the Time Value of Money


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1 8658d_c06.qxd 11/8/02 11:00 AM Page 251 mac62 mac62:1st Shift: 6 CHAPTER Accounting and the Time Value of Money he Magic of Interest T 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 taxfree 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. Interest rates 12% 8% 10% $35,000 $32,620 $30,000 $26,851 $25,000 $22,080 $20,000 $15,000 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. $10, End of Year 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. 251
2 8658d_c06.qxd 11/8/02 11:00 AM Page 252 mac62 mac62:1st Shift: PREVIEW OF CHAPTER 6 As indicated in the opening story, the timing of the returns on investments has an important 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. ACCOUNTING AND THE TIME VALUE OF MONEY Basic Time Value Concepts SingleSum Problems Annuities More Complex Situations Present Value Measurement Applications The nature of interest Simple interest Compound interest Fundamental variables Future value of a single sum Present value of a single sum Solving for other unknowns 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 Deferred annuities Valuation of longterm bonds Effective interest method of bond discount/premium amortization Choosing an appropriate interest rate Expected cash flow illustration BASIC TIME VALUE CONCEPTS OBJECTIVE Identify accounting topics where the time value of money is relevant. In accounting (and finance), the term time value of money is used to indicate a relationship 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 compare today s dollar and tomorrow s dollar on the same footing to compare apples to apples. We do that by using the concept of present value, which has many applications 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. 1 Some of the applications of present valuebased measurements to accounting topics are listed below, several of which are required in this textbook. 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. 252
3 8658d_c06.qxd 11/8/02 11:00 AM Page 253 mac62 mac62:1st Shift: Basic Time Value Concepts 253 PRESENT VALUEBASED ACCOUNTING MEASUREMENTS Notes. Valuing noncurrent receivables and payables that carry no stated interest rate or a lower than market interest rate. Leases. Valuing assets and obligations to be capitalized under longterm leases and measuring the amount of the lease payments and annual leasehold amortization. Pensions and Other Postretirement Benefits. Measuring service cost components of employers postretirement benefits expense and postretirement benefits obligation. LongTerm Assets. Evaluating alternative longterm investments by discounting future cash flows. Determining 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, liabilities, 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 longterm purchase contracts. In addition to accounting and business applications, compound interest, annuity, and present value concepts apply to personal finance and investment decisions. In purchasing 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 Interest is 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 period 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. 2 In 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. Lowrisk borrowers like Microsoft or Intel can 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 percentage points above the market rate if you can get a loan at all. 2 Federal law requires the disclosure of interest rates on an annual basis in all contracts. That is, instead of stating the rate as 1% per month, it must be stated as 12% per year if it is simple interest or 12.68% per year if it is compounded monthly.
4 8658d_c06.qxd 11/8/02 11:00 AM Page 254 mac62 mac62:1st Shift: 254 Chapter 6 Accounting and the Time Value of Money The amount of interest involved in any financing transaction is a function of three variables: VARIABLES IN INTEREST COMPUTATION 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. 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 OBJECTIVE Distinguish between simple and compound interest. Simple interest is 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 where Interest p i n p principal i rate of interest for a single period n number 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 p i n $1, $450 If you borrow $1,000 for 3 months at 15%, the interest is $37.50, computed as follows. Interest $1, $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 wonder and its power. The object of their affection is compound interest. Compound interest is computed on principal and on any interest earned that has not been paid or withdrawn. It is the return on (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% 3 Simple interest is traditionally expressed in textbooks in business mathematics or business finance as: I(interest) P(principal) R(rate) T(time).
