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\title{Derivatives Lab}
\author{Session 5}
\date{September 24, 2012}
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\begin{enumerate}

\item Suppose you want to immunize a liability at time $D$ with
two bonds of Macaulay durations $D_1$ and $D_2$, respectively.  Show
that the fraction $\omega_1$ and $\omega_2$ of the two bonds in the
initial portfolio need to satisfy
\begin{gather*}
  \omega_1 + \omega_2 = 1 \,, \\
  \omega_1 \, D_1 + \omega_2 \, D_2 = D \,.
\end{gather*}

\item Construct a ``financial perpetual motion machine'' as follows.
\begin{enumerate}
\item Sell a zero coupon bond with a maturity date of $2$ periods at
interest rate $r$.
\item From the proceeds of this transaction, buy zero coupon bonds at
the identical interest rate $r$ with a maturities of $1$ and $3$
periods in a combination which immunizes the liability at maturity
time of the initial zero coupon bond.
\item After one period, liquidate (convert to cash) all bonds at their
current market value.
\item Show that if the interest rate remained constant, you neither
gain nor lose money through these transactions.
\item Now assume a random change in interest rate during the first
period.  Repeat many times and display gains and losses.  Describe
what you see.
\end{enumerate}


\item Visit the web site of the European Central Bank (ECB) and look
for their yield curve data.  What data is contained in the files?

(This is to be submitted as a written answer!)

\item The ECB publishes spot rate data in a downloadable CSV file.
Find online documentation on how to read CSV files, read in the ECB
spot rates, and plot the yield curve.

\end{enumerate}

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