E6-1 Using interest tables. Simple 5–10E6-2 Simple and compound interest computations. Simple 5–10E6-3 Computation of future values and present values. Simple 10–15E6-4 Computation of future values and present values. Moderate 15–20E6-5 Computation of present value. Simple 10–15E6-6 Future value and present value problems. Moderate 15–20E6-7 Computation of bond prices. Moderate 12–17E6-8 Computations for a retirement fund. Simple 10–15E6-9 Unknown rate. Moderate 5–10E6-10 Unknown periods and unknown interest rate. Simple 10–15E6-11 Evaluation of purchase options. Moderate 10–15E6-12 Analysis of alternatives. Simple 10–15E6-13 Computation of bond liability. Moderate 15–20E6-14 Computation of pension liability. Moderate 15–20E6-15 Investment decision. Moderate 15–20E6-16 Retirement of debt. Simple 10–15E6-17 Computation of amount of rentals. Simple 10–15E6-18 Least costly payoff. Simple 10–15E6-19 Least costly payoff. Simple 10–15E6-20 Expected cash flows. Simple 5–10E6-21 Expected cash flows and present value. Moderate 15–20E6-22 Fair value estimate. Moderate 15–20
P6-1 Various time value situations. Moderate 15–20P6-2 Various time value situations. Moderate 15–20P6-3 Analysis of alternatives. Moderate 20–30P6-4 Evaluating payment alternatives. Moderate 20–30P6-5 Analysis of alternatives. Moderate 20–25P6-6 Purchase price of a business. Moderate 25–30P6-7 Time value concepts applied to solve business problems. Complex 30–35P6-8 Analysis of alternatives. Moderate 20–30P6-9 Analysis of business problems. Complex 30–35P6-10 Analysis of lease vs. purchase. Complex 30–35P6-11 Pension funding. Complex 25–30P6-12 Pension funding. Moderate 20–25P6-13 Expected cash flows and present value. Moderate 20–25P6-14 Expected cash flows and present value. Moderate 20–25P6-15 Fair value estimate. Complex 20–25
1. Identify accounting topics where the time value of money is relevant.2. Distinguish between simple and compound interest.3. 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.
1. (L.O. 1) Chapter 6 discusses the essentials of compound interest, annuities and presentvalue. These techniques are being used in many areas of financial reporting where therelative values of cash inflows and outflows are measured and analyzed. The materialpresented in Chapter 6 will provide a sufficient background for application of thesetechniques to topics presented in subsequent chapters.
2. Compound interest, annuity, and present value techniques can be applied to many ofthe items found in financial statements. In accounting, these techniques can be used tomeasure the relative values of cash inflows and outflows, evaluate alternative investmentopportunities, and determine periodic payments necessary to meet future obligations.Some of the accounting items to which these techniques may be applied are: (a) notesreceivable and payable, (b) leases, (c) pensions, (d) long-term assets, (e) sinkingfunds, (f) business combinations, (g) disclosures, and (h) installment contracts.
Nature of Interest
3. (L.O. 2) Interest is the payment for the use of money. It is normally stated as a per-centage of the amount borrowed (principal), calculated on a yearly basis. For example, anentity 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 iscomputed on the amount of the principal only. The formula for simple interest can beexpressed as p x i x n where p is the principal, i is the rate of interest for one period, and nis the number of periods.
Compound Interest
4. (L.O. 2) Compound interest is the process of computing interest on the principal plusany interest previously earned. Referring to the example in (3) above, if the loan was fortwo 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 mostcommon in business situations where large amounts of capital are financed over longperiods of time. Simple interest is applied mainly to short-term investments and debts duein one year or less. How often interest is compounded can make a substantial differencein the level of return achieved.
5. In discussing compound interest, the term period is used in place of years becauseinterest may be compounded daily, weekly, monthly, and so on. Thus, to convert theannual interest rate to the compounding period interest rate, divide the annual interestrate by the number of compounding periods in a year. Also, the number of periods overwhich interest will be compounded is calculated by multiplying the number of years involvedby the number of compounding periods in a year.
