A percentage (%) change in a bond price with respect to a change in interest rate is approximated by using duration and convexity, which is based on the Talor approximation with the first and second order terms.


We have calculated a bond duration and convexity in the previous posts

Taylor Approximation

When a function \(f(x)\) is given, at some point, the first order or linear approximation is as follows. \[\begin{align} f(x) \approx f(a) + f^{‘}(a)(x-a) \end{align}\] To get more precise function value, we use the second order or quadratic approximation with the previous first order one, which is a well-known result. \[\begin{align} f(x) \approx f(a) + f^{‘}(a)(x-a) + \frac{1}{2} f^{”}(a) (x-a)^2 \end{align}\] When we set \(x = a + \Delta a\), the following approximation is obtained. \[\begin{align} f(a + \Delta a) \approx f(a) + f^{‘}(a)\Delta a + \frac{1}{2} f^{”}(a) (\Delta a)^2 \end{align}\]

% change in bond price

Bond price with unit notional amount, coupon C, YTM y, annual frequency is as follows. \[\begin{align} P = P(y) = \sum_{t=1}^{T} \frac{CF_t}{(1+y)^t} \end{align}\] Expanding the price of the bond in a Taylor series around \(y\) results in \[\begin{align} P(y + \Delta y) \approx P(y) + \frac{\partial P}{\partial y}\Delta y + \frac{1}{2} \frac{d^2P}{d y^2} (\Delta y)^2 \\ \end{align}\] Dividing both sides by \(P(y)\) gives \[\begin{align} P(y + \Delta y)\frac{1}{P(y)} &\approx P(y)\frac{1}{P(y)} + \frac{\partial P}{\partial y}\Delta y \frac{1}{P(y)} \\ &+ \frac{1}{2} \frac{d^2P}{d y^2} (\Delta y)^2 \frac{1}{P(y)} \end{align}\] Moving \(P(y)\frac{1}{P(y)}\) from the right hand side to the left one, \[\begin{align} \frac{P(y + \Delta y) – P(y)}{P(y)} \approx \frac{\partial P}{\partial y}\Delta y \frac{1}{P(y)} + \frac{1}{2} \frac{d^2P}{d y^2} (\Delta y)^2 \frac{1}{P(y)} \end{align}\] Using the definitions of duration and convexity, we get \[\begin{align} \frac{\Delta P}{P} \approx -D \Delta y + \frac{1}{2} C (\Delta y)^2 \\ \\ D = -\frac{d P}{d y}\frac{1}{P}, \quad C = \frac{d^2P}{d y^2} \frac{1}{P} \end{align}\]
Finally, the % change in a bond price w.r.t a small interest rate change using duration and convexity (without higher order terms) is as follows.
\[\begin{align} \frac{\Delta P}{P} = -D \Delta y + \frac{1}{2} C (\Delta y)^2 \end{align}\]

R code

The following R code calculates the % change of a bond price w.r.t a interest rate change by using three methods : 1) full pricing, 2) approximation with duration, 3) approximation with duration and convexity.



From the folloiwng output which is the results of the above R code, we can find that the first and third rounded % change in bond prices to the nearest eighth have the same value while the second results have some little difference with full valuation. This means that when a convexity is considered, we can make this approximation more precise.

Concluding Remarks

From this post, using derivmkt R package, we can easily calculate a percent(%) change of a bond price with duration and convexity. Using this approximation, A % change in a bond portfolio is also calculated by using the same approximation with the present value weighted duration and convexity. \(\blacksquare\)