FOUNDATIONS OF CORPORATE FINANCE

Transcription

1 edition 2 FOUNDATIONS OF CORPORATE FINANCE Kent A. Hickman Gonzaga University Hugh O. Hunter San Diego State University John W. Byrd Fort Lewis College

2 chapter 4 Time Is Money 00 After learning from his attorney that Time is money, Stanley was eager to begin his lengthy prison term.

3 Chapter 4 Time Is Money 0 CHAPTER 4 IN FOCUS THE FINANCIAL BALANCE SHEET Fixed claims Residual claims Cash generated from operations The value of the firm depends on the size, timing, and riskiness of cash flows. Chapter 4 develops the mathematical techniques for valuing cash flows that occur at different points of time. The saying time is money could not be more true than it is in finance. People rationally prefer to collect money earlier rather than later. By delaying the receipt of cash, individuals forgo the opportunity to purchase desired goods or invest the funds to increase their wealth. The foregone interest, which could be earned if cash were received immediately, is called the opportunity cost of delaying its receipt. Individuals require compensation to reimburse them for the opportunity cost of not having the funds available for immediate investment purposes. This chapter describes how such opportunity costs are calculated. Because many business activities require computing a value today for a series of future cash flows, the techniques presented in this chapter apply not only to finance but also to marketing, manufacturing, and management. Here are examples of questions that the tools introduced in this chapter can help answer: How much should we spend on an advertising campaign today if it will increase sales by 5% in the future? Is it worth buying a new computerized lathe for $20,000 if the lathe reduces material waste by 5%? Which strategy should we employ, given their respective costs and estimated contributions to future earnings? What types of health insurance and retirement plans are best for our employees, given the amount of money we have available? Being able to give a value to cash to be received in the future, whether dividends from a share of stock, interest from a bond, or profits from a new product, is one of the primary skills needed to run a successful business. The material in this chapter provides an introduction to that skill. The Time Value of Money The opportunity cost of being a fulltime student is the salary one could earn without attending school. Stone Street Capital will buy your winning lottery ticket by paying you its present value. See their site at com. Suppose a friend owes you $00 and the payment is due today. You receive a phone call from this friend, who says she would like to delay paying you for one year. You may reasonably demand a higher future payment, but how much more should you receive? The situation is illustrated here using a time line.

4 02 Chapter 4 Time Is Money t = 0 t = = $00 FV =? Some web sites have present/future value calculators, such as the one at com. In this diagram now, the present time, is assigned t 5 0, or time zero. One year from now is assigned t 5. The present value of the cash payment is $00 and is denoted. Its future value at t 5 is denoted FV. To find the amount that you could demand for deferring receipt of the money by one year, you must solve for FV, the future value of $00 one year from now. The FV value will depend on the opportunity cost of forgoing immediate receipt of $00. You know, for instance, that if you had the money today you could deposit the $00 in a bank account earning 5% interest annually. However, you know from Chapter 2 that value depends on risk. In your judgment, your friend is less likely to pay you next year than is the bank. Therefore, you will increase the rate of interest to reflect the additional risk that you think is inherent in the loan to your friend. Suppose that you decide that a 0% annual rate of interest is appropriate. The amount of the future payment, FV, will be the original principal plus the interest that could be earned at the 0% annual rate. Algebraically, you can solve for FV, being careful always to convert percentages to decimals when doing arithmetic calculations, FV 5 $00 ($00)(.0) (4.) Factoring $00 from the right-hand side of Equation (4.), FV 5 $00(.0) (4.2) FV 5 $00(.0) (4.3) FV 5 $0 (4.4) You may demand a $0 payment at t 5 in lieu of an immediate $00 payment because these two amounts have equivalent value. Let s say that your friend agrees to this interest rate but asks to delay payment for 2 years. t = 0 t = t = 2 = $00 FV = $0 =? t = 0 FV = $00(.0)? Now we must find, the future value of the payment 2 years from today. Since we know FV 5 $0 and we know the interest rate is 0%, we can solve Chapter 6 is devoted to assessing risk and finding interest rates that reflect investors required returns given the riskiness of an investment. Hereafter, this chapter will treat interest rates as known, ignoring risk.

5 Chapter 4 Time Is Money 03 for by recognizing that will equal FV plus the interest that could be earned on FV during the second year. 5 FV FV (.0) (4.5) 5 $0 ($0)(.0) (4.6) 5 $0(.0) (4.7) 5 $0(.0) (4.8) 5 $2 (4.9) You may demand a $2 payment at t 5 2 because its time value is equivalent to either $0 at t 5 or $00 at t 5 0, given the 0% interest rate. The time value of money and the mathematics associated with it provide important tools for comparing the relative values of cash flows received at different times. For example, recall from Chapter that to increase shareholder wealth managers must make investments that have greater value than their costs. Often, such investments require an immediate cash outlay, like buying a new delivery truck. The investment (the truck) then produces cash flows for the corporation in the future (delivery fee income, increased sales, lower delivery costs, etc.). To determine whether the future cash flows have greater value than the initial cost of the truck, managers must be able to calculate the present value of the future stream of cash flows produced by this investment. Just as a hammer may be the most useful item in a carpenter s toolbox, time value of money mathematics is indispensable to a financial manager. Compound and Simple Interest The preceding section showed that, at a 0% annual interest rate, $00 today is equivalent to $0 a year from now and $2 in 2 years. t = 0 t = t = 2 = $00 FV = $00(.0) = $0 = $0(.0) = $00(.0) 2 = $2 This result may be generalized using the following formulas, FV 5 ( r) (4.0) 5 ( r) 2 (4.) where r is the interest rate. Now, let s expand Equation (4.): 5 ( r)( r) (4.2) 5 ( 2r r 2 ) (4.3) 5 ( 2r) (r 2 ) (4.4) Equation (4.4) is broken down in a special way. The first term on the right side of the equal sign, ( 2r), would yield $20 given the information we have used in our example. The second term, (r 2 ), yields $. The value $20 equals

