Learning Objectives

After studying this appendix, you should be able to:

[1] Distinguish between simple and compound interest.

[2] Identify the variables fundamental to solving present value problems.

[3] Solve for present value of a single amount.

[4] Solve for present value of an annuity.

[5] Compute the present value of notes and bonds.

Nature of Interest

Would you rather receive $1,000 today or a year from now? You should prefer to receive the $1,000 today because you can invest the $1,000 and earn interest on it. As a result, you will have more than $1,000 a year from now. What this example illustrates is the concept of the time value of money. Everyone prefers to receive money today rather than in the future because of the interest factor.

Interest is payment for the use of another person's money. It is the difference between the amount borrowed or invested (called the principal) and the amount repaid or collected. The amount of interest to be paid or collected is usually stated as a rate over a specific period of time. The rate of interest is generally stated as an annual rate.

The amount of interest involved in any financing transaction is based on three elements:

1. Principal (p): The original amount borrowed or invested.

2. Interest Rate (i): An annual percentage of the principal.

3. Time (n): The number of years that the principal is borrowed or invested.

Simple Interest

Simple interest is computed on the principal amount only. It is the return on the principal for one period. Simple interest is usually expressed as shown in Illustration G-1.

Illustration G-1
Interest computation

Compound Interest

Compound interest is computed on principal and on any interest earned that has not been paid or withdrawn. It is the return on the principal for two or more time periods. Compounding computes interest not only on the principal but also on the interest earned to date on that principal, assuming the interest is left on deposit.

To illustrate the difference between simple and compound interest, assume that you deposit $1,000 in Bank Two, where it will earn simple interest of 9% per year, and you deposit another $1,000 in Citizens Bank, where it will earn compound interest of 9% per year compounded annually. Also assume that in both cases you will not withdraw any interest until three years from the date of deposit. Illustration G-2 shows the computation of interest you will receive and the accumulated year-end balances.

Illustration G-2
Simple versus compound interest

Note in Illustration G-2 that simple interest uses the initial principal of $1,000 to compute the interest in all three years. Compound interest uses the accumulated balance (principal plus interest to date) at each year-end to compute interest in the succeeding year—which explains why your compound interest account is larger.

Obviously, if you had a choice between investing your money at simple interest or at compound interest, you would choose compound interest, all other things—especially risk—being equal. In the example, compounding provides $25.03 of additional interest income. For practical purposes, compounding assumes that unpaid interest earned becomes a part of the principal. The accumulated balance at the end of each year becomes the new principal on which interest is earned during the next year.

Illustration G-2 indicates that you should invest your money at the bank that compounds interest annually. Most business situations use compound interest. Simple interest is generally applicable only to short-term situations of one year or less.

Present Value Concepts

The present value is the value now of a given amount to be paid or received in the future, assuming compound interest. The present value is based on three variables: (1) the dollar amount to be received (future amount), (2) the length of time until the amount is received (number of periods), and (3) the interest rate (the discount rate). The process of determining the present value is referred to as discounting the future amount.

In this textbook, we use present value computations in measuring several items. For example, Chapter 15 computed the present value of the principal and interest payments to determine the market price of a bond. In addition, determining the amount to be reported for notes payable and lease liabilities involves present value computations.

Present Value of a Single Amount

To illustrate present value, assume that you want to invest a sum of money that will yield $1,000 at the end of one year. What amount would you need to invest today to have $1,000 one year from now? Illustration G-3 shows the formula for calculating present value.

Illustration G-3
Formula for present value

Thus, if you want a 10% rate of return, you would compute the present value of $1,000 for one year as follows.

We know the future amount ($1,000), the discount rate (10%), and the number of periods (1). These variables are depicted in the time diagram in Illustration G-4.

Illustration G-4
Finding present value if discounted for one period

If you receive the single amount of $1,000 in two years, discounted at 10% [PV = $1,000 ÷ (1 + .10)²], the present value of your $1,000 is $826.45 ($1,000 ÷ 1.21), as shown in Illustration G-5.

Illustration G-5
Finding present value if discounted for two periods

You also could find the present value of your amount through tables that show the present value of 1 for n periods. In Table 1 below, n (represented in the table's rows) is the number of discounting periods involved. The percentages (represented in the table's columns) are the periodic interest rates or discount rates. The 5-digit decimal numbers in the intersections of the rows and columns are called the present value of 1 factors.

When using Table 1 to determine present value, you multiply the future value by the present value factor specified at the intersection of the number of periods and the discount rate.

For example, the present value factor for one period at a discount rate of 10% is .90909, which equals the $909.09 ($1,000 × .90909) computed in Illustration G-4. For two periods at a discount rate of 10%, the present value factor is .82645, which equals the $826.45 ($1,000 × .82645) computed previously.

Note that a higher discount rate produces a smaller present value. For example, using a 15% discount rate, the present value of $1,000 due one year from now is $869.57, versus $909.09 at 10%. Also note that the further removed from the present the future value is, the smaller the present value. For example, using the same discount rate of 10%, the present value of $1,000 due in five years is $620.92, versus the present value of $1,000 due in one year, which is $909.09.

