the Time Value of Money

LEARNING OBJECTIVES

After studying this chapter, you should be able to:

Identify accounting topics where time value of money is relevant. Distinguish between

simple and compound interest.

Learn how to use appropriate compound interest tables. Identify variables

fundamental to solving interest problems. Solve future and present

value of 1 problems. Solve future value of

ordinary and annuity due problems.

Solve present value of ordinary and annuity due problems.

Solve present value problems related to deferred annuities and bonds.

Apply the expected cash flow approach to present value measurement.

Sidney Homer, author of A History of Interest Rates, wrote, “$1,000

invested at a mere 8 percent for 400 years would grow to $23 quadrillion— $5 million for every human on earth. But the first 100 years are the hardest.” This startling quote highlights the power of time and compounding interest on money. Equally significant, although not mentioned in the quote, is the fact that a small difference in the interest rate makes a big difference in the amount of monies accumulated over time. Taking a more realistic example, assume that you had $20,000 in a tax-free retirement account. Half the money is in stocks returning 12 percent and the other half in bonds earning 8 percent. Assuming reinvested profits and quarterly compounding, your bonds would be worth $22,080 after ten years, a doubling of their value. But your stocks, returning 4 percent more, would be worth $32,620, or triple your initial value. The following chart shows this impact.

251

C H A P T E R

6

C H A P T E R

6

Accounting and

the Time Value of Money

T

h e M a g i c o f I n t e r e s t

$32,620

$26,851

$22,080

Money received tomorrow is not the same as money received today. Business people are acutely aware of this timing factor, and they invest and borrow only after carefully analyzing the relative present or future values of the cash flows.

PREVIEW OF CHAPTER 6

PREVIEW OF CHAPTER 6

As indicated in the opening story, the timing of the returns on investments has an im-portant effect on the worth of the investment (asset), and the timing of debt repayments has a similarly important effect on the value of the debt commitment (liability). As a financial expert, you will be expected to make present and future value measurements and to understand their implications. The purpose of this chapter is to present the tools and techniques that will help you measure the present value of future cash inflows and outflows. The content and organization of the chapter are as follows.

BASIC TIME VALUE CONCEPTS

In accounting (and finance), the term time value of moneyis used to indicate a rela-tionship between time and money—that a dollar received today is worth more than a dollar promised at some time in the future. Why? Because of the opportunity to invest today’s dollar and receive interest on the investment. Yet, when you have to decide among various investment or borrowing alternatives, it is essential to be able to com-pare today’s dollar and tomorrow’s dollar on the same footing—to “comcom-pare apples to apples.” We do that by using the concept of present value, which has many applica-tions in accounting.

Applications of Time Value Concepts

Financial reporting uses different measurements in different situations. Present value is one of these measurements, and its usage has been increasing.1Some of the appli-cations of present value-based measurements to accounting topics are listed below, sev-eral of which are required in this textbook.

252

ACCOUNTING AND THE TIME VALUE OF MONEY

•Applications •The nature of interest •Simple interest •Compound interest •Fundamental variables Basic Time Value Concepts •Future value of a single sum •Present value of a single sum •Solving for other

unknowns Single-Sum Problems •Future value of ordinary annuity •Future value of annuity due • Illustrations of FV of annuity •Present value of ordinary annuity •Present value of annuity due •Illustrations of PV of annuity Annuities •Deferred annuities •Valuation of long-term bonds •Effective interest method of bond discount/premium amortization More Complex Situations •Choosing an appro-priate interest rate •Expected cash flow

illustration Present Value Measurement

O B J E C T I V E

1_{Many of the recent standards, such as FASB Statements No. 106, 107, 109, 113, 114, 116, 141,}

142,and 144,have addressed the issue of present value somewhere in the pronouncement or related basis for conclusions.

In addition to accounting and business applications, compound interest, annuity, and present value concepts apply to personal finance and investment decisions. In pur-chasing a home or car, planning for retirement, and evaluating alternative investments, you will need to understand time value of money concepts.

The Nature of Interest

Interestis payment for the use of money. It is the excess cash received or repaid over and above the amount lent or borrowed (principal). For example, if the Corner Bank lends you $1,000 with the understanding that you will repay $1,150, then the excess over $1,000, or $150, represents interest expense. Or if you lend your roommate $100 and then collect $110 in full payment, the $10 excess represents interest revenue.

The amount of interest to be paid is generally stated as a rate over a specific pe-riod of time. For example, if you used $1,000 for one year before repaying $1,150, the rate of interest is 15% per year ($150 $1,000). The custom of expressing interest as a percentage rate is an established business practice.2In fact, business managers make investing and borrowing decisions on the basis of the rate of interest involved rather than on the actual dollar amount of interest to be received or paid.

How is the interest rate determined? One of the most important factors is the level of credit risk (risk of nonpayment) involved. Other factors being equal, the higher the credit risk, the higher the interest rate. Low-risk borrowers like Microsoftor Intelcan probably obtain a loan at or slightly below the going market rate of interest. You or the neighborhood delicatessen, on the other hand, would probably be charged several per-centage points above the market rate—if you can get a loan at all.

Notes.Valuing noncurrent receivables and payables that carry no stated in-terest rate or a lower than market inin-terest rate.

Leases.Valuing assets and obligations to be capitalized under long-term leases and measuring the amount of the lease payments and annual leasehold amor-tization.

Pensions and Other Postretirement Benefits.Measuring service cost compo-nents of employers’ postretirement benefits expense and postretirement ben-efits obligation.

Long-Term Assets.Evaluating alternative long-term investments by discount-ing future cash flows. Determindiscount-ing the value of assets acquired under deferred payment contracts. Measuring impairments of assets.

Sinking Funds.Determining the contributions necessary to accumulate a fund for debt retirements.

Business Combinations. Determining the value of receivables, payables, lia-bilities, accruals, and commitments acquired or assumed in a “purchase.”

Disclosures.Measuring the value of future cash flows from oil and gas reserves for disclosure in supplementary information.

Installment Contracts.Measuring periodic payments on long-term purchase contracts.

PRESENT VALUE-BASED ACCOUNTING MEASUREMENTS

2_{Federal law requires the disclosure of interest rates on an annual basisin all contracts. That} is, instead of stating the rate as “1% per month,” it must be stated as “12% per year” if it is sim-ple interest or “12.68% per year” if it is compounded monthly.

The amount of interest involved in any financing transaction is a function of three variables:

O B J E C T I V E

Principal.The amount borrowed or invested.

Interest Rate.A percentage of the outstanding principal.

Time.The number of years or fractional portion of a year that the principal is outstanding.

VARIABLES IN INTEREST COMPUTATION

3

Simple interest is traditionally expressed in textbooks in business mathematics or business finance as: I(interest) P(principal) R(rate) T(time).

The larger the principal amount, or the higher the interest rate, or the longer the time period, the larger the dollar amount of interest.

Simple Interest

Simple interestis computed on the amount of the principal only. It is the return on (or growth of) the principal for one time period. Simple interest is commonly expressed as follows.3

Interest pin

where

pprincipal

i rate of interest for a single period

nnumber of periods

To illustrate, if you borrow $1,000 for 3 years with a simple interest rate of 15% per year, the total interest you will pay is $450, computed as follows.

