Convexity: the bend duration leaves out
Duration draws a straight line through a curve. The true price-yield relationship bends upward, so a bond gains more on a rally than it loses on an equal sell-off. That bend is worth paying for, and the market charges for it.
- Estimate a price change with duration alone and with duration plus convexity
- Compare both estimates with the exact repricing and see what each leaves out
- Read effective duration and convexity off three repriced values
- Explain the asymmetry between a rally and a sell-off, and why convexity costs yield
The intuition
Take a bond with a modified duration of 7 and a convexity of 60. Duration says a 100 bp rise costs 7% and a 100 bp fall gains 7%. But as yields rise, each extra basis point is working on a price that has already shrunk, so it costs a little less than the one before; as yields fall, each earns a little more. The correction is half the convexity times the move squared: 0.3% here, and it is added whichever way yields go. So the rise costs 6.7% and the fall gains 7.3%. The gain beats the loss by 0.6 points.
That asymmetry is why convexity is valuable: a long bond wins more on a rally than it loses on an equal sell-off. The market knows, so higher-convexity bonds trade at lower yields, and in a quiet year the low-convexity bond out-earns them through its extra yield. Convexity pays only when rates actually move. For a few basis points it barely matters; for the 100 to 200 bp moves of a bad year it is the difference between a good estimate and a bad one.
% change ≈ −D × Δy + ½ × C × Δy². The convexity term is positive for a rise or a fall. Gain on a fall minus loss on a rise ≈ C × Δy². Effective duration = (P₋ − P₊) ÷ (2 × P₀ × Δy); effective convexity = (P₊ + P₋ − 2P₀) ÷ (P₀ × Δy²), from three repriced values.
Why it works
- The conventions here: annual coupons, price per 100. Modified duration and convexity are analytic: convexity = Σ t(t+1) CFₜ ÷ (1 + y)^(t+2) ÷ price. Estimates use the printed D (two decimals) and C (one decimal); "actual" changes reprice the bond exactly. Effective measures use ±100 bp shifts from prices printed to three decimals.
- A second-order Taylor term. Duration is the slope of price against yield; convexity is the curvature. The estimate adds half the curvature times the move squared, and a square is positive whichever way the move goes.
- What drives it. Convexity grows with maturity and with the spread of the cash flows in time, and falls with coupon and with yield. Zeros and long bonds have the most; a barbell of short and long bonds has more than a bullet of the same duration.
- Effective measures. Reprice the bond up and down by the same shift. The spread of the two prices gives the duration; how far their average sits above today's price gives the convexity. It is how desks measure bonds whose cash flows change with rates, such as callables and mortgages, which the formula cannot handle.
- Negative convexity. When yields fall, homeowners refinance and issuers call, so the price gain is capped; when yields rise, prepayments slow and duration lengthens. Holders are paid a higher yield for being short that option.
- Run it backwards. For a stated loss, −D × Δy + ½ C × Δy² = −loss is a quadratic in Δy; take the smaller root. It is a little more than the duration-only answer, because convexity cushions the fall.
| Duration term: 7 × 0.01 | 7% |
| Convexity term: ½ × 60 × 0.01² | 0.3%, added either way |
| Yields rise: −7 + 0.3 | −6.7% |
| Yields fall: +7 + 0.3 | +7.3% |
| Gain minus loss: 60 × 0.01² | 0.6 points |
| Repriced at 100, 93.3 and 107.3: effective D = (107.3 − 93.3) ÷ (2 × 100 × 0.01) | 7.0 |
| Effective C = (93.3 + 107.3 − 200) ÷ (100 × 0.0001) | 60 |
The formulas
The curvature of price against yield.
Duration's straight line plus the bend. The second term is positive either way.
The duration terms cancel; twice the convexity term remains.
Both measures from three repriced values.
The smaller root of the quadratic.
Worked example
The duration term, the convexity term, and their sum, for the drawn move. The follow-up asks why the convexity term is added whichever way yields move.
See it move
Same bond. Change its maturity, coupon and yield, the size of the yield move, and which way it goes.
- Make the move bigger. The gain on a fall, the loss on a rise and the gap between them all grow, and the gap grows with the square of the move. Duration alone gets further from the exact answer.
- Lengthen the maturity. Duration and convexity both rise, and so does the asymmetry.
- Raise the coupon, or raise the yield. Duration and convexity both fall, and the asymmetry shrinks.
- Flip the direction and watch the three estimates: the duration bar changes sign, the convexity correction stays positive, and the exact repricing lands near the corrected estimate on either side.
Run it backwards
A risk system shows today's price and the prices if yields rise and fall 100 bp. What effective duration and convexity do those three numbers imply?
Duration from the spread between the two shifted prices, divided by twice the price and the shift. Convexity from how far the average of the shifted prices sits above today's price, divided by the price and the shift squared.
Desks use these when the cash flows themselves move with yields, as in callable bonds and mortgages. The formula assumes fixed cash flows; repricing with a model at shifted yields does not, and it is how negative convexity shows up.
Traps
Say it in the interview
“What is convexity and why does it matter?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- % change ≈ −D × Δy + ½ C × Δy²; the second term always adds.
- Gain on a fall beats loss on an equal rise by C × Δy².
- Longer, lower-coupon, lower-yield bonds have more convexity; it costs yield.
- Effective measures come from three repriced values and handle bonds whose cash flows move.