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

Given the situation, you are paying a perpetual lease of $1 million per year starting from year 3 onward. The interest rate is flat at 3.73% but expected to shift downward in the future.

Let's break down the present value (PV) of this lease payment stream:

### 1. **Calculate the Present Value of the Perpetuity:**
   - The payment starts at year 3 (i.e., at t=3), so we need to discount the perpetuity back to t=3.
   - The formula for the present value of a perpetuity starting at t=3 is:
     \[
     PV_{\text{at } t=3} = \frac{A}{r}
     \]
     where $A$ is the annual payment, and $r$ is the interest rate.

     Given:
     \[
     A = 1,000,000 \text{ dollars}, \quad r = 3.73\% = 0.0373
     \]

     The present value at $t = 3$ is:
     \[
     PV_{\text{at } t=3} = \frac{1,000,000}{0.0373} \approx 26,820,645.21 \text{ dollars}
     \]

### 2. **Discount this Value Back to Present (t=0):**
   - To find the present value at $t = 0$, we need to discount the value calculated above back 3 years.

   \[
   PV_{\text{at } t=0} = PV_{\text{at } t=3} \times \frac{1}{(1 + r)^3}
   \]
   \[
   PV_{\text{at } t=0} = 26,820,645.21 \times \frac{1}{(1.0373)^3} \approx 24,115,764.18 \text{ dollars}
   \]

### **Conclusion:**
The present value of your perpetual lease payments, assuming the interest rate remains at 3.73%, is approximately **$24,115,764.18** at time $t=0$.

### **Considerations:**
- If the interest rate shifts down as expected, the present value would increase because the discount factor decreases. This would make the cost of the perpetuity higher if recalculated under the lower rate.

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Do you want further details on how the interest rate shift could affect the present value? Or have any other questions?

### Relative Questions:
1. How would a 1% decrease in interest rates affect the present value of the lease?
2. What would be the impact of payments starting at a different time, say t=5, on the present value?
3. How does the present value calculation change if payments are not perpetual but end after a fixed period?
4. What is the effect of varying the lease amount on the present value calculation?
5. How would an increasing or decreasing payment structure impact the present value?

### Tip:
Always consider how changes in interest rates can significantly impact the present value of future cash flows, particularly in long-term financial decisions.

Learn how to calculate the present value of perpetual lease payments starting at year 3, with an annual payment of $1 million, under a flat interest rate of 3.73%. Explore the impact of interest rate changes on the present value. Suitable for advanced learners and professionals.

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