# Appen. A&B for Fund. of Int. Acctg. Summer 2002 Chapter 6: Accounting and the Time Value of Money6-1

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6

Accounting and the

Time Value of Money

CHAPTER STUDY OBJECTIVES

1.Identify accounting topics where the time value of money is relevant.

2.Distinguish between simple and compound interest.

3.Learn how to use appropriate compound interest tables.

4.Identify variables fundamental to solving interest problems.

5.Solve future and present value of 1 problems.

6.Solve future value of ordinary and annuity due problems.

7.Solve present value of ordinary and annuity due problems.

8.Solve present value problems related to deferred annuities and bonds.

9.Apply expected cash flows to present value measurement.

*10.Use a financial calculator to solve time value of money problems.

CHAPTER REVIEW

1.(S.O. 1) 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.

2.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. Some of the accounting items to which these techniques may be applied are: (a) notes receivable and payable, (b) leases, (c) pensions, (d) long-term assets, (e) sinking funds, (f) business combinations, (g) disclosures, and (h) installment contracts.

Nature of Interest

3.(S.O. 2) 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. For example, an entity may borrow \$5,000 from a bank at 7% interest. The yearly interest on this loan is \$350. If the loan is repaid in six months, the interest due would be 1/2 of \$350, or \$175. This type of interest computation is known as simple interest because the interest is computed on the amount of the principal only. The formula for simple interest can be expressed as p x i x n where p is the principal, i is the rate of interest for one period, and n is the number of periods.

Compound Interest

4.(S.O. 2) Compound interest is the process of computing interest on the principal plus any interest previously earned. Referring to the example in (2) above, if the loan was for two years with interest compounded annually, the second year's interest would be \$374.50 (principal plus first year's interest multiplied by 7%). Compound interest is most common in business situations where large amounts of capital are financed over long periods of time. Simple interest is applied mainly 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.

5.In discussing compound interest, the term period is used in place of years because interest may be compounded daily, weekly, monthly, and so on. Thus, 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. Also, 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.

Time Value of Money Tables

6.(S.O. 3) Compound interest tables have been developed to aid in the computation of present values and annuities. Examples of the five types of compound interest tables discussed in Appendix A are presented at the end of the appendix in this study guide. Careful analysis of the problem as to which compound interest tables will be applied is necessary to determine the appropriate procedures to follow.

7.The following is a summary of the contents of the five types of compound interest tables discussed in the appendix.

"Future value of 1" table. Contains the amounts to which 1 will accumulate if deposited now at a specified rate and left for a specified number of periods.

"Present value of 1" table. Contains the amount that must be deposited now at a specified rate of interest to amount to 1 at the end of a specified number of periods.

"Future value of an ordinary annuity of 1" table. Contains the amount to which periodic rents of 1 will accumulate if the rents are invested at a specified rate of interest and are continued 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.)

"Present value of an ordinary annuity of 1" table. 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.

"Present value of an annuity due of 1" table. 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.

8.(S.O. 4) Certain concepts are fundamental to all compound interest problems. These concepts

are:

a.Rate of Interest. The annual rate that must be adjusted to reflect the length of the compounding period if less than one year.

b.Number of Time Periods. The number of compounding periods (a period may be equal to or less than a year).

c.Future Amount. The value at a future date of a given sum or sums invested assuming compound interest.

d.Present Value. The value now (present time) of a future sum or sums discounted assuming compound interest.

9.(S.O. 5) The remaining appendix review paragraphs pertain to present values and future amounts. The text material covers the following six major time value of money concepts:

a.Future value of a single sum.

b.Present value of a single sum.

c.Future value of an ordinary annuity.

d.Future value of an annuity due.

e.Present value of an ordinary annuity.

f.Present value of an annuity due.

10.Single-sum problems generally fall into one of two categories. The first category consists of problems that require the computation of the unknown future amount 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.

Present Value

11.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 Amount of an Annuity

12.(S.O. 6) 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 amount 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

13.(S.O. 7) The present value of an annuity is a sum of money invested today at compound interest that will provide for a series of equal withdrawals for a specified number of future periods. 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 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.

