How is effective convexity computed for a putable bond and why is it positive and higher than straight?

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AcadiFi · Accepted Answer

Effective convexity uses the same tree-based approach as effective duration. Formula: EffConv = (V_plus + V_minus - 2*V_0) / (V_0 * delta_y^2). Example: a putable bond with V_0 = 99.60, shift rates up 25 bps to get V_plus = 98.75, shift down 25 bps to get V_minus = 100.55. EffConv = (98.75 + 100.55 - 2*99.60) / (99.60 * 0.0025^2) = (199.30 - 199.20) / (99.60 * 0.00000625) = 0.10 / 0.0006225 = 160.6. The straight bond of the same coupon and maturity might have EffConv of 22. The putable has much higher convexity because the put floors the downside — when rates rise, the bond can be put at 100, so V_plus is bounded. When rates fall, the put is out-of-the-money and the bond rises with the straight bond. The asymmetry between upside (full participation) and downside (limited to 100) creates positive convexity. Intuition via the price-yield curve: a putable bond's price-yield curve bends sharply to the right as rates rise (asymptoting to a floor near the put price), while following the straight bond on the downside. That strong curvature is high convexity. Contrast with callable bonds — their effective convexity can be negative in certain rate regions because rising rates allow the callable to act like a straight bond (no calls) while falling rates cap upside. In summary: putable bonds exhibit positive (often large) effective convexity, callable bonds exhibit low or negative effective convexity, and straight bonds exhibit moderate positive convexity. Portfolio managers use putables to add convexity without taking excessive duration, useful in hedging mortgage-backed securities that have negative convexity.

How is effective convexity computed for a putable bond and why is it positive and higher than straight?

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