Pure endowment (discounted survival): ${}_tE_x$

Computes the actuarial present value of a pure endowment, i.e., the expected present value of a payment of 1 made at time $t$ if and only if a life aged $x$ survives to age $x + t$: $${}_tE_x = v^t \cdot {}_tp_x.$$

Usage

t_Ex(
  lt,
  x,
  t,
  i,
  i_type = "effective",
  m = 1L,
  frac,
  tidy = FALSE,
  check = TRUE,
  tol = 1e-10
)

Arguments

lt: A lifetable object as produced by lifetable. Must contain columns x and lx.
x: Integer age(s) at which the endowment starts.
t: Nonnegative numeric duration(s) in years. Fractional durations are allowed and are handled through t_px.
i: Annual interest-rate input(s).
i_type: Character vector indicating the interest-rate type. Allowed values are "effective", "nominal_interest", "nominal_discount", and "force".
m: Positive integer vector giving the conversion frequency for nominal rates. Ignored for "effective" and "force".
frac: Fractional-age assumption passed to t_px: "UDD", "CF", "CML" (alias of CF), or "Balducci". If not specified and lt carries a frac attribute, that value is used.
tidy: Logical. If TRUE, returns a tibble with columns x, t, i, i_type, m, i_effective, frac, and nEx.
check: Logical. If TRUE, performs validity checks.
tol: Numeric tolerance for integer checks on x.

Value

Numeric vector of ${}_tE_x$, or a tibble if tidy = TRUE.

Details

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

The pure endowment is a fundamental building block in life contingency mathematics. It serves as the actuarial discount factor, combining financial discounting with mortality: $${}_tE_x = v^t \cdot {}_tp_x.$$

Discounted products use the original log accumulation rate, retaining representable values when the annual effective equivalent loses range. This is an actuarial present value, not a probability: it can exceed one with negative interest rates and is not capped. The equivalent rate in tidy output may round to -1 or overflow, without being used for valuation. In ordinary notation the discount factor is: $$v = (1+i_e)^{-1}.$$

Key identities involving ${}_nE_x$:

Actuarial accumulated value: $\ddot{s}_{x:\overline{n}|} = \ddot{a}_{x:\overline{n}|} / {}_nE_x$.
Endowment insurance decomposition: $A_{x:\overline{n}|} = A^1_{x:\overline{n}|} + {}_nE_x$.
Deferred annuity: ${}_{n|}\ddot{a}_x = {}_nE_x \cdot \ddot{a}_{x+n}$.

For a constant force of mortality $\mu$ and force of interest $\delta$: $${}_nE_x = e^{-n(\mu + \delta)}.$$

The variance of the pure endowment random variable is: $$\mathrm{Var}(Z) = v^{2n} \cdot {}_np_x \cdot {}_nq_x.$$

Examples

x  <- 0:5
lx <- c(100000, 99500, 99000, 98200, 97000, 95000)
lt <- lifetable(x = x, lx = lx, omega = 5, close = TRUE)

# Basic pure endowment: 3_E_0 = v^3 * 3_p_0
t_Ex(lt, x = 0, t = 3, i = 0.06)
#> [1] 0.8245061

# Verify manually:
(1.06)^(-3) * t_px(lt, x = 0, t = 3)
#> [1] 0.8245061

# Constant force of interest
lt_exp <- lifetable(x = 0:50, lx = 100000 * exp(-0.05 * (0:50)))
t_Ex(
  lt_exp,
  x = 30,
  t = 10,
  i = 0.10,
  i_type = "force"
)
#> [1] 0.2231302
exp(-1.5)
#> [1] 0.2231302

# Nominal annual interest rate convertible monthly
t_Ex(
  lt,
  x = 0,
  t = 3,
  i = 0.06,
  i_type = "nominal_interest",
  m = 12
)
#> [1] 0.8206033

# Vectorized: multiple ages at once
t_Ex(lt, x = c(0, 1, 2), t = 3, i = 0.05, tidy = TRUE)
#> # A tibble: 3 × 8
#>       x     t     i i_type        m i_effective frac    nEx
#>   <int> <dbl> <dbl> <chr>     <int>       <dbl> <chr> <dbl>
#> 1     0     3  0.05 effective     1        0.05 UDD   0.848
#> 2     1     3  0.05 effective     1        0.05 UDD   0.842
#> 3     2     3  0.05 effective     1        0.05 UDD   0.829

# Use in a tidy pipeline
if (requireNamespace("dplyr", quietly = TRUE)) {
  tibble::tibble(x = 0:4, t = c(5, 4, 3, 2, 1)) |>
    dplyr::mutate(pure_endow = t_Ex(lt, x = x, t = t, i = 0.06))
}
#> # A tibble: 5 × 3
#>       x     t pure_endow
#>   <int> <dbl>      <dbl>
#> 1     0     5      0.710
#> 2     1     4      0.756
#> 3     2     3      0.806
#> 4     3     2      0.861
#> 5     4     1      0.924

Pure endowment (discounted survival): \({}_tE_x\)

Usage

Arguments

Value

Details

See also

Examples