Interest pin $1,000 .15 3

$450

If you borrow $1,000 for 3 months at 15%, the interest is $37.50, computed as follows. Interest $1,000 .15 .25

$37.50

Compound Interest

John Maynard Keynes, the legendary English economist, supposedly called it magic. Mayer Rothschild, the founder of the famous European banking firm, is said to have proclaimed it the eighth wonder of the world. Today people continue to extol its won-der and its power. The object of their affection is compound interest.

Compound interestis computed on principal andon any interest earned that has not been paid or withdrawn. It is the return on (or growth of) 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 the Last National Bank, where it will earn simple interest of 9%

per year, and you deposit another $1,000 in the First State Bank, where it will earn com-pound interest of 9% per year comcom-pounded annually. Also assume that in both cases you will not withdraw any interest until 3 years from the date of deposit. The compu-tation of interest to be received and the accumulated year-end balance are indicated in Illustration 6-1.

Simple Interest Calculation

Simple Interest

Last National Bank

Year 1 $1,000.00 _× 9% $ 90.00 $1,090.00 Accumulated Year-end Balance Year 2 $1,000.00 _× 9% Year 3 $1,000.00 _× 9% $270.00 Compound Interest Calculation Compound Interest

First State Bank

Year 1 $1,000.00 _× 9% $ 90.00 $1,090.00 Accumulated Year-end Balance Year 2 $1,090.00 _× 9% 98.10 Year 3 $1,188.10 _× 9% 106.93 $295.03 $1,188.10 $1,188.10 $1,295.03 $1,180.00 $1,270.00 90.00 90.00 $25.03 Difference $25.03 Difference ILLUSTRATION 6-1 Simple vs. Compound Interest

Here is an illustration of the power of time and compounding interest on money. In 1626, Peter Minuit bought Manhattan Island from the Manhattoe Indians for $24 worth of trinkets and beads. If the Indians had taken a boat to Holland, invested the $24 in Dutch securities returning just 6 percent per year, and kept the money and in-terest invested at 6 percent, by 1971 they would have had $13 billion, enough to buy back Manhattan and still have a couple of billion dollars left for doodads (Forbes,June 1, 1971). By 2002, 376 years after the trade, the $24 would have grown to approximately $79 billion.

Spare change

What do the

numbers mean?

Note in the illustration above that simple interest uses the initial principal of $1,000 to compute the interest in all 3 years. Compound interest uses the accumu-lated 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 addi-tional interest revenue. For practical purposes compounding assumes that unpaid interest earned becomes a part of the principal, and the accumulated balance at the end of each year becomes the new principal sum on which interest is earned during the next year.

Compound interest is the typical interest computation applied in business situa-tions, particularly in our economy where large amounts of long-lived assets are used productively and financed over long periods of time. Financial managers view and evaluate their investment opportunities in terms of a series of periodic returns, each of which can be reinvested to yield additional returns. Simple interest is usually ap-plicable only to short-term investments and debts that involve a time span of one year or less.

Illustration 6-2 indicates the general format and content of these tables. It shows how much principal plus interest a dollar accumulates to at the end of each of five pe-riods at three different rates of compound interest.

O B J E C T I V E

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. (Table 6-1)

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. (Table 6-2)

Future Value of an Ordinary Annuity of 1 table.Contains the amounts to which periodic rents of 1 will accumulate if the payments (rents) are invested at the

end of each period at a specified rate of interest for a specified number of periods. (Table 6-3)

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 endof regular periodic intervals for the specified number of peri-ods. (Table 6-4)

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 peri-ods. (Table 6-5)

INTEREST TABLES AND CONTENTS

FUTURE VALUE OF 1 AT COMPOUND INTEREST (EXCERPT FROMTABLE6-1,PAGE303)

Period 9% 10% 11% 1 1.09000 1.10000 1.11000 2 1.18810 1.21000 1.23210 3 1.29503 1.33100 1.36763 4 1.41158 1.46410 1.51807 5 1.53862 1.61051 1.68506 ILLUSTRATION 6-2

Excerpt from Table 6-1

The compound tables are computed using basic formulas. For example, the for-mula to determine the future value factor (FVF) for 1 is:

FVFn,i(1i)n

where

FVFn,ifuture value factor for nperiods at iinterest

nnumber of periods

irate of interest for a single period

Compound Interest Tables (see pages 302—311)

Five different types of compound interest tables are presented at the end of this chap-ter. These tables should help you study this chapter as well as solve other problems involving interest. The titles of these five tables and their contents are:

Throughout the discussion of compound interest tables the use of the term periods

instead of yearsis intentional. Interest is generally expressed in terms of an annual rate, but in many business circumstances the compounding period is less than one year. In such circumstances the annual interest rate must be converted to correspond to the length of the period. The process is to convert the “annual interest rate” into the “compound-ing period interest rate” by dividing the annual rate by the number of compounding periods per year.

In addition, the number of periods is determined by multiplying the number of years involved by the number of compounding periods per year. To illustrate, assume that $1.00 is invested for 6 years at 8% annual interest compounded quarterly. Using Table 6-1 from page 302, we can determine the amount to which this $1.00 will ac-cumulate: Read the factor that appears in the 2% column on the 24th row—6 years 4 compounding periods per year, namely 1.60844, or approximately $1.61. Thus, the term periods, not years, is used in all compound interest tables to express the quan-tity of n.

The following schedule shows how to determine (1) the interest rate per com-pounding period and (2) the number of comcom-pounding periods in four situations of dif-fering compounding frequency.4

ILLUSTRATION 6-3

Accumulation of Compound Amounts

Beginning-of- Multiplier End-of-Period Formula Period Period Amount (1 i) Amount* (1 i)n

1 1.00000 1.09 1.09000 (1.09)1

2 1.09000 1.09 1.18810 (1.09)2

3 1.18810 1.09 1.29503 (1.09)3

*Note that these amounts appear in Table 6-1 in the 9% column.

4_{Because interest is theoretically earned (accruing) every second of every day, it is possible} to calculate interest that is compounded continuously. Computations involving continuous com-pounding are facilitated through the use of the natural, or Napierian, system of logarithms. As a practical matter, however, most business transactions assume interest to be compounded no more frequently than daily.

12% Annual Interest Rate Interest Rate per Number of over 5 Years Compounded Compounding Period Compounding Periods Annually (1) .121 .12 5 years 1 compounding

per year 5 periods Semiannually (2) .12 2 .06 5 years 2 compoundings

per year 10 periods Quarterly (4) .12 4 .03 5 years 4 compoundings

per year 20 periods Monthly (12) .12 12 .01 5 years 12 compoundings

per year 60 periods

ILLUSTRATION 6-4

Frequency of Compounding

The FVFn,iand other time value of money formulas are programmed into financial

calculators. The use of a financial calculator to solve time value of money problems is illustrated in Appendix 6A.

To illustrate the use of interest tables to calculate compound amounts, assuming an interest rate of 9%, the future value to which 1 accumulates (the future value fac-tor) is shown below.

Fundamental Variables

The following four variables are fundamental to all compound interest problems.

5_{The formula for calculating the effective rate in situations where the compounding} fre-quency (n) is greater than once a year is as follows.

Effective rate (1 i)n ₁

To illustrate, if the stated annual rate is 8% compounded quarterly (or 2% per quarter), the effective annual rate is:

Effective rate (1 .02)4 ₁ (1.02)4 ₁ 1.0824 1 .0824 8.24%

How often interest is compounded can make a substantial difference in the rate of return. For example, a 9% annual interest compounded dailyprovides a 9.42% yield, or a difference of .42%. The 9.42% is referred to as the effective yield.5The annual in-terest rate (9%) is called the stated, nominal, or face rate. When the compounding fre-quency is greater than once a year, the effective interest rate will always be greater than the stated rate.

