Second-order price approximation
Delta P / P approx -D_mod Delta y + 0.5 C (Delta y)^2Duration gives the slope and convexity corrects for curvature as the yield move grows.
Foundations
Duration and Convexity (Intuition)
Duration is first-order rate sensitivity; convexity is the second-order correction.
Mental model
Duration captures the first price response to rates and convexity captures how that response bends.
Duration approximates how bond price changes for small yield changes (linear sensitivity).
Convexity improves the approximation for larger yield moves (curvature).
Explore the mechanism
When does a linear duration estimate stop being accurate enough for a yield move?
A fixed coupon bond reacts to yield moves. Duration gives a first-order linear approximation to the price change; convexity adds a quadratic correction that matters more for larger shifts.
Numbers
Initial price P₀ ≈ 107.43
Macaulay duration D ≈ 3.73 years
Modified duration D* ≈ 3.63
Convexity C ≈ 17.2
Yield shift Δy ≈ 1.00%
Interpretation
For small yield moves, the orange duration-only bar is close to the blue exact bar. As you increase |Δy|, the gap grows and the green duration+convexity bar stays closer to the true price.
Duration gives a linear sensitivity; convexity corrects this for curvature in the price–yield relationship. Higher convexity improves the approximation but is itself a priced feature of the bond or the portfolio.
UNDERSTAND
What to notice
Higher convexity is generally valuable, but it is not free: it is priced.
The quantitative structure
Notation
P- bond price
D_mod- modified duration
C- convexity
Delta y- change in yield
A small example
A rate shock with convexity correction
A bond has modified duration 5 and convexity 30. Yields rise by 1%, so Delta y = 0.01.
- Duration contribution: -5 * 0.01 = -5.00%.
- Convexity contribution: 0.5 * 30 * 0.01^2 = 0.15%.
- Combine them: -5.00% + 0.15% = -4.85%.
ResultThe second-order estimate is a 4.85% price fall, slightly smaller than the duration-only estimate.
Where it appears
- bond portfolio hedging
- interest-rate stress testing
- comparing rate sensitivity across fixed-income instruments
Where the intuition breaks
What this view leaves out
- Duration and convexity are local approximations; large non-parallel curve moves, optionality, and changing cash flows can make them misleading.
Common mistakes
“Duration is the time remaining until a bond matures.”
Not quite.Duration is a price-sensitivity measure; maturity is only one input to it.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Duration and Convexity (Intuition)
Duration and Convexity (Intuition)YOU ARE HERE
Built onDiscounting and Present ValueDuration and convexity describe how discounted fixed-income values respond to changes in rates.
Used byScenario and Stress EngineDuration and convexity provide fast approximations of fixed-income losses under interest-rate shocks.
Supported byYield Curve BootstrappingCurve construction provides the rate term structure against which bond sensitivities are measured.
Related toGreeks and Risk FactorsDuration and convexity are fixed-income sensitivity measures analogous in spirit to first- and second-order Greeks.
Built on · Discounting and Present Value
Duration and convexity describe how discounted fixed-income values respond to changes in rates.
Explore concept →