Convexity Bonds
Convexity is a measure of the curvature in the relationship between a bond’s price and its yield. While Modified Duration provides a linear approximation of price change, convexity accounts for the fact that the price-yield relationship is curved, not straight. Bonds with higher convexity experience larger price gains when rates fall and smaller price losses when rates rise, compared to low-convexity bonds with the same duration.
What Is Convexity?
The price of a bond does not change in a perfectly straight line when interest rates change. As yields fall significantly, bond prices rise more than the linear (duration) estimate. As yields rise significantly, bond prices fall less than the linear estimate. This curvature is called convexity.
Convexity is always positive for regular bonds. A more convex bond is more desirable: it offers better protection against large rate increases and greater profit in rate decline scenarios.
Better Price Estimate Using Convexity
Price change (%) ≈ -Modified Duration x (change in yield) + 0.5 x Convexity x (change in yield)^2
The second term (convexity correction) improves the accuracy of the price change estimate, especially for large rate movements.
High vs Low Convexity Bonds
**High convexity bonds:**
– Long-maturity, low-coupon bonds
– Zero-coupon bonds
– Bonds with embedded call options or put options
**Low convexity bonds:**
– Short-maturity bonds
– High-coupon bonds
– Callable bonds (where the issuer can redeem early if rates fall, limiting price upside)
Why Convexity Matters for Investors
Given two bonds with the same modified duration, the more convex bond is superior: it gains more when rates fall and loses less when rates rise. Investors should be willing to pay a premium (accept slightly lower yield) for a bond with higher convexity.
Negative Convexity
Callable bonds and mortgage-backed securities can exhibit negative convexity at certain yield levels. When rates fall, the issuer exercises the call option, preventing the bond price from rising as much. Investors in callable bonds must account for this.
Practical Example
Bond A: Modified Duration 7, Convexity 60
Bond B: Modified Duration 7, Convexity 30
Rates fall by 2%:
– Duration estimate for both: +14% price gain
– Convexity correction for A: +0.5 x 60 x (0.02)^2 = +1.2%
– Convexity correction for B: +0.5 x 30 x (0.02)^2 = +0.6%
Bond A price rises 15.2%; Bond B rises 14.6%. Bond A’s higher convexity delivers a better outcome in the rate decline scenario.
Key Takeaways
– Convexity measures the curvature in the bond price-yield relationship beyond linear duration estimates
– Higher convexity is desirable: more gain when rates fall, less loss when rates rise
– Long-duration, low-coupon, and zero-coupon bonds have higher convexity
– Callable bonds have lower convexity (or negative convexity) due to the issuer’s call option
– Combined use of duration and convexity gives more accurate price change estimates for large rate movements