The schedule below shows how compounding for five different time periods affects the effective yield and the amount earned by an investment of $10,000 for one year.

ILLUSTRATION 6-5

Comparison of Different Compounding Periods

Compounding Periods Interest

Rate Annually Semiannually Quarterly Monthly Daily

8% 8.00% 8.16% 8.24% 8.30% 8.33% $800 $816 $824 $830 $833 9% 9.00% 9.20% 9.31% 9.38% 9.42% $900 $920 $931 $938 $942 10% 10.00% 10.25% 10.38% 10.47% 10.52% $1,000 $1,025 $1,038 $1,047 $1,052

The relationship of these four fundamental variables is depicted in the time diagram

on the following page.

Rate of Interest. This rate, unless otherwise stated, is an annual rate that must be adjusted to reflect the length of the compounding period if less than a year.

Number of Time Periods.This is the number of compounding periods. (A pe-riod may be equal to or less than a year.)

Future Value.The value at a future date of a given sum or sums invested as-suming compound interest.

Present Value. The value now (present time) of a future sum or sums dis-counted assuming compound interest.

FUNDAMENTAL VARIABLES

O B J E C T I V E

In some cases all four of these variables are known, but in many business situa-tions at least one variable is unknown. As an aid to better understanding the problems and to finding solutions, we encourage you to sketch compound interest problems in the form of the preceding time diagram.

SINGLE-SUM PROBLEMS

Many business and investment decisions involve a single amount of money that either exists now or will in the future. Single-sum problems can generally be classified into one of the following two categories.

Computing the unknown future valueof a known single sum of money that is in-vested now for a certain number of periods at a certain interest rate.

Computing the unknown present valueof a known single sum of money in the future that is discounted for a certain number of periods at a certain interest rate. When analyzing the information provided, you determine first whether it is a future value problem or a present value problem. If you are solving for a future value, all cash flows must be accumulatedto a future point. In this instance, the effect of interest is to increase the amounts or values over time so that the future value is greater than the present value. However, if you are solving for a present value, all cash flows must be discounted from the future to the present. In this case, the dis-counting reduces the amounts or values so that the present value is less than the future amount.

Preparation of time diagrams aids in identifying the unknown as an item in the fu-ture or the present. Sometimes it is neither a fufu-ture value nor a present value that is to be determined but, rather, the interest or discount rate or the number of compounding or discounting periods.

Future Value of a Single Sum

To determine the future valueof a single sum, multiply the future value factor by its present value (principal), as follows.

FVPV(FVFn,i)

where

FV future value

PVpresent value (principal or single sum)

FVFn,ifuture value factor for nperiods at iinterest

Present Value 2 Future Value 4 Number of Periods Interest 5 1 0 3 ILLUSTRATION 6-6

Basic Time Diagram

O B J E C T I V E

Using the formula, this investment problem is solved as follows. Future value PV(FVFn,i)

$50,000 (FVF5,11%)

$50,000 (1 .11)5 $50,000 (1.68506)

$84,253

To determine the future value factor of 1.68506 in the formula above, use a financial calculator or read the appropriate table, in this case Table 6-1 (11% column and the 5-period row).

This time diagram and formula approach can be applied to a routine business situation. To illustrate, Commonwealth Edison Company deposited $250 million in an escrow account with the Northern Trust Company at the beginning of 2002 as a commitment toward a power plant to be completed December 31, 2005. How much will be on deposit at the end of 4 years if interest is 10%, compounded semi-annually?

With a known present value of $250 million, a total of 8 compounding periods (42), and an interest rate of 5% per compounding period (.10 2), this problem can be time-diagrammed and the future value determined as follows.

1 Present Value PV = $50,000 5 4 0 2 3 Number of Periods n = 5 Interest Rate i = 11% Future Value FV FV = ?= ? FV= ? 5 4 6 PV = $250,000,000 n = 8 0 1 2 3 7 8 i = 5% FVFV FV= ?= ?= ? Future value $250,000,000 (FVF8,5%) $250,000,000 (1 .05)8 $250,000,000 (1.47746) $369,365,000

To illustrate, assume Brueggen Co. wants to determine the future value of $50,000 invested for 5 years compounded annually at an interest rate of 11%. In time-diagram form, this investment situation would appear as follows.

Using a future value factor found in Table 6-1 (5% column, 8-period row), we find that the deposit of $250 million will accumulate to $369,365,000 by December 31, 2005.

Present Value of a Single Sum

The Brueggen Co. example on page 260 showed that $50,000 invested at an annually compounded interest rate of 11% will be worth $84,253 at the end of 5 years. It follows, then, that $84,253, 5 years in the future is worth $50,000 now. That is, $50,000 is the

present valueof $84,253. The present value is the amount that must be invested now to produce the known future value. The present value is always a smaller amount than the known future value because interest will be earned and accumulated on the present value to the future date. In determining the future value, we move for-ward in time using a process of accumulation. In determining present value, we move backward in time using a process of discounting.

As indicated earlier, a “present value of 1 table” appears at the end of this chapter as Table 6-2. Illustration 6-7 demonstrates the nature of such a table. It shows the pres-ent value of 1 for five differpres-ent periods at three differpres-ent rates of interest.

ILLUSTRATION 6-7

Excerpt from Table 6-2 PRESENT VALUE OF 1 AT COMPOUND INTEREST

(EXCERPT FROMTABLE6-2,PAGE305)

Period 9% 10% 11% 1 0.91743 0.90909 0.90090 2 0.84168 0.82645 0.81162 3 0.77218 0.75132 0.73119 4 0.70843 0.68301 0.65873 5 0.64993 0.62092 0.59345

The present value of 1 (present value factor) may be expressed as a formula:

PVFn,i

where

PVFn,ipresent value factor for nperiods at iinterest

To illustrate, assuming an interest rate of 9%, the present value of 1 discounted for three different periods is as follows.

1

(1 i)n

The present value of any single sum (future value), then, is as follows.

PVFV(PVFn,i)

where

PVpresent value

FVfuture value

PVFn,ipresent value factor fornperiods at iinterest

ILLUSTRATION 6-8 Present Value of $1 Discounted at 9% for Three Periods Discount Formula Periods 1 (1 i)n Present Value* 1/(1i)n 1 1.00000 1.09 .91743 1/(1.09)1 2 1.00000 (1.09)2 .84168 1/(1.09)2 3 1.00000 (1.09)3 .77218 1/(1.09)3

Present value $2,000 (PVF3,8%) $2,000

Using the formula, this problem is solved as follows. Present value FV(PVFn,i)

$84,253 (PVF5,11%)

$84,253

$84,253 (.59345)

$50,000

To determine the present value factor of .59345, use a financial calculator or read the present value of a single sum in Table 6-2 (11% column, 5-period row).

The time diagram and formula approach can be applied in a variety of situations. For example, assume that your rich uncle proposes to give you $2,000 for a trip to Europe when you graduate from college 3 years from now. He proposes to finance the trip by investing a sum of money now at 8% compound interest that will provide you with $2,000 upon your graduation. The only conditions are that you graduate and that you tell him how much to invest now.

To impress your uncle, you might set up the following time diagram and solve this problem as follows. 1 (1 .11)5 Future Value $84,253 0 1 2 3 4 5 Interest Rate i = 11% Number of Periods n = 5 PV PV = ?= ? Present Value PV = ? FV = $2,000 i = 8% 0 1 2 3 n = 3 PV PV = ?= ? PV = ?

