\documentclass{article}
\input{15401-preamble}
\psetnum{2}
\date{2011/09/28}

\begin{document}
\begin{pset}
  
  \begin{problem}
    
    \begin{subproblem}
      The T-bill has a six-month interest rate of
      $\nicefrac{\$100000}{\$98058} -1 \approx 1.98\%$, so its effective
      annual rate is $1.0198^2 - 1\approx 4.00\%$.

      The coupon bond pays 2.1\% every six months, so its effective
      annual rate is $1.0021^2 - 1 = 4.2441\%$.

      The coupon bond has the higher EAR.
    \end{subproblem}

    \begin{subproblem}
      With a YTM of 8\%, the three-month interest rate is
      $1.08^{\nicefrac14} -1 \approx 1.94\%$. The sale price of the
      bond, therefore, will be $\frac{\$100000}{1.0194} \approx
      \$98094$. The holding period return is \$36.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}
      \begin{align*}
        PV_A &= \frac{\$160}{1.12} + \frac{\$160}{1.12^2}
        \frac{\$1160}{1.12^3} &\approx \$1096.07 \\
        PV_B &= \frac{\$40}{1.12} + \frac{\$40}{1.12^2}
        \frac{\$1040}{1.12^3} &\approx \$807.85 \\
        PV_C &= \frac{\$1000}{1.12^3} &\approx \$711.78
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      The Macaulay durations (not modified durations) of the three bonds  are:
      \begin{align*}
        D_A &= \frac{1}{PV_A}\left(\sum_{i=1}^3 i PV_i \right) &=
        \frac{1}{\$1096.07} \left(1 \cdot \frac{\$160}{1.12} + 2 \cdot
          \frac{\$160}{1.12^2} +
        3 \cdot \frac{\$1160}{1.12^3}\right) &\approx 2.623 \\
        D_B &= \frac{1}{PV_B}\left(\sum_{i=1}^3 i PV_i \right) &=
        \frac{1}{\$807.85} \left(1 \cdot \frac{\$40}{1.12} + 2 \cdot
          \frac{\$40}{1.12^2} +
        3 \cdot \frac{\$1040}{1.12^3}\right) &\approx 2.872 \\
        D_C &= \frac{1}{PV_C}\left(\sum_{i=1}^3 i PV_i \right) &=
        \frac{1}{\$711.78} \left(
        3 \cdot \frac{\$1000}{1.12^3}\right) &= 3
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      
      \begin{subsubproblem}
        Computing the price directly:
        \begin{align*}
          PV'_A &= \frac{\$160}{1.10} + \frac{\$160}{1.10^2}
          + \frac{\$1160}{1.10^3} &\approx \$1149.21 \\
          PV'_B &= \frac{\$40}{1.10} + \frac{\$40}{1.10^2}
          + \frac{\$1040}{1.10^3} &\approx \$850.78 \\
          PV'_C &= \frac{\$1000}{1.10^3} &\approx \$751.31
        \end{align*}
      \end{subsubproblem}

      \begin{subsubproblem}
        Using the duration approximation,
        \begin{align*}
          PV'_A &= PV_A (1 - D^*_A \Delta R) &\approx \$1096.07 \left(1 +
          \frac{2.623}{1.12} \cdot 0.02 \right) &\approx \$1147.41 \\
          PV'_B &= PV_B (1 - D^*_B \Delta R) &\approx \$807.85 \left(1 +
          \frac{2.872}{1.12} \cdot 0.02 \right) &\approx \$849.28 \\
          PV'_C &= PV_C (1 - D^*_C \Delta R) &\approx \$711.78 \left(1 +
          \frac{3}{1.12} \cdot 0.02 \right) &\approx \$749.91
        \end{align*}
      \end{subsubproblem}

      \begin{subsubproblem}
%        \frac{1}{\left(1+y\right)^2} \sum_{k=1}^{T}
%        \frac{C_k}{(1+y)^k} k \cdot (k+1)
        First, let's compute the convexity of the bonds:
        \begin{align*}
          V_{m_A} &= \frac{1}{PV_A \cdot (1+y)^2} \sum_{i=1}^ i \cdot
          (i+1) \cdot PV_i \\&\approx \frac{1}{\$1096.07 (1.12)^2} \left(1
            \cdot 1 \cdot 2 \cdot \frac{\$160}{1.12} + 2 \cdot 3 \cdot
            \frac{\$160}{1.12^2} +  3 \cdot 4
            \cdot \frac{\$1160}{1.12^3}\right) &\approx 7.971 \\
          V_{m_B} &= \frac{1}{PV_B \cdot (1+y)^2} \sum_{i=1}^ i \cdot
          (i+1) \cdot PV_i \\&\approx \frac{1}{\$807.85 (1.12)^2} \left(1
            \cdot 1 \cdot 2 \cdot \frac{\$40}{1.12} + 2 \cdot 3 \cdot
            \frac{\$40}{1.12^2} +  3 \cdot 4
            \cdot \frac{\$1040}{1.12^3}\right) &\approx 9.025 \\
          V_{m_C} &= \frac{1}{PV_C \cdot (1+y)^2} \sum_{i=1}^ i \cdot
          (i+1) \cdot PV_i \\&\approx \frac{1}{\$711.78 (1.12)^2} \left(
            3 \cdot 4
            \cdot \frac{\$1000}{1.12^3}\right) &\approx 9.566 
        \end{align*}

