\problemdata                                                             
        {Music Stars Contracts} % latex-friendly label for the prob.
        {Annuities} % The topic(s) of the problem content.  
        {Spring 00} % Source (if known)      
        {S01 Quiz 2, 1} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
Recently, David Bowie, a British music star, sold a portion of the
future profits from his music for \$55 million dollars to Prudential
Insurance.  This deal spawned others among superstar entertainers and
athletes.

(If you don't have a calculator, you don't have to compute numerical
answers. BUT, your answer should be in closed form, in particular, it
must not contain any summations.)

\begin{problemparts}

\problempart Manny Ramirez's contract with the Red Sox is that they will
pay him \$20 million a year for the next eight years. To be specific, assume
he will be paid \$20 million a year today (the first year) and on the same
day of each future year through the eighth year. A little-known provision of his
contract provides him with a ``bonus'' based on performance, which
will be paid out only once, on the same day in the ninth year. Manny has
offered to transfer his salary and bonus 
to a group of investors for \$160 million
dollars paid to him today. How large must the ``bonus'' be for the
investors to break even, assuming an interest rate of 10\% a year?
{\bf (5 points)}

%\vspace*{2.6in}
\solution{ 
Let x be the ``bonus'' Manny must recieve on the the ninth year.

\begin{eqnarray*}
160 million 
        & = & \sum_{i=0}^{7} \frac{20 \text{ million}}{(1+0.1)^i} + \frac{x}{(1+0.1)^8}\\
        & = &   \frac{20 \text{ million*}(1-\frac{1}{(1+0.1)^8})}{1-\frac{1}{(1+0.1)}} + \frac{x}{(1+0.1)^8}\\
        & = &   \$117.368 \text{ million} + x \cdot 0.4665 \\
\end{eqnarray*}

Solving the above equation for \emph{x}, we get \emph{x} = 91.4 million.

Common errors for this part include: not using the right indices in the sum, not discounting the bonus and assuming that it was paid today, and discounting the bonus by the wrong amount.}

\problempart Britney Spears figures that she can generate at least \$20
million from her music starting today and on
the same day of all future years for eternity.  What
should an investor who wants a 10\% return pay today for the rights to her
income? 
\medskip
\solution{ 

Let $M = \$20,000,000$.  Britney is worth

\begin{eqnarray*}
\sum_{i=0}^{\infty} \frac{M}{(1+0.1)^i}
        & = &   \frac{M}{1 - \frac{1}{1+0.1}} \\
        & = &   \$220 \text{ million}
\end{eqnarray*}

Since the investor wants a 10\% return, he should pay

\begin{eqnarray*}
\$220 \text{ million} / 1.1  & = &     \$200 \text{ million}
\end{eqnarray*}

The biggest mistake in this problem was not understanding what ``10\% return'' meant.  This simply means that the investor wants to make 10\% on however much he pays today, and so, $220\text{ million} = 110\%$ of the investment. 
}

\end{problemparts}
\end{problem}
                                 