To illustrate, what is the present value of $84,253 to be received or paid in 5 years discounted at 11% compounded annually? In time-diagram form, this problem is drawn as follows.

Advise your uncle to invest $1,587.66 now to provide you with $2,000 upon gradua-tion. To satisfy your uncle’s other condition, you must pass this course, and many more.

Solving for Other Unknowns in Single-Sum Problems

In computing either the future value or the present value in the previous single-sum illustrations, both the number of periods and the interest rate were known. In many business situations, both the future value and the present value are known, but the number of periods or the interest rate is unknown. The following two illustrations are single-sum problems (future value and present value) with either an unknown num-ber of periods (n) or an unknown interest rate (i). These illustrations and the accom-panying solutions demonstrate that if any three of the four values (future value, FV; present value, PV; number of periods,n; interest rate, i) are known, the remaining un-known variable can be derived.

Illustration—Computation of the Number of Periods

The Village of Somonauk wants to accumulate $70,000 for the construction of a veter-ans monument in the town square. If at the beginning of the current year the Village deposited $47,811 in a memorial fund that earns 10% interest compounded annually, how many years will it take to accumulate $70,000 in the memorial fund?

In this illustration, both the present value ($47,811) and the future value ($70,000) are known along with the interest rate of 10%. A time diagram of this investment prob-lem is as follows. i = 10% PV = $47,811 FV = $70,000 n n = ? = ? n = ?

Because both the present value and the future value are known, we can solve for the unknown number of periods using either the future value or the present value for-mulas as shown below.

ILLUSTRATION 6-9

Solving for Unknown Number of Periods

Future Value Approach Present Value Approach FVPV(FVFn,10%) PVFV(PVFn,10%) $70,000$47,811 (FVFn,10%) $47,811$70,000 (PVFn,10%) FVFn,10% 1.46410 PVFn,10% .68301 $47,811 _$70,000 $70,000 _$47,811

Using the future value factor of 1.46410, refer to Table 6-1 and read down the 10% column to find that factor in the 4-period row. Thus, it will take 4 years for the $47,811 to accumulate to $70,000 if invested at 10% interest compounded annually. Using the present value factor of .68301, refer to Table 6-2 and read down the 10% column to find again that factor in the 4-period row.

PV = $800,000 5 FV = $1,409,870 4 3 2 1 0 n = 5 ii = ? = ? i = ?

The unknown interest rate may be determined from either the future value ap-proach or the present value apap-proach as shown in Illustration 6-10.

ILLUSTRATION 6-10

Solving for Unknown Interest Rate

Future Value Approach Present Value Approach

FVPV(FVF5,i) PVFV(PVF5,i) $1,409,870$800,000 (FVF5,i) $800,000$1,409,870 (PVF5,i) FVF5,i 1.76234 PVF5,i .56743 $800,000 _$1,409,870 $1,409,870 _$800,000

Using the future value factor of 1.76234, refer to Table 6-1 and read across the 5-period row to find that factor in the 12% column. Thus, the $800,000 must be invested at 12% to accumulate to $1,409,870 in 5 years. And, using the present value factor of .56743 and Table 6-2, again find that factor at the juncture of the 5-period row and the 12% column.

ANNUITIES

The preceding discussion involved only the accumulation or discounting of a single principal sum. Individuals frequently encounter situations in which a series of dollar amounts are to be paid or received periodically, such as loans or sales to be repaid in installments, invested funds that will be partially recovered at regular intervals, or cost savings that are realized repeatedly. A life insurance contract is probably the most com-mon and most familiar type of transaction involving a series of equal payments made at equal intervals of time. Such a process of periodic saving represents the accumula-tion of a sum of money through an annuity. An annuity, by definition, requires that (1) the periodic payments or receipts (called rents) always be the same amount, (2) the

intervalbetween such rents always be the same, and (3) the interest be compounded

once each interval. The future value of an annuityis the sum of all the rents plus the accumulated compound interest on them.

Illustration—Computation of the Interest Rate

Advanced Design, Inc. wishes to have $1,409,870 for basic research 5 years from now. The firm currently has $800,000 to invest for that purpose. At what rate of interest must the $800,000 be invested to fund basic research projects of $1,409,870, 5 years from now?

O B J E C T I V E

Because the rents that comprise an ordinary annuity are deposited at the end of the period, they can earn no interest during the period in which they are originally de-posited. For example, the third rent earns interest for only two periods (periods four and five). Obviously the third rent earns no interest for the first two periods since it is not deposited until the third period. Furthermore, it can earn no interest for the third period since it is not deposited until the end of the third period. Any time the future value of an ordinary annuity is computed, the number of compounding periods will always be one less than the number of rents.

Although the foregoing procedure for computing the future value of an ordinary annuity will always produce the correct answer, it can become cumbersome if the num-ber of rents is large. A more efficient way of expressing the future value of an ordinary annuity of 1 is in a formula that is a summation of the individual rents plus the com-pound interest:

FVF-OAn,i

where

FVF-OAn,ifuture value factor of an ordinary annuity

i rate of interest per period

n number of compounding periods

For example, FVF-OA5,12%refers to the value to which an ordinary annuity of 1 will accumulate in 5 periods at 12% interest.

Using the formula above, tables have been developed similar to those used for the “future value of 1” and the “present value of 1” for both an ordinary annuity and an annuity due. The table in Illustration 6-12 is an excerpt from the “future value of an ordinary annuity of 1” table.

(1i)n 1

i

ILLUSTRATION 6-11

Solving for the Future Value of an Ordinary Annuity

END OF PERIOD IN WHICH $1.00 IS TO BE INVESTED

Value at End Present 1 2 3 4 5 of Year 5 $1.00 $1.57352 $1.00 1.40493 $1.00 1.25440 $1.00 1.12000 $1.00 1.00000 Total (future value of an ordinary annuity of $1.00 for 5 periods at 12%) $6.35285

It should be noted that the rents may occur at either the beginning or the end of the periods. To distinguish annuities under these two alternatives, an annuity is clas-sified as an ordinary annuity if the rents occur at the end of each period, and as an

annuity dueif the rents occur at the beginning of each period.

Future Value of an Ordinary Annuity

One approach to the problem of determining the future value to which an annuity will accumulate is to compute the value to which eachof the rents in the series will accu-mulate and then total their individual future values. For example, assume that $1 is deposited at the endof each of 5 years (an ordinary annuity) and earns 12% interest compounded annually. The future value can be computed as follows using the “future value of 1” table (Table 6-1) for each of the five $1 rents.

Using the formula, this investment problem is solved as follows. Future value of an ordinary annuity R(FVF-OAn,i)

$5,000 (FVF-OA5,12%)

$5,000

$5,000 (6.35285)

$31,764.25

We can determine the future value of an ordinary annuity factor of 6.35285 in the for-mula above using a financial calculator or by reading the appropriate table, in this case Table 6-3 (12% column and the 5-period row).

