Modified Duration and Convexity as Bond Interest-Rate Sensitivities
Summary
The document uses a fixed-coupon bond example to ask how to interpret Macaulay duration, modified duration, and convexity. It presents the bond price as the discounted value of coupon and principal cash flows, then identifies modified duration with the first-order sensitivity of price to yield and convexity with the curvature correction. These measures describe how a bond’s price responds approximately to a change in yield, rather than assigning the bond a single, universal risk percentage.
The stated example gives a bond price, duration measures, and a convexity adjustment, but its interpretation of a 0.01% yield move appears to mix percentage points with decimal yield units. Modified duration estimates the percentage price change for a yield change expressed as a decimal; convexity contributes a second-order term whose size also depends on the yield move. The document poses questions about risk and does not supply a full answer or define a standalone risk metric.
Key ideas
- Macaulay duration summarizes the timing of a bond’s discounted cash flows.
- Modified duration approximates the percentage price response to a small yield change.
- Convexity captures curvature and improves the price-change approximation for larger yield moves.
- The yield change must be expressed consistently as a decimal when applying modified duration.
- Duration alone is a sensitivity measure, not a complete or universal measure of bond risk.
Tags
Full text
# Modified Duration as interest risk
# Modified Duration as interest risk
I am new to bond pricing and I am studying the sensitivity measures of a bond (with discrete compounding) and even though I understand the mathematical concepts of modified duration and convexity adjustment since the first is the tangent line of the price-yield curve (1st derivative) and the second is the 2nd derivative. But I want to present you a toy example in order to understand my difficulty that I face in interpretation of these measures.
Let's say that we have a bond with annual coupons $c =40$ and 10 years of maturity $n=10$ ,principal of face value $P_p=1000$ and interest rate $y = 0.08$.
Pricing the bond :
$$\begin{align*} P &= \frac{ c}{(1+y)} +\frac{ c}{(1+y)^2} +\frac{ c}{(1+y)^3}+ \dots + \frac{ c+P_p}{(1+y)^n}\\ &= \frac cy \left[1-(1+y)^{-n}\right] + P_p(1+y)^{-n}\\ &= \frac{40}{0.08}\left(1-1.08^{-10}\right) + 1000\cdot 1.08^{-10}\\ &\approx 268.4033 + 463.1935\\ &\approx 731.5967 \end{align*}$$
The Macaulay duration is $D_{Mac} = 8.1184$ and the Modified duration $D_{m}=7.5171$.
The modified duration is expressed in years. But it is also a sensitivity measure of the bond and this comes from that if there is a change on interest rates of 0.01% the price of the bond will change (increase or decrease according to the sign) by $0.01 \times 7.5171 = 7.51\%$. The convexity adjustment is $C_{adj} = 57.691\%$.
And here are my questions:
1) This percentage (7.51%) is the risk of the bond ? Can i say for example that this bond with these characteristics has a risk of $7.51\%$?
2) Is the interest rate risk ?
3) What is the interpretation of the value $57.691\%$ in the convexity adjustment?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.