You purchase an office building at $5,000,000 with a loan to value ratio of 60.00%. Interest rates are at 5.25% and loan amortization periods are at 25 years. Assuming monthly loan payment of $17,977.43. You are considering selling this office building six years from now. Calculate the remaining loan balance outstanding at the time of sale at the end of year 6.

To calculate the remaining loan balance after 6 years, we will use the **amortization formula** for a loan, which is given by:

\[
B = P \times \frac{(1 + r)^n - (1 + r)^p}{(1 + r)^n - 1}
\]

Where:
- $ B $ is the remaining loan balance.
- $ P $ is the loan amount (principal).
- $ r $ is the monthly interest rate (annual rate divided by 12).
- $ n $ is the total number of payments (loan amortization period in months).
- $ p $ is the number of payments made after 6 years (payments per year times 6).

### Step 1: Identify the loan amount
The **loan-to-value ratio** (LTV) is 60%, meaning the loan covers 60% of the building's purchase price.

\[
P = 60\% \times 5,000,000 = 0.60 \times 5,000,000 = 3,000,000
\]

### Step 2: Monthly interest rate
The **annual interest rate** is 5.25%, so the **monthly interest rate** $ r $ is:

\[
r = \frac{5.25\%}{12} = \frac{0.0525}{12} = 0.004375
\]

### Step 3: Number of payments
The loan is amortized over **25 years**, so the total number of payments $ n $ is:

\[
n = 25 \times 12 = 300 \text{ payments}
\]

### Step 4: Payments made after 6 years
After 6 years, the number of payments $ p $ made is:

\[
p = 6 \times 12 = 72 \text{ payments}
\]

### Step 5: Calculate remaining loan balance
Now, we can calculate the remaining loan balance $ B $. I will calculate this for you.The remaining loan balance at the end of 6 years would be approximately **$2,590,379.16**.

Would you like further details on how this calculation works or have any other questions?

Here are 5 related questions to expand on this information:
1. How would the remaining loan balance change if the interest rate were lower?
2. What impact does making additional principal payments have on the remaining balance?
3. How can we calculate the total interest paid over the 6 years?
4. What is the effect of a shorter loan amortization period on monthly payments and the remaining balance?
5. How does the loan-to-value ratio affect the initial loan amount?

**Tip:** Making extra payments toward your principal can significantly reduce the overall interest paid on a loan.

Learn how to calculate the remaining loan balance after 6 years for a $5,000,000 office building purchase with a 60% loan-to-value ratio and a 5.25% interest rate. This step-by-step guide covers key amortization concepts, formulas, and detailed calculations suitable for college-level finance students.

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