To illustrate these computations in a business situation, assume that Hightown Elec-tronics decides to deposit $75,000 at the end of each 6-month period for the next 3 years for the purpose of accumulating enough money to meet debts that mature in 3 years. What is the future value that will be on deposit at the end of 3 years if the annual in-terest rate is 10%? (1.12)51 .12 0 1 2 3 4 5 n = 5 R = $5,000 $5,000 FV FV––0A0A = ? = ? Present Value i = 12% $5,000 $5,000 Future Value FV–OA = ? $5,000

Interpreting the table, if $1 is invested at the end of each year for 4 years at 11% in-terest compounded annually, the value of the annuity at the end of the fourth year will be $4.71 (4.70973 $1). Multiply the factor from the appropriate line and column of the table by the dollar amount of one rentinvolved in an ordinary annuity. The result: the accumulated sum of the rents and the compound interest to the date of the last rent.

The future value of an ordinary annuity is computed as follows. Future value of an ordinary annuity R(FVF-OAn,i)

where

Rperiodic rent

FVF-OAn,ifuture value of an ordinary annuity factor

for nperiods at iinterest

To illustrate, what is the future value of five $5,000 deposits made at the end of each of the next 5 years, earning interest of 12%? In time-diagram form, this problem is drawn as follows.

ILLUSTRATION 6-12

Excerpt from Table 6-3 FUTURE VALUE OF AN ORDINARY ANNUITY OF 1(EXCERPT FROMTABLE6-3,PAGE307)

Period 10% 11% 12% 1 1.00000 1.00000 1.00000 2 2.10000 2.11000 2.12000 3 3.31000 3.34210 3.37440 4 4.64100 4.70973 4.77933 5 6.10510 6.22780 6.35285*

Future value of an ordinary annuity R(FVF-OAn,i) $75,000 (FVF-OA6,5%)

$75,000

$75,000 (6.80191)

$510,143.25

Thus, six 6-month deposits of $75,000 earning 5% per period will grow to $510,143.25.

Future Value of an Annuity Due

The preceding analysis of an ordinary annuity was based on the assumption that the periodic rents occur at the endof each period. An annuity dueassumes periodic rents occur at the beginningof each period. This means an annuity due will accumulate in-terest during the first period, whereas an ordinary annuity rent will earn no inin-terest during the first period because the rent is not received or paid until the end of the pe-riod. In other words, the significant difference between the two types of annuities is in the number of interest accumulation periods involved.

If rents occur at the end of a period (ordinary annuity), in determining the future value of an annuitythere will be one less interest period than if the rents occur at the beginning of the period (annuity due). The distinction is shown in Illustration 6-13.

(1.05)61 _.05 i = 5% $75,000 0 1 2 3 4 5 6 $75,000 $75,000 $75,000 n = 6 R = $75,000 FV FV––0A0A = ? = ? Future Value FV–OA = ? $75,000 ILLUSTRATION 6-13

Comparison of the Future Value of an Ordinary Annuity with an Annuity Due

Future Value of an Annuity of 1 at 12%

Future value of an ordinary annuity (per Table 6-3)

Period 1 Period 2 Period 3 Period 4 Period 5

Period 1 Period 4 Period 5

No interest Interest Interest Interest Interest

Interest Interest Interest Interest Interest Period 2 Period 3

First deposit here

(No table provided) Ordinary annuity Ordinary annuity Annuity due Annuity due Ordinary annuity Annuity due 1.00000 2.12000 3.37440 4.77933 6.35285 7.11519 6.35285 4.77933 3.37440 2.12000 1.00000

First deposit here

Referring to the “future value of an ordinary annuity of 1” table for 8 periods at 12%, we find a factor of 12.29969. This factor is then multiplied by (1 .12) to arrive at the future value of an annuity due factor. As a result, the accumulated value on Howard’s eighteenth birthday is $11,020.52 as shown in Illustration 6-14.

In this example, because the cash flows from the annuity due come exactly one period earlier than for an ordinary annuity, the future value of the annuity due factor is exactly 12% higher than the ordinary annuity factor. For example, the value of an ordinary annuity factor at the end of period one at 12% is 1.00000, whereas for an annuity due it is 1.12000. Thus, the future value of an annuity due factor can be found by multiplying the future value of an ordinary annuity factor by 1 plus the interest rate.For example, to determine the future value of an annuity due interest factor for 5 periods at 12% compound interest, simply multiply the future value of an ordinary annuity interest factor for 5 periods (6.35285) by one plus the interest rate (1 .12), to arrive at 7.11519 (6.35285 1.12).

To illustrate the use of the ordinary annuity tables in converting to an annuity due, assume that Sue Lotadough plans to deposit $800 a year on each birthday of her son Howard, starting today, his tenth birthday, at 12% interest compounded annually. Sue wants to know the amount she will have accumulated for college expenses by her son’s eighteenth birthday. If the first deposit is made on Howard’s tenth birthday, Sue will make a total of 8 deposits over the life of the annuity (assume no deposit on the eigh-teenth birthday). Because all the deposits will be made at the beginning of the periods, they represent an annuity due.

ILLUSTRATION 6-14

Computation of Accumulated Value of Annuity Due

1. Future value of an ordinary annuity of 1 for 8 periods

at 12% (Table 6-3) 12.29969

2. Factor (1 .12) 1.12

3. Future value of an annuity due of 1 for 8 periods at 12% 13.77565

4. Periodic deposit (rent) $800

5. Accumulated value on son’s eighteenth birthday $11,020.52

Depending on the college he chooses, Howard may have only enough to finance his first year of school.

Illustrations of Future Value of Annuity Problems

In the foregoing annuity examples three values were known—amount of each rent, in-terest rate, and number of periods. They were used to determine the fourth value, fu-ture value, which was unknown. The first two fufu-ture value problems presented illustrate the computations of (1) the amount of the rents and (2) the number of rents. The third problem illustrates the computation of the future value of an annuity due.

4 i = 12%

n = 8

R = $800 $800 $800 $800 $800 $800 $800 $800

FV–AD = Future value of an annuity due

0 1 2 3 5 6 7 8 FV FV––ADAD = ? = ? Future Value FV–AD = ?

Using the formula for the future value of an ordinary annuity, the amount of each rent is determined as follows.

Future value of an ordinary annuity R(FVF-OAn,i)

$14,000 R(FVF-OA10,4%) $14,000 R(12.00611)

R

R $1,166.07

Thus, you must make 10 semiannual deposits of $1,166.07 each in order to accumulate $14,000 for your down payment.

Computation of the Number of Periodic Rents

Suppose that your company wishes to accumulate $117,332 by making periodic de-posits of $20,000 at the end of each year that will earn 8% compounded annually while accumulating. How many deposits must be made?

The $117,332 represents the future value of n(?) $20,000 deposits, at an 8% annual rate of interest. This problem appears in the form of a time diagram as follows.

$14,000 12.00611 Future Value FV–OA = $14,000 i = 4% n = 10

FV–OA = Future value of an ordinary annuity

0 1 2 3 4 5 6 7 8 9 10 ?? ?? ?? ?? ?? ?? ?? ?? ?? R R = = ? ? ? ? ? ? ? ? ? ??? R = 0 1 2 R = $20,000 Future Value FV–OA = $117,332 $20,000 $20,000 n i = 8% 3 n n = ? = ? n = ? Computation of Rent

Assume that you wish to accumulate $14,000 for a down payment on a condominium apartment 5 years from now; for the next 5 years you can earn an annual return of 8% compounded semiannually. How much should you deposit at the end of each 6-month period?

The $14,000 is the future value of 10 (5 2) semiannual end-of-period payments of an unknown amount, at an interest rate of 4% (8% 2). This problem appears in the form of a time diagram as follows.

Using the future value of an ordinary annuity formula, we obtain the following factor.

