Future Value of a Single Amount

The previous example was an attempt to determine what the future amount of $10,000 invested at 12% for three years would be, given a certain compounding pattern. This is an example of determining the future value of a single amount. Future value means the amount to which the investment will grow by a future date if interest is compounded. Single amount means that a lump sum was invested at the beginning of year 1 and was left intact for all three years. Thus there were no additional investments or withdrawals. These future value or compound interest calculations are important in many personal and business financial decisions. For example, an individual may be interested in determining how much an investment of $50,000 will amount to in five years if interest is compounded semiannually versus quarterly, or what rate of return compounded annually must be earned on a $10,000 investment if $18,000 is needed in seven years. All of these situations relate to determining the future value of a single amount.

One way to solve problems of this type is to construct tables similar to the one in Table 11.2. However, this method is time-consuming and not very flexible. Mathematical formulas also can be used. For example, the tables used in Table 11.1 and Table 11.2 and in Table 11.3 to determine the accumulated amount of a single deposit at different compounded rates are based on the following formula:

$\text{Accumulated amount}=p(1+i{)}^n$

where

p = principal amount

i = interest rate

n = number of compounding periods

That is, in the example of the $10,000 compounded annually for three years at 12%, the $14,049.28 can be determined by the following calculation:

$\mathrm{\$}14,049.28=\mathrm{\$}10,000(1+0.12{)}^3$

One of the simplest methods is to use tables that give the future value of $1 at different interest rates and for different periods. Essentially, these tables interpret the mathematical formula just presented for various interest rates and compounding periods for a principal amount of $1. Once the amount of $1 is known, it is easy to determine the amount for any principal amount by multiplying the future amount for $1 by the required principal amount. Most business calculators also have function keys that can be used to solve these types of problems.

To illustrate, Table 11.3, an excerpt from the future value of a single amount table (see Table III in Appendix B), shows the future value of $1 for 10 interest periods for interest rates ranging from 2% to 15%. Suppose that you want to determine the future value of $10,000 at the end of three years if interest is compounded annually at 12% (the previous example). In order to solve this, look down the 12% column in the table until you come to the third interest period. The factor from the table is 1.40493, which means that $1 invested today at 12% will accumulate to $1.405 at the end of three years. Because you are interested in $10,000 rather than $1, just multiply the factor of 1.40493 by $10,000 to determine the future value of the $10,000 principal amount. The amount is $14,049.30, which, except for a slight rounding error, is the same as was determined from Table 11.1.

The use of the future value table can be generalized by using the following formula:

$\text{Accumulated amount}=\text{Factor (from the table)}\times \text{Principal}$

$\mathrm{\$}\mathrm{14,049.30}=1.40493\times \mathrm{\$}\mathrm{10,000}$

This formula can be used to solve a variety of related problems. For example, as noted above, you may be interested in determining what rate of interest must be earned on a $10,000 investment if you want to accumulate $18,000 by the end of seven years. Or you may want to know the number of years an amount must be invested in order to grow to a certain amount. In all these cases, you have two of the three items in the formula, and you can solve for the third.

The future value of a lump sum can also be computed using a business calculator. The keystrokes described below are for a Hewlett-Packard 10B business calculator; similar keystrokes are used with other business calculators. To compute the future value of $10,000 three years from today, with the prevailing interest rate being 12 percent, make the following keystrokes:

Hewlett-Packard Keystrokes:

1. Always CLEAR ALL before doing anything else. With a Hewlett-Packard 10B, one does this by pressing the yellow key, then pressing “Input.” This has the effect of clearing out any information left over from a prior computation.

2. Always set P/YR (payments per year) to the correct number; 1 in this case. With a Hewlett-Packard 10B, one does this by pressing “1,” then pressing the yellow key, then pressing “PMT.” This has the effect of telling the calculator that each year is being viewed as a separate period. Interest can be compounded over different periods, such as monthly, quarterly, daily, or even continuously. Setting P/YR equal to 1 means that the interest rate is compounded annually. Sometimes, the default for this amount is “12” because the calculator is set to compute monthly payments. Until you are comfortable using your calculator, your best strategy is to set this amount to “1” in order to avoid having the calculator doing too many mysterious things automatically.

3. 10,000 Press PV

4. 3 Press N

5. 12 Press I/YR

6. Press FV for the answer of $14,049.28. (Note: Sometimes the answer is displayed as a negative number. The sign of the inputs and the answers is usually irrelevant, except for some calculations of interest rate and length of period. These cases will be illustrated later.)

Interest Compounded More Often Than Annually

As stated, interest usually is compounded more often than annually. In such situations, simply adjust the number of interest periods and the interest rate. If you want to know what $10,000 will accumulate to by the end of three years if interest is compounded quarterly at an annual rate of 12%, just look down the 3% column until you reach 12 periods (see Table III in Appendix B). The factor is 1.42576 and (employing the general formula) the accumulated amount is $14,257.60, determined as follows:

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

$\mathrm{\$}14,257.60=1.42576\times \mathrm{\$}10,000$

Calculating the same amount using a business calculator involves the following keystrokes:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. 10,000 Press PV

4. 12 Press N

5. 3 Press I/YR

6. Press FV for the answer of $14,257.61.

Determination of the Number of Periods or the Interest Rate

There are many situations in which the unknown variable is the number of interest periods that the dollars must remain invested or the rate of return (interest rate) that must be earned. For example, assume that you invest $5,000 today in a financial institution that will pay interest at 10% compounded annually. You need to accumulate $8,857.80 for a certain project. How many years does the investment have to remain in the savings and loan association? Using the general formula, the answer is six years, determined as follows:

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

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

$1.77156=\frac{\mathrm{\$}8,857.80}{\mathrm{\$}5,000.00}$

Looking down the 10% column in Table III (see Appendix B), the factor of 1.77156 appears at the sixth-period row. Because the interest is compounded annually, the sixth period is interpreted as six years. This example was constructed so that the factor equals a round number of periods. If it does not, interpolation is necessary. The examples, exercises, and problems in this book will not require interpolation.

The necessary business calculator keystrokes are as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. –5,000.00 Press PV

4. 8,857.80 Press FV

5. 10 Press I/YR

6. Press N for the answer of 6.00 years. For this calculation, it is important to input the initial investment amount of $5,000 as a negative number, representing a cash outflow. For most business calculators, if you forget to do this, you get the message “No Solution.”

You can use the same setup to determine the required interest rate. For example, assume that you invest $10,000 for eight years. What rate of return or interest rate compounded annually must you earn if you want to accumulate $30,590.32? Using the general formula, the answer is 15%, determined as follows:

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

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

$3.05902=\frac{\mathrm{\$}30,590.23}{\mathrm{\$}10,000.00}$

Looking across the eight-period row, you find the factor of 3.05902 at the 15% column.

The necessary business calculator keystrokes are as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. –10,000.00 Press PV

4. 30,590.32 Press FV

5. 8 Press N

6. Press I/YR for the answer of 15.00%. For this calculation, it is important to input the initial investment amount of $10,000 as a negative number, representing a cash outflow.

Exercises