What Is Bond Convexity?
Bond convexity is the curvature in the relationship between a bond's price and its yield, which duration's straight-line estimate misses. Because the relationship curves, a fall in rates gains more than an equal rise loses, so convexity works in the bondholder's favour.
Duration says a bond moves a fixed percentage for each point of rate change. That is a straight line drawn through something that curves, and the gap between the line and the curve has a name.
How it works
Price and yield move in opposite directions, but not proportionally. Plot price against yield and you get a curve that is steeper at low yields and flatter at high ones. Duration is the slope of that curve at one point — accurate for small moves, increasingly wrong for large ones.
Convexity measures the curvature. It is the second-order correction to duration’s first-order estimate, and for an ordinary bond it is positive.
Positive convexity means the asymmetry favours you. If rates fall one point you gain slightly more than duration predicted. If they rise one point you lose slightly less. Both errors point the same way, and that way is good for whoever owns the bond.
Nobody charges you for it. It falls out of the arithmetic of discounting future cash flows, and it is one of the few genuinely free things in a bond.
A worked example
Take a bond with a duration of 8.
Rates fall one point. Duration predicts +8%. The actual gain is a little more — say 8.4%.
Rates rise one point. Duration predicts −8%. The actual loss is a little less — say 7.6%.
That half a percent either way is convexity. On a one-point move it is a detail. The point is what happens when the move is larger.
On a three-point move the error is no longer a detail. Duration’s estimate is off by several percentage points, and a portfolio hedged on duration alone is no longer hedged. This is why the term shows up in fund reporting and almost never in retail conversation — it is invisible until the day it is not.
Where it turns against you
Negative convexity exists and it is the mirror image. A bond the issuer can repay early has a price that stops rising as rates fall, because the closer the call gets the less the extra yield is worth.
So gains are capped and losses are not. That is the same shape as selling an option, which is exactly what a callable bondholder has done — callable bond covers it in full.
Why it exists at all
It falls out of discounting. Each future payment is divided by a factor that compounds with the rate — and division by a growing number produces a curve, not a line. The further away the payment, the more pronounced the curvature.
So convexity rises with maturity, exactly as duration does. A thirty-year bond has both more duration and more convexity than a two-year one. The two travel together because they are the first and second derivatives of the same relationship.
A lower coupon also raises it, for the same reason it raises duration: more of the value sits at the far end where the curve is steepest.
This is why the asymmetry is described as free. Nobody built it in and no issuer is granting it as a concession — it is a consequence of the arithmetic everyone is using. The exception is where an issuer adds a right that truncates the curve, which is what a call provision does, and that right is paid for with a higher coupon rather than given away.
The original data
Use this site’s thirty-year fee measurement as a yardstick for what small percentages become: 5 basis points a year costs 1.5% of the final balance, 20 costs 5.8%, 75 costs 20.2%, 150 costs 36.5%.
Convexity’s correction is of that order on any single large move. A few tenths of a percent sounds negligible against a rate cycle, and it is the same size as the fee differences that decide thirty-year outcomes — which is the honest way to hold both figures at once.
And acting on it costs. A round trip on this site’s shared series is 0.0098, about 2% of the median bar range of 0.493. Restructuring a holding to capture a convexity difference pays that before capturing anything.
Where it shows up in practice: barbells against bullets. Two portfolios can hold identical duration and different convexity — one concentrated at a single maturity, the other split between very short and very long. The split version has more curvature, so it does better on large rate moves in either direction, and pays for it with a slightly lower yield.
When it fails
The characteristic failure is hedging a portfolio on duration alone and calling it hedged. Matching duration between a holding and a hedge neutralises the first-order move and leaves the curvature unmatched. For small changes nobody notices. For a large one the two sides diverge by an amount that was predictable from the start, and the position that was described as neutral turns out to have a direction.
A second failure is paying up for convexity. It is free in an ordinary bond and priced into anything structured to enhance it; buying the enhanced version means buying something else as well.
A third is assuming all bonds have it. Callables and mortgage-backed securities can have negative convexity, where the asymmetry runs the other way.
A fourth is quoting it without duration. Convexity is a correction to a number; on its own it says nothing about how far anything moves.
And a fifth is treating it as a reason to hold long bonds. The asymmetry is real and small; bond duration is the number that decides whether the holding is survivable.
Related
Bond duration covers the straight-line estimate convexity corrects. Interest rate covers the variable both are measured against. And callable bond covers where convexity turns negative.
Convexity is the rare case in finance where the correction to a simple model works in your favour rather than against it. Duration tells you what you will lose and quietly overstates it, and understates what you will gain. Almost every other approximation I have met does the opposite.
— Michael Whitman
This page is educational, not financial advice. Test every idea on your own charts before risking money.