The current term structure of interest rates is flat at 3.73%, and is expected to shift down. You have locked into a long-term lease on a real estate in Cambridge to pay $1 million a year forever, starting 3 years t=3 from now t=0

To analyze this scenario, we need to calculate the present value (PV) of the perpetual payments under the given conditions. 

### Key Information:
- **Term structure of interest rates**: Flat at 3.73% (implying that the discount rate $ r = 3.73\% $)
- **Annual payment**: $1,000,000 starting from year 3 ( $ t = 3 $ )
- **Duration of payments**: Forever (perpetual payments)

### Steps to Calculate Present Value (PV):

1. **Calculate the present value of the perpetual payments starting at year 3**:
   - The formula for the present value of a perpetuity starting at time $ t = 3 $ is:
   \[
   PV_{\text{start at } t=3} = \frac{P}{r} \times \frac{1}{(1 + r)^2}
   \]
   Where:
   - $ P = 1,000,000 $ (annual payment)
   - $ r = 3.73\% = 0.0373 $ (discount rate)

2. **Substitute the values into the formula**:
   \[
   PV_{\text{start at } t=3} = \frac{1,000,000}{0.0373} \times \frac{1}{(1 + 0.0373)^2}
   \]

3. **Simplify the expression**:
   \[
   PV_{\text{start at } t=3} = 26,810,584.96 \times \frac{1}{1.0772}
   \]
   \[
   PV_{\text{start at } t=3} = 26,810,584.96 \times 0.9283
   \]
   \[
   PV_{\text{start at } t=3} = 24,890,106.33
   \]

### Conclusion:
The present value of the perpetual payments starting at year 3, discounted at 3.73%, is approximately **$24.89 million**.

Would you like more details on this calculation, or have any questions?

### Related Questions:
1. How does the shift in the interest rate term structure affect the present value of the lease?
2. What would happen if the payments started earlier, say at $ t = 1 $?
3. How would inflation affect the real value of these payments over time?
4. Can we determine the impact of changing the interest rate on the present value?
5. How would a different payment frequency (e.g., monthly instead of yearly) change the calculations?

### Tip:
Always remember to adjust the present value formula to account for the start time of the payments when they begin at a future date.

Learn how to calculate the present value (PV) of perpetual lease payments starting from year 3, discounted at 3.73% interest rate. This problem involves applying the present value formula for a perpetuity and understanding the impact of interest rates on lease valuation.

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