% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/multilife_ch12_ext.R
\name{continuous_multilife_annuities}
\alias{continuous_multilife_annuities}
\alias{abarxy}
\alias{abarxybar}
\alias{abarx_y}
\alias{abary_x}
\title{Continuous multi-life annuities}
\usage{
abarxy(x, y, i, tbl = NULL, model = NULL, ...)

abarxybar(x, y, i, tbl = NULL, model = NULL, ...)

abarx_y(x, y, i, tbl = NULL, model = NULL, ...)

abary_x(x, y, i, tbl = NULL, model = NULL, ...)
}
\arguments{
\item{x}{Age of the first life. May be scalar or vector.}

\item{y}{Age of the second life. May be scalar or vector.}

\item{i}{Effective annual interest rate. May be scalar or vector.}

\item{tbl}{Optional life table object retained for backward compatibility.
Continuous calculations currently require \code{model}.}

\item{model}{Parametric survival model.}

\item{...}{Additional parameters passed to the survival functions.}
}
\value{
A numeric vector of actuarial present values.
}
\description{
Computes continuous joint-life, last-survivor, and reversionary whole-life
annuities for two independent lives.
}
\details{
\code{abarxy()} computes the joint-life annuity.

\code{abarxybar()} computes the last-survivor annuity.

\code{abarx_y()} computes the reversionary annuity payable to the second
life after the death of the first.

\code{abary_x()} computes the reversionary annuity payable to the first
life after the death of the second.

These functions require a parametric survival model.


Under independence, \code{abarxy()} represents the continuous joint-life
annuity, payable while both lives survive.

The last-survivor annuity satisfies
\deqn{\bar{a}_{\overline{xy}} =
\bar{a}_x + \bar{a}_y - \bar{a}_{xy}.}

The reversionary annuities are obtained as the difference between the
corresponding single-life annuity and the joint-life annuity.
}
\examples{
abarxy(
  x = 40,
  y = 50,
  i = 0.05,
  model = "uniform",
  omega = 100
)

abarxybar(
  x = 40,
  y = 50,
  i = 0.05,
  model = "uniform",
  omega = 100
)

}
