Lesson 2 of 5 · 13 min
Effective convexity: callable versus putable bonds
An embedded call caps a bond's price gains when rates fall and can make its effective convexity negative, while an embedded put cushions price losses when rates rise and keeps effective convexity positive.
In short
- Effective convexity = (PV₋ + PV₊ − 2PV₀) ÷ ((ΔCurve)² × PV₀), the curve-based version of approximate convexity.
- Callable bond price = non-callable price − value of the call (held by the issuer). The investor bears call risk.
- When rates fall, a callable bond's price gains are capped: its effective duration drops and its effective convexity can turn negative.
- Putable bond price = non-putable price + value of the put (held by the investor).
- When rates rise, the put limits price losses and lowers effective duration; a putable bond's effective convexity is always positive.
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