The following two demonstration problems (Illustrations G-6 and G-7) illustrate how to use Table 1.

Illustration G-6
Demonstration problem—Using Table 1 for PV of 1

Illustration G-7
Demonstration problem—Using Table 1 for PV of 1

Present Value of an Annuity

The preceding discussion involved the discounting of only a single future amount. Businesses and individuals frequently engage in transactions in which a series of equal dollar amounts are to be received or paid at evenly spaced time intervals (periodically). Examples of a series of periodic receipts or payments are loan agreements, installment sales, mortgage notes, lease (rental) contracts, and pension obligations. As discussed in Chapter 15, these periodic receipts or payments are annuities.

The present value of an annuity is the value now of a series of future receipts or payments, discounted assuming compound interest. In computing the present value of an annuity, you need to know (1) the discount rate, (2) the number of discount periods, and (3) the amount of the periodic receipts or payments.

To illustrate how to compute the present value of an annuity, assume that you will receive $1,000 cash annually for three years at a time when the discount rate is 10%. Illustration G-8 (page G6) depicts this situation, and Illustration G-9 (page G6) shows the computation of its present value.

Illustration G-8
Time diagram for a three-year annuity

Illustration G-9
Present value of a series of future amounts computation

This method of calculation is required when the periodic cash flows are not uniform in each period. However, when the future receipts are the same in each period, there are two other ways to compute present value. First, you can multiply the annual cash flow by the sum of the three present value factors. In the previous example, $1,000 × 2.48686 equals $2,486.86. The second method is to use annuity tables. As illustrated in Table 2, these tables show the present value of 1 to be received periodically for a given number of periods.

Table 2 shows that the present value of an annuity of 1 factor for three periods at 10% is 2.48685.¹ (This present value factor is the total of the three individual present value factors, as shown in Illustration G-9.) Applying this amount to the annual cash flow of $1,000 produces a present value of $2,486.85.

The following demonstration problem (Illustration G-10) illustrates how to use Table 2.

Illustration G-10
Demonstration problem—Using Table 2 for PV of an annuity of 1

Time Periods and Discounting

In the preceding calculations, the discounting was done on an annual basis using an annual interest rate. Discounting may also be done over shorter periods of time such as monthly, quarterly, or semiannually.

When the time frame is less than one year, you need to convert the annual interest rate to the applicable time frame. Assume, for example, that the investor in Illustration G-8 received $500 semiannually for three years instead of $1,000 annually. In this case, the number of periods becomes six (3 × 2), the discount rate is 5% (10% ÷ 2), the present value factor from Table 2 is 5.07569, and the present value of the future cash flows is $2,537.85 (5.07569 × $500). This amount is slightly higher than the $2,486.86 computed in Illustration G-9 because interest is paid twice during the same year. Therefore, interest is earned on the first half-year's interest.

Computing the Present Value of a Long-Term Note or Bond

The present value (or market price) of a long-term note or bond is a function of three variables: (1) the payment amounts, (2) the length of time until the amounts are paid, and (3) the discount rate. Our illustration uses a five-year bond issue.

The first variable—dollars to be paid—is made up of two elements: (1) a series of interest payments (an annuity), and (2) the principal amount (a single sum). To compute the present value of the bond, we must discount both the interest payments and the principal amount—two different computations. The time diagrams for a bond due in five years are shown in Illustration G-11 (page G8).

When the investor's market interest rate is equal to the bond's contractual interest rate, the present value of the bonds will equal the face value of the bonds. To illustrate, assume a bond issue of 10%, five-year bonds with a face value of $100,000 with interest payable semiannually on January 1 and July 1. If the discount rate is the same as the contractual rate, the bonds will sell at face value. In this case, the investor will receive the following: (1) $100,000 at maturity, and (2) a series of ten $5,000 interest payments [($100,000 × 10%) ÷ 2] over the term of the bonds. The length of time is expressed in terms of interest periods—in this case—10, and the discount rate per interest period, 5%. The following time diagram (Illustration G-12) depicts the variables involved in this discounting situation.

Illustration G-11
Present value of a bond time diagram

Illustration G-12
Time diagram for present value of a 10%, five-year bond paying interest semiannually

Illustration G-13 shows the computation of the present value of these bonds.

Illustration G-13
Present value of principal and interest—face value

Now assume that the investor's required rate of return is 12%, not 10%. The future amounts are again $100,000 and $5,000, respectively, but now a discount rate of 6% (12% ÷ 2) must be used. The present value of the bonds is $92,639, as computed in Illustration G-14.

Illustration G-14
Present value of principal and interest—discount

Conversely, if the discount rate is 8% and the contractual rate is 10%, the present value of the bonds is $108,111, computed as shown in Illustration G-15.

Illustration G-15
Present value of principal and interest—premium

The above discussion relies on present value tables in solving present value problems. Many people use spreadsheets such as Excel or financial calculators (some even on websites) to compute present values, without the use of tables. Many calculators, especially financial calculators, have present value (PV) functions that allow you to calculate present values by merely inputting the proper amount, discount rate, and periods, and then pressing the PV key. Appendix H illustrates how to use a financial calculator in various business situations.

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