% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/annuity_relationships.R
\name{annuity_relationships}
\alias{annuity_relationships}
\alias{annuity_identity_ax}
\alias{annuity_identity_adotx}
\alias{annuity_identity_abarx}
\alias{annuity_identity_axn}
\alias{annuity_identity_adotxn}
\alias{annuity_identity_abarxn}
\alias{annuity_identity_nax}
\alias{annuity_identity_nadotx}
\alias{annuity_identity_nabarx}
\title{Annuity-insurance relationships (Chapter 8)}
\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{
\item{x}{Age.}

\item{i}{Effective annual interest rate.}

\item{model}{Survival model.}

\item{...}{Additional model parameters passed to the survival model.}

\item{n}{Term in years.}

\item{k_max}{Maximum summation horizon for non-terminating models.}

\item{tol}{Truncation tolerance for non-terminating models.}
}
\value{
Numeric vector.
}
\description{
This file provides the core Chapter 8 identities linking annual and
continuous annuity functions to the corresponding insurance functions.

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

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

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

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

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

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

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

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

Computes \eqn{{}_{n|}\bar{a}_x = {}_nE_x\, \bar{a}_{x+n}}.
}
\details{
Included identities:
\itemize{
  \item whole life immediate: \eqn{a_x = (v - A_x)/d}
  \item whole life due: \eqn{\ddot{a}_x = (1 - A_x)/d}
  \item whole life continuous: \eqn{\bar{a}_x = (1 - \bar{A}_x)/\delta}
  \item temporary immediate: \eqn{a_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d - 1 + {}_nE_x}
  \item temporary due: \eqn{\ddot{a}_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d}
  \item temporary continuous: \eqn{\bar{a}_{x:\overline{n}|} = (1 - \bar{A}_{x:\overline{n}|})/\delta}
  \item deferred immediate: \eqn{{}_{n\mid}a_x = {}_nE_x a_{x+n}}
  \item deferred due: \eqn{{}_{n\mid}\ddot{a}_x = {}_nE_x \ddot{a}_{x+n}}
  \item deferred continuous: \eqn{{}_{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
\eqn{a_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d - 1 + {}_nE_x}.
}
