Answer: effective duration.

## Explanation For bonds that do not have a well-defined internal rate of return (such as callable bonds, putable bonds, or bonds with embedded options), **effective duration** is the most appropriate measure of interest rate risk. ### Key Concepts: 1. **Effective Duration**: - Measures the sensitivity of a bond's price to changes in the benchmark yield curve - Accounts for changes in expected cash flows due to embedded options - Calculated using: $$\text{Effective Duration} = \frac{P_- - P_+}{2 \times P_0 \times \Delta y}$$ where: - $P_-$ = price when yield decreases by $\Delta y$ - $P_+$ = price when yield increases by $\Delta y$ - $P_0$ = initial price - $\Delta y$ = change in yield 2. **Modified Duration**: - Measures price sensitivity assuming cash flows do not change with yield changes - Appropriate for option-free bonds - Derived from Macaulay duration: $$\text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \frac{y}{m}}$$ where $y$ = yield to maturity, $m$ = number of compounding periods per year 3. **Macaulay Duration**: - Weighted average time to receive cash flows - Not a direct measure of price sensitivity - Calculated as: $$\text{Macaulay Duration} = \frac{\sum_{t=1}^n \frac{t \times CF_t}{(1+y)^t}}{P}$$ ### Why Effective Duration is Correct: - Bonds without well-defined internal rates of return typically have embedded options - Embedded options cause cash flows to change with interest rate movements - Effective duration accounts for these changing cash flows - Modified and Macaulay durations assume fixed cash flows, making them inappropriate for such bonds ### Example: For a callable bond: - When interest rates fall, the issuer may call the bond, changing the cash flow pattern - Effective duration captures this optionality - Modified duration would overestimate price appreciation potential

Author: LeetQuiz .