What is Annuity Formula?

Understanding the Annuity Formula

The annuity formula is a fundamental equation in finance used to calculate the periodic payment amount for loans, mortgages, and annuities. It's also known as the loan payment formula or the present value of annuity formula.

An annuity is a series of equal payments made at regular intervals. The annuity formula helps determine how much each payment should be, given the principal amount, interest rate, and number of payments.

The Annuity Formula

The formula for calculating the periodic payment (PMT) is:

PMT = P × [r(1+r)^n] / [(1+r)^n - 1]

Where:

• PMT = Periodic payment amount
• P = Principal (loan amount or present value)
• r = Periodic interest rate (annual rate / number of periods per year)
• n = Total number of payments

How It Works

The annuity formula calculates the fixed payment needed to pay off a loan over a specified period, including both principal and interest. Each payment includes:

• A portion that goes toward paying interest
• A portion that goes toward reducing the principal

Example Calculation

Calculate the monthly payment for a $200,000 mortgage at 4% annual interest for 30 years:

1. Principal (P) = $200,000
2. Monthly rate (r) = 4% / 12 = 0.003333
3. Number of payments (n) = 30 × 12 = 360
4. PMT = 200,000 × [0.003333(1.003333)^360] / [(1.003333)^360 - 1]
5. Result: PMT = $954.83 per month

Key Characteristics

• Fixed Payments: Each payment is the same amount
• Amortization: Early payments are mostly interest; later payments are mostly principal
• Total Cost: The total amount paid exceeds the principal due to interest

Applications

The annuity formula is used for:

• Mortgage payments
• Auto loans
• Personal loans
• Retirement annuity calculations
• Lease payments