6. (L.O. 3) Compound interest tables have been developed to aid in the computation ofpresent values and annuities. Careful analysis of the problem as to which compound interesttables 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:
“Future value of 1” table. Contains the amounts to which 1 will accumulate if depositednow at a specified rate and left for a specified number of periods. (Table 1)
“Present value of 1” table. Contains the amount that must be deposited now at a specifiedrate of interest to equal 1 at the end of a specified number of periods. (Table 2)
“Future value of an ordinary annuity of 1” table. Contains the amount to whichperiodic rents of 1 will accumulate if the rents are invested at the end of each period at aspecified rate of interest for a specified number of periods. (This table may also be usedas a basis for converting to the amount of an annuity due of 1.) (Table 3)
“Present value of an ordinary annuity of 1” table. Contains the amounts that must bedeposited now at a specified rate of interest to permit withdrawals of 1 at the end ofregular periodic intervals for the specified number of periods. (Table 4)
“Present value of an annuity due of 1” table. Contains the amounts that must bedeposited now at a specified rate of interest to permit withdrawals of 1 at the beginning ofregular periodic intervals for the specified number of periods. (Table 5)
8. (L.O. 4) Certain concepts are fundamental to all compound interest problems. Theseconcepts are:
a. Rate of Interest. The annual rate that must be adjusted to reflect the length of thecompounding period if less than a year.
b. Number of Time Periods. The number of compounding periods (a period may beequal to or less than a year).
c. Future Amount. The value at a future date of a given sum or sums investedassuming compound interest.
d. Present Value. The value now (present time) of a future sum or sums discountedassuming compound interest.
9. (L.O. 5) The remaining review paragraphs pertain to present values and future values.The text material covers the following six major time value of money concepts:
10. Single-sum problems generally fall into one of two categories. The first category consistsof problems that require the computation of the unknown future value of a known singlesum of money that is invested now for a certain number of periods at a certain interestrate. The second category consists of problems that require the computation of theunknown present value of a known single sum of money in the future that is discountedfor 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 toproduce a known future value. This is the opposite of the compound interest discussion inwhich the present value was known and the future value was determined. An example ofthe type of question addressed by the present value method is: What amount must beinvested today at 6% interest compounded annually to accumulate $5,000 at the end of10 years? In this question the present value method is used to determine the initial dollaramount to be invested. The present value method can also be used to determine thenumber of years or the interest rate when the other facts are known.
Future Value of an Annuity
12. (L.O. 6) An annuity is a series of equal periodic payments or receipts called rents. Anannuity requires that the rents be paid or received at equal time intervals, and thatcompound 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 ordinaryannuity. If rents occur at the beginning of each time period, it is an annuity due. Thus, indetermining the amount of an annuity for a given set of facts, there will be one lessinterest period for an ordinary annuity than for an annuity due.
Present Value of an Annuity
13. (L.O. 7) The present value of an annuity is the single sum that, if invested at compoundinterest now, would provide for a series of equal withdrawals for a certain number of futureperiods. If the annuity is an ordinary annuity, the initial sum of money is invested atthe beginning of the first period and withdrawals are made at the end of each period. Ifthe annuity is an annuity due, the initial sum of money is invested at the beginning of thefirst period and withdrawals are made at the beginning of each period. Thus, the first rentwithdrawn 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 oneless discount period for an annuity due than for an ordinary annuity.
14. (L.O. 8) A deferred annuity is an annuity in which two or more periods have expired beforethe rents will begin. For example, an ordinary annuity of 10 annual rents deferred fiveyears means that no rents will occur during the first five years, and that the first of the10 rents will occur at the end of the sixth year. An annuity due of 10 annual rents deferredfive years means that no rents will occur during the first five years, and that the first of the10 rents will occur at the beginning of the sixth year. The fact that an annuity is a deferredannuity affects the computation of the present value. However, the future value of adeferred annuity is the same as the future value of an annuity not deferred becausethere 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 lifeof the bond, and (2) the principal (face value) paid at maturity. At the date of issue, bondbuyers determine the present value of these two cash flows using the market rate ofinterest.
16. (L.O. 9) Concepts Statement No. 7 introduces an expected cash flow approach that usesa range of cash flows and incorporates the probabilities of those cash flows to providea more relevant measurement of present value. The FASB takes the position that aftercomputing the expected cash flows, a company should discount those cash flows by therisk-free rate of return, which is defined as the pure rate of return plus the expectedinflation rate.
