What is the modified duration of this liability? (**Hint**: You need to use the definition of modified duration and differentiate the price (i.e., present value) of this liability.)

To calculate the modified duration of the liability, we need to follow these steps:

### Key Concepts:
- **Modified Duration (D\(_\text{mod}\))** measures the sensitivity of the price (present value) of a bond or liability to changes in interest rates. It is a useful measure for understanding interest rate risk.
- The formula for modified duration is:
  \[
  D_{\text{mod}} = \frac{D_{\text{Mac}}}{1 + r}
  \]
  where \( D_{\text{Mac}} \) is the **Macaulay Duration**.

### Step 1: Calculate the Macaulay Duration (D\(_\text{Mac}\))
For a perpetuity, the Macaulay Duration \( D_{\text{Mac}} \) is given by:
\[
D_{\text{Mac}} = \frac{1 + r}{r}
\]
However, in this case, the payments start at \( t = 3 \). So, we need to adjust the formula to reflect this delay.

The adjusted formula for \( D_{\text{Mac}} \) for a perpetuity starting at \( t = 3 \) is:
\[
D_{\text{Mac}} = 2 + \frac{1}{r}
\]
Where:
- The "2" accounts for the fact that the payments start at \( t = 3 \) instead of \( t = 0 \).
- \( r \) is the discount rate, which is 3.73% or 0.0373.

### Step 2: Calculate the Macaulay Duration (D\(_\text{Mac}\))
Substitute \( r = 0.0373 \) into the formula:
\[
D_{\text{Mac}} = 2 + \frac{1}{0.0373}
\]
\[
D_{\text{Mac}} = 2 + 26.8106
\]
\[
D_{\text{Mac}} = 28.8106
\]

### Step 3: Calculate the Modified Duration (D\(_\text{mod}\))
Now, use the formula for modified duration:
\[
D_{\text{mod}} = \frac{28.8106}{1 + 0.0373}
\]
\[
D_{\text{mod}} = \frac{28.8106}{1.0373}
\]
\[
D_{\text{mod}} = 27.7703
\]

### Conclusion:
The modified duration of this liability is approximately **27.77 years**.

This means that for a 1% change in the interest rate, the present value of this liability would change by approximately 27.77%. 

Would you like more details or have any further questions?

### Related Questions:
1. How does the modified duration compare to the Macaulay duration for this liability?
2. What impact does a higher discount rate have on the modified duration?
3. Can modified duration be used to estimate the percentage change in the liability's present value for small interest rate changes?
4. What would be the modified duration if the payments started immediately at \( t = 0 \)?
5. How does modified duration affect the overall interest rate risk management strategy?

### Tip:
Modified duration is particularly useful for understanding interest rate sensitivity, especially when managing long-term liabilities or assets.

Learn how to calculate the modified duration of a liability using the Macaulay Duration and interest rate sensitivity concepts. Follow a detailed step-by-step guide with a practical example. Suitable for advanced learners and professionals in finance.

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