Present Value of an Annuity

Definition

The present value of an annuity (PVA) is the current worth of regular cash flows to be received at a specific date in the future based on the interest rate, which is also called the required rate of return. Coupon payments of a fixed-rate bond and amortized loans are common examples of annuities.

Formula

Basically, annuities can be classified as two types: ordinary annuities and annuities due. The difference between them is when payment is made.

If a payment is made at the end of each period, we have an ordinary annuity, and its present value can be found as follows:

present-value-of-an-ordinary-annuity-1

Here P is payment or cash flow per period, I represents the interest rate per period, and N is the number of periods.

An annuity due is taking place in case payments are made at the beginning of each period (in advance). In such a case, its present value can be calculated as follows:

present-value-of-an-annuity-due-2

In turn, the equation describing the relationship between the present value of an annuity due and an ordinary annuity is as follows:

present-value-of-an-annuity-3

Examples

A private investor is going to buy a 3-year fixed-rate bond with a semiannual coupon payment of $500. Such cash flows are an example of an annuity due because coupon payments are regularly made at fixed intervals at the end of each half-year. Let’s find their present value if the semiannual required rate of return is 5.25%. The discounting of cash flows is shown in the chart below.

PV-Ordinary-Annuity-1

The present value of the first coupon payment received at the end of the first half-year is $475.06. The other cash flows are treated the same way.

PV1 = $500/(1+0.0525)1 = $475.06

PV2 = $500/(1+0.0525)2 = $451.36

PV3 = $500/(1+0.0525)3 = $428.85

PV4 = $500/(1+0.0525)4 = $407.46

PV5 = $500/(1+0.0525)5 = $387.13

PV6 = $500/(1+0.0525)6 = $367.82

So, the total present value of all cash flows is $2,517.68. We can also get the same value using the formula of present value of an ordinary annuity mentioned above.

PVA-1

Let’s assume a scenario where coupon payments are made in advance (at the beginning of each half-year). In this case, we have an annuity due, and its present value is equal to the sum of present values of all cash flows (shown on the chart below).

PV-Annuity-Deu-2

The present value of the first coupon payment is $500 because it is received immediately (zero point in the chart above). The second coupon payment has a present value of $475.06. The other coupon payments are treated the same way.

PV1 = $500/(1+0,0425)0 = $500.00

PV2 = $500/(1+0,0425)1 = $475.06

PV3 = $500/(1+0,0425)2 = $451.36

PV4 = $500/(1+0,0425)3 = $428.85

PV5 = $500/(1+0,0425)4 = $407.46

PV6 = $500/(1+0,0425)5 = $387.13

The total present value of all cash is $2,649.86. We can get the same amount using the formula of present value of an annuity due.

PVA-2PVA-3

Calculator and Tables

You can also calculate the present value of an annuity using our online calculator or tables.