Deferred Annuities

14.(S.O. 8) A deferred annuity is an annuity in which two or more periods must pass, after it has been arranged, 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 amount of a deferred annuity is the same as the amount of an annuity not deferred because there is no accumulation or investment on which interest may accrue.

15.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.

16.(S.O. 9) Concepts Statement No. 6 introduces an expected cash flow approach that uses arrange 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 the expected cash flow are computed, they should be discounted by the risk-free rate of return, which is defined as the pure rate of return plus the expected inflation rate.

Financial Calculators

*17.(S.O. 10) Business professionals, after mastering the above concepts, will often use a financial (business) calculator to solve time value of money problems. When using financial calculators, the five most common keys used to solve time value of money problems are:

where:

N =number of periods.

I =interest rate per period (some calculators use I/YR or i).

PV =present value (occurs at the beginning of the first period).

PMT =payment (all payments are equal, and none are skipped).

FV =future value (occurs at the end of the last period).

DEMONSTRATION PROBLEMS

1.Compute the future amount of 10 periodic payments of \$5,000 each made at the beginning of each period and compounded at 12%.

Solution:

1.Future amount of ordinary annuity for 10 periods of 12% (Table 6-3)17.54874

2.Factor (1 + .12) x 1.12

3.Future amount of annuity due for 10 periods of 12%19.65459

4.Periodic payment (rent)\$ 5,000

5.Future amount\$98,272.95

Solution:

2. Compute the present value of 14 receipts of \$800 each received at the beginning of each period, discounted at 10% compound interest.

Solution:

This is the present value of an annuity due of \$800 payments for 14 periods at 10%.

1.Present value of an annuity due for 14 periods at 10% (Table 6-5)8.10336

2.Periodic receipt (rent)x \$800

3.Present value\$6,482,69

Solution:

3.How much must be invested at the end of each year to accumulate a fund of \$50,000 at the end of 10 years, if the fund earns 9% interest, compounded annually?

Solution:

Known final amount (a)\$ 50,000

Divide (a) by the amount of an ordinary annuity of \$1 for

10 years at 9% (Table 6-3) 15.19293

The result is the periodic rent that would accumulate \$50,000 at the end of

10 years at 9% interest\$ 3,291

Solution:

4.An asset has a cash price of \$9,593.37. The purchaser agrees to pay \$2,000 down and 4 annual payments of \$2,500 at the end of each year. Assuming compounding on an annual basis, what is the stated interest rate of this transaction?

Solution:

Cash price\$9,593.37

Down payment 2,000.00

Net amount due\$7,593.37

\$7,593.37  \$2,500 = 3.03735

Go to Table 6-4 and find the factor 3.03735 in row 4 and read up to the top of the column to find the appropriate interest rate which is 12%.

5.A fund of \$25,000 is deposited in a savings account earning a 12% stated rate but interest is compounded quarterly (3%). What is the maximum amount that could be withdrawn quarterly at the end of each quarter for the next 10 years?

Solution:

PV=R (PVF - OAn,i)

\$25,000=R (PVF - OA40, 3%)

\$25,000=R (23.11477)

\$1,081.56=R

Thus, \$1,081.56 can be withdrawn at the end of each quarter for the next 10 years. The solution requires that the 12% interest rate be divided by 4 and that the 10 years be multiplied by 4 due to the quarterly compounding. Use the 3% column in Table 6-4 for 40 periods.