6 04 Chapter 4 Time Is Money Remember the good deal made by Peter Minuit in 624 when he bought Manhattan Island? In 2000, that $24 would be worth $2.7 trillion if it had been invested at 7%, compounded annually. your original principal ($00) plus the amount of interest earned ($20) if your friend paid simple interest. For example, if you withdraw interest earned during each year at the end of that year, you would earn simple interest. In this case, you would receive $0 interest payments at the end of years and 2, totaling $20. If, on the other hand, your friend credited (but did not pay) interest to you every year, then you would earn interest during year 2 on the interest credited to you at the end of year. Earning interest on previously earned interest is known as compounding. Thus, you would earn an extra dollar, a total of $2, over the 2-year period with interest compounded annually. The example assumed annual compounding since nearly all transactions are now based on compound rather than simple interest. Not all compounding is done on an annual basis, however. Sometimes interest is added to an account every 6 months (semiannual compounding). Other contracts call for quarterly, monthly, or daily compounding. As you will see, the frequency of compounding can make a big difference when the time value of money is calculated. The Time Value of a Single Cash Flow Continuing our example, let us suppose that your friend who wishes to delay paying you agrees to a 0% annual rate of interest over the two-year period and will allow you to compound interest semiannually. What will you be paid in 2 years given this agreement? Semiannual compounding means that interest will be credited to you every 6 months, based on half of the annual rate. In effect you will be earning a 5% semiannual rate of interest over four 6-month periods. In other words, the periodic interest rate will be half the annual rate because you are using semiannual compounding and you will be earning interest for four time periods (n 5 through 4), each period being one-half year long. The new situation is illustrated next. 6 months year years 2 years n = 0 n = n = 2 n = 3 n = 4 2 = $00 FV FV 3 FV 4 Here, FV is the future value of the $00 at the end of period (the first six months). As before, FV equals the $00 beginning principal plus interest earned over the 6 months at the 5% semiannual interest rate. FV 5 $00 $00(.05) (4.5) FV 5 $00(.05) (4.6) FV 5 $05 (4.7) Therefore, at the end of period (at n 5 ) the principal balance you are owed will be $05. will be equal to the principal at the beginning of period 2 plus interest earned during period 2.

7 Chapter 4 Time Is Money 05 5 $05 $05(.05) (4.8) 5 $05(.05) (4.9) 5 $0.25 (4.20) Note that we could substitute [$00(.05)] for $05 in Equation (4.9). Doing so, could be reexpressed as follows: 5 $05(.05) (4.9) 5 [$00(.05)](.05) (4.2) 5 $00(.05) 2 (4.22) Following this pattern, finding FV 3 and FV 4 is straightforward. FV 3 5 $00(.05) 3 (4.23) FV 3 5 $5.76 (4.24) FV 4 5 $00(.05) 4 5 $2.55 (4.25) Equation (4.25) gives the answer we seek. The future value at the end of four 6- month periods is $2.55. Changing from annual compounding to semiannual compounding has increased the future value of your friend s obligation to you by 55. The additional interest earned from semiannual compounding, 55, doesn t seem like much but imagine a firm borrowing $00 million; then the compounding period annual, semiannual, quarterly can turn into tens of thousands of dollars. The Future Value of a Single Cash Flow The pattern established here may be generalized into the formula for the future value of a single cash flow using compound interest. FV n 5 ( r) n (4.26) where FV n 5 the future value at the end of n time periods 5 the present value of the cash flow r 5 the periodic interest rate which equals the annual nominal rate divided by the number of compounding periods per year, annual nominal rate r 5 number of periods per year n 5 the number of compounding periods until maturity, or n 5 (number of years until maturity) (compounding periods per year) It is critical when using this formula to be certain that r and n agree with each other. If, for example, you are finding the future value of $00 after 6 years and the annual rate is 8%, compounded monthly, then the appropriate r is.5% per month (8% %) and n is 72 months (6 years times 2 months per year 5 72 months). Students often adjust the interest rate and then forget to adjust the number of periods (or vice versa)! The answer to this problem is

8 06 Chapter 4 Time Is Money EXHIBIT 4. THE FUTURE VALUE OF $00 AFTER 6 YEARS USING 8% INTEREST PER YEAR WITH DIFFERENT COMPOUNDING PERIODS COMPOUNDING ASSUMPTION N R FV N Annual compounding $ Semiannual compounding $28.27 Quarterly compounding $ Monthly compounding $292.2 Weekly compounding $ Daily compounding 2, $ FV $00a b 632 FV 72 5 $00(.05) 72 5 $ For simple interest, without compounding, the future value is simply equal to the annual interest earned times the number of years, plus the original principal. The formula for the future value of a single cash flow using simple interest is FV s n 5 (n)( )(r) 5 ( nr) (4.27) where FV s n 5 the future value at the end of n periods using simple interest n 5 the number of periods until maturity (Generally n simply equals the number of years, because there is no adjustment for compounding periods.) r 5 the periodic rate (which also usually equals the annual rate because there is no adjustment for compounding periods) For the previous example, the future value of $00 invested for 6 years in an account paying 8% per year using simple interest is FV s 6 5 $00[ (6)(0.8)] 5 $ Thus, monthly compounding yielded a future value after 6 years of $292.2, or $84.2 more than simple interest in this example. Exhibit 4. illustrates the future value of $00, bearing 8% annual interest, with different compounding assumptions. Be sure that you can replicate the solutions illustrated here using your calculator. 2 Simple interest calculations are abandoned at this point because they are so uncommon. The Present Value of a Cash Flow We have solved for the future value of a current cash flow. Often, we must solve for the present value of a future cash flow, solving for PV rather than FV. 2 To do these problems, you must master the y x key (or similar key) on your calculator. In this case, y corresponds to ( r) in the formula and x corresponds to n in the formula. Be sure your n and r agree (e.g., both are monthly, yearly, etc.) and always be sure you express percentages as decimals before doing any calculations. You should practice with your calculator until your answers match those given in Exhibit 4..

