How do you correct a duration estimate with convexity?
Duration is the first-order term. Convexity is the second. Add them together and your estimate stops drifting.
The two-term formula
%ΔP ≈ − Modified duration × Δy + ½ × Convexity × (Δy)²
And convexity itself is a present-value-weighted average of t(t+1), scaled by the price and the yield:
Convexity = [ Σ t(t+1) × PV(CF_t) ] ÷ [ Price × (1 + y ÷ k)² × k² ]
It reads more forbiddingly than it computes. For our 5-year, 5% par bond — annual coupons, so k = 1 and the k terms disappear — the numerator Σ t(t+1) × PV(CF_t) comes to 26,389, and the denominator is €1,000 × 1.05² = 1,102.5:
Convexity ≈ 23.9
Mind that k, exactly as you did for Macaulay duration. With k coupon periods a year you run the sum in periods, discounting at the periodic yield, so the answer arrives in periods² — and it has to be divided by k² before it can sit beside an annual Δy. Restate this same bond semiannually (10 periods of 2.5%) and the sum gives a period convexity of 90.4; ÷ k² = 4 turns that into an annual convexity of 22.6. Skip the division on a semiannual payer — which is most of the government bond market — and you overstate convexity, and the correction it buys you, by a factor of four.
Watch the correction land
Modified duration is 4.33. For a 100bp rise:
- Duration term: −4.3294 × 0.01 = −4.33%
- Convexity term: ½ × 23.9 × (0.01)² = +0.12%
- Combined estimate: −4.21% → €957.90
- True repriced value: €957.88
Two cents out on a €1,000 bond. For a 100bp fall: +4.33% + 0.12% = +4.45% → €1,044.49 against a true €1,044.52. Even at ±200bp the two-term estimate lands within about twenty cents of a full repricing.
Three properties that explain everything
- The convexity term is always positive for an ordinary bond, because t(t+1) is positive for every cash flow. That is the algebraic statement of last lesson's asymmetry: the correction adds value whether rates rise or fall.
- It is symmetric. (Δy)² is the same for +1% and −1%, so the same +0.12% is added in both directions — reducing the loss and increasing the gain.
- It scales with the square of the move. At 25bp the adjustment is 0.0075% — pure noise. At 300bp it is 1.08% — the difference between a usable number and a bad one.
More convexity comes from the same things that give more duration: longer maturity, lower coupon, lower yield. For plain bonds convexity rises roughly with the square of duration, which is why long zeros are both the most sensitive and the most curved instruments around.
When convexity turns negative
Attach an option that belongs to the issuer and the geometry inverts. A callable bond cannot rise far above its call price — as rates fall, the growing likelihood of a call pins the price down, and the price-yield curve bends the wrong way. Mortgage-backed securities behave the same way, because homeowners refinance when rates fall.
With negative convexity, every conclusion flips: duration now overstates the gain when rates fall and understates the loss when they rise. Capped upside, uncapped downside — exactly the asymmetry you met in the yield-to-call lesson, now visible in the curvature.
Because the closed-form formulas assume fixed cash flows, option-bearing bonds are measured with effective duration and effective convexity instead: reprice the bond under a small up-shift and a small down-shift of the whole curve with the option modelled, and read the sensitivity from the two prices. This is a description of how instruments behave, not a view on any of them.
Try it now
- Apply the two-term formula to the house bond at ±200bp and compare with a full repricing (€918.00 and €1,091.59). How close does the second term get you?
- Two bond funds report durations of 3 and 12. Without computing anything, say which one carries far more convexity and why the difference is roughly sixteen-fold rather than four-fold.
- Now see the bend in real prices. The chart below is five years of a long-Treasury fund — the high-convexity end of step 2. Find its largest fall and its largest rise and note that they are not the same size. Then reason about a callable bond, whose call schedule is written in its prospectus: what does a call feature do to the shape of that relationship above the call price?