What it means
If you plot a bond's price against interest rates, you do not get a straight line, you get a curve that bends upwards. Duration is the slope of that curve at one point, which is a fine approximation for tiny rate moves but increasingly wrong as moves get larger.
Convexity measures the bend, and adding it to a duration estimate produces a far more accurate price prediction. The practical consequence is asymmetry.
For a bond with positive convexity, a 1% fall in yields produces a bigger price gain than the price loss caused by a 1% rise in yields. That asymmetry is a genuine benefit, which is why investors will accept a slightly lower yield on a more convex bond.
Convexity is generally higher for bonds with longer maturities, lower coupons and lower yields. A thirty-year zero-coupon bond is extremely convex, while a two-year bond paying a high coupon is nearly linear over normal rate ranges.
Portfolio managers deliberately build convexity into a portfolio when they expect large rate moves but are unsure of the direction. Some bonds have negative convexity, and this is where the concept earns its keep.
Callable bonds and mortgage-backed securities can be repaid early, so when rates fall their prices stop rising because the issuer or borrower simply refinances. The holder gets the downside of rising rates without the matching upside, which is a poor trade unless the yield compensates for it.
For a non-specialist, the takeaway is that duration alone understates losses in a rate selloff for some instruments and understates gains in a rally for others. Anyone quoting a portfolio's rate sensitivity from duration alone is giving you a first approximation.
Convexity is the second-order correction that tells you how reliable that approximation is.
In practice
Real-world examples
Example 1
A pension fund comparing two bonds with identical yields and durations picks the one with higher convexity, reasoning that it will perform better whichever way rates move. The trade-off is that the more convex bond usually trades at a slightly richer price.
Example 2
A treasury team at an insurance company reports rate sensitivity to its board using duration only. After a sharp rate move, the actual portfolio loss comes in smaller than forecast, and the team adds convexity to its reporting so future estimates are closer to reality.
Example 3
An investor buys a mortgage-backed security for its attractive yield without noticing the negative convexity. When rates drop, homeowners refinance, the bonds are repaid early, and the investor is left reinvesting the cash at the new lower rates rather than enjoying a price gain.
Think of it
Convexity is how bond prices curve with rate changes-beneficial curvature.
Formula
Calculation
Approximate price change % = (-Modified duration x Change in yield) + (0.5 x Convexity x Change in yield squared)
A bond portfolio worth $1,000,000 has a modified duration of 7 and a convexity of 60. Interest rates rise by 1%, which is 0.01 in decimal terms.
Duration effect = -7 x 0.01 = -0.07, or -7.0%
Convexity effect = 0.5 x 60 x (0.01 x 0.01) = 0.5 x 60 x 0.0001 = 0.003, or +0.3%
Total estimated price change = -7.0% + 0.3% = -6.7%
In money terms, the duration-only estimate predicts a loss of 7.0% x $1,000,000 = $70,000, while adding convexity predicts a loss of 6.7% x $1,000,000 = $67,000. The convexity adjustment is worth $3,000, and it works in the investor's favour in both directions: if yields fell by 1% instead, the estimated gain would be 7.0% + 0.3% = 7.3%, not 7.0%.Case study
Seen in the real world
This case is illustrative and fictional. Aldergate Mutual, an invented insurance company, held a $400,000,000 bond portfolio and managed interest rate risk entirely through duration targets, keeping portfolio duration close to the duration of its liabilities.
During a period of unusually large rate swings, the actual profit and loss on the portfolio kept diverging from what the duration model predicted, sometimes by several million dollars a quarter. An analyst discovered that roughly a fifth of the portfolio sat in callable bonds with negative convexity, while the liabilities behaved like long, highly convex instruments. The two sides had matching durations but very different curvature.
In this fictional example, Aldergate rebuilt its reporting to match convexity as well as duration, and cut its callable holdings. The lesson is that matching the slope of two curves is not the same as matching their shape.
Watch out
Common mistakes
- Relying on duration alone to estimate losses from a large rate move, which overstates the damage on ordinary bonds and understates it on callable ones.
- Assuming all bonds have positive convexity, when callable bonds and mortgage-backed securities frequently do not.
- Treating high convexity as free, when the market prices it in and the investor typically gives up a small amount of yield to get it.
Questions
People also ask
Is high convexity always desirable?
Generally yes for a bondholder, but it is paid for in a lower yield, so it is only worthwhile if rates are likely to move sharply.
Does convexity matter for small rate changes?
Barely, because the effect depends on the square of the rate change, so it stays negligible until moves get large.
How is convexity reported?
Usually as a single number per bond or portfolio alongside duration, and it is used in the same estimation formula rather than read on its own.