9 Chapter 4 Time Is Money 07 Time Value Table 4. Equation (4.26), FV n 5 ( r) n, is sometimes written FV n 5 (FVIF n,r ) In this form, ( r) n is replaced by FVIF n,r, an acronym for future value interest factor for n periods at a periodic rate of r. Values for commonly encountered interest rates and periods have been calculated for FVIF and are given in Table A.. A portion of Table A. is reproduced here. N % 2% 3% 4% 5% Suppose you wanted to find the future value of an $800 deposit in a bank that pays 2% annual interest, compounded quarterly, when the money will be left in the account year; then PV would equal $800, r 5 3% per quarter, and n 5 4. Note that FVIF 4,3% is highlighted in the table and equals.26. Thus, FV 4 5 $800(.26) 5 $ The use of tables, such as Table A. for solving time value problems was very popular prior to the availability of inexpensive calculators. Such tables are limited, however, in their accuracy and because it is impossible to tabulate all possible rates of interest and all possible periods. Suppose, for example, you are going to receive a bonus of $,000 in one year. You could really use some cash today and are able to borrow from a bank that would charge you an annual interest rate of 2%, compounded monthly. You decide to borrow as much as you can now such that you will still be able to pay off the loan in one year using the $,000 bonus. In essence, you wish to solve for the present value of a $,000 future value, knowing the interest rate (2% per year, compounded monthly) and the term of the loan ( year, or 2 monthly compounding periods). Following is a time line illustrating the problem. n = 0 n = 2 =? r = 0.0 = $,000 In this case n 5 2, r 5 %, and is known, whereas is unknown. We may still use Equation (4.26), substituting in the known quantities and using some algebra.

10 08 Chapter 4 Time Is Money Compounding and discounting are flip sides of the same coin. Compounding is used to express a value at a future date given a rate of interest. Discounting involves expressing a future value as an equivalent amount at an earlier date. FV n 5 ( r) n (4.26) $,000 5 (.0) 2 (4.28) 5 $,000(.0) 22 5 $,000 (4.29) $ (4.30) You could borrow $ today and fully pay off the loan, given the bank s terms, in one year using your $,000 bonus. Equation (4.29) may be generalized into the formula for the present value of a single cash flow with compound interest. Solving for the present value of a future cash flow is also known as discounting. This formula is also called the discounting formula for a single future cash flow. 5 FV n s rd 2n 5 FV n (4.3) s rd n The variables, FV n, n, and r are defined exactly as they are in the future value formula because both formulas are really the same, just solved for different unknowns. Exhibit 4.2 solves for the present, or discounted, value of a $,000 cash flow to be received in year at a 2% per year discount rate using different compounding periods. You should be able to replicate these solutions on your calculator. 3 Present and future value formulas are very useful because they may be used to solve a variety of problems. Suppose you make a $500 deposit in a bank today and you want to know how long it will take your account to double in value, assuming that the bank pays 8% interest per year, compounded annually. Here, you are solving for the number of time periods. You may substitute the known quantities 5 $500, FV n 5 $,000, r into either formula and solve for n: r = 0.08 = $500 FV n = $,000 n =? EXHIBIT 4.2 THE PRESENT VALUE OF $,000 TO BE RECEIVED IN YEAR,DISCOUNTED AT A 2% ANNUAL RATE WITH DIFFERENT COMPOUNDING PERIODS COMPOUNDING ASSUMPTION N R Compounded annually 0.2 $ Compounded semiannually $ Compounded quarterly $ Compounded monthly $ Compounded weekly $ Compounded daily $ To find the present values on your calculator, you may use the y x key with a negative exponent or with the x key, recalling that a negative exponent means finding the reciprocal, or over the quantity.

11 Chapter 4 Time Is Money 09 Time Value Table 4.2 Equation (4.3), 5 FV n ( r) 2n, is sometimes written: 5 FV n [PVIF n,r ] Here PVIF n,r stands for the present value interest factor for n periods at a periodic rate of r. For many common rates of interest and periods, PVIF is tabulated in Table A.2. A part of Table A.2 is reproduced here. N % 2% 3% 4% 5% 6% PVIF n,r is nothing more than another way of writing ( r) 2n, and Table A.2 simply tabulates these values. Suppose you wish to know the present value of $2,000, which will be received in 2 years, if the discount rate is 5% per year, compounded annually. is equal to $2,000 and PVIF n52,r55%, highlighted above, equals (PVIF n,r ) 5 $2,000(0.907) 5 $,84. 5 FV n ( r) 2n (4.3) $500 5 $,000(.08) 2n (4.32) (.08) n 5 $,000/$500 (4.33) (.08) n 5 2 (4.34) At this point, without using logarithms 4 you must use trial and error to solve for n. Suppose you try n 5 0 as your first guess for n: (.08) This value yields a number higher than our objective of 2. Therefore, try n 5 9, because a lower value of n will yield a lower answer: (.08) which is close enough. In 9 years the balance in your account will double. Suppose the account earned 8% per year compounded monthly. To find the time until the account s balance doubled, you would convert the interest rate to reflect monthly compounding sr d and solve for the number of compounding periods. 4 To solve using logarithms, use the natural log key on your calculator [LN] and the following formula: n 5 ln3fv>pv4 ln3 r4

