How to Calculate Loan Payments by Hand

The Loan Payment Formula Explained

To calculate the monthly payment on a fixed-rate loan, you can use a standard amortization formula. This formula works for loans where the interest rate stays the same throughout the term, such as many personal loans and auto loans.

The formula is: M = P × [r(1+r)^n] / [(1+r)^n - 1]. In this formula, M is the monthly payment, P is the principal (the initial loan amount), r is the monthly interest rate (as a decimal), and n is the total number of monthly payments.

This formula spreads the loan amount and interest evenly across the term, so each payment covers both principal and interest. It is widely used by financial institutions and is the standard method for fixed-rate installment loans.

• M = monthly payment
• P = principal, or amount borrowed
• r = monthly interest rate (annual rate divided by 12)
• n = number of monthly payments (term in years × 12)

Sources: TransUnion, Bankrate, TD Bank

Step 1: Convert Annual Rate to Monthly Rate

Interest rates on loans are usually quoted as an annual percentage rate (APR). However, because payments are made monthly, you need the monthly interest rate for the formula. To convert, divide the annual rate by 12.

Sources: TransUnion, Bankrate

Step 2: Convert Term to Number of Payments

Next, determine the total number of monthly payments you will make over the life of the loan. Multiply the number of years by 12, because there are 12 months in a year.

• Example: For a 5-year loan, n = 5 × 12 = 60 payments.
• For a 3-year loan, n = 3 × 12 = 36 payments.

Sources: TransUnion, Bankrate

Step 3: Worked Example with $20,000 at 6% for 5 Years

Let's calculate the monthly payment for a loan of $20,000 at an annual interest rate of 6%, with a 5-year term. We'll use the formula step by step.

First, convert the annual rate to a monthly rate: r = 6% ÷ 12 = 0.5% = 0.005. Then, find the number of payments: n = 5 × 12 = 60.

Now plug these values into the formula: M = 20000 × [0.005(1.005)^60] / [(1.005)^60 - 1].

1. Calculate (1 + r)^n. For this example, (1.005)^60 ≈ 1.348850.
2. Multiply the result by r: 1.348850 × 0.005 = 0.00674425.
3. Divide by [(1 + r)^n - 1]: 1.348850 - 1 = 0.348850. So 0.00674425 ÷ 0.348850 ≈ 0.019333.
4. Finally, multiply by the principal: 20000 × 0.019333 ≈ 386.66.

Sources: calculator-calculatrice.ca

Step 4: Handling Fees in the Calculation

Some lenders add fees or charges before the loan is made. These fees increase the effective loan amount you need to finance. In the formula, you should use the total amount that includes these fees, not just the amount you receive.

• That higher principal is then used in the formula, which increases your monthly payment.

Sources: TD Bank

Common Mistakes to Avoid

Mistakes often happen when the interest rate or term is not converted correctly, or when fees are ignored. Here are the most frequent pitfalls:

• Using the annual rate directly in the formula instead of dividing by 12.
• Using the number of years instead of the number of monthly payments.
• Forgetting to include upfront fees in the principal.
• Rounding intermediate numbers too early, which can change the final result by a few cents.

Sources: TransUnion, Bankrate, TD Bank