% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/annuity_annual.R
\name{annuity_annual}
\alias{annuity_annual}
\alias{ax}
\alias{adotx}
\alias{abarx}
\alias{axn}
\alias{adotxn}
\alias{abarxn}
\alias{nax}
\alias{nadotx}
\alias{nabarx}
\alias{sxn}
\alias{sdotxn}
\alias{sbarxn}
\title{Annual annuity functions (Chapter 8)}
\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{
\item{x}{Age.}

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

\item{model}{Survival model.}

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

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

\item{tol}{Truncation tolerance for non-terminating models.}

\item{n}{Term in years.}
}
\value{
Numeric vector.
}
\description{
Annual whole life, temporary, deferred, and actuarial accumulated value
annuity functions in immediate, due, and continuous forms.

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

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

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

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

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

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

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

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

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

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

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

Computes \eqn{\bar{s}_{x:\overline{n}|} = \bar{a}_{x:\overline{n}|} / {}_nE_x}.
}
\details{
Naming convention follows Chapter 8 notation:
\itemize{
  \item \code{ax()} = \eqn{a_x}
  \item \code{adotx()} = \eqn{\ddot{a}_x}
  \item \code{abarx()} = \eqn{\bar{a}_x}
  \item \code{axn()} = \eqn{a_{x:\overline{n}|}}
  \item \code{adotxn()} = \eqn{\ddot{a}_{x:\overline{n}|}}
  \item \code{abarxn()} = \eqn{\bar{a}_{x:\overline{n}|}}
  \item \code{nax()} = \eqn{{}_{n|}a_x}
  \item \code{nadotx()} = \eqn{{}_{n|}\ddot{a}_x}
  \item \code{nabarx()} = \eqn{{}_{n|}\bar{a}_x}
  \item \code{sxn()} = \eqn{s_{x:\overline{n}|}}
  \item \code{sdotxn()} = \eqn{\ddot{s}_{x:\overline{n}|}}
  \item \code{sbarxn()} = \eqn{\bar{s}_{x:\overline{n}|}}
}

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