Future value of an ordinary annuity R(FVF-OAn,i)

$117,332 $20,000 (FVF-OAn,8%) FVF-OAn,8% 5.86660

Using Table 6-3 and reading down the 8% column, we find 5.86660 in the 5-period row. Thus, five deposits of $20,000 each must be made.

Computation of the Future Value

Walter Goodwrench, a mechanic, has taken on weekend work in the hope of creating his own retirement fund. Mr. Goodwrench deposits $2,500 today in a savings account that earns 9% interest. He plans to deposit $2,500 every year for a total of 30 years. How much cash will have accumulated in Mr. Goodwrench’s retirement savings ac-count when he retires in 30 years? This problem appears in the form of a time diagram as follows. $117,332 $20,000 $2,500 i = 9% 0 1 2 29 30 R = $2,500 $2,500 $2,500 n = 30 FV FV––ADAD = ? = ? Future Value FV–AD = ?

1. Future value of an ordinary annuity of 1 for 30 periods at 9% 136.30754

2. Factor (1 .09) 1.09

3. Future value of an annuity due of 1 for 30 periods at 9% 148.57522

4. Periodic rent $2,500

5. Accumulated value at end of 30 years $371,438 ILLUSTRATION 6-15

Computation of

Accumulated Value of an Annuity Due

Using the “future value of an ordinary annuity of 1” table, the solution is com-puted as follows.

Present Value of an Ordinary Annuity

The present value of an annuity is the single sumthat, if invested at compound in-terest now, would provide for an annuity (a series of withdrawals) for a certain num-ber of future periods. In other words, the present value of an ordinary annuity is the present value of a series of equal rents to be withdrawn at equal intervals.

One approach to finding the present value of an annuity is to determine the pres-ent value of each of the rpres-ents in the series and then total their individual prespres-ent val-ues. For example, an annuity of $1 to be received at the endof each of 5 periods may be viewed as separate amounts; the present value of each is computed from the table of present values (see pages 304–305), assuming an interest rate of 12%.

O B J E C T I V E

This computation tells us that if we invest the single sum of $3.60 today at 12% interest for 5 periods, we will be able to withdraw $1 at the end of each period for 5 periods. This cumbersome procedure can be summarized by:

PVF-OAn,i

The expression PVF-OAn,i refers to the present value of an ordinary annuity of 1

fac-tor for nperiods at iinterest. Using this formula, present value of ordinary annuity ta-bles are prepared. An excerpt from such a table is shown below.

1 (1 1 i)n i ILLUSTRATION 6-16

Solving for the Present Value of an Ordinary Annuity

END OF PERIOD IN WHICH $1.00 IS TO BE RECEIVED Present Value at Beg. of Year 1 1 2 3 4 5 $0.89286 $1.00 .79719 $1.00 .71178 $1.00 .63552 $1.00 .56743 $1.00

$3.60478 Total (present value of an ordinary annuity of $1.00 for five periods at 12%)

ILLUSTRATION 6-17

Excerpt from Table 6-4 PRESENT VALUE OF AN ORDINARY ANNUITY OF 1

(EXCERPT FROMTABLE6-4,PAGE309)

Period 10% 11% 12% 1 0.90909 0.90090 0.89286 2 1.73554 1.71252 1.69005 3 2.48685 2.44371 2.40183 4 3.16986 3.10245 3.03735 5 3.79079 3.69590 3.60478*

*Note that this annuity table factor is equal to the sum of the present value of 1 factors shown in Illustration 6-16.

The general formula for the present value of any ordinary annuity is as follows. Present value of an ordinary annuity R(PVF-OAn,i)

where

Rperiodic rent (ordinary annuity)

PVF-OAn,ipresent value of an ordinary annuity of 1

for nperiods at iinterest

To illustrate, what is the present value of rental receipts of $6,000 each to be re-ceived at the end of each of the next 5 years when discounted at 12%? This problem may be time-diagrammed and solved as follows.

0 1 2 3 4 5 n = 5 R = $6,000 $6,000 $6,000 $6,000 i = 12% $6,000 PV PV––OAOA = ? = ? Present Value PV–OA = ?

Present value of an ordinary annuity R(PVF-OAn,i)

$6,000 (PVF-OA5,12%) $6,000 (3.60478)

$21,628.68

The present value of the 5 ordinary annuity rental receipts of $6,000 each is $21,628.68. Determining the present value of the ordinary annuity factor 3.60478 can be accom-plished using a financial calculator or by reading the appropriate table, in this case Table 6-4 (12% column and 5-period row).

Time value of money concepts also can be relevant to public policy debates. For exam-ple, several states are evaluating how to receive the payments from tobacco companies as settlement for a national lawsuit against the companies for the health-care costs of smoking.

The State of Wisconsin is due to collect 25 years of payments totaling $5.6 billion. The state could wait to collect the payments, or it can sell the payments to an invest-ment bank (a process called securitization) and receive a lump-sum payment today of $1.26 billion. Assuming a discount rate of 8% and that the payments will be received in equal amounts, the present value of the tobacco payment annuity is:

$5.6 billion 25 $224 million 10.67478* $2.39 billion

*PV-OA(i8%, n25)

Why are some in the state willing to take just $1.26 billion today for an annuity, the present value of which is almost twice that value? One reason is that Wisconsin faces a hole in its budget today that can be plugged in part by the lump-sum payment. Also, some believe that the risk of getting paid by the tobacco companies in the future makes it prudent to get paid today.

If this latter reason has merit, then the present value computation above should have been based on a higher interest rate. Assuming a discount rate of 15%, the present value of the annuity is $1.448 billion ($5.6 billion 25 $224 million; $224 million 6.46415), which is much closer to the lump-sum payment offered to the State of Wisconsin.

What do the

numbers mean?

Up in smoke

Present Value of an Annuity Due

In the discussion of the present value of an ordinary annuity, the final rent was dis-counted back the same number of periods that there were rents. In determining the present value of an annuity due, there is always one fewer discount period. This dis-tinction is shown graphically in Illustration 6-18.

Because each cash flow comes exactly one period sooner in the present value of the annuity due, the present value of the cash flows is exactly 12% higher than the pres-ent value of an ordinary annuity. Thus, the present value of an annuity due factor can be found by multiplying the present value of an ordinary annuity factor by 1 plus the interest rate.

To determine the present value of an annuity due interest factor for 5 periods at 12% interest, take the present value of an ordinary annuity for 5 periods at 12% inter-est (3.60478) and multiply it by 1.12 to arrive at the present value of an annuity due, 4.03735 (3.60478 1.12). Because the payment and receipt of rentals at the beginning of periods (such as leases, insurance, and subscriptions) are as common as those at the end of the periods (referred to as “in arrears”), we have provided present value annu-ity due factors in the form of Table 6-5.

ILLUSTRATION 6-18

Comparison of Present Value of an Ordinary Annuity with an Annuity Due

ILLUSTRATION 6-19

Computation of Present Value of an Annuity Due

1. Present value of an ordinary annuity of 1 for 4 periods at 11% (Table 6-4) 3.10245

2. Factor (1 .11) 1.11

3. Present value of an annuity due of 1 for 4 periods at 11% 3.44372

4. Periodic deposit (rent) $4,800,000

5. Present value of payments $16,529,856

Present Value of an Annuity of 1 at 12%

Present value of an ordinary annuity (per Table 6-4) Present value of annuity due (per Table 6-5)

Period 1 Period 2 Period 3 Period 4 Period 5

Period 1 Period 4 Period 5

Discount

Discount

.89286 1.69005 2.40183 3.03735 3.60478

Period 2 Period 3

Discount Discount Discount Discount

Discount Discount Discount No discount

Rent at beginning of period

Rent at end of period

Annuity due Annuity due Annuity due Ordinary annuity Ordinary annuity Ordinary annuity 1.00000 1.89286 2.69005 3.40183 4.03735

Space Odyssey, Inc., rents a communications satellite for 4 years with annual rental payments of $4.8 million to be made at the beginning of each year. If the relevant an-nual interest rate is 11%, what is the present value of the rental obligations?

