Present value of an annuity

Annuity means a stream or series of equal payments; for example, you have made an investment that will generate an interest income of $5,000 for you at the end of each year for five years. The income of $5,000 at the end of each year is an annuity.

This article explains the computation of present value of an annuity. If you want to learn the computation of present value of a single sum to be received or paid in future, read ‘present value of a single payment in future’ article.

Capital investments usually involve in generating series of cash flows and managers need to take into account the present value of such series of cash flows (i.e., annuities) to know the true profitability of an investment project.

The formula to compute present value of an annuity is given below:

Formula:

Where;

A = Present value of an annuity
R = Amount of an annuity
i = Rate of interest
n = Number of periods

Example 1:

A company has made an investment in government bonds. The bonds will generate an interest income of $25,000 each year for 5 years. The interest rate is 10% compounded annually.

Required: Compute present value of the stream of interest income for 5 years.

Solution:

= $25,000 × [(1 + 0.1)⁵ – 1 / 0.1(1 + 0.1)⁵]
= $25,000 × [(1.1)⁵ – 1 / 0.1(1.1)⁵]
= $25,000 × [1.61051 – 1 / 0.1(1.61051)]
= $25,000 × [0.61051 / 0.161051]
= $25,000 × [3.791]
= $94,775

At 10% interest compounded annually, the present value of this annuity is $94,775.

Use of present value of an annuity of $1 in arrears table:

The above computations may be complex for some people. Alternatively, we can compute present value of an annuity using  present value of an annuity of $1 in arrears table. This table contains the present value of $1 to be received each year over a series of years at various interest rates.

= $25,000 × [(1 + 0.1)⁵ – 1 / 0.1(1 + 0.1)⁵]
= $25,000 × 3.791*
= $94,775

*Value of [(1 + 0.1)⁵ – 1 / 0.1(1 + 0.1)⁵] from present value of an annuity of $1 in arrears table: 5 periods; 10% interest rate.

Example 2:

A company expects a series of 24 monthly receipts of $3,600 each. The first payment will be received 1 month from today. Determine the present value of this series assuming an interest rate of 12% per year compounded semiannually.

Solution:

= $3,600 × [(1 + 0.06)²⁴ – 1 / 0.1(1 + 0.06)²⁴]
= $3,600 × 12.550*
= $45,180

i = 12%/2 = 6% (annual interest rate has been divided by 2 because interest is compounded semiannually)

*Value of [(1 + 0.06)²⁴ – 1 / 0.1(1 + 0.06)²⁴] from present value of an annuity of $1 in arrears table: 24 periods; 6% interest rate.