5 8658d_c06.qxd 11/8/02 11:00 AM Page 255 mac62 mac62:1st Shift: Basic Time Value Concepts 255 per year, and you deposit another $1,000 in the First State Bank, where it will earn compound interest of 9% per year compounded annually. Also assume that in both cases you will not withdraw any interest until 3 years from the date of deposit. The computation of interest to be received and the accumulated yearend balance are indicated in Illustration 61. ILLUSTRATION 61 Simple vs. Compound Interest Last National Bank First State Bank Simple Interest Calculation Simple Interest Accumulated Yearend Balance Compound Interest Calculation Compound Interest Accumulated Yearend Balance Year 1 $1, % $ $1, Year 1 $1, % $ $1, Year 2 $1, % $1, Year 2 $1, % $1, Year 3 $1, % $ $1, $25.03 Difference Year 3 $1, % $ $1, 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 accumulated balance (principal plus interest to date) at each yearend to compute interest in the succeeding year which explains why your compound interest account is larger. Obviously if you had a choice between investing your money at simple interest or at compound interest, you would choose compound interest, all other things especially risk being equal. In the example, compounding provides $25.03 of additional interest 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 situations, particularly in our economy where large amounts of longlived 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 applicable only to shortterm investments and debts that involve a time span of one year or less. Spare change 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 interest 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. What do the numbers mean?
6 8658d_c06.qxd 11/8/02 11:00 AM Page 256 mac62 mac62:1st Shift: 256 Chapter 6 Accounting and the Time Value of Money OBJECTIVE Learn how to use appropriate compound interest tables. Compound Interest Tables (see pages ) Five different types of compound interest tables are presented at the end of this chapter. 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: INTEREST TABLES AND CONTENTS 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 61) 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 62) 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 63) Present Value of an Ordinary Annuity of 1 table. Contains the amounts that must be deposited now at a specified rate of interest to permit withdrawals of 1 at the end of regular periodic intervals for the specified number of periods. (Table 64) Present Value of an Annuity Due of 1 table. Contains the amounts that must be deposited now at a specified rate of interest to permit withdrawals of 1 at the beginning of regular periodic intervals for the specified number of periods. (Table 65) Illustration 62 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 periods at three different rates of compound interest. ILLUSTRATION 62 Excerpt from Table 61 FUTURE VALUE OF 1 AT COMPOUND INTEREST (EXCERPT FROM TABLE 61, PAGE 303) Period 9% 10% 11% The compound tables are computed using basic formulas. For example, the formula to determine the future value factor (FVF) for 1 is: FVF n,i (1 i) n where FVF n,i future value factor for n periods at i interest n number of periods i rate of interest for a single period
7 8658d_c06.qxd 11/8/02 11:00 AM Page 257 mac62 mac62:1st Shift: Basic Time Value Concepts 257 The FVF n,i and 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 factor) is shown below. Beginningof Multiplier EndofPeriod Formula Period Period Amount (1 i) Amount* (1 i) n (1.09) (1.09) (1.09) 3 ILLUSTRATION 63 Accumulation of Compound Amounts *Note that these amounts appear in Table 61 in the 9% column. Throughout the discussion of compound interest tables the use of the term periods instead of years is 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 compounding 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 61 from page 302, we can determine the amount to which this $1.00 will accumulate: Read the factor that appears in the 2% column on the 24th row 6 years 4 compounding periods per year, namely , or approximately $1.61. Thus, the term periods, not years, is used in all compound interest tables to express the quantity of n. The following schedule shows how to determine (1) the interest rate per compounding period and (2) the number of compounding periods in four situations of differing compounding frequency. 4 12% Annual Interest Rate Interest Rate per Number of over 5 Years Compounded Compounding Period Compounding Periods Annually (1) years 1 compounding per year 5 periods Semiannually (2) years 2 compoundings per year 10 periods Quarterly (4) years 4 compoundings per year 20 periods Monthly (12) years 12 compoundings per year 60 periods ILLUSTRATION 64 Frequency of Compounding 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 compounding 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.