Solution:

GLOSSARY

Annuity. / A series of equal dollar amounts (rents) that are paid or received periodically at equal intervals of time.
Annuity due. / An annuity whereby each rent is payable (receivable) at the beginning of the period.
Compound interest. / Interest accrues on the unpaid interest of past periods as well as on the principal.
Credit risk rate of
interest. / The amount of interest that depends on the financial stability, profitability, etc., of a business enterprise.
Deferred annuity. / An annuity in which the rents begin after a specified number of periods.
Expected inflation rate
of interest. / The amount of interest that is based on inflationary or deflationary expectations.
Future value. / The value at a future date of a given sum or sums invested assuming compound interest.
Future value of 1 table. / Contains the amounts to which 1 will accumulate if deposited now at a specified rate and left for a specified number of periods.
Future value of an ordinary
annuity of 1 table. / Contains the amounts to which periodic rents of 1 will accumulate if the payments are invested at the end of each period at a specified rate of interest for a specified number of periods.
Interest. / Payment for the use of money.
Ordinary annuity. / An annuity whereby each rent is payable (receivable) at the end of the period.
Present value. / The value now (present time) of a future sum or sums discounted assuming compound interest.
Present value of 1 table. / Contains the amounts that must be deposited now at a specified rate of interest to equal 1 at the end of a specified number of periods.
Present value of an annuity
due of 1 table. / 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.
Present value of an ordinary
annuity of 1 table. / 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.
Principal. / The amount borrowed or invested.
Pure rate of interest. / The amount at lender would charge if there were no possibilities of default and no expectation of inflation.
Simple interest. / Interest on principal only, regardless of interest that may have accrued in the past.

CHAPTER OUTLINE

Fill in the outline presented below.

(S.O. 1) Present Value-Based Accounting Measurements

Nature of Interest

(S.O. 2) Simple Interest

Compound Interest

(S.O. 4) Fundamental Variables

(S.O. 5) Future Amount of a Single Sum

Present Value of a Single Sum

(S.O. 6) Future Amount of an Ordinary Annuity

Chapter Outline(continued)

Future Amount of an Annuity Due

(S.O. 7) Present Value of an Ordinary Annuity

Present Value of an Annuity Due

(S.O. 8) Deferred Annuities and Bonds

(S.O. 9) Expected Cash Flows

*(S.O. 10) Use of a Financial Calculator

REVIEW QUESTIONS AND EXERCISES

TRUE-FALSE

Indicate whether each of the following is true (T) or false (F) in the space provided.

_____1.(S.O. 1) Present value techniques can be used in valuing receivables and payables that carry no stated interest rate.

_____2.(S.O. 2) The amount of interest on a \$1,000, 6%, 6-month note is the same as the amount of interest on a \$1,000, 3%, 1-year note.

_____3.(S.O. 2) In the formula for compound interest, the number of periods refers to the number of months an obligation will be outstanding.

_____4.(S.O. 2) The major difference between compound interest and simple interest lies in the fact that compound interest is computed twice each year, whereas simple interest is computed only once.

_____5.(S.O. 2) The growth in principal is the same under both compound and simple interest if only one compounding period is involved.

_____6.(S.O. 3) If interest is compounded quarterly and the annual interest rate is 8%, the compounding period interest rate is 4%.

_____7.(S.O. 4) Present value is the amount that must be invested now to produce a known future amount.

_____8.(S.O. 6) An annuity requires that periodic rents always be the same even though the interval between the rents may vary.

_____9.(S.O. 6) An annuity is classified as an ordinary annuity if the rents occur at the end of the period; it is classified as an annuity due if the rents occur at the beginning of the period.

_____10.(S.O. 6) The ordinary annuity table may be used to compute the periodic rents when the desired future amount and the present value of the annuity are not known.

_____11.(S.O. 6) Periodic interest earnings under an ordinary annuity will always be lower by one period's interest than the interest earned by an annuity due.

_____12.(S.O. 7) The present value of an ordinary annuity is the present value of series of rents to be made at equal intervals in the future.

_____13.(S.O. 7) The number of rents exceeds the number of discount periods under the present value of an ordinary annuity.

_____14.(S.O. 7) The future amount of a deferred annuity is normally greater than the future amount of an annuity not deferred.

_____15.(S.O. 7) The valuation of a sum as of an earlier date involves a determination of present value; the valuation of a sum as of a later date involves a determination of an amount.

MULTIPLE CHOICE

Select the best answer for each of the following items and enter the corresponding letter in the space provided.

_____1.(S.O. 3) Which of the following tables would show the largest value for an interest rate of 10% for 8 periods?