Ordinary Annuity Present Value Formulas

Formulas for Calculating
The Present Value of an Ordinary Annuity

1) Solving the Present Value

A friend offers to buy your car if he can pay you $100 per month for 3 years at an annual interest rate of 7.5%
What is the present value of all these payments?

The payment per period ('p') is $100
the total number of periods ('n') is:
12 periods per year for 3 years, equals 12*3 = 36
the interest rate is
.075 ÷ 12 = 0.00625

Putting these numbers into the formula:
Present Value = 100 * [(1-(1.00625)^-36)/.00625]
Present Value = 100 * [(1-0.799075542783109)/.00625]
Present Value = 100 * [(0.200924457216891)/.00625]
Present Value = 100 * 32.1479131547025
Present Value = 3,214.79

2) Solving the Periodic Payment

(Using the data from question 1)
A 3 year 7.5% monthly annuity has a present value of $3,214.79.
What was the periodic payment?

The total number of periods ('n') is:
12 periods per year for 3 years, equals 12*3 = 36
the interest rate is
.075 ÷ 12 = 0.00625
the present value is $3,214.79

Putting these numbers into the formula:
Payment= 3,214.79 ÷ [(1-(1.00625)^-36) / 0.00625]
Payment= 3,214.79 ÷ ((1-0.799075542783109) / 0.00625)
Payment= 3,214.79 ÷ (0.200924457216891 / 0.00625)
Payment= 3,214.79 ÷ 32.1479
Payment= 100.00

3) Solving for Years

(Using the data from question 1)
A 7.5% annuity with $100 monthly payments has a present value of $3,214.79
How many years is this annuity for?

the interest rate is
.075 ÷ 12 = 0.00625
the present value is $3,214.79
the monthly payment =100

Putting these numbers into the formula:
n=[Log(100) ― Log((100 -(0.00625)*3,214.79))] / Log(1.00625)
n=[Log(100) ― Log((100 -(20.0924375)))] / Log(1.00625)
n=[Log(100) ― Log((79.9075625))] / Log(1.00625)
n=(2 -1.9025878832)/ 0.00270589337592
n=0.0974121168 / 0.00270589337592
n=35.9999834682621
n=36 total number of periods
This is a monthly annuity so, dividing 36 by 12 equals
3 years.

4) Solving for Rate
The present value of an ordinary annuity formula cannot be solved for rate.
The present value calculator solves for rate by using a trial and error process.

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