# What One Sum Is Worth Today

How Much Does a Single Amount Mean at Present?

The present value of a single quantity is the monetary value of a future promise to pay or receive that amount at a given interest rate.

## The Value of a Single Sum at the Present Moment

In both business and everyday life, there are situations when estimating the present value of a future lump sum payment is desirable.

Take the hypothetical case of calculating the present discounted worth of \$15,000 received after 5 years at a rate of return of 12%.

How much would you have to invest now at 12% compounded annually to have \$15,000 in your account after 5 years?

These kind of inquiries and issues are referred to as “present value of a single amount problems.” For this reason, it is important to calculate the present value of a future cash payment.

We all know deep down that the future value will be higher than the present value.

If you were offered the option of collecting \$12,000 now or in two years, you would likely choose the former.

The reason for this is that if you invest the \$12,000, it will grow to more than \$12,000 at the end of the 2 years.

One alternative interpretation is that you would prefer to get a lesser sum now than \$12 000 in two years due to the time value of money.

The rate of return or interest you receive determines how much of a loss you are willing to take.

Present Value (PV) = the amount now discounted to the future

Value in the future (or future value) = present value multiplied by a constant.

i = Percentage interest rate

n = The interval in time after which the payment will be made

Example

Let’s say that after 5 years, the corporation anticipates receiving \$8,000. Use the market interest rate of 12%, compounded yearly, to determine the present value of this money.

Solution

The solution is to use the above formula for present value. Five periods (n) and a rate of interest (i) of 12% are used here. As a result, here is how we arrive at the PV:

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

These calculations show that in five years, you will receive \$8,000, but its present value is only \$4,540.

Investing the \$4,540 at 12% p.a. compounded annually reveals that the amount will increase to \$8,000 after 5 years.

The interest rate is referred to as the discount rate when dealing with present value calculations. This is because we are applying a discount rate to a value in the distant future. “Discounted present value problems” is a term used by some to describe what is more often known as “present value problems.”

The basic formula we derived for calculating the future value of a single amount can also be used to tackle situations involving the present value of an amount.

Using the same scenario as before, let’s say your 5-year goal is to accumulate \$15,000. How much do you need to invest today to get a return of 12% a year, compounded?

The result is \$8,511.40 if you use the standard formula for the present value table. Here’s how it works:

Amount accumulated = Multiplier x Initial Value

Amount borrowed = Accumulated sum / Factor

= \$15,000 / 1.7623

= \$8,511.40

At a 12% yearly compounded interest rate, it would be the same whether you received \$8,511.40 today or \$15,000 after 5 years.

This finding suggests that an investor would be wise to accept an offer to invest \$8,000 in a venture that promises a return of \$15,000 after five years if the venture has a minimum rate of return of 12%.

The reason is that if you invest the \$15,000 at 12%, in a year’s time it will be worth \$8,511.45, but you will only have to pay out \$8,000.

## Methods for Calculating Future Value

Present value tables can be used in place of future value tables, eliminating the need to make any revisions to the underlying calculation.

Present value tables are derived using the same mathematical formula as future value tables. Because of the connection between the two values, the present value table is the mirror image of the future value table.

A portion of a present value table is shown below as an example. Similar to a futures value table, this one computes values using the formula:

Value at this time = Factor times total accumulated sum

Present value of \$15,000 expected in 5 years (compounded annually at 12%) can be calculated by looking down the 12% column and multiplying that factor by \$15,000.

That brings us to a total of \$8,511.45. We get this conclusion by considering:

Value at this time = Factor times total accumulated sum

= 0.56743 x \$15,000

= \$8,511.45 Variables with Different Present Values

Given knowledge of two of the three variables, the general formula can be applied to solve other variants, as demonstrated by the case of future value.

To illustrate, let’s say you’re interested in the required interest rate (compounded semi-annually) to grow your \$7,049.60 investment into \$10,000 in three years.

The formula for arriving at this figure yields a result of 6% semiannually (or 12% annually):

Value at this time = Factor times total accumulated sum

Multiplier = Current Value / Total Amount

= \$7,049.60 / \$10,000.00

= 0.70496

The intersection of the sixth row and the 6% column in the present value table gives us the value 0.70496. The annual rate is 12% due to the semi-annual compounding of interest.

## Calculating the Difference Between an Amount’s Future Value and Its Present Value

It’s important to keep in mind the difference between present value and future value issues when beginning to work with time value of money challenges.

## Timelines are a useful tool for making this kind of assessment.

Below is a timeline depicting the above example, in which we determined the worth of \$10,000 after three years of compounding at a rate of 12%.