Present Value of a Single Amount

In many business and personal situations, you are interested in determining the value today of receiving a fixed 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 five 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 five years? This is a problem of determining the present value of a single amount, because you are interested in knowing the present value, or the value today, of receiving a set sum in the future.

Intuitively, it is clear 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 two 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 by the end of two 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.

In present value problems, the interest rate often is called the discount rate. This is because a future value is being discounted back to the present. Present value problems are sometimes called discounted present value problems.

One way to solve present value problems is to use the general formula previously developed for future value problems. For example, returning to the previous example, assume that at the end of five 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 Table III (see Appendix B), the answer is $8,511.41, determined as follows:

$\text{Accumulated amount}=\text{Factor}\times \text{Principal}$

$\text{Principal}=\frac{\text{Accumulated amount}}{\text{Factor}}$

$\mathrm{\$}8,511.41=\frac{\mathrm{\$}15,000}{1.76234}$

This is equivalent to saying that at a 12% interest rate, compounded annually, it does not matter whether you receive $8,511.41 today or $15,000 at the end of five years. Thus, if someone offered you an investment at a cost of $8,000 that would return $15,000 at the end of five years, you would take it if the minimum rate of return were 12%. This is because at 12%, the $15,000 is actually worth $8,511.41 today. Therefore, your smaller investment of $8,000, for the same amount of $15,000 in five years, would earn more than the 12% interest.

Present Value Tables

Rather than using future value tables and making the necessary adjustments to the general formula, you 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 values. Because of the relationship between future and present values, the present value table is the inverse of the future value table. Table 11.4 presents an excerpt from the present value tables (see Table II in Appendix B). The table works the same way the future value table does, except that the general formula is:

$\text{Present value}=\text{Factor}\times \text{Accumulated amount}$

For example, if you want to use the table to determine the present value of $15,000 to be received at the end of five years, compounded annually at 12%, simply look down the 12% column and multiply that factor by $15,000. Thus the answer is $8,511.45,1 determined as follows:

$\text{Present value}=\text{Factor}\times \text{Accumulated amount}$

$\mathrm{\$}8,511.45=.56743\times \mathrm{\$}15,000$

The present value of a single amount can also be computed using a business calculator as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. 15,000 Press FV

4. 5 Press N

5. 12 Press I/YR

6. Press PV for the answer of $8,511.40.

Other Present Value Situations

As in the future value case, you can use the general formula to solve other variations, as long as you 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 by the end of three years, with an investment of $7,903.10 today. The answer is 4% semiannually, or 8% annually, determined as follows:

$\text{Present value}=\text{Factor}\times \text{Accumulated amount}$

$\text{Factor}=\frac{\text{Present value}}{\text{Accumulated amount}}$

$.79031=\frac{\mathrm{\$}7,903.10}{\mathrm{\$}10,000.00}$

Looking across the sixth-period row in Table 11.4, you come to .79031 in the 4% column. Because interest is compounded semiannually, the annual rate is 8%.

The necessary business calculator keystrokes are as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. –7,903.10 Press PV

4. 10,000.00 Press FV

5. 6 Press N

6. Press I/YR for the answer of 4.00% every six months, or 8.00% compounded semiannually.

The Distinction between Future Value and Present Value

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 you determined the future value of $10,000 compounded annually at 12% for three years is as follows:

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

Exercises