\problemdata      
        {annuities-1} % latex-friendly label for the prob.
        {Annuities} % The topic(s) of the problem content.
        {?} % Source (if known)
        {S02 CP6W} % Usage (list ones you are aware of).
        {Christos Kapoutsis, S02} % Last revision info (author, date).




\begin{problem}
Suppose you were about to enter college today and a college loan
officer offered you the following deal: \$25,000 at the start of each
year for four years to pay for your college tuition and an option of
choosing one of the following repayment plans:
\begin{description}
\item[Plan A:] 
Wait four years, then repay \$20,000 at the start of
each year for the next ten years.
\item[Plan B:] 
Wait five years, then repay \$30,000 at the start of
each year for the next five years.
\end{description}
Suppose the annual interest rate paid by banks is 7\% and does not change
in the future.
\begin{problemparts}

\problempart
Assuming that it's no hardship for you to meet the terms of either payback
plan, which one is a better deal?  (You will need a calculator.)

\solution{  
\$1 today will be worth $\$1.07$ next year, and $\$1.07^2$ the
year after, \etc
So set $r = \frac{1}{1.07}$.  Then:
\begin{eqnarray*}
\lefteqn{\text{current value of Plan A}}\\
  & = & \sum_{y=4}^{13} 20000 \cdot r^{y}  \\
  & = & \sum_{y=0}^{9} 20000 \cdot r^{y+4}  \\
  & = & r^4 \cdot \sum_{y=0}^{9} 20000 \cdot r^y  \\
  & = & 20000r^4 \cdot \sum_{y=0}^{9} r^y \\
  & = & 20000r^4 \cdot \frac{1 - r^{10}}{1 - r} \\
  & = & \$114,66.69
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{\text{current value of Plan B}}\\
  & = & \sum_{y=5}^{9} 30000 \cdot r^{y}  \\
  & = & \sum_{y=0}^{4} 30000 \cdot r^{y+5}  \\
  & = & r^5 \cdot \sum_{y=0}^{4} 30000 \cdot r^y  \\
  & = & 30000r^5 \cdot \sum_{y=0}^{4} r^y \\
  & = & 30000r^5 \cdot \frac{1 - r^{5}}{1 - r} \\
  & = & \$93,840.63
\end{eqnarray*}
You should clearly take Plan B. You will be paying back much less in
today's dollars.
}

\problempart
What is the loan officer's effective profit (in today's
dollars) on the loan?

\solution{
The value of the money you are given is:
\begin{eqnarray*}
Loan & = & \sum_{y=0}^{3} 25000 \cdot r^{y}  \\
  & = & 25000 \cdot \sum_{y=0}^{3} r^y \\
  & = & 25000 \cdot \frac{1 - r^{4}}{1 - r} \\
  & = & \$90,607.90
\end{eqnarray*}

Therefore, the loan officer's profit is effectively $\$3,233$. (Or 
$\$24,059$ if we are not on the ball).
}
\end{problemparts}
\end{problem}
