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