Answer: less than the effective duration of an option-free bond.

## Explanation When interest rates are increasing, the effective duration of a bond with an embedded put option is **less than** the effective duration of an option-free bond. ### Key Concepts: 1. **Embedded Put Option**: A put option gives the bondholder the right to sell the bond back to the issuer at a predetermined price (usually par) before maturity. 2. **Effective Duration**: Measures the sensitivity of a bond's price to changes in interest rates, accounting for embedded options. 3. **Impact of Rising Interest Rates**: - When interest rates rise, bond prices fall - For a putable bond, as rates rise, the put option becomes more valuable because the bondholder can put the bond back to the issuer at the strike price (usually par) rather than selling it at the lower market price - This put option acts as a floor on the bond's price decline 4. **Duration Comparison**: - **Option-free bond**: Price declines more sharply as rates rise (higher duration) - **Putable bond**: Price decline is limited by the put option (lower duration) - Therefore, the putable bond has **lower effective duration** than an otherwise identical option-free bond when rates are rising ### Mathematical Insight: Effective duration for a putable bond is calculated as: ``` Effective Duration = (P_- - P_+) / (2 × P_0 × Δy) ``` Where: - P_- = price when yields decrease - P_+ = price when yields increase - P_0 = initial price - Δy = change in yield For putable bonds, P_+ is higher than for option-free bonds (due to the put protection), resulting in a smaller numerator and thus lower effective duration. ### Why Not Other Options: - **B (same)**: Incorrect because the embedded put option changes the bond's price sensitivity - **C (greater)**: Incorrect - put options reduce duration, not increase it, when rates are rising **Answer: A** - The put option provides downside protection, limiting price decline and reducing effective duration compared to an option-free bond.

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