Financial Calculators
*17. Business professionals, after mastering the above concepts, will often use a financial(business) calculator to solve time value of money problems. When using financialcalculators, the five most common keys used to solve time value of money problems are:
N I PV PMT FV
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).
This chapter can be covered in two to three class sessions. Most students have had previousexposure to single sum problems and ordinary annuities, but annuities due and deferredannuities will be new material for most students. The first class session can be used fordiscussing Illustration 6-5.
TEACHING TIP
Illustration 6-5 can be distributed to students as a self-contained 6-page handout. It uses10 sample problems to demonstrate a 4-step solution method that can be used to solve anyof the problems discussed in the chapter.
Some students with a background in math or finance courses may prefer to use exponentialformulas rather than interest tables to find interest factors. Other students with sophisticatedcalculators may prefer to “let the calculator do the work.” Remind students that whether theyuse interest tables, exponential formulas, or internal calculator routines, they cannot solveproblems correctly unless they can correctly identify the type of problem, the number ofperiods, and the interest rate involved. Students often have no difficulty with problems that areworded: “At 6%, what is the present value of an annuity due of 20 payments of $10,000 each?”but they may not know how to proceed if the same problem is worded: “What amount must bedeposited now in an account paying 12% if it is desired to make 20 semiannual withdrawals of$10,000 each, beginning today?” Emphasize to students the importance of properly setting upthe problem.
The second and third class sessions can be used for determining solutions to more complexproblems, including deferred annuities, bond valuation and other accounting applications.Some of the journal entries for the accounting applications can be discussed briefly.
The following lecture outline is appropriate for this chapter.
A. (L.O. 1) Basic Time Value Concepts.
1. Discuss the importance of the time value of money.
2. Describe accounting applications of time value concepts: long-term assets, pensions,leases, long-term notes.
3. Describe personal applications of time value concepts: purchasing a home, planning forretirement, evaluating alternative investments.
B. (L.O. 1) Nature of Interest.
1. Interest is payment for the use of money. It is the excess cash received or repaid overand above the principal (amount lent or borrowed).
2. Interest rates are stated on an annual basis unless indicated otherwise.
Illustration 6-1 can be used to distinguish between simple interest and compound interest.
1. Simple interest is computed on the amount of the principal only.
2. Simple interest = p X i X n where
p = principal.i = rate of interest for a single period.n = number of periods.
D. (L.O. 2) Compound Interest.
1. Compound interest is computed on the principal and on any interest earned that hasnot been paid or withdrawn.
2. Discuss the power of time and compounding. (E.g., “What do the numbers mean?”on text page 313 indicates that at 5% compounded annually, $1,000 grows to$23,839 in 65 years. At 5% simple interest, $1,000 would grow to only $4,250 in65 years.) $4,250 = $1,000 + ($1,000 X .05 X 65).
3. The term period should be used instead of years.
a. Interest may be compounded more than once a year:
If interest is Number of compoundingcompounded periods per year Annually 1Semiannually 2Quarterly 4Monthly 12
b. Adjustment when interest is compounded more than once a year.
(1) Compute the compounding period interest rate: divide the annual interestrate by the number of compounding periods per year.
(2) Compute the total number of compounding periods: multiply the number ofyears by the number of compounding periods per year.
1. The tables contain interest factors that simplify the computation of compound interest.Example: If $1,000 is deposited today at 9% compound interest, the balance in 3 yearscan be determined:
a. By repetitive calculation—
First year: $1,000 + ($1,000 X .09) = $1,090.Second year: $1,090 + ($1,090 X .09) = $1,188.Third year: $1,188 + ($1,188 X .09) = $1,295 (rounded).
b. By use of exponential formulas—
$1,000 X (1.09)3 = $1,295 (rounded).
c. By use of financial calculators or spreadsheet programs.
d. By obtaining the 1.29503 interest factor from Table 6-1 for 3 periods at 9% andperforming the appropriate computation—
$1,000 X 1.29503 = $1,295.03.
2. Describe the five interest tables provided in the text:
a. Table 6-1: Future Value of 1.
b. Table 6-2: Present Value of 1.
c. Table 6-3: Future Value of an Ordinary Annuity of 1.
d. Table 6-4: Present Value of an Ordinary Annuity of 1.
e. Table 6-5: Present Value of an Annuity Due of 1.
