Convexity
Convexity is the second-order measure of a bond's price sensitivity to changes in yield, capturing the curvature of the price-yield relationship that modified duration (a linear approximation) fails to capture. Positive convexity means that bond price gains from falling yields are larger than price losses from rising yields of the same magnitude — a desirable asymmetric return profile.
Key takeaways
- Duration approximates bond price change linearly; convexity corrects for the curvature: ΔP/P ≈ −D·Δy + ½·C·(Δy)².
- All non-callable bonds exhibit positive convexity: the price-yield curve bows upward, with larger gains for yield decreases than losses for yield increases.
- Callable bonds and mortgage-backed securities exhibit negative convexity in certain yield ranges because the issuer's prepayment or call option truncates price appreciation.
- Higher convexity is generally desirable (all else equal) because it provides asymmetric payoff — investors pay for it through lower yields.
- In options terms, owning a bond is equivalent to being long an asset with positive convexity; convexity is analogous to Gamma in option theory.
Explanation
The price-yield relationship for a standard bond is a curve, not a line. Modified duration describes the slope of the tangent to that curve at the current yield level, providing a first-order approximation: ΔP/P ≈ −MD × Δy. But as yield changes become large, the tangent line increasingly diverges from the actual curve. Convexity captures this second-order effect:
ΔP/P ≈ −MD × Δy + ½ × C × (Δy)²
where MD is modified duration and C is convexity. For a standard bullet bond with N semi-annual coupon periods:
C = [Σ t(t+1)·CF_t / (1+y)^t] / [P × (1+y)²]
where CF_t is the cash flow at period t, y is the yield per period, and P is the current price. This formula weights each cash flow by t(t+1) — cash flows further in the future contribute more to convexity, which is why longer-maturity, lower-coupon bonds have higher convexity.
The investment significance of convexity is its asymmetric payoff. Consider two bonds with identical durations: Bond A with convexity of 150 (units: years²) and Bond B with convexity of 90. For a 1% (100 bps) parallel yield shift: using C=150 for Bond A, the convexity correction is ½ × 150 × (0.01)² = 0.0075 (75 bps of additional price appreciation for a yield decline, or 75 bps less price decline for a yield increase). Bond A outperforms Bond B by 30 bps (the convexity differential of 60 × ½ × 0.01²) in both directions of yield movement. Convexity is thus 'free money' in a volatile rate environment — the bond with higher convexity outperforms regardless of the direction of rate moves. Investors pay for this convexity in the form of a lower starting yield (the convexity premium).
Negative convexity, exhibited by callable bonds and mortgage-backed securities, means the price-yield curve bows downward in certain ranges. When yields decline (prices rise) toward the call price or prepayment threshold, price appreciation is capped because issuers will call or prepay, reinvesting at lower rates. This creates a ceiling on bond prices at high yield levels that is not offset by equivalent upside at low yields — the opposite of positive convexity's desirable asymmetry.
Formula
ΔP/P ≈ −MD × Δy + ½ × C × (Δy)² | Convexity = [Σ t(t+1)·CF_t/(1+y)^t] / [P × (1+y)²]
Example
A portfolio manager holds a 10-year Treasury bond with a modified duration of 8.5 and convexity of 80. The 10-year yield falls from 4.50% to 3.50% (100 bps decline). Duration-only estimated price change: −8.5 × (−0.01) = +8.5%. Convexity correction: ½ × 80 × (0.01)² = +0.40%. Total estimated price change: +8.5% + 0.40% = +8.90%. For a 100 bps yield increase (from 4.50% to 5.50%): Duration estimate: −8.5%. Convexity correction: +0.40% (convexity always adds positively). Total: −8.10%. The convexity advantage is clear: the bond gains 8.90% when yields fall but loses only 8.10% when yields rise by the same amount — a 40-basis-point asymmetric advantage attributable to convexity in each direction of movement.
Related terms
Basis Bond Bullet Bond Current Yield Duration Equity Tranche Investment Grade Modified Duration Mortgage Backed Security Negative Convexity Nob Spread Premium