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Bond convexity measures how sensitive a bond’s price is to changes in interest rates. It complements duration, which captures the relationship only linearly. Convexity accounts for the actual curvature of that relationship and provides a more accurate estimate of price behavior.
To understand convexity, it’s best to start with duration. Duration estimates how much a bond’s price changes in response to a 1% change in interest rates. If a bond has a duration of 5 years, a 1% increase in rates would reduce its price by approximately 5%.
But that estimate is linear. In reality, the price-interest-rate relationship forms a curve, not a straight line. Convexity captures curvature and corrects the error duration introduces when interest rate movements are large.
Convexity can be positive or negative. Each type has different implications for the bond investor.
The bond price rises more than duration predicts when rates fall, and falls less than expected when rates rise. This is favorable for the investor because it amplifies gains and cushions losses.
The bond price behaves oppositely: it rises less when rates fall and falls more when rates rise. Callable bonds typically exhibit negative convexity.
Convexity is expressed as a number. Its use complements duration in a straightforward process.
Check the bond’s duration. This is the percentage change in price in response to a 1% movement in interest rates.
Check the convexity. It adjusts that estimate by capturing the relationship's actual curvature.
Apply both. Price change ≈ (-Duration × Δ interest rates) + (0.5 × Convexity × Δ interest rates²).
Compare bonds. At equal duration, the bond with higher positive convexity offers better protection against rate hikes.
Institutional investors routinely use this information when constructing fixed-income portfolios.
Two bonds have a duration of 7 years. Bond A has a convexity of 60, and Bond B has a convexity of 30. If interest rates rise by 2%, duration predicts a 14% decline for both. However, the convexity adjustment reduces the decline for A to approximately 11.6% and that for B to 12.8%. The higher the positive convexity, the smaller the actual loss in the event of rate hikes.
Convexity is an advanced concept and gives rise to specific misunderstandings.
Ignoring convexity and relying solely on duration.
Failing to distinguish between positive and negative convexity.
Assuming thata longer duration always implies a greater risk.
This information provides direct advantages when analyzing bonds.
Comparing bonds with the same duration but different interest rate sensitivity.
Estimate potential losses more accurately.
Build portfolios that are more resilient to sudden shifts in interest rates.
Convexity accounts for what duration doesn’t capture: the actual curvature between price and interest rate. A bond with high positive convexity performs better in the face of large interest rate movements. Including it in your analysis distinguishes a basic estimate from an accurate assessment.