F. (L.O. 4) Variables in Compound Interest Problems.
1. Describe the four fundamental variables in compound interest problems:
TEACHING TIP
Illustration 6-2 depicts a time diagram that identifies the four variables that are fundamentalto all compound interest problems. Illustration 6-3 can be used to show students how thefour fundamental variables relate to the time value of money concepts.
a. Rate of interest: an annual rate, adjusted to reflect the length of the compoundingperiod.
b. Number of time periods: the number of compounding periods.
c. Future value: the value at a future date of a given sum(s) invested assumingcompound interest.
d. Present value: the value now of a future sum(s) discounted assuming compoundinterest.
G. Steps in Solving Compound Interest Problems.
TEACHING TIP
Use Illustration 6-4 to discuss the 5-step solution method that can be used to solve anycompound interest problem.
1. Emphasize the importance of performing Steps 1 and 2 correctly. Whether studentsuse interest tables, exponential formulas, or financial calculators, they cannot solveproblems correctly unless they can correctly identify the type of problem, the number ofperiods, and the interest rate involved.
H. (L.O. 5) Single-Sum Problems.
TEACHING TIP
Problems 1, 2, and 3 in Illustration 6-5 demonstrate single-sum problem situations.
1. Formula for future value:
Future value = present value (or amount)X future value factor for n periods at i %
FV = PV(FVFn, i)
2. Formula for present value:
Present value = future value x present value factor for n periods at i %
PV = FV(PVFn, i)
3. Point out that the present value is always a smaller quantity than the future value.
4. The process of finding the future value is called accumulation. The process of findingthe present value is called discounting.
5. The factors in Table 6-2 are the reciprocal of corresponding factors in Table 6-1.Therefore, all single-sum problems can be solved by using either Table 6-1 or 6-2. Forexample, if the future value is known and the present value is to be solved for, thepresent value can be found:
a. by multiplying the known future value by the appropriate factor from Table 6-2, or
b. by dividing the known future value by the appropriate factor from Table 6-2.
I. (L.O. 6 and 7) Ordinary Annuities.
1. Annuity problems involve a series of equal periodic payments or receipts called rents.
a. In an ordinary annuity the rents occur at the end of each period. The first rentwill occur one period from now.
b. In an annuity due the rents occur at the beginning of each period. The first rentwill occur now.
TEACHING TIP
Problems 4, 5, and 8 in Illustration 6-5 demonstrate ordinary annuity problem situations.
2. Formula for future value of an ordinary annuity:
Future value of ordinary annuity (FVOA)
= periodic rent X future value of ordinary annuity factor for n periods at i %
FVOA = R (FVF–OAn, i)
3. Formula for present value of an ordinary annuity:
Present value of ordinary annuity (PVOA)
= periodic rent X present value of ordinary annuity factor for n periods at i %
PVOA = R (PVF–OAn, i)
4. Point out that the present value of an ordinary annuity is always smaller than the futurevalue of a similar annuity.
5. The factors in Tables 6-3 and 6-4 are not reciprocals of each other.
6. In annuity problems, the rents, interest payments, and number of periods must all bestated on the same basis. For example, if interest is compounded semiannually, thenn = the number of semiannual rents paid or received, i = the annual interest rate dividedby 2, and R = the amount of rent paid or received every 6 months.
7. Some confusion may arise in annuity problems because of two different meanings ofthe word “period.”
a. For the purpose of looking up interest factors, n equals the number of “periods”and is always equal to the number of rents.
b. In the phrase “when computing the future value of an ordinary annuity the numberof compounding periods is one less than the number of rents,” the term “periods”refers to “compounding periods” or “interest-bearing periods.” This refers to thenumber of times interest is earned on the principal and any accumulated interest.This usage of the term “period” is useful for distinguishing between ordinary annui-ties and annuities due. This usage is intended to explain why the adjustment offactors from Table 6-3 is done the way it is when the problem involves the futurevalue of an annuity due.
J. (L.O. 6 and 7) Annuities Due.
TEACHING TIP
Problems 6 and 7 in Illustration 6-5 demonstrate annuity due problem situations.
1. Formula for future value of annuity due:
Future value of annuity due (FVAD) = periodic rent x future value ofordinary annuity factor for n periods ati % X (1 + i)
FVAD = R (FVF–OAn, i ) X (1 + i)
a. An interest table is not provided for the future value of an annuity due.
b. Example: At 9%, what is the future value of an annuity due of 7 payments of$3,000 each?