This problem is time-diagrammed as follows.

$4.8M

i = 11%

n = 4

PV–AD = the present value of an annuity due

0 1 2 3 4 $4.8M $4.8M PV PV––ADAD = ? = ? Present Value PV–AD = ? R = $4.8M

This problem is solved in the following manner.

Since we have Table 6-5 for present value of an annuity due problems, we can also lo-cate the desired factor 3.44372 and compute the present value of the lease payments to be $16,529,856.

Illustrations of Present Value of Annuity Problems

The following three illustrations demonstrate the computation of (1) the present value, (2) the interest rate, and (3) the amount of each rent.

Computation of the Present Value of an Ordinary Annuity

You have just won a lottery totaling $4,000,000 and learned that you will be paid the money by receiving a check in the amount of $200,000 at the end of each of the next 20 years. What amount have you really won? That is, what is the present value of the $200,000 checks you will receive over the next 20 years? A time diagram of this envi-able situation is as follows (assuming an appropriate interest rate of 10%).

$200,000 0 1 2 19 20 R = $200,000 $200,000 $200,000 i = 10% n = 20 PV PV––OAOA = ? = ? PV–OA = ?

The present value is determined as follows:

Present value of an ordinary annuity R(PVF-OAn,i)

$200,000 (PVF-OA20,10%) $200,000 (8.51356)

$1,702,712

As a result, if the state deposits $1,702,712 now and earns 10% interest, it can withdraw $200,000 a year for 20 years to pay you the $4,000,000.

Computation of the Interest Rate

Many shoppers make purchases by using a credit card. When you receive the invoice for payment you may pay the total amount due or you may pay the balance in a cer-tain number of payments. For example, if you receive an invoice from Visa with a bal-ance due of $528.77 and you are invited to pay it off in 12 equal monthly payments of $50 each, with the first payment due one month from now, what rate of interest would you be paying?

The $528.77 represents the present value of the 12 payments of $50 each at an un-known rate of interest. This situation in the form of a time diagram appears as follows.

n = 12 Present Value PV–OA = $528.77 R = $50 1 0 2 3 4 5 6 7 8 9 10 11 12 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 $50 ii = ? = ? i = ?

The rate is determined as follows.

Present value of an ordinary annuity R(PVF-OAn,i)

$528.77 $50 (PVF-OA12,i)

(PVF-OA12,i) 10.57540

$528.77

Referring to Table 6-4 and reading across the 12-period row, we find 10.57534 in the 2% column. Since 2% is a monthly rate, the nominal annual rate of interest is 24% (12 2%), and the effective annual rate is 26.82413% [(1 .02)121]. Obviously, you’re better off paying the entire bill now if you possibly can.

Computation of a Periodic Rent

Norm and Jackie Remmers have saved $18,000 to finance their daughter Dawna’s col-lege education. The money has been deposited in the Bloomington Savings and Loan Association and is earning 10% interest compounded semiannually. What equal amounts can their daughter withdraw at the end of every 6 months during the next 4 years while she attends college, without exhausting the fund? This situation is time-diagrammed as follows.

The answer is not determined simply by dividing $18,000 by 8 withdrawals because that would ignore the interest earned on the money remaining on deposit. Tak-ing into consideration that interest is compounded semiannually at 5% (10% 2) for 8 periods (4 years 2), and using the same present value of an ordinary annuity for-mula, we determine the amount of each withdrawal that she can make as follows.

Present value of an ordinary annuity R(PVF-OAn,i)

$18,000 R(PVF-OA8,5%)

$18,000 R(6.46321)

R$2,784.99

MORE COMPLEX SITUATIONS

Often it is necessary to use more than one table to solve time value problems. The busi-ness problem encountered may require that computations of both present value of a single sum and present value of an annuity be made. Two common situations are:

Deferred annuities.

Bond problems.

Deferred Annuities

Adeferred annuityis an annuity in which the rents begin after a specified number of periods. A deferred annuity does not begin to produce rents until 2 or more periods have expired. For example, “an ordinary annuityof six annual rents deferred 4 years” means that no rents will occur during the first 4 years, and that the first of the six rents will occur at the end of the fifth year. “An annuity dueof six annual rents deferred 4 years” means that no rents will occur during the first 4 years, and that the first of six rents will occur at the beginning of the fifth year.

R R = ? = ? ?? ?? ?? ?? ?? ?? ?? n = 8 i = 5% 0 1 2 3 4 5 6 7 8 Present Value PV–OA = $18,000R = ? ? ? ? ? ? ? ? O B J E C T I V E Solve present value problems related to deferred annuities and bonds.

The value accumulated is determined by using the standard formula for the future value of an ordinary annuity:

Future value of an ordinary annuity R(FVF-OAn,i)

$80,000 (FVF-OA3,12%)

$80,000 (3.37440)

$269,952

Present Value of a Deferred Annuity

In computing the present value of a deferred annuity, the interest that accrues on the original investment during the deferral period must be recognized.

To compute the present value of a deferred annuity, we compute the present value of an ordinary annuity of 1 as if the rents had occurred for the entire period, and then subtract the present value of rents which were not received during the deferral period. We are left with the present value of the rents actually received subsequent to the de-ferral period.

To illustrate, Tom Bytehead has developed and copyrighted a software computer program that is a tutorial for students in advanced accounting. He agrees to sell the copyright to Campus Micro Systems for six annual payments of $5,000 each. The pay-ments are to begin 5 years from today. Given an annual interest rate of 8%, what is the present value of the six payments?

This situation is an ordinary annuity of 6 payments deferred 4 periods. The fol-lowing time diagram helps to visualize this sales agreement.

Future Value of a Deferred Annuity

In the case of the future value of a deferred annuity the computations are relatively straightforward. Because there is no accumulation or investment on which interest may accrue, the future value of a deferred annuity is the same as the future value of an an-nuity not deferred. That is, the deferral period is ignored in computing the future value. To illustrate, assume that Sutton Corporation plans to purchase a land site in 6 years for the construction of its new corporate headquarters. Because of cash flow prob-lems, Sutton is able to budget deposits of $80,000 that are expected to earn 12% annu-ally only at the end of the fourth, fifth, and sixth periods. What future value will Sut-ton have accumulated at the end of the sixth year?

A time diagram of this situation is as follows.

i = 12%

R = $80,000 $80,000

n = 3

(first 3 periods are ignored)

0 1 2 3 4 5 6 FV FV––OAOA = ? = ? Future Value FV–OA = ? $80,000 i = 8% n = 6 $5,000 $5,000 $5,000 $5,000 $5,000 0 1 2 3 4 5 6 7 8 9 10 n = 4 R = $5,000 PV PV = ? = ? PV = ?

Two options are available to solve this problem. The first is to use only Table 6-4 as follows.

The subtraction of the present value of an annuity of 1 for the deferred periods elimi-nates the nonexistent rents during the deferral period and converts the present value of an ordinary annuity of $1 for 10 periods to the present value of 6 rents of $1, deferred 4 periods.