8 8658d_c06.qxd 11/8/02 11:00 AM Page 258 mac62 mac62:1st Shift: 258 Chapter 6 Accounting and the Time Value of Money How often interest is compounded can make a substantial difference in the rate of return. For example, a 9% annual interest compounded daily provides a 9.42% yield, or a difference of.42%. The 9.42% is referred to as the effective yield. 5 The annual interest rate (9%) is called the stated, nominal, or face rate. When the compounding frequency 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 65 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 OBJECTIVE Identify variables fundamental to solving interest problems. Fundamental Variables The following four variables are fundamental to all compound interest problems. FUNDAMENTAL VARIABLES 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 period may be equal to or less than a year.) Future Value. The value at a future date of a given sum or sums invested assuming compound interest. Present Value. The value now (present time) of a future sum or sums discounted assuming compound interest. The relationship of these four fundamental variables is depicted in the time diagram on the following page. 5 The formula for calculating the effective rate in situations where the compounding frequency (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) %
9 8658d_c06.qxd 11/8/02 11:00 AM Page 259 mac62 mac62:1st Shift: SingleSum Problems 259 Present Value Interest Future Value ILLUSTRATION 66 Basic Time Diagram Number of Periods 4 5 In some cases all four of these variables are known, but in many business situations 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. SINGLESUM PROBLEMS Many business and investment decisions involve a single amount of money that either exists now or will in the future. Singlesum problems can generally be classified into one of the following two categories. Computing the unknown future value of a known single sum of money that is invested now for a certain number of periods at a certain interest rate. Computing the unknown present value of a known single sum of money in the future that is discounted for a certain number of periods at a certain interest rate. 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 accumulated to 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 discounting 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 future or the present. Sometimes it is neither a future 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. OBJECTIVE Solve future and present value of 1 problems. Future Value of a Single Sum To determine the future value of a single sum, multiply the future value factor by its present value (principal), as follows. FV PV (FVF n,i ) where FV future value PV present value (principal or single sum) FVF n,i future value factor for n periods at i interest
10 8658d_c06.qxd 11/8/02 11:00 AM Page 260 mac62 mac62:1st Shift: 260 Chapter 6 Accounting and the Time Value of Money 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 timediagram form, this investment situation would appear as follows. Present Value PV = $50,000 Interest Rate i = 11% Future Value FV =? Number of Periods n = Using the formula, this investment problem is solved as follows. Future value PV (FVF n,i ) $50,000 (FVF 5,11% ) $50,000 (1.11) 5 $50,000 ( ) $84,253 To determine the future value factor of in the formula above, use a financial calculator or read the appropriate table, in this case Table 61 (11% column and the 5period 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, How much will be on deposit at the end of 4 years if interest is 10%, compounded semiannually? With a known present value of $250 million, a total of 8 compounding periods (4 2), and an interest rate of 5% per compounding period (.10 2), this problem can be timediagrammed and the future value determined as follows. PV = $250,000,000 i = 5% FV =? n = 8 Future value $250,000,000 (FVF 8,5% ) $250,000,000 (1.05) 8 $250,000,000 ( ) $369,365,000
11 8658d_c06.qxd 11/8/02 11:00 AM Page 261 mac62 mac62:1st Shift: SingleSum Problems 261 Using a future value factor found in Table 61 (5% column, 8period row), we find that the deposit of $250 million will accumulate to $369,365,000 by December 31, 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 value of $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 forward 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 62. Illustration 67 demonstrates the nature of such a table. It shows the present value of 1 for five different periods at three different rates of interest. PRESENT VALUE OF 1 AT COMPOUND INTEREST (EXCERPT FROM TABLE 62, PAGE 305) ILLUSTRATION 67 Excerpt from Table 62 Period 9% 10% 11% The present value of 1 (present value factor) may be expressed as a formula: 1 PVF n,i (1 i) n where PVF n,i present value factor for n periods at i interest To illustrate, assuming an interest rate of 9%, the present value of 1 discounted for three different periods is as follows. Discount Formula Periods 1 (1 i) n Present Value* 1/(1 i) n /(1.09) (1.09) /(1.09) (1.09) /(1.09) 3 ILLUSTRATION 68 Present Value of $1 Discounted at 9% for Three Periods *Note that these amounts appear in Table 62 in the 9% column. The present value of any single sum (future value), then, is as follows. PV FV (PVF n,i ) where PV present value FV future value PVF n,i present value factor for n periods at i interest
