\problemdata     
{S01-PS5-1} 
{Annuities}              
{S98-PS5, S01-PS5} 		
{S01 Karger}

\begin{problem}



A college loan officer offers prospective M.Eng students the following
deal.  He will give them \$23,100 at the start of each year for five years
starting today to pay for their college tuition.  He offers the following
plan for repayment:\\
Wait five years, then repay \$$x.00$ at the start
of each year for the next ten years.

Suppose the annual interest rate is 8\%.  What value of $x$ will make the 
present value of the repayment equal to the present value of the loan?

\solution {
The formula for finding the present value of an annuity (given in lecture)
is
\[V = m \sum_{j=0}^{n-1}{1 \over (1+p)^j}\]
where $V$ is the present value, $m$ is the dollar amount of each payment, $n$ is
the number of years of the annuity, and $p = 0.08$ is the current
interest rate.

So the present value of what the college loan officer offers is
\begin{eqnarray*}
V & = & 23100 \cdot {\displaystyle\sum_{j=0}^4(\frac{1}{1+p})^j} \\
 & = & 23100 \cdot {\displaystyle\frac {(1 - \frac{1}{(1 + p)^5})} {(1 -
 \frac{1}{1 + p})}} \\
 & = & 23100 \cdot 4.312\\
 & = & \$99,610.13.
\end{eqnarray*}

The value, $V'$, of the repayment is
\[V' = x \cdot \sum_{j=5}^{14} \frac{1}{(1+p)^j}.\]

Since we want the present values to be equal, let $V' = V$, and solve for
$x$:
\begin{eqnarray*}
x & = & {\$99,610.13 \over { (\frac{1}{1+p})^5} \cdot (\sum_{j=0}^9(\frac{1}{1+p})^j) }\\
 & = & \frac {\$99,610.13} { (\frac{1} {1+p})^5 \cdot { {(1 - \frac{1}{(1+p)^{10}})} \over {(1 - \frac{1}{1+p})}}}\\
 & = & \frac  {\$99,610.13}{0.6805 \cdot 7.246}\\
 & = & \$20,196.25.
\end{eqnarray*}

So, rounding off to the nearest dollar, $x=\$20, 196$.
}

\end{problem}
