Loan Payment Calculator

Calculate your monthly loan payments, total payment, and total interest using our comprehensive Loan Payment Calculator.

* All amounts are in U.S. dollars.

Step 1: Enter Loan Details

Enter the total principal amount of the loan.

Enter the interest rate as a percentage (e.g., 5 for 5%).

Enter the duration of the loan in years.

Formula Used:
Monthly Payment \( M \) is calculated as:

\[ M = P \times \frac{r(1+r)^n}{(1+r)^n – 1} \]

where \( P \) is the loan amount, \( r \) is the monthly interest rate (annual rate divided by 12 and converted to decimal), and \( n \) is the total number of payments (years × 12).


Example:
For a loan amount of \$10,000 at an annual interest rate of 5% over 5 years, the monthly payment is approximately \$188.71.

Loan Payment Calculator (In-Depth Explanation)

Loan Payment Calculator (In-Depth Explanation)

When you take out a loan—whether it’s a mortgage, car loan, or any installment-based borrowing— you often repay the principal plus interest in fixed periodic payments (e.g., monthly). This process is typically governed by an amortization formula, which ensures you pay back both principal and interest over a specified term.

Below, we’ll break down the standard formula for calculating a constant periodic payment for a loan with a fixed interest rate. We’ll explain each part and how you can use it to figure out the recurring payment amount.

1. The Standard Loan Payment Formula

Suppose you borrow an amount (called the principal) $P$ at an annual interest rate $r$. For many loans, this rate is converted to a periodic or monthly rate $i$ when payments are monthly:

  • $P$: The initial principal (the amount borrowed).
  • $r$: Annual nominal interest rate (e.g., 6% per year).
  • $i$: Periodic interest rate, often $r/12$ if payments are monthly.
  • $n$: Total number of payments (for a 30-year monthly mortgage, $n=30\times12=360$).

The formula for the fixed payment $M$ (sometimes called the mortgage-style formula) is:

$$M = P \times \frac{i \,(1 + i)^{n}}{(1 + i)^{n} – 1}.$$

This formula ensures that by making the payment $M$ each period (month, typically), you will pay off the entire loan (principal + interest) exactly by the end of the term.

2. Breaking Down the Formula

Let’s interpret each component of $$ M = P\, \frac{i(1+i)^n}{(1+i)^n – 1}. $$

  • $(1+i)^n$: This is the compound growth factor for interest over $n$ periods. It’s how much $1$ of principal would grow to if left to accumulate at rate $i$ for $n$ compounding periods (like months).
  • $i (1+i)^n$: This part is tied to how each payment must offset not only the interest accruing each period but also gradually repay the principal.
  • Denominator $(1+i)^n – 1$: This term normalizes the payment so that the sum of all payments covers the original principal $P$ by the end of $n$ periods. The difference $(1+i)^n – 1$ effectively captures the total effect of interest growth minus the initial principal value.
  • Multiplying by $P$: Finally, we scale by the principal to get the actual payment. If $P$ is larger, the periodic payment $M$ must also be larger, all else being equal.

Putting it all together: each payment covers the interest that has accrued plus a portion of principal, so that after $n$ payments the debt is fully amortized to zero.

3. Applying the Formula

To actually use the formula, you need to:

  1. Convert your annual percentage rate (APR) to a periodic rate. For example, if $r=0.06$ (6% annual), and you have 12 payments per year, then $i=r/12=0.06/12=0.005$ (0.5% per month).
  2. Determine the total number of payments $n$. For a 30-year monthly loan, $n=30\times12=360$.
  3. Plug $i$ and $n$ into the formula along with the principal $P$.
  4. Compute the result to find the periodic payment $M$.

Example: Monthly Payment on a $200{,}000 Loan

Suppose you borrow $ \$200{,}000$ at an annual nominal rate of 6%, with monthly compounding, over 30 years (360 months total).

  • Principal, $P = 200{,}000$
  • Annual nominal interest, $r = 0.06$
  • Monthly interest, $i = r/12 = 0.06/12 = 0.005$
  • Total payments, $n = 360$

Using the payment formula:

$$M = 200{,}000 \times \frac{0.005 \times (1 + 0.005)^{360}}{(1 + 0.005)^{360} – 1}.$$

Evaluate step by step:

  • $(1 + i)^{360} = (1.005)^{360} \approx 6.0226$ (using a calculator).
  • Numerator: $0.005 \times 6.0226 = 0.030113$.
  • Denominator: $6.0226 – 1 = 5.0226$.

So inside the fraction we have $\tfrac{0.030113}{5.0226} \approx 0.005996$. Multiply by $200{,}000$:

$$M \approx 200{,}000 \times 0.005996 = 1{,}199.2 \text{ (dollars per month)}.$$

Hence, the monthly payment is about \$1,199.20. Over 360 payments, this repays the entire \$200,000 principal plus the interest accrued.

Key Points & Interpretation:
  • Fixed Payment Amortization: Each payment is the same, but early on, more of your payment goes to interest; later, more goes to principal.
  • Total Interest Paid: Over the life of the loan, you can compute total interest by $M\times n – P$.
  • Effect of Interest Rate & Term: Higher interest or longer terms increase total interest cost, though monthly payments may be lower if the term is longer.
  • Annual vs. Monthly Rates: Always ensure consistent units (annual rate to monthly if payments are monthly) or adapt to quarterly/weekly as needed.

By understanding and using this loan payment formula, you can estimate monthly mortgage or car-loan payments, see how much interest is paid over time, and make informed financial decisions about different loan structures.