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Benefit reserves for single-life insurance

Source: R/reserve_x.R
reserve_x.Rd

Computes terminal benefit reserves at selected integer policy durations by the prospective or recursive method.

Usage

reserve_x(
  lt,
  x,
  i,
  i_type = "effective",
  m = 1L,
  type = c("whole", "term", "endowment"),
  n = Inf,
  h = 0L,
  benefit = 1,
  P = NULL,
  k = 1L,
  frac = c("UDD", "CF", "CML", "Balducci"),
  timing = c("due", "immediate"),
  premium_start = c("issue", "deferred"),
  n_prem = NULL,
  woolhouse = c("none", "first", "second"),
  t = NULL,
  method = c("prospective", "recursive"),
  output = c("summary", "value", "audit", "table"),
  tidy = NULL,
  check = TRUE,
  ...
)

Arguments

lt

A life table containing columns x and lx, or a single-life contract created with life_contract.

x

Integer actuarial age at issue. 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 interest rates.

type

Insurance type: "whole", "term", or "endowment".

n

Insurance term in years. Use Inf for whole-life insurance.

h

Nonnegative integer deferment period in years.

benefit

Positive insurance benefit.

P

Optional annualized premium \(P^{(k)}\). If NULL, it is calculated with premium_x by the equivalence principle.

k

Positive integer. Number of premium payments per year.

frac

Fractional-age assumption for exact k-thly premium annuities.

timing

Premium timing. Currently only "due" is supported, which gives reserves immediately before any premium payable at duration t.

premium_start

Start of premium payments: "issue" or "deferred".

n_prem

Premium-paying term in years. A fractional term is permitted when n_prem * k is an integer.

woolhouse

Woolhouse approximation order for the premium annuity: "none", "first", or "second".

t

Optional integer vector of policy durations. If NULL, all supported integer durations are returned.

method

Reserve method: "prospective" or "recursive". The recursive method currently supports annual premiums only (k = 1).

output

Output level. "summary" returns a compact reserve schedule, "value" returns a named numeric vector, and "audit" returns the reserve 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.

...

Transitional compatibility for older calls using mortality_table, age, rate, rate_type, insurance_type, term_years, premium, premium_term_years, and durations.

Value

For output = "summary", a tibble with at most six columns: duration, attained age, reserve, annualized premium, premium per payment, and method.

For output = "value", a named numeric vector.

For output = "audit", a long-format tibble containing future benefit APV, future premium APV, reserve, and premium amounts at each duration.

Details

The prospective method supports annual and true k-thly premiums payable in advance. The premium input P is always interpreted as the annualized premium \(P^{(k)}\). The amount paid at each premium date is \(P^{(k)} / k\).

At integer duration \(t\), conditional on survival to age \(x+t\), the prospective reserve is $$ {}_tV_x = \operatorname{APV}_t(\text{future benefits}) - P^{(k)} \operatorname{APV}_t(\text{future premium annuity}). $$

The premium annuity is normalized to an annual payment rate of 1, so each installment has amount \(1/k\). Therefore, the multiplier in the reserve formula is the annualized premium \(P^{(k)}\), not the installment \(P^{(k)}/k\).

Reserves are measured immediately before any premium payable at duration \(t\). This is the standard fully discrete terminal-reserve convention for premiums payable in advance.

For annual premiums, the recursive method uses $$ {}_{t+1}V_x = \frac{ ({}_tV_x + P_t)(1+i) - b_{t+1}q_{x+t} }{p_{x+t}}. $$

The implementation evaluates its algebraically equivalent backward form $${}_tV_x = v(p_{x+t}{}_{t+1}V_x + q_{x+t}b_{t+1}) - P_t.$$ This avoids amplification by repeated division by small survival probabilities. The boundary value is the prospective value at the last supported positive cohort within the contract horizon; it is not generally zero for whole-life insurance. Already defined maturity payments are retained. Unrequested extinct conditioning states are not traversed. A requested intermediate reserve conditional on an extinct cohort remains undefined.

Tables must have nonincreasing finite lx on an annual age grid. The recursive calculation retains the original log accumulation rate and forms death decrements before dividing survivor counts. Positive survival is obtained directly from counts, not as \(1-q_x\). The whole-life output horizon remains the last recorded age with positive lx; finite coverage must be supported by recorded table ages. These local conventions are retained separately from the implicit final interval in insurance APVs. t indexes durations of one contract, not independent policy scenarios. No clipping or zeroing of small/negative reserves is applied. Nonfinite future APV components do not in general identify their difference; exact recovery of arbitrary cancellations between infinite components is not promised.

See also

premium_x, insurance_x, annuity_x, life_contract, add_insurance, add_premium_schedule

Other life-contingencies: add_insurance(), add_premium_schedule(), annuity_multi(), annuity_x(), annuity_xy(), insurance_variable_k(), insurance_x(), insurance_xy(), life_contract(), premium_gross(), premium_x(), premium_xy(), reserve_xy(), simulate_annuity_x(), simulate_insurance_x()

Examples

lt <- data.frame(
  x = 40:90,
  lx = round(100000 * exp(-0.018 * (0:50)^1.35))
)
lt$lx[nrow(lt)] <- 0

reserve_x(
  lt = lt,
  x = 40,
  i = 0.05,
  type = "term",
  n = 20,
  benefit = 100000,
  k = 12,
  n_prem = 10,
  t = c(0, 5, 10, 15, 20)
)
#> # A tibble: 5 × 6
#>       t   age reserve premium_annualized premium_per_payment method     
#>   <int> <int>   <dbl>              <dbl>               <dbl> <chr>      
#> 1     0    40      0               5841.                487. prospective
#> 2     5    45  17829.              5841.                487. prospective
#> 3    10    50  36402.              5841.                487. prospective
#> 4    15    55  24485.              5841.                487. prospective
#> 5    20    60      0               5841.                487. prospective

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
  ) |>
  reserve_x(
    t = c(0, 5, 10, 15, 20),
    output = "summary"
  )
#> # A tibble: 5 × 6
#>       t   age reserve premium_annualized premium_per_payment method     
#>   <int> <int>   <dbl>              <dbl>               <dbl> <chr>      
#> 1     0    40      0               5841.                487. prospective
#> 2     5    45  17829.              5841.                487. prospective
#> 3    10    50  36402.              5841.                487. prospective
#> 4    15    55  24485.              5841.                487. prospective
#> 5    20    60      0               5841.                487. prospective