lifecontingencies (version 1.4.3)

endowment_trio: Endowment, insurance, pure endowment, and survival annuity APVs (shared topic)

Description

This help page groups four classical life-contingency present values:

• Exn: pure endowment, pays 1 at time n if alive.

• Axn: term/whole life insurance, pays 1 at death within n years (or up to limiting age if n is missing), with fractional claim timing.

• AExn: n-year endowment insurance, i.e. Axn + Exn.

• axn: survival annuity (immediate/due), with deferment m and k payments per year.

Usage

Exn(actuarialtable, x, n, i = actuarialtable@interest, type = "EV", power = 1)
axn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  m,
  k = 1,
  type = "EV",
  power = 1,
  payment = "advance",
  ...
)
Axn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  m,
  k = 1,
  type = "EV",
  power = 1,
  ...
)
AExn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  k = 1,
  type = "EV",
  power = 1
)

Value

A numeric value (or vector for vectorized inputs): the APV in expected value, or one simulated realization when type="ST".

Arguments

actuarialtable

A lifetable or actuarialtable object.

x

Attained age at inception.

n

Contract length in years. If missing, for Exn and Axn it is set to pmax(ceiling((getOmega(actuarialtable)+1 - x - m)*k)/k, 0); for AExn it defaults to getOmega(actuarialtable) - x - 1. (See function-specific details below.)

i

Annual effective interest rate. Defaults to actuarialtable@interest.

type

Output type: "EV" (expected value, default) or "ST" (one stochastic realization via rLifeContingencies).

power

Power of the discounted payoff before expectation (default 1).

m

Deferment (years). Default 0. Vector accepted. (Axn/axn)

k

Fractional periods per year (\(k \ge 1\)). Default 1. Must be scalar. (Axn/axn/AExn insurance leg)

payment

Payment timing for annuities: "advance" (aka due) or "immediate" (aka arrears). (axn)

...

Extra args forwarded to mortality helpers (pxt, qxt), e.g. fractional assumptions. (Axn)

<code>Exn</code> — Pure endowment

Computes the actuarial present value (APV) of a pure endowment that pays 1 at time n provided survival to x+n.

Details

Exn: \(E_x^n = v^n \, {}_np_x\) with \(v=(1+i)^{-1}\).

Axn: With fractional claims, $$A_{\overline{n}|}^{(k)} = \sum_{j=1}^{nk} v^{t_j + 1/k}\; {}_{t_j}p_x\; q_{x+t_j}^{(1/k)},$$ where \(t_j = m + (j-1)/k\), computed via pxt(...) and qxt(..., t=1/k).

AExn: returns Axn(...) + Exn(...) with aligned arguments.

axn: Survival annuity with payment timing "immediate" (arrears) or "due" (advance), deferment m and k payments per year (see function-specific parameters).

axn — Survival annuity (immediate/due), with deferment m and k fractional payments. For type="EV" the annuity is computed as $$a_{\overline{n}|}^{(k)} = \sum_{j=1}^{nk} \frac{1}{k}\, v^{t_j}\, {}_{t_j}p_x,$$ where \(t_j\) are the payment times depending on payment and m.

Axn — Life insurance (term / whole life), fractional claim times. Vectorized in x, n, m. k must be scalar.

AExn — n-year endowment insurance, computed as Axn + Exn.

References

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., Nesbitt, C. J. (1997). Actuarial Mathematics, 2nd ed., SOA.

See Also

Axn, AExn, axn, Exn

Other life-contingency APVs: arithmetic_variation_insurances

Examples

## Common setup used in legacy docs
data(soaLt)
soa08Act <- with(soaLt, new("actuarialtable", interest=0.06, x=x, lx=Ix, name="SOA2008"))

## Exn (pure endowment)
Exn(soa08Act, x=30, n=35)

## Axn (term / whole life insurance)
# 10-year term, semiannual claims:
Axn(soa08Act, x=50, n=10, k=2)
# Whole life (n inferred), monthly:
Axn(soa08Act, x=30, k=12)

## AExn = Axn + Exn  (legacy book-check)
AExn(soa08Act, x=35, n=30, i=0.06)
Exn(soa08Act, x=35, n=30, i=0.06) + Axn(soa08Act, x=35, n=30, i=0.06)

## axn (survival annuity, legacy example)
# Life-long annuity for age 65:
axn(soa08Act, x=65)

## axn specific legacy examples
# Immediate (arrears) vs due (advance), quarterly, 15-year term deferred 5 years:
axn(soa08Act, x=60, n=15, m=5, k=4, payment="immediate")
axn(soa08Act, x=60, n=15, m=5, k=4, payment="due")
# Vectorization over x/n:
axn(soa08Act, x=c(60,65), n=c(10,20), k=12, payment="due")