Alternatively, the present value of the 6 rents could be computed using both Table 6-2 and Table 6-4. One can first discount the annuity 6 periods, but because the annu-ity is deferred 4 periods, the present value of the annuannu-ity must then be treated as a fu-ture amount to be discounted another 4 periods. A time diagram illustrates this two-step process as follows.

Step 1: Present value of

an ordinary annuityR(PVF-OAn,i) $5,000 (PVF-OA6,8%)

$5,000 (4.62288)

(Table 6-4 Present value of an ordinary annuity)

$23,114.40 Step 2: Present value FV(PVFn,i)

$23,114.40 (PVF4,8%) $23,114.40 (.73503)

(Table 6-2 Present value of a single sum)

$16,989.78

The present value of $16,989.78 computed above is the same result although computed differently from the first illustration.

Valuation of Long-Term Bonds

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 is-sue, bond buyers determine the present value of these two cash flows using the mar-ket rate of interest.

ILLUSTRATION 6-20

Computation of the Present Value of a Deferred Annuity

1. Each periodic rent $5,000

2. Present value of an ordinary annuity of 1 for total periods (10)

[number of rents (6) plus number of deferred periods (4)] at 8% 6.71008 3. Less: Present value of an ordinary annuity of 1 for the

number of deferred periods (4) at 8% 3.31213

4. Difference 3.39795

5. Present value of six rents of $5,000 deferred 4 periods $16,989.75

$5,000 $5,000 $5,000 $5,000 $5,000 FV (PVFn, i) R (PVF–OAn, i) 1 2 3 4 5 6 7 8 9 10 0 $5,000 PV PV = ? = ? PVPV––OA OA = ?= ? PV = ? PV–OA = ?

The periodic interest payments represent an annuity, and the principal represents a single-sum problem. The current market value of the bonds is the combined present values of the interest annuity and the principal amount.

To illustrate, Alltech Corporation on January 1, 2003, issues $100,000 of 9% bonds due in 5 years with interest payable annually at year-end. The current market rate of interest for bonds of similar risk is 11%. What will the buyers pay for this bond issue?

The time diagram depicting both cash flows is shown below.

The present value of the two cash flows is computed by discounting at 11% as follows.

By paying $92,608.10 at date of issue, the buyers of the bonds will realize an effective yield of 11% over the 5-year term of the bonds. This is true because the cash flows were discounted at 11%.

Effective Interest Method of Amortization of

Bond Discount or Premium

In the Alltech Corporation bond issue, in Illustration 6-21, the bonds were issued at a discount computed as follows.

4 n = 5 i = 11% 2 3 5 0 1 $9,000 $9,000 $9,000 $9,000 $9,000 Interest $100,000 Principal PV PV PV PV––OAOA PV PV–OA ILLUSTRATION 6-21 Computation of the Present Value of an Interest-Bearing Bond

1. Present value of the principal:FV(PVF5,11%) $100,000 (.59345) $59,345.00 2. Present value of the interest payments:R(PVF-OA5,11%) $9,000 (3.69590) 33,263.10 3. Combined present value (market price)—carrying value of bonds $92,608.10

ILLUSTRATION 6-22

Computation of Bond Discount

Maturity value (face amount) of bonds $100,000.00 Present value of the principal $59,345.00

Present value of the interest 33,263.10

Proceeds (present value and cash received) 92,608.10

Discount on bonds issued $ 7,391.90

Use of Spreadsheets to Calculate Bond

Amortization

This discount of $7,391.90 under acceptable accrual accounting is amortized (written off) over the life of the bond issue to interest expense.

The profession’s preferred procedure for amortization of a discount or premium is the effective interest method (also called present value amortization). Under the ef-fective interest method:

Bond interest expense is computed first by multiplying the carrying value of the bonds at the beginning of the period by the effective interest rate.

The bond discount or premium amortization is then determined by comparing the bond interest expense with the interest to be paid.

The computation of the amortization is depicted graphically as follows.

Bond Interest Expense Effective Interest Rate Carrying Value of Bonds at Beginning of Period × Amortization Amount Bond Interest Paid

Face Amount of Bonds ×

–

=

The effective interest method produces a periodic interest expense equal to a constant

percentage of the carrying value of the bonds. Since the percentage used is the

effec-tive rate of interest incurred by the borrower at the time of issuance, the effeceffec-tive in-terest method results in matching expenses with revenues.

The effective interest method of amortization can be illustrated using the data from the Alltech Corporation example, where $100,000 face value of bonds were issued at a discount of $7,391.90, resulting in a carrying value of $92,608.10. The effective interest amortization schedule is shown in Illustration 6-24.

SCHEDULE OF BOND DISCOUNT AMORTIZATION 5-YEAR, 9% BONDSSOLD TOYIELD11%

Cash Bond Carrying

Interest Interest Discount Value Date Paid Expense Amortization of Bonds

1/1/04 $92,608.10 12/31/04 $9,000a _$10,186.89b _$1,186.89c _93,794.99d 12/31/05 9,000 10,317.45 1,317.45 95,112.44 12/31/06 9,000 10,462.37 1,462.37 96,574.81 12/31/07 9,000 10,623.23 1,623.23 98,198.04 12/31/08 9,000 10,801.96 1,801.96 100,000.00 $45,000 $52,391.90 $7,391.90 a $100,000 .09 $9,000 c $10,186.89 $9,000 $1,186.89 b $92,608.10 .11 $10,186.89 d $92,608.10 $1,186.89 $93,794.99 ILLUSTRATION 6-24 Effective Interest Amortization Schedule ILLUSTRATION 6-23 Amortization Computation

The amortization schedule illustrated above is used for note and bond transactions in Chapters 7 and 14.

PRESENT VALUE MEASUREMENT

In the past, most accounting calculations of present value were based on the most likely cash flow amount. Concepts Statement No.76_{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.

To illustrate the expected cash flow model, assume that there is a 30% probability that future cash flows will be $100, a 50% probability that they will be $200, and a 20% probability that they will be $300. In this case, the expected cash flow would be $190 [($100 0.3) ($200 0.5) ($300 0.2)]. Under traditional present value approaches, the most likely estimate ($200) would be used, but that estimate does not consider the different probabilities of the possible cash flows.

6_{“Using Cash Flow Information and Present Value in Accounting Measurements,”}

State-ment of Financial Accounting Concepts No. 7(Norwalk, Conn.: FASB, 2000).

O B J E C T I V E

Apply the expected cash flow approach to present value measurement.

The FASB takes the position that after the expected cash flows 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. The Board notes that the expected cash flow framework adjusts for credit risk because it incorporates the probability of receipt or

??THREE COMPONENTS OF

Pure Rate of Interest (2%–4%).This would be the amount a lender would

charge if there were no possibilities of default and no expectation of in-flation.

Expected Inflation Rate of Interest (0%–?).Lenders recognize that in an

in-flationary economy, they are being paid back with less valuable dollars. As a result, they increase their interest rate to compensate for this loss in pur-chasing power. When inflationary expectations are high, interest rates are high.

Credit Risk Rate of Interest (0%–5%).The government has little or no credit

risk (i.e., risk of nonpayment) when it issues bonds. A business enterprise, however, depending upon its financial stability, profitability, etc., can have a low or a high credit risk.

THREE COMPONENTS OF INTEREST

Choosing an Appropriate Interest Ra