\problemdata      
        {?} % latex-friendly label for the prob.
        {Annuities} % The topic(s) of the problem content.
        {?} % Source (if known)
        {Fall 2001, quiz 3} % Usage (list ones you are aware of).
        {Christos Kapoutsis, S02} % Last revision info (author, date).


\begin{problem}
Suppose that you plan to purchase an annuity that pays you $m$ dollars per
year for $n$ years, starting $d$ years from now.  Suppose the interest
rate you could earn from a bank is such that \$$r$ deposited today will grow
to be $\$1$ in one year, \eg if the interest rate is 5\%, then $r =
1/1.05$.  Assume this rate will remain fixed forever.  Find a closed-form
formula in $m,n,d$ and $r$ for the current value of this annuity.


\solution{
\begin{eqnarray*}
\text{Current price of the annuity}
& = & \sum_{i=d}^{n+d-1} mr^i\\
& = & \sum_{i=0}^{n+d -1}mr^i - \sum_{i=0}^{d-1}mr^i\\
& = & \frac{m ((r^{n+d} -1) - (r^{d}-1))}{r-1}\\
& = & \frac{mr^d(1- r^n)}{1-r}
\end{eqnarray*}
}


\iffalse
\ppart Find a closed-form formula for the same annuity if instead of
$n$ years, it pays forever.
\solution{$\frac{mr^d}{1-r}$}
\eparts
\fi

\end{problem}