        Now, using the duration + convexity approximation,

        \begin{align*}
          PV'_A &= PV_A \left(1 - D^*_A \Delta R + V_{m_A} \frac{(\Delta R)^2}{2}\right) &\approx \$1096.07 \left(1 +
          \frac{2.623}{1.12} \cdot 0.02 + 7.971 \cdot \frac{0.02^2}{2}
        \right) &\approx \$1149.16 \\
          PV'_B &= PV_B \left(1 - D^*_B \Delta R + V_{m_B}
            \frac{(\Delta R)^2}{2}\right) &\approx \$807.85 \left(1 +
          \frac{2.872}{1.12} \cdot 0.02 + 9.025 \cdot \frac{0.02^2}{2}
        \right) &\approx \$850.74 \\
        PV'_C &= PV_C \left(1 - D^*_C \Delta R + V_{m_C} \frac{(\Delta R)^2}{2}\right) &\approx \$711.78 \left(1 +
          \frac{3}{1.12} \cdot 0.02 \right) &\approx \$751.27
        \end{align*}

        The approximation with just duration is fairly good --- giving
        an error of approximately 0.3\% --- because the change in
        interest rate is small. However, the approximation with
        duration and convexity is better, with an error below 0.01\%
        for all three bonds.
      \end{subsubproblem}

      \begin{subsubproblem}
        The durations are now
      \begin{align*}
        D'_A &= \frac{1}{PV'_A}\left(\sum_{i=1}^3 i PV_i \right) &=
        \frac{1}{\$1149.21} \left(1 \cdot \frac{\$160}{1.10} + 2 \cdot
          \frac{\$160}{1.10^2} +
        3 \cdot \frac{\$1160}{1.10^3}\right) &\approx 2.632 \\
        D'_B &= \frac{1}{PV'_B}\left(\sum_{i=1}^3 i PV_i \right) &=
        \frac{1}{\$850.78} \left(1 \cdot \frac{\$40}{1.10} + 2 \cdot
          \frac{\$40}{1.10^2} +
        3 \cdot \frac{\$1040}{1.10^3}\right) &\approx 2.876 \\
        D'_C &= \frac{1}{PV'_C}\left(\sum_{i=1}^3 i PV_i \right) &=
        \frac{1}{\$751.31} \left(
        3 \cdot \frac{\$1000}{1.10^3}\right) &= 3
      \end{align*}
      \end{subsubproblem}
    \end{subproblem}
  \end{problem}

  \begin{problem}

    \begin{subproblem}
      The implied forward rate is
      \[ \frac{1.0550^2}{1.0525} -1 \approx 5.75\% \]
      which is higher than the bank's rate of 5.5\%.
    \end{subproblem}

    \begin{subproblem}
      We'd like to sell short 10,000 1-year STRIPS (face value \$10
      million), which we'll pay off with the expected \$10 million
      account in one year. This will provide immediate income of
      $\frac{\$10,000,000}{1.0525} \approx \$9,501,187$. We'll use
      those proceeds to purchase 2-year STRIPS (9,501.187 of them,
      assuming we can trade fractional amounts).

      In two years, the 2-year STRIPS will return $9,501,187 \cdot
      (1.055)^2 \approx 10,575,059$ --- which, sure enough, is the
      5.75\% return we wanted on the \$10,000,000 receivable.
    \end{subproblem}
  \end{problem}


  \begin{problem}
    The duration of a set of bonds is the weighted average of their
    durations, and the duration of a zero-coupon bond is its maturity,
    so we want portfolio weights of 70\% 6-year bonds and 30\% 16-year
    bonds to give a portfolio duration of 9 years.

    The price of a 6-year zero-coupon bond is $\frac{\$100}{1.075^6}
    \approx \$64.80$ and that of a 16-year bond is
    $\frac{\$100}{1.075^16} \approx \$31.44$. Valerie should purchase
    $70\% \cdot \$500,000 = \$350,000$ worth of the 6-year bonds, and
    $\$150,000$ of the 16-year bonds. That amounts to 5402 6-year
    bonds, and 4771 16-year bonds.
  \end{problem}
\end{pset}
\end{document}
