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