Bond duration and convexity
Price a coupon bond from its yield, then measure how much that price will move when yields change. Duration gives the first-order sensitivity, convexity the curvature — and the gap between them is why bond maths is not linear.
| Period | Cash flow | Present value | Weight |
|---|---|---|---|
| 3 | $27.50 | $25.65 | 2.41% |
| 6 | $27.50 | $23.92 | 2.25% |
| 9 | $27.50 | $22.31 | 2.10% |
| 12 | $27.50 | $20.81 | 1.96% |
| 15 | $27.50 | $19.41 | 1.83% |
| 18 | $27.50 | $18.10 | 1.70% |
| 20 | $1,027.50 | $645.69 | 60.73% |
The final period carries the face value as well as the last coupon, which is why its weight jumps.
P = Σ c/(1+y/f)ᵗ + F/(1+y/f)ⁿ · f = coupons per year
D = Σ (t/f)·PVₜ / P · in years
D* = D/(1+y/f) · the price sensitivity coefficient
ΔP/P ≈ −D*Δy + ½C(Δy)² · convexity improves the estimate
A clear path from inputs to results.
Duration is the present-value-weighted time to a bond’s cash flows, and it doubles as the first-order sensitivity of price to yield. Convexity is the curvature that makes the second-order term matter — the reason price rises more on a yield fall than it loses on an equal rise.
How to use
- Enter the face value, coupon rate and yield to maturity.
- Set the years to maturity and the coupon frequency.
- Read the clean price — above par if the coupon exceeds the yield.
- Check Macaulay duration in years and modified duration as the sensitivity.
- Use the ±1% shock check to see convexity in dollars.
The formula
Reference: Fabozzi, "Bond Markets, Analysis and Strategies", ch. 4–5.
Worked example
5.5% coupon, 4.7% yield, 10 years, semi-annual
- Face $1,000.
- Coupon 5.5%, yield 4.7%.
- 10 years, semi-annual coupons.
- 1
Per-period coupon: 1,000 × 5.5% / 2 = $27.50; period yield 2.35%.
- 2
Discount 20 periods plus the face value: price = $1,063.25.
- 3
Macaulay duration: 7.90 years. Modified: 7.90 / 1.0235 = 7.72.
- 4
A 1% yield rise moves price by about −7.72%, or −$82.10.
- 5
The ±1% check shows a larger gain on the fall than the loss on the rise.
Price $1,063.25 · Macaulay 7.90 yr · modified 7.72 · convexity positive.
Interpreting the result
Duration falls as the coupon rises, because more of the value arrives early. A zero-coupon bond has duration equal to maturity, which is the maximum for a given maturity. Modified duration is what you use for hedging: a portfolio with modified duration 7.72 loses about 7.72% for a one-point parallel rise in yields.
Assumptions & limitations
- A flat yield curve and a parallel shift when yields change.
- No default risk, no embedded options, no liquidity premium.
- Coupons are reinvested at the same yield, which is what makes realised return equal YTM.
- Day-count and settlement conventions are ignored; accrued interest is not separated.
Frequently asked questions
What is the difference between Macaulay and modified duration?
Macaulay is a time in years — the weighted average wait for cash flows. Modified divides it by (1 + y/f) to give the percentage price change per unit change in yield, which is the number used for hedging.
What is convexity for?
It corrects the duration estimate for curvature. For large yield moves the linear duration approximation drifts, and convexity captures the second-order term. Higher convexity is always favourable for a given duration.
How do I immunise a liability?
Match the portfolio’s duration to the liability’s horizon and rebalance as time passes. Duration matching protects against small parallel shifts, not against curve reshaping.
Why does duration matter for a bond fund?
Because it tells you the fund’s interest-rate risk. A fund with duration 6 loses roughly 6% if yields rise one percentage point, regardless of the bonds’ maturities.
Can duration be negative?
For a plain bond, no. For mortgage-backed securities and callable bonds with negative convexity, effective duration can behave oddly, and for some hedged positions it can be negative.
Methodology last reviewed 2026-10-18. Results are informational and not financial or engineering advice — see our disclaimer.