12 0 Chapter 4 Time Is Money The difference between compounding frequencies offered at various banks makes shopping around worthwhile whether you are a borrower or a saver. 5 FV n ( r) 2n (4.3) $500 5 $,000(.00667) 2n (4.35) (.00667) n 5 2 (4.36) Using trial and error, you get the answer n This should be interpreted as 05 months because you are dealing with monthly compounding periods. Thus, in 8.75 years the account will double in value when using monthly rather than annual compounding. This example illustrates an important lesson. It takes less time to achieve a desired amount of wealth with more frequent compounding at a given nominal interest rate. It is no surprise that borrowers prefer less frequent compounding, while savers (or lenders) prefer compounding as frequently as possible. Another type of problem is solving for the interest rate. This time let s suppose that an investment costing $200 will make a single payment of $275 in 5 years. What is the interest rate such an investment will yield? Substitute 5 $200, FV 5 5 $275, and n 5 5 into the formula and solve for r. = 200 FV 5 = 275 r =? As you have seen, the frequency of compounding is important. Truth-in-lending laws now require that financial institutions reveal the effective annual percentage rate (EAR) to customers so that the true cost of borrowing is explicitly stated. Before this legislation, banks could quote customers annual interest rates without revealing the compounding period. Such a lack of disclosure can be costly to borrowers. For example, borrowing at a 2% yearly rate from bank A may be more costly than borrowing from bank B, which charges 2.% yearly, if bank A compounds interest daily and bank B compounds semiannu- 5 FV n ( r) 2n (4.3) $200 5 $275( r) 25 s rd 5 5 $275 $200 (4.37) (4.38) ( r) (4.39) r 5 (.375) /5 (4.40) r 5 (.375) 0.20 (4.4) r 5 (.375) (4.42) r (4.43) The answer, r , is based on an annual compound rate, because we assumed n 5 5 years. It is also expressed as a decimal and could be reexpressed as a percentage, 6.576% per year compounded annually. Effective Annual Percentage Rate Be sure to read Problem 4 in this chapter. It describes the rule of 72, a very useful method for estimating the effect of compound interest.

13 Chapter 4 Time Is Money ally. Both 2% and 2.% are nominal rates they reveal the rate in name only but not in terms of the true economic cost. To find the effective annual rate, you must divide the nominal annual percentage rate (APR) by the number of compounding periods per year and add. Then raise this sum to an exponent equal to the number of compounding periods per year. Finally, subtract from this result: EAR 5 a APR cp (4.44) CP b 2 For our example, EAR A 5 a (4.45) 365 b % EAR B 5 a (4.46) 2 b % Thus, if you are a borrower, you would prefer to borrow from bank B despite its higher APR. The lower EAR translates into a lower cost over the life of the loan. The disclosure of EARs makes comparison shopping for rates much easier. Continuous Compounding Suppose you were lucky enough to find an investment that yielded 00% interest annually. Without compounding, you would double your money in year. Keep in mind that, expressed as a decimal, 00% interest is.00, so r 5.00 and the -year future value would be FV 5 $(.00) 5 $(2) 5 $2. Now, assume you found a bank that offered 00% nominal annual interest with your choice of compounding periods. Exhibit 4.3 shows the -year future value of $, deposited in such accounts with various compounding frequencies. Notice that, as expected, the future value of the deposit increases as the frequency of compounding increases until the increase appears to level off at daily compounding. The same phenomenon appears in Exhibit 4.. There appears to be a limit to the benefit of increasing the frequency of compounding. In fact, there is such a limit; it is known as continuous compounding, meaning that interest is credited to the depositor s account constantly (every microsecond, so to speak). The future value and present value formulas for continuous compounding are For those students who may want to study options and other derivatives in the future, continuous compounding will be very useful. EXHIBIT 4.3 FV OF $ DEPOSITED IN AN ACCOUNT FOR YEAR BEARING A 00% NOMINAL ANNUAL RATE OF INTEREST AT VARIOUS COMPOUNDING FREQUENCIES COMPOUNDING FREQUENCY FORMULA FV Annual FV 5 $(2) $2.00 Semiannual FV 5 $(.5) 2 $2.25 Quarterly FV 5 $(.25) 4 $2.44 Monthly FV 5 $(.0833) 2 $2.6 Weekly FV 5 $(.0923) 52 $2.69 Daily FV 5 $(.00274) 365 $2.746 Every hour FV 5 $(.0004) 8,760 $2.78

