Present value of a single payment in future

The value of money changes over time. The value of a dollar in hand today is more than the value of a dollar to be received a year from now because if you have a dollar in hand today you can invest it elsewhere and earn some interest on it.

The present value of an amount means today’s value of the amount to be received at a point of time in future.

The concept of present value is frequently used in capital budgeting techniques and its understanding is, therefore, very important for managers, business owners and others involved in making capital budgeting decisions. Examples of capital budgeting techniques that take into account the present value of money are ‘net present value method’, ‘internal rate of return method’ and ‘discounted payback method’.

The present value is computed either for a single payment or for a series of payments (known as annuity) to be received in future. This article explains the computation of the present value of a single payment to be received at a single point of time in future. To understand the computation of the present value of a series of payments to be received in future, read ‘present value of an annuity’ article.

The present value of a single payment in future can be computed either by using present value formula or by using a table known as present value of $1 table. Both the methods are equivalent and produce the same answer.

Present value formula:

The formula to calculate present value of a single sum is give below:

Where;

• PV = Present value of the amount
• FV = Future value of the amount (amount to be received in future)
• i = Interest rate in percentage
• n = Number of periods after which the amount will be received in future

The following examples explain the computation of the present value of a single payment.

Example 1:

A company is expecting to receive $5,000 four years from now. Compute present value of this sum if the current market interest rate is 10% and the interest is compounded annually.

Solution:

To find out the present value, the amount of $5,000 to be received in future would be discounted using the given interest rate of 10%. We can do so using the present value formula given above or present value of $1 table. Both the methods are given below:

(1) Use of present value formula:

Number of periods (n) = 4 and Interest rate (i) = 10% or 0.1

By putting the values of ‘n’ and ‘i’ into the present value of a single sum formula:

PV = FV × 1/(1 + i)ⁿ

= $5,000 × 1/(1 + 10%)⁴

= $5,000 × 1/(1 + 0.1)⁴

= $5,000 × 1/(1.1)⁴

= $5,000 × 1/1.4641

=$5,000 × 0.683

= $3,415

The amount of $5,000 to be received after four years has a present value of $3,415. It means if the amount of $3,415 is invested today @10% per year compounded annually, it will grow to $5,000 in 4 years.

Alternatively, we can compute the present value using the factor from present value of $1 table. It is shown below:

(2) Use of present value of $1 table:

Number of periods (n) = 4 and Interest rate (i) = 10% or 0.1

= $5,000 × 0.683*

= $3,415

*The factor from present value of $1 table: 4th period; 10% interest rate.

Example 2:

A company wants to accumulate $600,000 in 5 years. The interest rate is 12% compounded semiannually.

Required:

1. What amount should be invested today?
2. How much interest will be earned in five years?

Solution:

(1) Amount to be invested today:

The interest is compounded semiannually therefore the number of years would be multiplied by 2 to obtain ‘n’ and the interest rate would be divided by 2 to obtain ‘i’:
n = 5 × 2 = 10
i = 12%/2 = 6% or 0.06

= $600,000 × 0.558*

= $334,800

*The factor from present value of $1 table: 10th period; 6% rate

(2) The amount of interest that will be earned over 5 year period:

= $600,000 – $334,800

= $265,200

At 12% interest per year compounded semiannually, the company needs to invest $334,000 today to accumulate $600,000 in 5 years. The total interest income of $265,200 will be earned over the period.