800          8              0.0833                  1                12
#> 2       1200         12              0.0833                  1                12
#> # ℹ 6 more variables: P <dbl>, premium_annualized <dbl>,
#> #   premium_per_payment <dbl>, premium_annuity_scale <dbl>,
#> #   mc_equivalence_residual <dbl>, premium <dbl>

# Here P is annualized because the annuity uses payment = 1 / 12.

# With payment = 1 at each monthly date, P is the monthly amount,
# while premium_annualized is twelve times P.
simulated_unit_payments <- tibble::tibble(
  pv_benefit = c(800, 1200),
  pv_annuity = c(96, 144),
  payment_per_payment = 1,
  payment_annualized = 12,
  payments_per_year = 12
)

simulated_unit_payments |>
  mc_premium()
#> # A tibble: 2 × 11
#>   pv_benefit pv_annuity payment_per_payment payment_annualized payments_per_year
#>        <dbl>      <dbl>               <dbl>              <dbl>             <dbl>
#> 1        800         96                   1                 12                12
#> 2       1200        144                   1                 12                12
#> # ℹ 6 more variables: P <dbl>, premium_annualized <dbl>,
#> #   premium_per_payment <dbl>, premium_annuity_scale <dbl>,
#> #   mc_equivalence_residual <dbl>, premium <dbl>