Modified duration is a core concept in finance tutoring, especially in financial math and fixed‑income modeling. Students often learn duration as a measure of interest rate sensitivity but struggle to understand why duration alone systematically underestimates actual bond price changes. This page explains what modified duration is, how it works, and why it fails for larger yield movements.
The formula for modified duration is:
\[ D_{mod} = \frac{D_{mac}}{1+y} \]
where \(D_{mac}\) is Macaulay duration and \(y\) is yield.
Why does modified duration underestimate price changes? Because it assumes the price–yield curve is a straight line. In reality, the curve is convex—bond prices fall at a decreasing rate when yields rise and rise at an increasing rate when yields fall. Duration captures only the first‑order (linear) effect, ignoring curvature. This is why duration underestimates price increases when yields fall and overestimates price decreases when yields rise.
- Start with the bond price formula. \[ P = \sum_{t=1}^{T} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^T} \]
- Compute Macaulay duration. Weighted average time to receive cash flows.
- Convert to modified duration. \[ D_{mod} = \frac{D_{mac}}{1+y} \]
- Use duration to estimate price changes. \[ \frac{\Delta P}{P} \approx -D_{mod} \Delta y \]
- Recognize the limitation. Duration assumes a linear approximation of a nonlinear curve.
- Add convexity for accuracy. \[ \frac{\Delta P}{P} \approx -D_{mod}\Delta y + \frac{1}{2}Cvx(\Delta y)^2 \]
- Interpret the correction term. Convexity is always positive for standard bonds, so it increases the estimated price gain when yields fall and reduces the estimated loss when yields rise.
Suppose a bond has:
- Price: 100
- Modified duration: 5
- Convexity: 30
Consider a 1% (0.01) drop in yields.
\[ \frac{\Delta P}{P} \approx -5(0.01) = +5\% \]
\[ \frac{\Delta P}{P} \approx -5(0.01) + \frac{1}{2}(30)(0.01)^2 \] \[ = 0.05 + 0.0015 = 0.0515 = 5.15\% \]
The true price increase is larger than the duration-only estimate. This difference grows as yield changes become larger.
- Assuming duration is accurate for large yield changes.
- Ignoring convexity when modeling interest rate risk.
- Confusing Macaulay duration with modified duration.
- Believing duration always overestimates price changes—it underestimates for yield drops.
- Using duration for callable bonds (negative convexity distorts results).
Modified duration is essential for understanding interest rate sensitivity, but it is only a first‑order approximation. Convexity is required for accurate pricing, especially when yield changes are large or when modeling risk for long‑term or low‑coupon bonds. Mastering duration and convexity is crucial for fixed‑income modeling, CFA preparation, and graduate‑level finance.
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Why does modified duration underestimate bond price changes?
Answer First
Modified duration underestimates bond price changes because it assumes the price‑yield curve is a straight line. In reality, the curve is convex. As a result, duration overstates losses when yields rise and understates gains when yields fall.
Problem Setup
Modified duration is defined as: \[ D_{\text{mod}} = \frac{D_{\text{mac}}}{1+y}, \] and approximates price changes using: \[ \frac{\Delta P}{P} \approx -D_{\text{mod}} \Delta y. \] But the true price‑yield curve is curved, not linear.
Step-by-Step Explanation
1. Duration is a first‑derivative (slope) measure
Duration measures the slope of the price‑yield curve at a single point. It assumes the curve is linear for small yield changes.
2. The price‑yield curve is convex
The true relationship is curved: \[ \frac{\partial^2 P}{\partial y^2} > 0. \] This curvature means duration is only accurate for tiny yield changes.
3. Duration underestimates gains when yields fall
Because the curve bends upward, the actual price increase is larger than the linear estimate.
4. Duration overestimates losses when yields rise
The curve flattens as yields rise, so the actual price drop is smaller than the duration estimate.
5. When duration fails completely
- Large yield changes (anything above ~50 bps).
- Long‑maturity bonds with high curvature.
- Low‑coupon bonds (more convexity).
- Callable or mortgage‑backed securities (negative convexity).
Intuition
Duration is like estimating a curve using a tangent line. It works for tiny movements but fails for larger ones. Convexity is the “curvature correction” that makes the estimate accurate.
Common Exam Mistakes
- Using duration alone for large rate changes.
- Ignoring convexity when comparing long‑term bonds.
- Assuming duration is always accurate (it rarely is beyond small Δy).
- Confusing modified duration with Macaulay duration.
Final Summary
Modified duration underestimates price changes because it ignores convexity. It is a linear approximation of a curved relationship, making it reliable only for small yield changes and standard bonds.
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