Duration Is Not How Long You Hold the Bond
It is a sensitivity measure that happens to be quoted in years, and confusing the two is the standard beginner error.
Price and yield move in opposite directions
A bond's coupon is fixed at issue. If market rates rise above it, nobody will pay full price for your below-market income stream, so the price falls until the total return matches what is available elsewhere. That inverse relationship is the whole of bond mathematics.
Bond price
P = Σ (C/m) ÷ (1 + y/m)ᵗ + M ÷ (1 + y/m)ⁿ
- C/m = coupon per period, m = periods per year
- M = face value repaid at maturity
- Solving this for y given P is YTM — no closed form, so it is done numerically
What duration really measures
Macaulay duration is the weighted average time to receive the bond's cash flows, weighted by present value. It comes out in years, which is where the confusion starts.
Modified duration is the number that matters: the approximate percentage price change for a one percentage point change in yield. A modified duration of 7.8 means a 100bp rise in rates costs roughly 7.8% of the price.
A useful sanity check: for a zero-coupon bond, Macaulay duration equals maturity exactly, because there is only one cash flow. Set the coupon to 0% above and confirm it. Any coupon at all pulls duration below maturity, since some money arrives earlier.
Why convexity exists
The price/yield relationship is a curve, and duration is the straight line tangent to it. For small moves the line is close enough. For large ones it drifts, and it drifts in a direction that is systematically favourable to the bondholder.
Second-order price change
ΔP/P ≈ −D* × Δy + ½ × C × (Δy)²
- D* = modified duration (first-order, the straight line)
- C = convexity (second-order, the curve)
- The convexity term is always positive for a plain bond
Because the correction is positive in both directions, positive convexity means you gain slightly more when rates fall than you lose when they rise by the same amount. Look at the ±200bp rows in the shock table: duration alone materially understates the gain, and adding convexity closes almost all of the error.
Clean price, dirty price and what you actually pay
Bonds are quoted at the clean price, excluding interest that has accrued since the last coupon. What settles is the dirty price, clean plus accrued — because the seller is owed the interest earned during their holding period.
Quoting cleanly avoids a sawtooth pattern in the price chart that would otherwise reset at every coupon date and obscure genuine market moves.