\problemdata                                                             
        {expection and sums} % latex-friendly label for the prob.
        {expection} % The topic(s) of the problem content.  
        {} % Source (if known)                                          
        {S01 Final, 16} % Usage (list ones you are aware of). 
        {Tina Wang, S02} % Last revision info (author, date).   
                                                                         
                                                                         
\begin{problem}           
Short-term Capital Management (STCM) wants you to invest in a
fund with the following rules: You invest one million dollars in their
Forward Looking Internet Package (FLIP). Each year, the value of your
FLIP will double or halve (each with 50\% probability).  Once the
doubling/halving has been determined in a given year, STCM will pay
you one tenth of the current market value of your investment out of
their own pockets (i.e., that money they pay you every year is not
deducted from your investment). [In this problem, you can assume that
the interest rate is 0\%, i.e., the present value of an investment is
always equal to the value of the investment.]
\begin{problemparts}
\problempart What is the expected market value of your investment at
the end of $k$ years?

\solution{  If the starting capital is~$D$, the value of the
  investment after $k$~years is $D \cdot 2^{X_1} \cdot 2^{X_2} \cdots
  2^{X_k}$ where $X_1$~is $1$ with probabilty $1/2$ and $-1$ with
  probability $1/2$. Since the $X_i$~are independent, the expectation
  of the products is the products of the expectations. Moreover,
  $\Ex(2^{X_i}) = (2^1 + 2^{-1})/2 = 5/4$. To sum up, the expected
  value after $k$~years is $10^6 \cdot (5/4)^k$.}


\problempart Give a closed form expression for the total amount of
money you expect to receive over the 10~years.

\solution{ 
  \begin{displaymath}
    10^5 \sum_{k=1}^{10} (5/4)^k = 5 \cdot 10^5 \bigl((5/4)^{10}-1\bigr)
    = \frac{43585245}{1048576} \cdot 10^5
  \end{displaymath}}



\problempart What is the expected market value of your investment at the end
of $k$~years if your annual payment is taken from your investment
instead of from STCM's pockets?

\solution{  $10^6 \cdot (0.9\cdot5/4)^k = 10^6 \cdot (9/8)^k$.}


\problempart Adam Smith analyzes the above questions the following
way.  Let $x_i=1$ if our money doubles after year $i$, and $-1$
otherwise.  Then the value of our investment after year $k$ is $ 10^6
\cdot 2^{x_0+x_1+\cdots+x_k}$. Note that
$E[x_i]=0$.  It follows that $2^{E[\sum x_i]}=1$, which implies that
the expected value of our investment in year $k$ is equal to our
original investment.  What is wrong with this analysis?

\solution{  $\Ex(2^{x_0+x_1+\cdots+x_k}) \neq 2^{E[\sum x_i]}$.}
\end{problemparts}
\end{problem}

                                 