Skip to contents

Annual annuity functions (Chapter 8)

annuity_annual.Rd

Annual whole life, temporary, deferred, and actuarial accumulated value annuity functions in immediate, due, and continuous forms.

Computes the annual whole life annuity-immediate \(a_x = \sum_{t=1}^{\infty} v^t \, {}_t p_x\).

Computes the annual whole life annuity-due \(\ddot{a}_x = \sum_{t=0}^{\infty} v^t \, {}_t p_x = 1 + a_x\).

Computes the continuous whole life annuity \(\bar{a}_x = \int_0^{\infty} v^t \, {}_t p_x \, dt\).

Computes the annual temporary annuity-immediate \(a_{x:\overline{n}|} = \sum_{t=1}^{n} v^t \, {}_t p_x\).

Computes the annual temporary annuity-due \(\ddot{a}_{x:\overline{n}|} = \sum_{t=0}^{n-1} v^t \, {}_t p_x\).

Computes the continuous temporary annuity \(\bar{a}_{x:\overline{n}|} = \int_0^n v^t \, {}_t p_x \, dt\).

Computes the annual deferred whole life annuity-immediate \({}_{n|}a_x = {}_nE_x \, a_{x+n}\).

Computes the annual deferred whole life annuity-due \({}_{n|}\ddot{a}_x = {}_nE_x \, \ddot{a}_{x+n}\).

Computes the continuous deferred whole life annuity \({}_{n|}\bar{a}_x = {}_nE_x \, \bar{a}_{x+n}\).

Computes \(s_{x:\overline{n}|} = a_{x:\overline{n}|} / {}_nE_x\).

Computes \(\ddot{s}_{x:\overline{n}|} = \ddot{a}_{x:\overline{n}|} / {}_nE_x\).

Computes \(\bar{s}_{x:\overline{n}|} = \bar{a}_{x:\overline{n}|} / {}_nE_x\).

Usage

ax(x, i, model, ..., k_max = 5000, tol = 1e-12)

adotx(x, i, model, ..., k_max = 5000, tol = 1e-12)

abarx(x, i, model, ..., tol = 1e-10)

axn(x, n, i, model, ...)

adotxn(x, n, i, model, ...)

abarxn(x, n, i, model, ...)

nax(x, n, i, model, ..., k_max = 5000, tol = 1e-12)

nadotx(x, n, i, model, ..., k_max = 5000, tol = 1e-12)

nabarx(x, n, i, model, ..., tol = 1e-10)

sxn(x, n, i, model, ...)

sdotxn(x, n, i, model, ...)

sbarxn(x, n, i, model, ...)

Arguments

x

Age.

i

Effective annual interest rate.

model

Survival model.

...

Additional model parameters passed to the survival model.

k_max

Maximum summation horizon for non-terminating models.

tol

Truncation tolerance for non-terminating models.

n

Term in years.

Value

Numeric vector.

Details

Naming convention follows Chapter 8 notation:

  • ax() = \(a_x\)

  • adotx() = \(\ddot{a}_x\)

  • abarx() = \(\bar{a}_x\)

  • axn() = \(a_{x:\overline{n}|}\)

  • adotxn() = \(\ddot{a}_{x:\overline{n}|}\)

  • abarxn() = \(\bar{a}_{x:\overline{n}|}\)

  • nax() = \({}_{n|}a_x\)

  • nadotx() = \({}_{n|}\ddot{a}_x\)

  • nabarx() = \({}_{n|}\bar{a}_x\)

  • sxn() = \(s_{x:\overline{n}|}\)

  • sdotxn() = \(\ddot{s}_{x:\overline{n}|}\)

  • sbarxn() = \(\bar{s}_{x:\overline{n}|}\)

These functions work directly from the Chapter 5 survival model functions and the Chapter 7 pure endowment function nEx().