12 8658d_c06.qxd 11/8/02 11:00 AM Page 262 mac62 mac62:1st Shift: 262 Chapter 6 Accounting and the Time Value of Money To illustrate, what is the present value of $84,253 to be received or paid in 5 years discounted at 11% compounded annually? In timediagram form, this problem is drawn as follows. Present Value PV =? Interest Rate i = 11% Future Value $84, Number of Periods n = 5 Using the formula, this problem is solved as follows. Present value FV (PVF n,i ) $84,253 (PVF 5,11% ) $84,253 1 (1.11) 5 $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 62 (11% column, 5period 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. PV =? i = 8% FV = $2, n = 3 Present value $2,000 (PVF 3,8% ) 1 $2,000 (1.08) 3 $2,000 (.79383) $1,587.66
13 8658d_c06.qxd 11/8/02 11:00 AM Page 263 mac62 mac62:1st Shift: SingleSum Problems 263 Advise your uncle to invest $1, now to provide you with $2,000 upon graduation. To satisfy your uncle s other condition, you must pass this course, and many more. Solving for Other Unknowns in SingleSum Problems In computing either the future value or the present value in the previous singlesum 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 singlesum problems (future value and present value) with either an unknown number of periods (n) or an unknown interest rate (i). These illustrations and the accompanying 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 unknown 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 veterans 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 problem is as follows. PV = $47,811 i = 10% FV = $70,000 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 formulas as shown below. Future Value Approach Present Value Approach FV PV (FVF n,10% ) PV FV (PVF n,10% ) $70,000 $47,811 (FVF n,10% ) $47,811 $70,000 (PVF n,10% ) $70,000 $47,811 FVF n,10% PVF n,10% $47,811 $70,000 ILLUSTRATION 69 Solving for Unknown Number of Periods Using the future value factor of , refer to Table 61 and read down the 10% column to find that factor in the 4period 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 62 and read down the 10% column to find again that factor in the 4period row.
14 8658d_c06.qxd 11/8/02 11:00 AM Page 264 mac62 mac62:1st Shift: 264 Chapter 6 Accounting and the Time Value of Money 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? A time diagram of this investment situation is as follows. PV = $800,000 i =? FV = $1,409, n = The unknown interest rate may be determined from either the future value approach or the present value approach as shown in Illustration ILLUSTRATION 610 Solving for Unknown Interest Rate Future Value Approach Present Value Approach FV PV (FVF 5,i ) PV FV (PVF 5,i ) $1,409,870 $800,000 (FVF 5,i ) $800,000 $1,409,870 (PVF 5,i ) $1,409,870 $800,000 FVF 5,i PVF 5,i $800,000 $1,409,870 Using the future value factor of , refer to Table 61 and read across the 5period 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 and Table 62, again find that factor at the juncture of the 5period 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 common 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 accumulation 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 interval between such rents always be the same, and (3) the interest be compounded once each interval. The future value of an annuity is the sum of all the rents plus the accumulated compound interest on them.
15 8658d_c06.qxd 11/8/02 11:00 AM Page 265 mac62 mac62:1st Shift: Annuities 265 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 classified as an ordinary annuity if the rents occur at the end of each period, and as an annuity due if 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 each of the rents in the series will accumulate and then total their individual future values. For example, assume that $1 is deposited at the end of 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 61) for each of the five $1 rents. OBJECTIVE Solve future value of ordinary and annuity due problems. END OF PERIOD IN WHICH $1.00 IS TO BE INVESTED Value at End Present of Year 5 $1.00 $ $ $ $ $ Total (future value of an ordinary annuity of $1.00 for 5 periods at 12%) $ ILLUSTRATION 611 Solving for the Future Value of an Ordinary Annuity 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 deposited. 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 number 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 compound interest: (1 i) n 1 FVFOA n,i i where FVFOA n,i future value factor of an ordinary annuity i rate of interest per period n number of compounding periods For example, FVFOA 5,12% 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 612 is an excerpt from the future value of an ordinary annuity of 1 table.