14 2 Chapter 4 Time Is Money FV n 5 (e rn ) (4.47) 5 FV n (e rn ) (4.48) The letter e is one of those special numbers in mathematics that is assigned its own name. (Another one is p, which you may remember is approximately equal to 3.4.) The number e is approximately equal to 2.72 (more precisely, 5 it is approximately equal to ). That value is the same number that seemed to be the limit of compounding benefit in Exhibit 4.3. Go back to Exhibit 4. and find the future value in year 6 of $00 deposited at 8%, compounded daily. That answer is $ Now find FV 6 if $00 is deposited at 8%, compounded continuously: FV 6 5 ($00)[e (0.8)(6) ] 5 ($00)( ) 5 $ (4.49) In the continuous compounding formulas, r is always equal to the annual rate of interest and n is always equal to the number of years. Thus, to find the present value of $,000 to be received in 8 months, discounted at 6% per year, compounded continuously, you would use r and n 5.5: 5 FV.5 e 2(0.6)(.5) (4.50) 5 $,000( ) (4.5) 5 $ (4.52) Valuing Multiple Cash Flows Many problems in finance involve finding the time value of multiple cash flows. Consider the following problem. A charity has the opportunity to purchase a used mobile hot dog stand being sold at an auction. The charity would use the hot dog stand to raise money at special events held in the summer of each year (at the county fair and at baseball and soccer games). The old hot dog stand will only last 2 years and then will be worthless. The charity estimates that, after all operating expenses, the stand will produce cash flows of $,000 in both June and July in each of the next 2 years and cash flows of $,500 in each of the next two Augusts. The auction takes place January, and the charity requires that its fundraising projects return 2% on their invested funds. How much should the charity bid for the hot dog stand? The strategy for solving this problem is shown in Exhibit 4.4. The present value of the stream of cash flows the stand is expected to produce is found by applying Equation (4.3) to each of the six future cash flows. Note that % is used as the periodic rate (2% per year 4 2 months) because cash flows are spaced in monthly intervals. The charity should bid a maximum of $6,53.07 for the hot dog stand. Given the level of expected cash flows, paying more than this amount would result in the charity earning a lower return than its 2% objective. The hot dog stand example illustrates the general formula for finding the present value of any cash flow stream, 5 e is the base of the natural log and is defined as e 5 lim a x xsq x b

15 Chapter 4 Time Is Money 3 EXHIBIT 4.4 N 5 0 N 5 6 N 5 7 N 5 8 N 5 8 N 5 9 N 5 20 June July August June July August $,000 $,000 $,500 $,000 $,000 $,500 $ $,000(.0) 26 $ $,000(.0) 27 $, $,500(.0) 28 $ $,000(.0) 28 $ $,000(.0) 29 $, $,500(.0) 220 $6, Total Present Value where 5 CF s rd CF 2 s rd 2... n 5 the number of compounding periods from time 0 (4.53) CF n 5 the cash flow to be received exactly n compounding periods from time 0 (e.g., CF is the cash flow received at the end of period, etc.) r 5 the periodic interest rate CF N N s rd 5 CF n N a s rd n N 5 the number of periods until the last cash flow The future value formula for a cash flow stream is also found by finding the future value of each individual cash flow and summing. Terms in the formula are defined as in the present value formula. FV N 5 CF ( r) N2 CF 2 ( r) N22... CF N (4.54) You may question why in Equation (4.54) the first term is raised to the exponent N 2 and why the last term is not multiplied by an interest factor. This situation may be clarified by using a time line. The last cash flow (CF N ) occurs at the end of the last time period and therefore earns no interest. n5 0 2 N CF CF 2 CF N CF 2 ( + r) N 2 + FV N of CF 2 CF ( + r) N + FV N of CF = FV N of cash flow stream As the time line shows, CF will earn interest for N 2 periods, but CF N earns no interest and is simply added to the other sums to find the total future value.

16 4 Chapter 4 Time Is Money By convention, we assume that the cash flows from investments do not start immediately but are deferred until the end of the first period. This is not always the case, however. Practitioners must carefully analyze any problem to be certain exactly when cash flows will occur. A time line is a useful aid in modeling when the cash flows from a project will occur. Perpetuities The British government and the Canadian Pacific Railroad have issued perpetual securities known as consuls. Some special patterns of cash flows are frequently encountered in finance. Furthermore, the nature of these patterns allows the general formulas to be simplified to a more concise form. The first special case is that of perpetuities. These are cash flow streams where equal cash flow amounts are uniformly spaced in time (every year, or every month, etc.). Perpetuity means that these payments continue forever. To illustrate, suppose an investment is expected to pay $50 every year forever. Investors require a return of 0% on this investment. What should be its current price? Recognizing that today s price should equal the present value of the investment s future cash flows, the problem may be illustrated using a time line CF = $50 CF 2 = $50 CF 3 = $50 CF 00 = $50 The arrow indicates that these cash flows continue into the future indefinitely. This poses a problem: If there are an infinite number of cash flows, how can we find all of their present values? Let s consider the algebraic expression of this problem. 5 $50 (4.55) s.0d $50 s.0d... $50 2 s.0d $50 c (4.56) s.0d s.0d... 2 s.0d... d 00 The expression in brackets in Equation (4.56) is known as the sum of a geometric series. 6 Fortunately, there is a technique for summing such series. The appendix to this chapter shows how to sum a geometric series and then derives the following formula for the present value of a perpetuity. 5 CF (4.57) r Note that there is no subscript attached to CF because all the cash flows are the same. Therefore, there is no need to distinguish CF from CF 2, and so on. Let s apply the formula to the example. CF 5 $50, r 5 0.0, and 5 $ $500 (4.58) 6 A geometric series is a series in which each successive term is a constant factor times the term preceding it. For example, the series, 2, 4, 8, 6,..., is a geometric series because each term is multiplied by 2 to arrive at the next term. In Equation (4.56) the factor is.0, because multiplying any term times.0 will yield the next term in the series.

