# Present Value of an Annuity

An annuity is a series of evenly-spaced equal payments made for a certain amount of time. There are two basic types of annuities: ordinary annuity and annuity due. Ordinary annuity is the one in which the periodic payments are made at the end of each period while annuity due is the one in which the periodic payments occur at the beginning of each period.

The present value an annuity is the sum of the periodic payments each discounted at the given rate of interest to reflect the time value of money. Alternatively defined, the present value of an annuity is the amount which if invested at the start of first period at the given annual percentage rate will equate the sum of the amount invested and the compound interest earned on the investment with the product of number of the periodic payments and the face value of each payment.

## Formula

Although the present value (PV) of an annuity can be calculated by discounting each periodic payment separately to the starting point and then adding up all the discounted figures, it is more convenient to use the following formulas.

 PV of an Ordinary Annuity = R × 1 − (1 + i)-n i

Since cash flows occur one period earlier in case of an annuity due, the present value of annuity due can be determined by reversing discounting for one period. The relationship between present value of an ordinary annuity and present value of an annuity due is given by the following expression:

PV of Annuity Due = PV of Ordinary Annuity × (1 + i)

Substituting the expression for present value of ordinary annuity, we get the following equation:

 PV of an Annuity Due = R × 1 − (1 + i)-n × (1 + i) i

Where,
i is the interest rate per compounding period;
n are the number of compounding periods; and
R is the fixed periodic payment.

## Examples

Example 1: Calculate the present value on Jan 1, 2011 of an annuity of \$500 paid at the end of each month of the calendar year 2011. The annual interest rate is 12%.

Solution

```We have,
Periodic Payment       R  = \$500
Number of Periods      n  = 12
Interest Rate          i  = 12%/12 = 1%
Present Value         PV  = \$500 × (1-(1+1%)^(-12))/1%
= \$500 × (1-1.01^-12)/1%
≈ \$500 × (1-0.88745)/1%
≈ \$500 × 0.11255/1%
≈ \$500 × 11.255
≈ \$5,627.54```

Example 2: A certain amount was invested on Jan 1, 2010 such that it generated a periodic payment of \$1,000 at the beginning of each month of the calendar year 2010. The interest rate on the investment was 13.2%. Calculate the original investment and the interest earned.

Solution

```Periodic Payment       R  = \$1,000
Number of Periods      n  = 12
Interest Rate          i  = 13.2%/12 = 1.1%
Original Investment       = PV of annuity due on Jan 1, 2010
= \$1,000 × (1-(1+1.1%)^(-12))/1.1% × (1+1.1%)
= \$1,000 × (1-1.011^-12)/0.011 × 1.011
≈ \$1,000 × (1-0.876973)/0.011 × 1.011
≈ \$1,000 × 0.123027/0.011 × 1.011
≈ \$1,000 × 11.184289 × 1.011
≈ \$11,307.32
Interest Earned           ≈ \$1,000 × 12 − \$11,307.32
≈ \$692.68```