% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/chapter14_multiple_decrement.R
\name{AS_path}
\alias{AS_path}
\title{Projected asset share path \eqn{{}_{k}AS}}
\usage{
AS_path(AS0, G, r, e, b1, b2, q1, q2, p_tau, i, b3 = NULL)
}
\arguments{
\item{AS0}{Initial asset share \eqn{{}_{0}AS}.}

\item{G}{Level annual premium.}

\item{r}{Numeric vector of percent-of-premium expense factors.}

\item{e}{Numeric vector of fixed contract expenses.}

\item{b1}{Numeric vector of Cause 1 benefit amounts.}

\item{b2}{Numeric vector of Cause 2 benefit amounts.}

\item{q1}{Numeric vector of Cause 1 decrement probabilities.}

\item{q2}{Numeric vector of Cause 2 decrement probabilities.}

\item{p_tau}{Numeric vector of in-force probabilities.}

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

\item{b3}{Optional numeric vector of survival benefit amounts payable at the
end of year \eqn{k} conditional on survival through year \eqn{k}. Defaults
to a zero vector.}
}
\value{
A data frame with columns \code{k} and \code{AS}.
}
\description{
Computes projected asset shares recursively using Equations (14.5b) and
(14.6b) of Chapter 14, with optional support for a survival benefit payable
at the end of year \eqn{k}.
}
\details{
For policy year \eqn{k},
\deqn{
[{}_{k-1}AS + G(1-r_k) - e_k](1+i)
= b_k^{(1)} q_{x+k-1}^{(1)}
+ b_k^{(2)} q_{x+k-1}^{(2)}
+ p_{x+k-1}^{(\tau)} \left(b_k^{(3)} + {}_{k}AS\right)
}
so that
\deqn{
{}_{k}AS =
\frac{[{}_{k-1}AS + G(1-r_k) - e_k](1+i)
- b_k^{(1)} q_{x+k-1}^{(1)}
- b_k^{(2)} q_{x+k-1}^{(2)}}{p_{x+k-1}^{(\tau)}} - b_k^{(3)}
}
}