17 Chapter 4 Time Is Money 5 Annuities Of all the special patterns of cash flow streams, annuities are the most common. As we shall see, millions of fixed-rate home mortgages are annuities. Automobile loans often fit the annuity pattern, as do many leases. An annuity is a stream of equally sized cash flows, equally spaced in time, which end after a fixed number of payments. Thus, annuities are like perpetuities, except they do not go on forever. The present value of an annuity can be found by summing the present values of all the individual cash flows. N CF 5 a (4.59) n5 s rd n 5 CF s rd CF s rd... CF 2 s rd N Here N is the number of cash flows being paid and CF is the uniform amount of each cash flow. Solving for using Equation (4.59) would be a time-consuming problem if n were large. However, because the right-hand side of the equation is yet another geometric series, it can be simplified to yield the formula for finding the present value of an annuity. Retirement payments, bond interest payments, home mortgages, automobile loan payments, and lottery jackpot payoffs all often come in the form of annuities. 5 xcfc a 2 3> x rcn 4 b r (4.60) To convince you that Equations (4.59) and (4.60) are equivalent, let s work an example using both approaches. Suppose you wished to know the present value of a stream of $50 payments made semiannually over the next 2 years. The first payment is scheduled to begin 6 months from today. The annual rate of interest is 0%. The problem is illustrated with a time line. n = 0 n = n = 2 n = 3 n = 4 =? $50 $50 $50 $50 (st $50) (2nd $50) (3rd $50) + (4th $50) = of the annuity Using Equation (4.59), and recognizing that r 5 5% semiannually, this problem may be solved as follows: 5 $50 (4.6) s.05d $50 s.05d $50 2 s.05d $50 3 s.05d 4 5 $47.69 $45.35 $43.92 $4.35 (4.62) 5 $77.30 (4.63)

18 6 Chapter 4 Time Is Money TIME VALUE TABLE 4.3 Equation (4.60) may be written 5 (CF)(PVIFA N,r ) where PVIFA N,r is an acronym for present value interest factor for an annuity of N payments at a periodic rate of r%. Table A.3 tabulates values of PVIFA N,r for many common rates of interest and payment periods. A portion of Table A.3 is reproduced here. N % 2% 3% 4% 5% The problem in the text, an annuity paying $50 semiannually for 2 years and a 5% semiannual discount rate, can be solved using the table: 5 $50(PVIFA N 5 4,r 5 5% ) 5 $50(3.546) 5 $77.30 Alternatively, Equation (4.60) could be used to solve the same problem. 5 xcfc x 2 3> x rcn 4c r 5 x$50c 3 2 3> x.05c > x c4 5 x$50c s$50ds d x$50cx c $ (4.60) (4.64) (4.65) (4.66) (4.67) (4.68) 5 $77.30 (4.69) All of the steps are shown to aid you in following the calculations. It may appear that using Equation (4.60) is just as time-consuming as using Equation (4.59), but consider the work involved had there been 300 payments rather than 4. The problem just solved is an example of an ordinary annuity because cash flows commence at the end of the first period. Most loans require interest payments at the end of each period. Rent, on the other hand, is usually payable in advance. Annuities in which cash flows are made at the beginning of each period are called annuities due. Let s change the example we just worked slightly to require that the cash flows be made at the beginning of each period.

19 Chapter 4 Time Is Money 7 n = 0 n = n = 2 n = 3 n = 4 Ordinary $50 $50 $50 $50 n = 0 n = n = 2 n = 3 n = 4 Due $50 $50 $50 $50 The time line shows that in a four-payment annuity due, each payment occurs one period sooner than in an otherwise similar ordinary annuity. Because of this characteristic, each cash flow is discounted for one less period when finding the PV of an annuity due. The formula for finding the present value of an annuity due, PV due 0 5 xcfc a 2 3> x rcn 4 b x rc (4.70) r is simply the formula for an ordinary annuity times r, which adjusts for one less discounting period. Thus, it is usually easier to find the PV of an ordinary annuity and multiply times r when solving for the PV of an annuity due. PV due 0 5 PV ord 0 x rc (4.7) Now suppose you save $00 each month for 2 years in an account paying 2% interest annually, compounded monthly. What will be the balance in the account at the end of 2 years if you make your first deposit at the end of this month? n = 0 n = n = 2 n = 23 n = 24 $00 $00 $00 $00 4 In this case we are trying to solve for the future value of an ordinary annuity. 4 5 $00(.0) 23 $00(.0) $00 (4.72) Solving our problem in this manner would take considerable time. Fortunately, the future value of an annuity is also a geometric series, which can be simplified. The formula for the future value of an ordinary annuity is FV N 5 scfd s rdn 2 (4.73) r Substituting the values for our example into Equation (4.73) yields the solution: 4 5 s$00d s.0d24 2 (4.74) $2, (4.75)

