Bond Convexity Explained

Duration tells you how sensitive a bond’s price is to a small change in yield. Convexity tells you why that sensitivity is not constant — and why that asymmetry generally works in the bondholder’s favour. Understanding convexity helps explain why two bonds with identical durations can behave very differently when interest rates move by a large amount.

A quick recap of duration

Duration is a measure of a bond’s price sensitivity to yield changes. Specifically, modified duration approximates the percentage change in a bond’s price for a 1% (100 basis point) move in yield. A bond with a modified duration of 7 will lose roughly 7% of its price if yields rise by 1%.

Duration works well as an approximation for small yield changes. But for larger moves — say, yields jumping 200 or 300 basis points — the linear approximation starts to break down. The actual price change is not perfectly proportional to the yield change. Convexity captures the curvature that duration misses.

What convexity actually measures

If you plot a bond’s price against its yield, you get a curve, not a straight line. Duration is the slope of that curve at any given point. Convexity measures how the slope itself changes as yields move — the rate of change of duration with respect to yield.

Mathematically, convexity is the second derivative of the price-yield relationship (where duration is the first derivative). In practice, you rarely need to calculate it by hand — bond analytics platforms compute it for you. What matters is the interpretation.

For most standard (non-callable) bonds, convexity is positive. This means:

• When yields fall, prices rise more than duration alone would predict.
• When yields rise, prices fall less than duration alone would predict.

Positive convexity is therefore a desirable property. It creates an asymmetric payoff: you gain more from a favourable rate move than you lose from an equivalent unfavourable one.

A numerical example

Consider a hypothetical 10-year bond with a coupon of 7%, priced at par, with a modified duration of approximately 7 and a convexity of 60.

If yields fall by 2% (200 basis points):

• Duration-only estimate: +14% price increase.
• Convexity adjustment: adds roughly +1.2% (convexity × 0.5 × yield change squared = 60 × 0.5 × 0.04).
• Better estimate: approximately +15.2%.

If yields rise by 2%:

• Duration-only estimate: −14% price decrease.
• Convexity adjustment: still adds +1.2%.
• Better estimate: approximately −12.8%.

The convexity adjustment always adds to the price, regardless of the direction of the yield move. This is what “positive convexity benefits the bondholder” means in practice.

Why convexity increases with maturity

Longer-dated bonds have higher convexity than shorter-dated bonds with similar coupons. The intuition is that the cash flows of a longer bond are spread further into the future, making the price-yield relationship more curved.

Zero-coupon bonds have the highest convexity for a given maturity among standard bonds, because all cash flow is concentrated at maturity — the price-yield curve is maximally curved.

Lower-coupon bonds also have higher convexity than higher-coupon bonds of the same maturity, because more of the total value comes from the distant principal repayment rather than the nearer coupon payments.

Callable bonds: negative convexity at low yields

Callable bonds — bonds that the issuer can redeem before maturity at a specified price — exhibit a property called negative convexity in certain yield environments.

Here is why. As yields fall, a standard bond’s price rises toward (and potentially above) its call price. At that point, the issuer has an incentive to call the bond — retire the expensive debt and refinance at lower rates. Investors know this, so the bond’s price gets capped near the call price. The price cannot rise freely the way a non-callable bond’s price would.

The result is that, in a falling-rate environment, the callable bond’s price rises less than a comparable non-callable bond. The upside is truncated. Meanwhile, in a rising-rate environment, the callable bond behaves similarly to a non-callable bond — prices fall. The asymmetry works against the bondholder: you keep the downside but surrender the upside.

This is negative convexity: price rises are smaller than duration predicts, price falls are roughly as large as duration predicts.

In India, callable provisions appear in some corporate NCDs and in certain government-linked bonds. The offer document will specify the call date, call price, and conditions. Read this section carefully before investing, as it materially affects the potential return profile.

Convexity as a tiebreaker between similar bonds

When two bonds have similar YTMs and similar durations, the one with higher convexity is generally preferable — you are getting better asymmetric protection without giving up yield. In practice, the market prices convexity: higher-convexity bonds typically yield slightly less, because investors accept a lower yield in exchange for better price behaviour in large rate moves.

The decision of whether the convexity premium is worth the yield sacrifice is a portfolio-level question that depends on your view of rate volatility and your investment horizon.

Practical relevance for Indian retail investors

Most retail investors in India hold bonds to maturity, which reduces (but does not eliminate) the relevance of convexity. If you never sell before maturity and the issuer does not call the bond, you receive exactly the promised cash flows regardless of price movements in the interim.

Convexity becomes most relevant when:

• You are investing in long-dated G-Secs (10–40 year maturities) and may need to sell before maturity.
• You are comparing a callable NCD against a non-callable NCD from the same issuer.
• You are building a bond portfolio and want to understand how it will behave if the RBI makes large rate moves.
• You are using the screener to filter bonds by modified duration and want to understand why similarly-duration bonds can behave differently.

Summary

Duration gives a linear approximation of how bond prices respond to yield changes. Convexity corrects for the curvature of the actual price-yield relationship. Positive convexity (characteristic of most standard bonds) means gains from falling yields exceed losses from rising yields of the same magnitude — an asymmetry that benefits the bondholder. Callable bonds can exhibit negative convexity when trading near the call price, reversing this asymmetry. Higher convexity is generally better, all else being equal, but the market prices it through modestly lower yields on more convex bonds.

Further reading

• Bond Duration Explained →
• Why Bond Prices Fall When Rates Rise →
• Callable and Puttable Bonds Explained →
• YTM Calculator →
• Browse bonds by duration in the screener →