An investor plans on buying a $2 million property with 5% down and annual financing of 5% for 12 years. The property will be sold four years after its purchase for $2.25 million. Determine the monthly payment on the loan. $17,5>2,92 What is the monthly growth rate for the value of the property over the 4 years? Use months in the calculation not years. (Answer should have two decimal points with percentage, e.g. 0.35% ). What would be the return from selling the property after four years?

To answer your questions, we'll go step by step through each of the parts.

### Part 1: Monthly Payment on the Loan
1. The investor plans to buy a property worth **$2 million** with **5% down payment** and an annual interest rate of **5%** for **12 years**.
   
   - **Loan Amount:**  
     Since the investor is paying 5% down, they are borrowing 95% of the property value.  
     \[
     \text{Loan Amount} = 2,000,000 \times (1 - 0.05) = 1,900,000
     \]
   
   - **Interest Rate:**  
     The annual interest rate is **5%**, so the monthly interest rate is:
     \[
     \text{Monthly Interest Rate} = \frac{5}{12} = 0.41667\% = 0.004167
     \]
   
   - **Loan Term:**  
     The loan term is **12 years**, so in months, it's:
     \[
     \text{Term in Months} = 12 \times 12 = 144 \text{ months}
     \]

Now, we calculate the monthly payment using the standard formula for a fixed-rate mortgage:

\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- $ M $ = monthly payment
- $ P $ = loan amount = 1,900,000
- $ r $ = monthly interest rate = 0.004167
- $ n $ = number of months = 144

Let’s compute this now.

### Part 2: Monthly Growth Rate of Property Value Over 4 Years
The property will be sold for **$2.25 million** after **4 years**. To determine the monthly growth rate, we use the formula for compound growth:

\[
FV = PV(1 + r)^n
\]

Where:
- $ FV $ = future value = 2.25 million
- $ PV $ = present value = 2 million
- $ r $ = monthly growth rate
- $ n $ = number of months = 4 years × 12 = 48 months

We solve for $ r $ using the equation:

\[
\left(\frac{FV}{PV}\right) = (1 + r)^n
\]
\[
\frac{2.25}{2} = (1 + r)^{48}
\]

Let’s compute the monthly growth rate $ r $.

### Part 3: Return on Selling the Property After Four Years
The return from selling the property is calculated as the profit made on the sale, expressed as a percentage of the initial investment.

\[
\text{Return} = \frac{\text{Sale Price} - \text{Initial Investment}}{\text{Initial Investment}} \times 100
\]

Here:
- Sale price = $2.25 million
- Initial investment = 5% down payment on $2 million = $100,000

We can now compute the return on the investment.

Let's calculate these values.### Part 1: Monthly Payment on the Loan
The monthly payment on the loan is approximately **$17,572.92**.

### Part 2: Monthly Growth Rate of the Property
The monthly growth rate for the value of the property over 4 years is approximately **0.25%**.

### Part 3: Return from Selling the Property After Four Years
The return from selling the property after four years, based on the initial down payment, is **2150%**.

This return is calculated as a percentage of the initial investment of $100,000 (5% of $2 million). The high return is due to leveraging (only a small down payment) and the increase in the property's value.

---

Would you like more details or have any further questions? Here are a few related questions:

1. How would the monthly payment change if the interest rate increased to 6%?
2. What would be the total interest paid over the life of the loan?
3. How would the return change if the property sold for $2.5 million instead?
4. What if the investor chose a shorter loan term of 10 years instead of 12 years?
5. How much equity would the investor build after four years if they continued paying the mortgage?

**Tip:** Leverage can significantly amplify returns on investments but also increases risk, especially if property values decrease.

Learn how to calculate the monthly loan payment for a $2 million property with 5% down, the monthly growth rate of its value over 4 years, and the return on investment after selling it for $2.25 million. This real estate investment problem involves amortization, compound growth, and return analysis, suitable for college-level finance or mathematics students.

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