You are considering a luxury apartment building project that requires a capital investment of 12,500,000. The building has 50 units. You expect the maintenance cost for the apartment building to be 250,000 in the first year, $300,000 in the second year, and increasing by 50,000 in subsequent years. The cost to hire a manager for the building is estimated to be 80,000 per year. After five years of operation, the apartment building can be sold for 14,000,000. What is the annual rent per apartment unit that will provide a return on investment of 15% per year? Assume the building will remain fully occupied during the five years.
(provide your answer in the nearest dollars, i.e., no decimal places)

To find the annual rent per apartment unit that will provide a return on investment (ROI) of 15% per year, we need to calculate the cash flows for each of the five years and the final sale value, then determine the rent that would satisfy the given ROI.

### Step 1: Define the Cash Flows

1. **Initial Investment:** 
   - The capital investment is $12,500,000.

2. **Operating Costs:**
   - **Maintenance Costs:**
     - Year 1: $250,000
     - Year 2: $300,000
     - Year 3: $350,000
     - Year 4: $400,000
     - Year 5: $450,000
   - **Manager Salary:**
     - $80,000 per year for all 5 years.

   So, total operating costs per year are:
   - Year 1: $250,000 + $80,000 = $330,000
   - Year 2: $300,000 + $80,000 = $380,000
   - Year 3: $350,000 + $80,000 = $430,000
   - Year 4: $400,000 + $80,000 = $480,000
   - Year 5: $450,000 + $80,000 = $530,000

3. **Revenue from Rent:**
   - Let $ R $ be the annual rent per apartment unit.
   - Total rent revenue per year = $ 50R $.

4. **Sale Value:**
   - The building can be sold for $14,000,000 after 5 years.

### Step 2: Calculate the Present Value (PV) of Costs and Revenues

The goal is to ensure the Net Present Value (NPV) of the project equals zero when applying the 15% required return on investment.

Using the formula for Present Value (PV):
\[
PV = \frac{Cash \ Flow}{(1 + r)^n}
\]
where $ r = 15\% $ or 0.15, and $ n $ is the year number.

**Year 1:**
- Revenue: $ \frac{50R}{(1.15)^1} $
- Costs: $ \frac{330,000}{(1.15)^1} $

**Year 2:**
- Revenue: $ \frac{50R}{(1.15)^2} $
- Costs: $ \frac{380,000}{(1.15)^2} $

**Year 3:**
- Revenue: $ \frac{50R}{(1.15)^3} $
- Costs: $ \frac{430,000}{(1.15)^3} $

**Year 4:**
- Revenue: $ \frac{50R}{(1.15)^4} $
- Costs: $ \frac{480,000}{(1.15)^4} $

**Year 5:**
- Revenue: $ \frac{50R}{(1.15)^5} $
- Costs: $ \frac{530,000}{(1.15)^5} $
- Sale Value: $ \frac{14,000,000}{(1.15)^5} $

### Step 3: Set Up the Equation

The equation for NPV = 0 is:
\[
NPV = -12,500,000 + \left(\frac{50R - 330,000}{1.15}\right) + \left(\frac{50R - 380,000}{(1.15)^2}\right) + \left(\frac{50R - 430,000}{(1.15)^3}\right) + \left(\frac{50R - 480,000}{(1.15)^4}\right) + \left(\frac{50R - 530,000 + 14,000,000}{(1.15)^5}\right) = 0
\]

### Step 4: Solve for $ R $

We'll solve this equation for $ R $ (the annual rent per apartment unit).

Let me calculate the value.The annual rent per apartment unit that will provide a 15% return on investment is approximately **$41,373**.

Would you like to go into more detail on how this was calculated, or do you have any other questions?

Here are 5 related questions for further exploration:
1. How would changes in the expected sale price after 5 years affect the required annual rent?
2. What would the impact be if the maintenance costs increase faster than expected?
3. How would the required rent change if the building were only 90% occupied?
4. What if the return on investment goal was set at 10% instead of 15%?
5. How would different financing options (e.g., loans) affect the required rent?

**Tip:** Understanding the sensitivity of your investment to changes in key variables (like occupancy rates or costs) is crucial for robust financial planning.

Learn how to calculate the annual rent per apartment unit that provides a 15% return on investment (ROI) for a luxury apartment building project. This problem involves determining the net present value (NPV) of cash flows over five years, considering initial investment, maintenance costs, manager salary, and eventual sale value.

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