20 8 Chapter 4 Time Is Money TIME VALUE TABLE 4.4 Equation (4.73) may be written, FV N 5 (CF)(FVIFA N,r ) where FVIFA N,r is an acronym for future value interest factor for an annuity of N payments at a periodic rate of r%. Values for common interest rates and payment periods are calculated for FVIFA and given in Table A.4 in the back of the text. A portion of Table A.4 is reproduced here. N % 2% 3% 4% Suppose you planned to deposit $00 in an account at the end of each of the next 4 years. If the account paid 3% annually, the balance you should have in your account at the end of the 4-year period is determined by the cash flow of $00 per period, N 5 4, and r 5 3%. FV 4 5 $00(FVIFA N 5 4,r 5 3% ) 5 $00(4.84) 5 $48.40 If the first deposit were made immediately, our problem would be one of finding the future value of an annuity due. n = 0 n = n = 2 n = 23 n = 24 $00 $00 $00 $00 4 Note that both the present value as well as the future value of an annuity due is always larger than an otherwise similar ordinary annuity. Each cash flow in an annuity due earns one additional period s interest compared to the future value of an ordinary annuity. Thus, the future value of an annuity due is equal to the future value of an ordinary annuity times r. FV due N 5 FV ord N x rc (4.76) FV due 24 5 ($2,697.35)(.0) (4.77) FV due 24 5 $2, (4.78) The future value of the deposits would therefore increase to $2, if they were made at the beginning of each period. Finally, note that the adjustment from an ordinary annuity to an annuity due is the same whether you are solving for PV or FV [compare Equations (4.76) and (4.7)].

21 Chapter 4 Time Is Money 9 Loan Amortization: An Annuity Application Many loans, such as home mortgages, require a series of equal payments made to the lender. Each payment is for an amount large enough to cover both the interest owed for the period as well as some principal. In the early stages of the loan, most of each payment covers interest owed by the borrower and very little is used to reduce the loan balance. Later in the loan s life, the small principal reductions have added up to a sum that has significantly reduced the amount owed. Thus, as time passes, less of each payment is applied toward interest and increasing amounts are paid on the principal. This type of loan is called an amortized loan. The final payment just covers both the remaining principal balance and the interest owed on that principal. An amortized loan is a direct application of the present value of an annuity. The original amount borrowed is the present value of the annuity ( ), while loan payments are the annuity s cash flows (CFs). If you borrow $00,000 to buy a house, what will your monthly payments be on a 30-year mortgage if the interest rate is 9% per year? For this problem the formula for finding the present value of an annuity is used [Equation (4.60)]. The present value is the loan amount ( 5 $00,000), there are 360 payments (N 5 360), and the monthly interest rate is 0.75% (9% 4 2 months). The payment amount (CF) is determined as follows: Many students will get jobs in banking. For them, working with loan amortizations will be an everyday occurrence. 5 xcfc x 2 3> x rcn 4c r $00,000 5 xcfc 2 3> x.0075c (4.60) (4.79) $00,000 5 xcfcx 2 > c (4.80) $00,000 5 xcfc x0.9324c (4.8) $00,000 5 (CF) (24.289) (4.82) CF 5 $ (4.83) A stream of 360 monthly payments of $ will cover the interest owed each month and will pay off all of the $00,000 loan as well. Of the first payment $ will be used to pay the interest owed the lender for the use of $00,000 during the first month at the 0.75% monthly rate. $54.62 of the first payment will be applied toward the principal. Thus, for the second month of the loan only $99, is owed. This reduces the amount of interest owed during the second month to $ and increases the second month s principal reduction to $ This pattern continues until the last payment when only a $ principal balance is remaining. The last month s interest on this balance is $5.99. Therefore, the last $ payment will just pay off the loan and pay the last month s interest too. Figure 4. illustrates how the amount of each payment applied toward principal increases over time, with a corresponding decrease in interest expense. Exhibit 4.5 is an amortization table showing principal and interest payments on a 5-year, $0,000 loan, amortized using a 0% rate compounded annually.

22 20 Chapter 4 Time Is Money FIGURE 4. AMORTIZED LOAN 30-year, monthly payments, $00,000 loan at 9% per year $ Total loan payment $54.62 Principal reduction $ $ Interest paid Payment number 360 $5.99 EXHIBIT 4.5 LOAN AMORTIZATION TABLE $0,000 LOAN, 5-YEAR AMORTIZATION, 0% INTEREST, COMPOUNDED ANNUALLY I II III IV V BEGINNING ENDING PRINCIPAL TOTAL PRINCIPAL PRINCIPAL PERIOD BALANCE PAYMENT INTEREST REDUCTION BALANCE $0,000 $2, $,000 3 $, $8, $ 8, $2, $ $,80.77 $6, $ 6, $2, $ $,98.94 $4, $ 4, $2, $ $2,80.4 $2, $ 2,398.8 $2, $ $2, Each period s beginning balance equals the prior period s ending balance. The loan balance at the beginning of the first year of this loan is obviously equal to the total loan amount. 2 This is solved by using equation (5.60). 3 Interest paid each period equals the rate times the period s beginning principal balance: Column III 5 (Column I)(r). 4 Principal reduction each period equals the total payment less the amount applied toward interest: Column IV 5 (Column II) 2 (Column III). 5 Ending principal balance equals the beginning balance minus the period s principal reduction: Column V 5 (Column I) 2 (Column IV)

