Net premium for two-life insurance by the equivalence principle

Computes the net benefit premium for a joint-life or last-survivor insurance contract.

Usage

premium_xy(
  lt,
  x = NULL,
  y = NULL,
  i = NULL,
  i_type = "effective",
  m = 1L,
  type = c("whole", "term", "endowment", "pure_endowment"),
  benefit = 1,
  n = Inf,
  h = 0L,
  k = 1L,
  frac = c("UDD", "CF", "CML", "Balducci"),
  timing = c("due", "immediate"),
  premium_start = c("issue", "deferred"),
  n_prem = NULL,
  status = c("joint", "last"),
  woolhouse = c("none", "first", "second"),
  output = c("value", "summary", "audit", "table"),
  tidy = NULL,
  check = TRUE,
  tol = 1e-10,
  ...
)

Arguments

lt: A life table, a list of two life tables list(lt_x, lt_y), or a two-life contract created with life_contract.
x: Integer actuarial age for the first life. Optional for a life_contract.
y: Integer actuarial age for the second life. Optional for a life_contract.
i: Numeric scalar. Annual interest-rate input. Optional for a life_contract.
i_type: Interest-rate type: "effective", "nominal_interest", "nominal_discount", or "force".
m: Positive integer. Conversion frequency for nominal rates.
type: Insurance type: "whole", "term", "endowment", or "pure_endowment".
benefit: Nonnegative insurance benefit.
n: Insurance term in years after deferment. Use Inf for whole-life insurance.
h: Nonnegative integer deferment period in years.
k: Positive integer. Number of premium payments per year.
frac: Fractional-age assumption: "UDD", "CF", "CML", or "Balducci".
timing: Premium timing: "due" or "immediate".
premium_start: Start of premiums: "issue" or "deferred".
n_prem: Premium-paying term in years. A fractional value is permitted when n_prem * k is an integer.
status: Two-life status: "joint" or "last". For a life_contract, the value is inferred from lives unless supplied explicitly.
woolhouse: Woolhouse approximation order for the premium annuity: "none", "first", or "second".
output: Output level. "value" returns the annualized premium; "summary" returns a compact one-row result; "audit" returns the equivalence components in long format. "table" is accepted as a deprecated alias for "summary".
tidy: Deprecated compatibility argument. TRUE maps to output = "summary" and FALSE to output = "value".
check: Logical scalar. If TRUE, validates inputs.
tol: Numeric tolerance for integer-grid checks.
...: Transitional compatibility for older calls using mortality_table, age_x, age_y, rate, rate_type, insurance_type, term_years, deferment_years, payments_per_year, premium_timing, premium_term_years, and cohort.

Value

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

For output = "summary", a one-row tibble with six columns: annualized premium, premium per payment, payment frequency, APV of benefits, APV of the premium annuity, and equivalence residual.

For output = "audit", a long-format tibble.

Details

For premiums payable $k$ times per year, the function distinguishes between the annualized premium $P^{(k)}$ and the amount paid at each installment, $P^{(k)} / k$.

Let $Z$ denote the present-value random variable of the two-life insurance benefit and let $Y^{(k)}$ denote the present value of the contingent premium annuity normalized to an annual payment rate of 1. The loss at issue is $$ L_0 = Z - P^{(k)}Y^{(k)}. $$

The equivalence principle gives $$ P^{(k)} = \frac{\operatorname{APV}(\text{benefits})} {\operatorname{APV}(\text{premium annuity})}. $$

Since annuity_xy assigns amount $1/k$ to each k-thly payment, this quotient is the annualized premium. The installment is $$ P_{\text{per payment}} = \frac{P^{(k)}}{k}. $$

Standard whole-life, term, and endowment benefits are valued through insurance_xy. A pure endowment is valued directly as the discounted benefit times status survival at maturity, retaining the issue-age conditioning, including for deferred last-survivor benefits. The identity endowment minus term remains a mathematical check, rather than the numerical calculation. Tables and horizons are validated without inventing terminal mortality. The premium annuity must have a positive finite representable APV; the output is one contract, not parallel scenarios.

A contract assembled with pipes may be valued directly:


life_contract(...) |>
  add_insurance(...) |>
  add_premium_schedule(...) |>
  premium_xy(output = "summary")

Explicit arguments supplied to premium_xy() override values stored in the contract components.

Examples

lt <- data.frame(
  x = 40:100,
  lx = round(100000 * exp(-0.012 * (0:60)^1.35))
)
lt$lx[nrow(lt)] <- 0

premium_xy(
  lt = lt,
  x = 60,
  y = 62,
  i = 0.05,
  type = "term",
  status = "joint",
  benefit = 100000,
  n = 20,
  k = 12,
  n_prem = 10,
  output = "summary"
)
#> # A tibble: 1 × 6
#>   premium_annualized premium_per_payment payments_per_year apv_benefits
#>                <dbl>               <dbl>             <int>        <dbl>
#> 1             11955.                996.                12       63020.
#> # ℹ 2 more variables: apv_premium_annuity <dbl>, equivalence_residual <dbl>

life_contract(
  lt = lt,
  lives = "joint",
  x = 60,
  y = 62,
  i = 0.05
) |>
  add_insurance(
    type = "term",
    benefit = 100000,
    n = 20
  ) |>
  add_premium_schedule(
    k = 12,
    n_prem = 10
  ) |>
  premium_xy(output = "summary")
#> # A tibble: 1 × 6
#>   premium_annualized premium_per_payment payments_per_year apv_benefits
#>                <dbl>               <dbl>             <int>        <dbl>
#> 1             11955.                996.                12       63020.
#> # ℹ 2 more variables: apv_premium_annuity <dbl>, equivalence_residual <dbl>