## 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:

Here P is payment or cash flow per period, r 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:

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

## 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.

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.

PV_{1} = $500 ÷ (1+0.0525)^{1} = $475.06

PV_{2} = $500 ÷ (1+0.0525)^{2} = $451.36

PV_{3} = $500 ÷ (1+0.0525)^{3} = $428.85

PV_{4} = $500 ÷ (1+0.0525)^{4} = $407.46

PV_{5} = $500 ÷ (1+0.0525)^{5} = $387.13

PV_{6} = $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.

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).

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.

PV_{1} = $500 ÷ (1+0,0425)^{0} = $500.00

PV_{2} = $500 ÷ (1+0,0425)^{1} = $475.06

PV_{3} = $500 ÷ (1+0,0425)^{2} = $451.36

PV_{4} = $500 ÷ (1+0,0425)^{3} = $428.85

PV_{5} = $500 ÷ (1+0,0425)^{4} = $407.46

PV_{6} = $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.

or