Annuity Payments

Payments in Time Value of Money formulas are a series of equal, evenly-spaced cash flows of an annuity such as payments for a mortgage or monthly receipts from a retirement account.

Payments must:

be the same amount each period
occur at evenly spaced intervals
occur exactly at the beginning or end of each period
be all inflows or all outflows (payments or receipts)
represent the payment during one compounding (or discount) period

Calculate Payments When Present Value Is Known

The Present Value is an amount that you have now, such as the price of property that you have just purchased or the value of equipment that you have leased. When you know the present value, interest rate, and number of periods of an ordinary annuity, you can solve for the payment with this formula:

payment = PVoa / [(1- (1 / (1 + i)ⁿ )) / i]

Where:
   PVoa = Present Value of an ordinary annuity (payments are made at the end of each period)
   i = interest per period
   n = number of periods

Example: You can get a $150,000 home mortgage at 7% annual interest rate for 30 years. Payments are due at the end of each month and interest is compounded monthly. How much will your payments be?

PVoa = 150,000, the loan amount
i = .005833 interest per month (.07 / 12)
n = 360 periods (12 payments per year for 30 years)

payment = 150,000 / [(1 - ( 1 / (1.005833)³⁶⁰)) / .005833] = 997.95

Calculate Payments When Future Value Is Known

The Future Value is an amount that you wish to have after a number of periods have passed. For example, you may need to accumulate $20,000 in ten years to pay for college tuition. When you know the future value, interest rate, and number of periods of an ordinary annuity, you can solve for the payment with this formula:

payment = FVoa / [((1 + i)ⁿ - 1 ) / i]

Where:
   FVoa = Future Value of an ordinary annuity (payments are made at the end of each period)
   i = interest per period
   n = number of periods

Example: In 10 years, you will need $50,000 to pay for college tuition. Your savings account pays 5% interest compounded monthly. How much should you save each month to reach your goal?

FVoa = 50,000, the future savings goal
i = .004167 interest per month (.05 / 12)
n = 120 periods (12 payments per year for 10 years)

payment = 50,000 / [(1.004167¹²⁰ - 1) / .004167] = 321.99

Concepts