$3,000 X 9.20044 X 1.09 = $30,085.44
2. Formula for present value of annuity due:
Present value of annuity due (PVAD) = periodic rent x present value ofannuity due factor for n periods at i %
a. The present value of an annuity due is always smaller than the future value ofa similar annuity due.
b. The future value (present value) of an annuity due is always larger than the futurevalue (present value) of a similar ordinary annuity with the same interest rate andnumber of rents.
K. (L.O. 8) Deferred Annuities.
TEACHING TIP
Problems 9 and 10 in Illustration 6-5 demonstrate deferred annuity problem situations.
1. A deferred annuity does not begin to produce rents until two or more periods haveexpired.
2. A deferred annuity problem can occur in either an ordinary annuity situation or anannuity due situation.
a. In order to keep the presentation straightforward, only the ordinary annuity situationhas been illustrated in the text and in Illustration 6-5.
b. The differences between the two situations are as follows:
Ordinary Annuity Annuity Due ofof n Rents Deferred n Rents Deferred for y Periods for y Periods
First rent occurs (y + 1) periods y periods fromfrom now now
Last rent occurs (y + n) periods (y + n – 1) periodsfrom now from now
Future value is immediately after one period aftermeasured as of the last rent the last rent
c. If a deferred annuity involves solving for a present value, the distinction betweenan ordinary annuity and an annuity due has no practical significance. (i.e., seeProblem 10 in Illustration 6-5. This can be set up either as the present value of anordinary annuity of 4 rents deferred 3 periods, as was done in Illustration 6-5, or itcan be set up as the present value of an annuity due of 4 rents deferred 4 periods.If the latter is done, different combinations of factors will be used, but the sameanswer will be obtained.)
d. However, if a deferred annuity involves solving for a future value, the distinctionbetween an ordinary annuity and an annuity due is important. The followingformula is required:
FVAD – d = R (FVF–OAn, i ) X (1 + i ) where
FVAD – d = future value of an annuity due ofn rents deferred for y periods
R = periodic rents
(FVF–OAn, i ) = factor from Table 6-3 for n periods at i %
(1 + i ) = 1 plus the interest rate
To illustrate, suppose Problem 9 in Illustration 6-5 had been worded as follows:
As of the beginning of his first year in college, a student plans to deposit $1,000 inan 8% account at the end of his third, fourth, and fifth years in school. What will bethe balance in the account one year after the last deposit?
The time diagram for this revised problem is:
The revised problem involves solving for the future value of an annuity due of3 rents deferred for 3 periods. The solution is
FVAD – d = R (FVF–OA3,.8%) (1.08) = $1,000 X 3.24640 X 1.08
= $3,506.11.
To sum up: In solving for the future value of a deferred annuity—
(1) If the future value is to be determined immediately after the last rent, theproblem may be thought of as an ordinary annuity. The computation describedin the text and in Illustration 6-5 is adequate.
(2) If the future value is to be determined one period after the last rent, theproblem should be thought of as an annuity due. An adjustment must bemade to accumulate interest for one more period.
1. Discuss the distinction between the stated interest rate and the market or effectiveinterest rate:
a. The stated interest rate is used to determine the periodic amount of cash interestpaid.
b. The market or effective interest rate is used to value the bonds. This is the ratewhich is looked up in the present value tables.
2. The example in the text demonstrates valuation of bonds which pay interest annually.
TEACHING TIP
Illustration 6-6 provides an example of a bond valuation problem in which bond interest ispaid semiannually.
M. (L.O. 9) Expected Cash Flow Approach.
1. Introduced by Concepts Statement No. 7.
2. Uses a range of cash flows and their related probabilities to provide a more relevantmeasurement of present value.
3. Choosing an appropriate interest rate:
a. is not always obvious.
b. three components of interest:
(1) pure rate of interest (2%–4%).
(2) expected inflation rate of interest (0%–?%).
(3) credit risk rate of interest (0%–5%).
TEACHING TIP
Use Illustration 6-7 to provide a basis for discussing how to apply the expected cash flowapproach.
4. After computing the expected cash flows, a company discounts these cash flows bythe risk-free rate of return. This rate is the pure rate of return plus the expectedinflation rate.
