Loan Amortization Formula Step by Step Math

Loan amortization is the process of spreading loan repayments over a fixed timeline through equal monthly installments. Each monthly payment is split into two parts: Interest (the lender's fee) and Principal (reducing your outstanding balance).

This guide breaks down the mathematical amortization formula step-by-step, showing how to manually construct an amortization schedule using our Mortgage Overpayment Calculator.

The Standard Amortization Formula

To calculate the fixed monthly payment ($PMT$) on an amortizing loan:

$$PMT = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}$$

Where:

• $P$ = Principal loan amount (e.g., $200,000)
• $r$ = Periodic monthly interest rate ($\text{Annual Rate} \div 12$)
• $n$ = Total number of monthly payments ($\text{Loan Term in Years} \times 12$)

Step-by-Step Worked Example

Calculate the monthly payment for a $100,000 loan at a 6.0% annual interest rate over 30 years ($n = 360$ months):

Step 1: Calculate monthly interest rate ($r$)

$$r = \frac{6.0%}{12} = \frac{0.06}{12} = 0.005$$

Step 2: Calculate $(1 + r)^n$

$$(1 + 0.005)^{360} = (1.005)^{360} \approx 6.022575$$

Step 3: Solve the PMT equation

$$PMT = 100,000 \times \frac{0.005 \times 6.022575}{6.022575 - 1}$$ $$PMT = 100,000 \times \frac{0.030113}{5.022575} = 100,000 \times 0.0059955 = \mathbf{$599.55}$$

How Interest vs Principal Shifts Over Time

During early loan months, interest dominates the payment. As the principal drops, monthly interest charges decrease, accelerating principal paydown.

Month 1:   $599.55 Payment ($500.00 Interest + $99.55 Principal)
Month 180: $599.55 Payment ($332.10 Interest + $267.45 Principal)
Month 360: $599.55 Payment ($2.98 Interest + $596.57 Principal)

Automate Amortization & Overpayments

Generate full payment schedules and calculate early payoff savings using our free Mortgage Overpayment Calculator. Explore additional tools in our Finance Category.

For loan payoff acceleration methods, see Pay Off Mortgage 5 Years Early.