\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}