## What is the present value of a single amount? – Definition

The value of a future promise to pay or receive a single amount at a specified interest rate is called the **present value of a single amount**.

## Explanation

In many business and personal situations, we are interested in determining the value today of receiving a set single amount at some time in the future. For example, assume that you want to know the value today of receiving $15,000 at the end of 5 years if a rate of return of 12% is earned. Another way of asking this question is, what is the amount that would have to be invested today at 12% compounded annually if you wanted to receive $15,000 at the end of 5 years? These are present-value-of-a-single-amount problems, because we are interested in knowing the present value, or the value today, of receiving a set sum in the future.

Intuitively, we know that the present value will be less than the future value. For example, if you had the choice of receiving $12,000 today or in 2 years, you would take the $12,000 today. This is because you can invest the $12,000 so that it will accumulate to more than $12,000 at the end of 2 years. Another way of looking at this is to say that because of the time value of money, you would take an amount less than $12,000 if you could receive it today, instead of $12,000 in two years. The amount you would be willing to accept depends on the interest rate or the rate of return you receive.

### Formula to Calculate the present value of a single amount

**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 amount will be received in future

### Example

A Company is expecting to receive $8,000 after 5 years from now. Calculate the present value of this sum if the current market interest rate is 12% and the interest is compounded annually.

### Solution

Using the above present value formula:

Number of periods (n) = 5

Interest rate (i) = 12%

PV = FV x 1 / (1+i)^{n}

= 8,000 x 1 / (1+12%)^{5 }

= 8,000 x 1 / (1+0.12)^{5 }

= 8,000 x 1 / (1.12)^{5 }

= 8,000 x 1 / 1.7623

= 8,000 x 0.5674

= $4,540

The amount of $8,000 to be recieved after 5 years has a present value of $4,540. This example shows if the $4,540 is invested today @ 12% interest rate per year compounded annually, it will grow to 8,000 after 5 years.

In present-value situations, the interest rate is often called the discount rate. This is because we are discounting a future value back to the present. Some individuals refer to present-value problems as discounted present-value problems.

One way to solve present-value problems is to use the general formula we developed for the future value of a single amount problems, For example, returning to the previous example, assume that at the end of 5 years, you wish to have $15,000, If you can earn 12% compounded annually, how much do you have to invest today? Using the general formula for Present Value Table; the answer is $8,511.40, determined as follows:

Accumulated Amount = Factor x Principal

Principal = Accumulated amount / Factor

= $15,000 / 1.7623

= $8,511.40

This equivalent to saying that at a 12% interest rate compounded annually, it does not matter whether you receive $8,511.40 today or $15,000 at the end of 5 years. Thus if someone offered you an investment at a cost of $8,000 that would return $15,000 at the end of 5 years, you would take it if the minimum rate of return were 12%. This is because at 12% the $15,000 is actually worth $8,511.45 today, but you would need to make an outlay of only $8,000.

## Using Present-Value Tables

Rather than using future-value tables and making the necessary adjustments to the general formula, we can use present-value tables. As is the case with future-value tables, present-value tables are based on the mathematical formula used to determine present value. Because of the relationship between future and present values, the present-value table is the inverse of the future-value table.

The below example presents an excerpt from the present-value tables. The table works the same as the future-value table does, except that the general formula is:

**Present value = Factor x Accumulated amount**

For example, if we want to use the table to determine the present value of $15,000 to be received at the end of 5 years, compounded annually at 12%, we simply look down the 12% column and multiply that factor by $15,000. Thus the answer is $8,511.45, determined as follows:

Present value = Factor x Accumulated amount

= 0.56743 x $15,000

= $8,511.45

## Other Present-Value Situations

As we did in the future value case, we can use the general formula to solve other variations, as long as we know two of the three variables. For example, assume that you want to know what interest rate compounded semiannually you must earn If you want to accumulate $10,000 at the end of 3 years, with an Investment of $7,049.60 today. The answer is 6% semiannually or 12% annually, determined follows:

Present value = Factor x Accumulated amount

Factor = Present value / Accumulated amount

= $7,049.60 / $10,000.00

= 0.70496

Looking across the sixth-period row, we come to .70496 at the 6% column. Because interest is compounded semiannually, the annual rate is 12%.

### Distinguishing between Future Value and Present Value of a single amount

In beginning to work with time-value-of-money problems, you should be careful to distinguish between present-value and future-value problems. One way to do this is to use timelines to analyze the situation. For example, the timeline relating to the example in which we determined the future value of $10,000 compounded at 12% for 3 years is as follows:

But the timeline relating to the present value of $15,000 discounted back at 12% for 5 years is: