Answer: less than the Macaulay duration.

## Explanation In a positive interest rate environment, the modified duration of an option-free bond is **less than** the Macaulay duration. ### Key Concepts: 1. **Macaulay Duration**: Measures the weighted average time until cash flows are received, expressed in years. 2. **Modified Duration**: Measures the price sensitivity of a bond to changes in interest rates. It is calculated as: \[ \text{Modified Duration} = \frac{\text{Macaulay Duration}}{1 + \frac{YTM}{m}} \] where: - YTM = Yield to Maturity - m = Number of compounding periods per year 3. **Relationship**: Since the denominator \(1 + \frac{YTM}{m}\) is **greater than 1** in a positive interest rate environment (YTM > 0), the modified duration will always be **less than** the Macaulay duration. ### Why Option A is Correct: - When interest rates are positive, YTM > 0 - Therefore, \(1 + \frac{YTM}{m} > 1\) - Dividing Macaulay duration by a number greater than 1 results in a smaller value - Thus: Modified Duration < Macaulay Duration ### Additional Notes: - This relationship holds true for all option-free bonds in positive interest rate environments - Only when interest rates are zero would modified duration equal Macaulay duration - Modified duration provides a more practical measure for interest rate risk management as it directly estimates percentage price change for a given change in yield

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