16 8658d_c06.qxd 11/8/02 11:00 AM Page 266 mac62 mac62:1st Shift: 266 Chapter 6 Accounting and the Time Value of Money ILLUSTRATION 612 Excerpt from Table 63 FUTURE VALUE OF AN ORDINARY ANNUITY OF 1 (EXCERPT FROM TABLE 63, PAGE 307) Period 10% 11% 12% * *Note that this annuity table factor is the same as the sum of the future values of 1 factors shown in Illustration Interpreting the table, if $1 is invested at the end of each year for 4 years at 11% interest compounded annually, the value of the annuity at the end of the fourth year will be $4.71 ( $1). Multiply the factor from the appropriate line and column of the table by the dollar amount of one rent involved 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 (FVFOA n,i ) where R periodic rent FVFOA n,i future value of an ordinary annuity factor for n periods at i interest 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 timediagram form, this problem is drawn as follows. Present Value i = 12% R = $5,000 $5,000 $5,000 $5,000 Future Value FV OA FV 0A =? $5, n = 5 Using the formula, this investment problem is solved as follows. Future value of an ordinary annuity R (FVFOA n,i ) $5,000 (FVFOA 5,12% ) $5,000 (1.12) $5,000 ( ) $31, We can determine the future value of an ordinary annuity factor of in the formula above using a financial calculator or by reading the appropriate table, in this case Table 63 (12% column and the 5period row). To illustrate these computations in a business situation, assume that Hightown Electronics decides to deposit $75,000 at the end of each 6month 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 interest rate is 10%?
17 8658d_c06.qxd 11/8/02 11:00 AM Page 267 mac62 mac62:1st Shift: Annuities 267 The time diagram and formula solution are as follows. R = $75,000 $75,000 i = 5% $75,000 $75,000 $75,000 Future Value FV OA FV 0A =? $75, n = 6 Future value of an ordinary annuity R (FVFOA n,i ) $75,000 (FVFOA 6,5% ) $75,000 (1.05) $75,000 ( ) $510, Thus, six 6month deposits of $75,000 earning 5% per period will grow to $510, 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 end of each period. An annuity due assumes periodic rents occur at the beginning of each period. This means an annuity due will accumulate interest during the first period, whereas an ordinary annuity rent will earn no interest during the first period because the rent is not received or paid until the end of the period. 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 annuity there 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 First deposit here Future Value of an Annuity of 1 at 12% ILLUSTRATION 613 Comparison of the Future Value of an Ordinary Annuity with an Annuity Due Ordinary annuity Future value of an ordinary annuity (per Table 63) First deposit here Period 1 Period 2 Period 3 Period 4 Period 5 No interest Interest Interest Interest Interest Annuity due Period 1 Period 2 Period 3 Period 4 Period 5 Interest Interest Interest Interest Interest (No table provided)
18 8658d_c06.qxd 11/8/02 11:00 AM Page 268 mac62 mac62:1st Shift: 268 Chapter 6 Accounting and the Time Value of Money 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 , whereas for an annuity due it is 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 ( ) by one plus the interest rate (1.12), to arrive at ( ). 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 eighteenth birthday). Because all the deposits will be made at the beginning of the periods, they represent an annuity due. i = 12% R = $800 $800 $800 $800 $800 $800 $800 $800 Future Value FV AD =? n = 8 FV AD = Future value of an annuity due Referring to the future value of an ordinary annuity of 1 table for 8 periods at 12%, we find a factor of 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, as shown in Illustration ILLUSTRATION 614 Computation of Accumulated Value of Annuity Due 1. Future value of an ordinary annuity of 1 for 8 periods at 12% (Table 63) Factor (1.12) Future value of an annuity due of 1 for 8 periods at 12% Periodic deposit (rent) $ Accumulated value on son s eighteenth birthday $11, 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, interest rate, and number of periods. They were used to determine the fourth value, future value, which was unknown. The first two future 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.
