Gross premium under a simple expense-loaded equivalence principle

Converts a net-premium result from premium_x or premium_xy into an expense-loaded gross premium.

Usage

premium_gross(
  prem,
  alpha = 0,
  beta = 0,
  gamma = 0,
  k = NULL,
  output = c("value", "summary", "audit", "table"),
  tidy = NULL,
  check = TRUE,
  tol = 1e-10,
  ...
)

Arguments

prem: A one-row data frame or tibble. The preferred input is premium_x(..., output = "summary") or premium_xy(..., output = "summary"). It must contain an annualized net premium and the APV of the normalized premium annuity.
alpha: Nonnegative numeric scalar. Initial acquisition expense as a multiple of one gross premium installment. The expense at issue is $\alpha G^{(k)} / k$.
beta: Numeric scalar in $[0,1)$. Proportional expense charged on every gross premium installment.
gamma: Nonnegative numeric scalar. Fixed monetary expense incurred at every premium-payment date. Its APV is $\gamma k a^{(k)}$, because a_premiums is normalized with payments of $1/k$.
k: Optional positive integer payment frequency. By default it is inferred from payments_per_year or k in prem; if neither is present, k = 1 is used.
output: Output level. "value" returns the annualized gross premium, "summary" returns a compact one-row tibble, and "audit" returns the extended-equivalence components in long format. "table" is accepted as a compatibility alias for "summary".
tidy: Deprecated compatibility argument. TRUE maps to output = "summary" and FALSE to output = "value".
check: Logical scalar. If TRUE, validates the inputs.
tol: Nonnegative finite numeric tolerance. Monetary consistency is checked relatively to the compared amounts, without a one-currency-unit floor; frequency and integer checks use absolute tolerance.
...: Transitional compatibility. The deprecated argument payments_per_year is mapped to k.

Value

For output = "value", a numeric scalar containing the annualized gross premium $G^{(k)}$.

For output = "summary", a one-row tibble with six columns: annualized gross premium, gross premium per payment, annualized net premium, annualized loading, payment frequency, and equivalence residual.

For output = "audit", a long-format tibble with the APVs of gross premiums, benefits, and each expense component.

Details

The function preserves the package-wide distinction between an annualized premium and the amount paid at each of the k payment dates.

Let $a^{(k)}$ denote the APV of the premium annuity returned by premium_x or premium_xy. That annuity has $k$ payments per year, each of amount $1/k$; hence it represents an annual payment rate of 1.

If $P^{(k)}$ is the annualized net premium and $G^{(k)}$ is the annualized gross premium, the extended equivalence equation implemented is $$ G^{(k)}a^{(k)} = P^{(k)}a^{(k)} + \alpha\frac{G^{(k)}}{k} + \beta G^{(k)}a^{(k)} + \gamma k a^{(k)}. $$

Therefore, $$ G^{(k)} = \frac{P^{(k)} + k\gamma} {(1-\beta)-\alpha/(k a^{(k)})}. $$

The actual premium installment is $$ G_{\mathrm{per\ payment}} = \frac{G^{(k)}}{k}. $$

For k = 1, this reduces to the previous annual formula: $$ G = \frac{P_{\mathrm{net}}+\gamma} {(1-\beta)-\alpha/a}. $$

This intentionally simple model assumes that gamma is incurred at the same dates and under the same contingency as premium payments. Expenses with a different timing or contingency require their own APV and are not represented by gamma.

Preferred column names in prem are premium_annualized, premium_per_payment, payments_per_year, and apv_premium_annuity. Legacy annual tables using P, premium, P_net, a_premiums, or apv_premiums remain supported.

Legacy premium columns are accepted automatically only when k = 1, because their unit is ambiguous for subannual premiums.

When annualized and per-payment premiums are both supplied, both must be valid and agree in their units. A supplied payment frequency is validated even if k is explicit. Acquisition shares and expense products retain representable values without overflowing $k a^{(k)}$ first. The loading is evaluated directly from the expense equation; a small positive loading can be reported even when the rounded gross and net premiums coincide. The equivalence residual uses the same equation factored in annuity units, avoiding subtraction of overflowing APVs where possible; it is not forced to zero. Individual audit APVs may still be infinite. No finite result is promised when the annualized premium itself is outside numeric range.

Examples

net <- tibble::tibble(
  premium_annualized = 1200,
  premium_per_payment = 100,
  payments_per_year = 12,
  apv_premium_annuity = 10
)

premium_gross(
  net,
  alpha = 0.5,
  beta = 0.05,
  gamma = 20,
  output = "summary"
)
#> # A tibble: 1 × 6
#>   gross_premium_annualized gross_premium_per_payment net_premium_annualized
#>                      <dbl>                     <dbl>                  <dbl>
#> 1                    1522.                      127.                   1200
#> # ℹ 3 more variables: loading_annualized <dbl>, payments_per_year <int>,
#> #   equivalence_residual <dbl>

# Pipeline from a net-premium calculation
if (FALSE) { # \dontrun{
life_contract(
  lt = lt,
  lives = "single",
  x = 40,
  i = 0.05
) |>
  add_insurance(
    type = "term",
    benefit = 100000,
    n = 20
  ) |>
  add_premium_schedule(
    k = 12,
    n_prem = 10
  ) |>
  premium_x(output = "summary") |>
  premium_gross(
    alpha = 0.5,
    beta = 0.05,
    gamma = 20,
    output = "summary"
  )
} # }