Amortized Loan with Fixed Payment Formula

An amortized loan is a type of loan where the borrower makes regular payments over time, which includes both interest and principal. The fixed payment formula for an amortized loan allows you to determine the constant payment amount that covers both the interest and the principal repayment. This formula is essential for understanding how your payments will be structured throughout the life of the loan.

Understanding Amortized Loans
An amortized loan is repaid through regular payments that cover both interest and principal over the life of the loan. Unlike interest-only loans, where payments only cover the interest, amortized loans gradually reduce the principal balance with each payment. This means that each payment includes a portion that goes toward interest and a portion that reduces the principal.

The Fixed Payment Formula
The fixed payment formula for an amortized loan is:

$P=\frac{r\cdot PV}{1-(1+r{)}^{-n}}$P=1−(1+r)−nr⋅PV​

Where:

• $P$P = Fixed monthly payment
• $r$r = Monthly interest rate (annual rate divided by 12)
• $PV$PV = Present value or loan amount
• $n$n = Total number of payments (loan term in months)

How to Use the Formula

1. Determine the Monthly Interest Rate
   Convert the annual interest rate into a monthly rate by dividing by 12. For example, if your annual interest rate is 6%, your monthly interest rate would be $\frac{0.06}{12}=0.005$120.06​=0.005.

2. Calculate the Total Number of Payments
   If you have a 30-year mortgage, for instance, the total number of payments would be $30\times 12=360$30×12=360 months.

3. Input Values into the Formula
   Substitute your monthly interest rate, loan amount, and the total number of payments into the formula to find the fixed monthly payment.

Example Calculation
Let's say you have a loan amount of $200,000 with an annual interest rate of 5% for a 30-year term.

1. Monthly interest rate: $\frac{0.05}{12}=0.004167$120.05​=0.004167
2. Total number of payments: $30\times 12=360$30×12=360 months

Plug these values into the formula:

$P=\frac{0.004167\cdot 200,000}{1-(1+0.004167{)}^{-360}}$P=1−(1+0.004167)−3600.004167⋅200,000​

$P=\frac{833.40}{1-(1.004167{)}^{-360}}$P=1−(1.004167)−360833.40​

$P=\frac{833.40}{1-0.2314}$P=1−0.2314833.40​

$P=\frac{833.40}{0.7686}$P=0.7686833.40​

$P=1084.29$P=1084.29

So, your fixed monthly payment would be approximately $1084.29.

Amortization Schedule
An amortization schedule breaks down each payment into interest and principal components over the life of the loan. At the beginning of the loan term, a larger portion of each payment goes toward interest, with a smaller portion reducing the principal. As the loan progresses, the interest portion decreases, and the principal portion increases.

Creating an Amortization Schedule
To create an amortization schedule, you need to calculate the interest and principal portions for each payment. Here’s a simplified table to illustrate the first few payments:

Payment #	Payment Amount	Interest Portion	Principal Portion	Remaining Balance
1	$1,084.29	$833.33	$250.96	$199,749.04
2	$1,084.29	$832.29	$252.00	$199,497.04
3	$1,084.29	$831.24	$253.05	$199,244.00

Benefits of Amortized Loans

• Predictable Payments: You know exactly how much you need to pay each month, making budgeting easier.
• Gradual Equity Build-Up: As you pay down the principal, you build equity in your asset (e.g., a home).
• Reduced Interest Over Time: As your loan balance decreases, so does the amount of interest you pay over time.

Conclusion
The fixed payment formula for an amortized loan is a powerful tool for managing loans. By understanding and using this formula, you can predict your monthly payments and plan your finances effectively. Whether you're taking out a mortgage, auto loan, or personal loan, knowing how to calculate your fixed payments can help you stay on top of your financial commitments.