The chapter discusses compound interest, annuities, and present value. These techniques are used in financial reporting to analyze cash inflows and outflows.
Introduction to Time Value of Money
Present Value and Future Value of a Single Amount
Future Value of Ordinary and Annuity Due
Present Value of Ordinary Annuity and Annuity Due
Present Value Bond Calculation
Chapter 6 discusses the essentials of compound interest, annuities, and present value. These techniques are being used in many areas of financial reporting where the relative values of cash inflows and outflows are measured and analyzed. The material presented in Chapter 6 will provide a sufficient background for application of these techniques to topics presented in subsequent chapters.
Compound interest, annuity, and present value techniques can be applied to many of the items found in financial statements. In accounting, these techniques can be used to measure the relative values of cash inflows and outflows, evaluate alternative investment opportunities, and determine periodic payments necessary to meet future obligations. It is frequently used when market-based fair value information is not readily available. Some of the accounting items to which these techniques may be applied are: (a) notes receivable and payable, (b) leases, (c) pensions and other postretirement benefits, (d) long-term assets, (e) stock-based compensation, (f) business combinations, (g) disclosures, and (h) environmental liabilities.
Nature of Interest
Interest is the payment for the use of money. It is normally stated as a percentage of the amount borrowed (principal), calculated on a yearly basis.
Simple interest is computed on the amount of the principal only. The formula for simple interest can be expressed as p × i × n where p is the principal, i is the rate of interest for one period, and n is the number of periods.
Compound interest is the process of computing interest on the principal plus any interest previously earned. Compound interest is common in business situations where capital is financed over long periods of time. Simple interest is applied to short-term investments and debts due in one year or less. How often interest is compounded can make a substantial difference in the level of return achieved, or the cost of borrowing.
In discussing compound interest, the term period is used in place of years because interest may be compounded daily, weekly, monthly, and so on. To convert the annual interest rate to the compounding period interest rate, divide the annual interest rate by the number of compounding periods in a year. The number of periods over which interest will be compounded is calculated by multiplying the number of years involved by the number of compounding periods in a year.
Compound Interest Tables
Compound interest tables have been developed to aid in the computation of present values and annuities. Careful analysis of the problem as to which compound interest tables will be applied is necessary to determine the appropriate procedures to follow. The contents of the five types of compound interest tables follow:
Future value of 1. Contains the amounts to which 1 will accumulate if deposited now at a specified rate and left for a specified number of periods. (Table 6-1)
Present value of 1. Contains the amount that must be deposited now at a specified rate of interest to equal 1 at the end of a specified number of periods. (Table 6-2)
Future value of an ordinary annuity of 1. Contains the amount to which periodic rents of 1 will accumulate if the rents are invested at the end of each period at a specified rate of interest for a specified number of periods. (This table may also be used as a basis for converting to the amount of an annuity due of 1.) (Table 6-3)
Present value of an ordinary annuity of 1. Contains the amounts that must be deposited now at a specified rate of interest to permit withdrawals of 1 at the end of regular periodic intervals for the specified number of periods. (Table 6-4)
Present value of an annuity due of 1. Contains the amounts that must be deposited now at a specified rate of interest to permit withdrawals of 1 at the beginning of regular periodic intervals for the specified number of periods. (Table 6-5)
Certain concepts are fundamental to all compound interest problems. These concepts are:
- Rate of Interest. The annual rate that must be adjusted to reflect the length of the compounding period if less than a year.
- Number of Time Periods. The number of compounding periods (a period may be equal to or less than a year).
- Future Amount. The value at a future date of a given sum or sums invested assuming compound interest.
- Present Value. The value now (present time) of a future sum or sums discounted assuming compound interest.
The remaining concepts in this chapter cover the following six major time value of money concepts:
- Future value of a single sum.
- Present value of a single sum.
- Future value of an ordinary annuity.
- Future value of an annuity due.
- Present value of an ordinary annuity
- Present value of an annuity due.
Single-sum problems generally fall into one of two categories. The first category consists of problems that require the computation of the unknown future value of a known single sum of money that is invested now for a certain number of periods at a certain interest rate. The second category consists of problems that require the computation of the unknown present value of a known single sum of money in the future that is discounted for a certain number of periods at a certain interest rate.
The concept of present value is described as the amount that must be invested now to produce a known future value. This is the opposite of the compound interest discussion in which the present value was known and the future value was determined. An example of the type of question addressed by the present value method is: What amount must be invested today at 6% interest compounded annually to accumulate $5,000 at the end of 10 years? In this question the present value method is used to determine the initial dollar amount to be invested. The present value method can also be used to determine the number of years or the interest rate when the other facts are known.
Future Value of an Annuity
An annuity is a series of equal periodic payments or receipts called rents. An annuity requires that the rents be paid or received at equal time intervals, and that compound interest be applied. The future value of an annuity is the sum (future value) of all the rents (payments or receipts) plus the accumulated compound interest on them. If the rents occur at the end of each time period, the annuity is known as an ordinary annuity. If rents occur at the beginning of each time period, it is an annuity due. Thus, in determining the amount of an annuity for a given set of facts, there will be one less interest period for an ordinary annuity than for an annuity due.
Present Value of an Annuity
The present value of an annuity is the present value of a series of equal rents, to withdraw at equal interests. If the annuity is an ordinary annuity, the initial sum of money is invested at the beginning of the first period and withdrawals are made at the end of each subsequent period. If the annuity is an annuity due, the initial sum of money is invested at the beginning of the first period and withdrawals are made at the beginning of each period. Thus, the first rent withdrawn in an annuity due occurs on the day after the initial sum of money is invested. When computing the present value of an annuity, for a given set of facts, there will be one less discount period for an annuity due than for an ordinary annuity.
A deferred annuity is an annuity in which two or more periods have expired before the rents will begin. For example, an ordinary annuity of 10 annual rents deferred five years means that no rents will occur during the first five years, and that the first of the
10 rents will occur at the end of the sixth year. An annuity due of 10 annual rents deferred five years means that no rents will occur during the first five years, and that the first of the 10 rents will occur at the beginning of the sixth year. The fact that an annuity is a deferred annuity affects the computation of the present value. However, the future value of a deferred annuity is the same as the future value of an annuity not deferred because there is no accumulation or investment on which interest may accrue.
A long-term bond produces two cash flows: (1) periodic interest payments during the life of the bond, and (2) the principal (face value) paid at maturity. At the date of issue, bond buyers determine the present value of these two cash flows using the market rate of interest.
Concepts Statement No. 7 introduces an expected cash flow approach that uses a range of cash flows and incorporates the probabilities of those cash flows to provide
a more relevant measurement of present value. The FASB takes the position that after computing the expected cash flows, a company should discount those cash flows by the risk-free rate of return, which is defined as the pure rate of return plus the expected inflation rate.