19 8658d_c06.qxd 11/8/02 11:00 AM Page 269 mac62 mac62:1st Shift: Annuities 269 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 6month period? The $14,000 is the future value of 10 (5 2) semiannual endofperiod 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. Future Value i = 4% FV OA = $14,000 R =?????????? n = 10 FV OA = Future value of an ordinary annuity 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 (FVFOA n,i ) $14,000 R (FVFOA 10,4% ) $14,000 R( ) $14,000 R R $1, Thus, you must make 10 semiannual deposits of $1, 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 deposits 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. R = $20,000 i = 8% $20,000 $20,000 Future Value FV OA = $117, n =? 3 n
20 8658d_c06.qxd 11/8/02 11:00 AM Page 270 mac62 mac62:1st Shift: 270 Chapter 6 Accounting and the Time Value of Money Using the future value of an ordinary annuity formula, we obtain the following factor. Future value of an ordinary annuity R (FVFOA n,i ) $117,332 $20,000 (FVFOA n,8% ) $117,332 FVFOA n,8% $20,000 Using Table 63 and reading down the 8% column, we find in the 5period 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 account when he retires in 30 years? This problem appears in the form of a time diagram as follows. i = 9% R = $2,500 $2,500 $2,500 $2,500 Future Value FV AD =? n = 30 Using the future value of an ordinary annuity of 1 table, the solution is computed as follows. ILLUSTRATION 615 Computation of Accumulated Value of an Annuity Due 1. Future value of an ordinary annuity of 1 for 30 periods at 9% Factor (1.09) Future value of an annuity due of 1 for 30 periods at 9% Periodic rent $2, Accumulated value at end of 30 years $371,438 OBJECTIVE Solve present value of ordinary and annuity due problems. Present Value of an Ordinary Annuity The present value of an annuity is the single sum that, if invested at compound interest now, would provide for an annuity (a series of withdrawals) for a certain number 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 present value of each of the rents in the series and then total their individual present values. For example, an annuity of $1 to be received at the end of 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 ), assuming an interest rate of 12%.
21 8658d_c06.qxd 11/8/02 11:00 AM Page 271 mac62 mac62:1st Shift: Annuities 271 END OF PERIOD IN WHICH $1.00 IS TO BE RECEIVED Present Value at Beg. of Year $ $ $ $ $ $1.00 $ Total (present value of an ordinary annuity of $1.00 for five periods at 12%) ILLUSTRATION 616 Solving for the Present Value of an Ordinary Annuity 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: 1 1 (1 i) n PVFOA n,i i The expression PVFOA n,i refers to the present value of an ordinary annuity of 1 factor for n periods at i interest. Using this formula, present value of ordinary annuity tables are prepared. An excerpt from such a table is shown below. PRESENT VALUE OF AN ORDINARY ANNUITY OF 1 (EXCERPT FROM TABLE 64, PAGE 309) ILLUSTRATION 617 Excerpt from Table 64 Period 10% 11% 12% * *Note that this annuity table factor is equal to the sum of the present value of 1 factors shown in Illustration The general formula for the present value of any ordinary annuity is as follows. Present value of an ordinary annuity R (PVFOA n,i ) where R periodic rent (ordinary annuity) PVFOA n,i present value of an ordinary annuity of 1 for n periods at i interest To illustrate, what is the present value of rental receipts of $6,000 each to be received at the end of each of the next 5 years when discounted at 12%? This problem may be timediagrammed and solved as follows. Present Value PV OA =? i = 12% R = $6,000 $6,000 $6,000 $6,000 $6, n = 5
22 8658d_c06.qxd 11/13/02 3:44 PM Page 272 mac62 Pdrive 03:es%0:wiley:8658d: :ch06:text_s: 272 Chapter 6 Accounting and the Time Value of Money Present value of an ordinary annuity R (PVFOA n,i ) $6,000 (PVFOA 5,12% ) $6,000 ( ) $21, The present value of the 5 ordinary annuity rental receipts of $6,000 each is $21, Determining the present value of the ordinary annuity factor can be accomplished using a financial calculator or by reading the appropriate table, in this case Table 64 (12% column and 5period row). Up in smoke What do the numbers mean? Time value of money concepts also can be relevant to public policy debates. For example, several states are evaluating how to receive the payments from tobacco companies as settlement for a national lawsuit against the companies for the healthcare 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 investment bank (a process called securitization) and receive a lumpsum 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 * $2.39 billion *PVOA (i 8%, n 25) 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 lumpsum 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 ), which is much closer to the lumpsum payment offered to the State of Wisconsin. Present Value of an Annuity Due In the discussion of the present value of an ordinary annuity, the final rent was discounted 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 distinction is shown graphically in Illustration 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 present 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% interest ( ) and multiply it by 1.12 to arrive at the present value of an annuity due, ( ). 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 annuity due factors in the form of Table 65.
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