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Annuity-insurance relationships (Chapter 8)

annuity_relationships.Rd

This file provides the core Chapter 8 identities linking annual and continuous annuity functions to the corresponding insurance functions.

Computes \(a_x = (v - A_x)/d\).

Computes \(\ddot{a}_x = (1 - A_x)/d\).

Computes \(\bar{a}_x = (1 - \bar{A}_x)/\delta\).

Computes \(a_{x:\overline{n}|} = \ddot{a}_{x:\overline{n}|} - 1 + {}_nE_x\) together with \(\ddot{a}_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d\).

Computes \(\ddot{a}_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d\).

Computes \(\bar{a}_{x:\overline{n}|} = (1 - \bar{A}_{x:\overline{n}|})/\delta\).

Computes \({}_{n|}a_x = {}_nE_x\, a_{x+n}\).

Computes \({}_{n|}\ddot{a}_x = {}_nE_x\, \ddot{a}_{x+n}\).

Computes \({}_{n|}\bar{a}_x = {}_nE_x\, \bar{a}_{x+n}\).

Usage

annuity_identity_ax(x, i, model, ...)

annuity_identity_adotx(x, i, model, ...)

annuity_identity_abarx(x, i, model, ...)

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

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

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

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

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

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

Arguments

x

Age.

i

Effective annual interest rate.

model

Survival model.

...

Additional model parameters passed to the survival model.

n

Term in years.

k_max

Maximum summation horizon for non-terminating models.

tol

Truncation tolerance for non-terminating models.

Value

Numeric vector.

Details

Included identities:

  • whole life immediate: \(a_x = (v - A_x)/d\)

  • whole life due: \(\ddot{a}_x = (1 - A_x)/d\)

  • whole life continuous: \(\bar{a}_x = (1 - \bar{A}_x)/\delta\)

  • temporary immediate: \(a_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d - 1 + {}_nE_x\)

  • temporary due: \(\ddot{a}_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d\)

  • temporary continuous: \(\bar{a}_{x:\overline{n}|} = (1 - \bar{A}_{x:\overline{n}|})/\delta\)

  • deferred immediate: \({}_{n\mid}a_x = {}_nE_x a_{x+n}\)

  • deferred due: \({}_{n\mid}\ddot{a}_x = {}_nE_x \ddot{a}_{x+n}\)

  • deferred continuous: \({}_{n\mid}\bar{a}_x = {}_nE_x \bar{a}_{x+n}\)

These are wrapper functions that evaluate the Chapter 8 relationships using the Chapter 7 insurance functions already implemented in the package.

Hence \(a_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d - 1 + {}_nE_x\).