Why Zero-Coupon Bonds Have Higher Normalized Convexity

Summary

The document resolves an apparent conflict between the intuition that higher coupon rates mean higher bond convexity and the claim that zero-coupon bonds have the greatest convexity. The key is the definition: convexity is the second derivative of bond price with respect to yield divided by the bond price. Comparing raw curvature across bonds with different prices does not compare convexity on an equal basis.

To make the comparison meaningful, normalize prices so the bonds pass through the same price at the same yield. Under that comparison, lower-coupon bonds have greater convexity, with zero-coupon bonds highest. The explanation frames convexity as sensitivity per dollar invested rather than per bond. It is a concise conceptual answer; it does not specify maturity, yield conventions, or other assumptions that could matter in broader comparisons.

Key ideas

• Bond convexity is the second derivative of price with respect to yield, normalized by price.
• Raw price curvature can mislead when comparing bonds with different price levels.
• Comparing bonds at the same price and yield shows greater convexity for lower coupons.
• A zero-coupon bond has the highest convexity under the comparison described.

Tags

# A very simple question about convexity of a bond


# A very simple question about convexity of a bond












I was always under the impression that, ceteris paribus, higher the coupon rate, higher the convexity of the bond.

But Investopedia says the following:

"zero-coupon bonds have the highest degree of convexity".

Isn't that wrong?

Please correct me if I am missing something here.

## Answer by nbbo2 (score 6, accepted)

https://quant.stackexchange.com/a/35623

The chart you posted does not give a correct visual representaion of convexity . Convexity is not $\frac{\partial^2 P}{\partial y^2}$ but $\frac{1}{P}\frac{\partial^2 P}{\partial y^2}$. So you have to normalize for P. The 4 curves you plot have very different P.

When the curves are redrawn normalized so they go through the same point $(y_0,P_0)$ you will see that smaller coupons have bigger convexity. And zero coupons the biggest of all.

Put differently: convexity is measured "per dollar invested" not "per bond".

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.