dstefan
07-08-2008, 02:22 PM
Chapter 5, Problem 9
A five year bond worth 101 has duration 1.5 years and convexity equal to 2.5. Use both the formula
\label{ch5_ex9_taylor_connection_duration}
\frac{\Delta B}{B} ~\approx~ -D \Delta y,
which does not include any convexity adjustment, and the formula
\label{ch5_ex9_taylor_connection_duration_convexity}
\frac{\Delta B}{B} ~\approx~ -D \Delta y ~+~ \frac{1}{2} C (\Delta y)^2,
to find the price of the bond if the yield increases by ten basis points (i.e., 0.001), fifty basis points, one percent, and two percent, respectively.
Solution:
The approximate values B_{new,D} and B_{new,D,C} of the bond corresponding to the new yield obtained with formulas (\ref{ch5_ex9_taylor_connection_duration}) and (\ref{ch5_ex9_taylor_connection_duration_convexity}), respectively, are
B_{new,D} = B~(1 - D \Delta y)
B_{new,D,C} = B \left( 1 -D \Delta y + \frac{C}{2} (\Delta y)^2 \right)
where B = 101, D = 1.5, and C = 2.5.
The following approximate values are obtained for different values of \Delta y:
\begin{tabular}{|c|c|c|c|}\hline \Delta y & B_{new,D} & B_{new,D,C} & \\ \hline 0.0010 & 100.8485 & 100.8486 & 0.0001% \\ \hline 0.0050 & 100.2425 & 100.2457 & 0.0031% \\ \hline 0.01 & 99.4850 & 99.4976 & 0.0127% \\ \hline 0.02 & 97.9700 & 98.0205 & 0.0515% \\ \hline \end{tabular}
The last column of the table represents the percent difference between the approximate value using duration alone, and the approximate value using both duration and convexity, i.e.,
\frac{B_{new,D,C}-B_{new,D}}{B_{new,D}}
A five year bond worth 101 has duration 1.5 years and convexity equal to 2.5. Use both the formula
\label{ch5_ex9_taylor_connection_duration}
\frac{\Delta B}{B} ~\approx~ -D \Delta y,
which does not include any convexity adjustment, and the formula
\label{ch5_ex9_taylor_connection_duration_convexity}
\frac{\Delta B}{B} ~\approx~ -D \Delta y ~+~ \frac{1}{2} C (\Delta y)^2,
to find the price of the bond if the yield increases by ten basis points (i.e., 0.001), fifty basis points, one percent, and two percent, respectively.
Solution:
The approximate values B_{new,D} and B_{new,D,C} of the bond corresponding to the new yield obtained with formulas (\ref{ch5_ex9_taylor_connection_duration}) and (\ref{ch5_ex9_taylor_connection_duration_convexity}), respectively, are
B_{new,D} = B~(1 - D \Delta y)
B_{new,D,C} = B \left( 1 -D \Delta y + \frac{C}{2} (\Delta y)^2 \right)
where B = 101, D = 1.5, and C = 2.5.
The following approximate values are obtained for different values of \Delta y:
\begin{tabular}{|c|c|c|c|}\hline \Delta y & B_{new,D} & B_{new,D,C} & \\ \hline 0.0010 & 100.8485 & 100.8486 & 0.0001% \\ \hline 0.0050 & 100.2425 & 100.2457 & 0.0031% \\ \hline 0.01 & 99.4850 & 99.4976 & 0.0127% \\ \hline 0.02 & 97.9700 & 98.0205 & 0.0515% \\ \hline \end{tabular}
The last column of the table represents the percent difference between the approximate value using duration alone, and the approximate value using both duration and convexity, i.e.,
\frac{B_{new,D,C}-B_{new,D}}{B_{new,D}}