A $3,000,000 apartment complex loan is to be paid off in 10 years by making 10 equal annual payments. How much is each payment if the interest rate is 4.5% compounded annually?

(a) State the type.

amortizationpresent value of an annuity    ordinary annuitysinking fundpresent value


(b) Answer the question. (Round your answer to the nearest cent.)

### (a) **State the type:**
The problem involves paying off a loan (principal) by making equal annual payments over a set period of time, with a given interest rate. This is an example of an **amortization** problem, where the loan is gradually paid off through periodic payments that include both principal and interest.

### (b) **Answer the question:**

The formula for calculating the periodic payment for an amortized loan is:

\[
PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}}
\]

Where:
- $ P $ is the loan amount (present value) = \$3,000,000
- $ r $ is the periodic interest rate = $ \frac{4.5\%}{100} = 0.045 $
- $ n $ is the total number of periods = 10 years (since it's compounded annually, $ n = 10 $)

Let’s calculate the payment step-by-step.

1. **Periodic Interest Rate**: Since the interest is compounded annually, $ r = 0.045 $.
2. **Number of Periods**: $ n = 10 $.
   
Now, plug these values into the formula:

\[
PMT = \frac{3,000,000 \cdot 0.045}{1 - (1 + 0.045)^{-10}}
\]

Let's break this down:

1. $ (1 + 0.045)^{-10} = (1.045)^{-10} \approx 0.64393 $
2. $ 1 - 0.64393 = 0.35607 $
3. $ \frac{0.045}{0.35607} \approx 0.12636 $
4. Finally, $ 3,000,000 \times 0.12636 = 379,084.13 $

So, the annual payment (rounded to the nearest cent) is:

\[
\boxed{379,084.13}
\]

### Summary:
- **Type**: Amortization
- **Annual payment**: \$379,084.13

Would you like more details or have any questions about the solution?

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Here are five related questions:
1. How would the payment change if the interest rate were 5% instead of 4.5%?
2. What would the total interest paid over the 10 years be?
3. How is an ordinary annuity different from an amortized loan?
4. If the loan were paid monthly instead of annually, how would the calculation change?
5. How would the total payment change if the loan term were extended to 15 years?

**Tip**: When dealing with loan amortization, the periodic payment includes both the interest and a portion of the principal, and as the loan progresses, the interest portion decreases while the principal portion increases.

A $3,000,000 apartment complex loan is to be paid off in 10 years by making 10 equal annual payments. How much is each payment if the interest rate is 4.5% compounded annually?

Learn how to calculate the annual payment for a $3,000,000 loan with a 4.5% interest rate compounded annually. This problem involves amortization, compound interest, and loan payment calculations. The final annual payment is $379,084.13. Suitable for college or advanced high school students.

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