23 Chapter 4 Time Is Money 2 Summary Chapter 4 has covered much of the topic of the time value of money. Next, the concepts and techniques covered here will be applied to finding the value of stocks, bonds, and other securities. Before that, however, it is best to practice the newly acquired skills. The authors cannot overemphasize the importance of mastering time value mathematics. Therefore, as you do your homework, make sure you feel confident in your ability. If not, now is a good time to seek out a quantitatively oriented friend or to investigate your instructor s office hours. Key Terms opportunity cost time line present value future value principal interest time value of money simple interest compounding periodic interest rate discounting discount rate effective annual percentage rate (EAR) nominal rates nominal annual percentage rate (APR) continuous compounding perpetuities geometric series annuity ordinary annuity annuities due amortized loan FVIF (See Time Value Table 4.) PVIF (See Time Value Table 4.2) FVIFA (See Time Value Table 4.4) PVIFA (See Time Value Table 4.3) annual percentage yield (APY) (See Problem 6.) rule of seventy-two (See Problem 4.) Key Formulas future value of a single cash flow with simple interest with compound interest FVn s 5 ( nr) FV n 5 ( r) n with continuous compounding FV n 5 (e rn ) present value of a single cash flow with compound interest 5 FV n ( r) 2n with continuous compounding 5 FV n (e 2rn ) EAR formula EAR 5 a APR cp CP b 2 general formula for finding the present value of a cash flow stream 5 CF s rd CF 2 s rd 2... CF N s rd N future value formula for a cash flow stream FV N 5 CF ( r) N2 CF 2 ( r) N22... CF N

24 22 Chapter 4 Time Is Money formula for the present value of a perpetuity present value of an annuity ordinary annuity e annuity due future value of an annuity ordinary annuity PV ord 5 CF r 0 5 scfd a 2 3>s rdn 4 b r PV due 0 5 PV ord 0 s rd FV ord N 5 scfd s rdn 2 r annuity due FV due N 5 FV ord N s rd Questions. Suppose you own some land, purchased by your father 20 years ago for $5,000. You are able to trade this land for a brand new Corvette sports car. What economic opportunity might you forego if you proceed with the trade? How would you estimate the opportunity cost of proceeding with the trade? 2. The Corvette dealership from Question is also willing to trade the car for an IOU you own that promises to pay you $2,000 at the end of each year for the next 0 years and $20,000 when it matures at the end of the 0-year period. Investors are currently valuing such IOUs using a 6% discount rate. What economic opportunity might you lose if you make the trade? How would you calculate the opportunity cost of the trade? 3. If the market for new automobiles and the real estate and bond markets are all efficient, what do you think you would discover about the opportunity costs of the trade in Questions and 2? 4. If you won the lottery, would you prefer a $ million cash settlement or $00,000 a year for 0 years (all else the same, such as taxes)? Why? 5. Would a borrower prefer that interest be compounded annually or semiannually, all else the same? 6. When interest rates rise, what happens to the present value of a fixed stream of cash flows? 7. Describe the essential differences between the following cash flow stream patterns. annuity versus perpetuity annuity due versus ordinary annuity 8. Why have many mortgage companies preferred to make home loans that have adjustable interest rates? 9. If you purchase a home and have the choice of making either an $800 payment each month on the mortgage or $400 payments on the first and fifteenth of every month, which option will allow you to pay off the loan more rapidly, assuming the same APR applies to both options? 0. XYZ, Inc., is signing a -year agreement with Acme Health Services to provide XYZ s employees with weekly aerobics classes at the firm s factory. Acme charges a flat fee for their services, with three payment options. XYZ can choose to pay Acme either $2,000 immediately, $,00 per month for

25 Chapter 4 Time Is Money 23 2 months, or $3,500 at the end of the year. How would you decide which payment option XYZ should choose?. Match each cash flow stream described with the time line illustrating the problem. Some of the diagrams do not match any of the cash flows described. a. A vacation cabin can be leased to provide its owner with cash flows of $4,000 per year for 5 years. Lease payments will be made at the beginning of each year starting immediately. The cabin s owner requires an 8% annual return on the investment in the cabin. What is the present value of the lease to the owner? b. Professor Hickman s daughter is 62 years old. She is expected to begin college in 2 years. Hickman estimates that college tuition costs will be $20,000 a year for each of the 4 years at that future time. Tuition is paid at the beginning of each year. What should be Hickman s savings goal if he wishes to have a balance in savings to fund his daughter s tuition expenses? The savings account is expected to earn 6% per year from now until the end of her 4 years at college. c. Referring to part b, Hickman plans to invest $200 immediately and at the end of each month in a mutual fund that is expected to earn 8% per year compounded monthly. Will he be able to fund his daughter s tuition with this savings plan? d. Your rich aunt passed away leaving you a financial security that pays $500 every 6 months forever. Honest John s Lending Emporium will loan you money with the security as collateral using a 20% annual discount rate, compounded semiannually. How much can you borrow against this financial security from Honest John? e. You want to save $30,000 for a down payment on a house. You wish to purchase the house 0 years from now. You can earn 9% per year on your savings and you plan to save an equal amount at the end of each month over this period. How much should you save to meet your objective? I r = 0% II r = 20% III 4,000 4,000 4,000 4,000 4,000 r = 8%

26 PV 2 r = 6% 24 Chapter 4 Time Is Money IV 4,000 4,000 4,000 4,000 4,000 r = 8% V r = 0.667% FV 44 VI ,000 20,000 20,000 20,000 Demonstration Problems. Suppose $,000 is deposited in an account that has an adjustable interest rate. During the first year, the rate is 0%, compounded annually. The second year the rate changes to 8%, compounded quarterly. What is the account s balance after 2 years? solution Illustrate the problem with a time line. n = 0 n = n = 2 = $,000 r = 0%/year FV r = 8%/year compounded quarterly =? Algebra Solution Set up the problem mathematically and solve: 5 FV (.02) 4 FV 5 (.0) Substituting

27 Chapter 4 Time Is Money 25 5 (.0)(.02) 4 5 $,000(.0)(.02) 4 5 $,000(.0)(.08243) 5 $,000(.90675) 5 $,90.68 Table Solution Solve the problem using the appropriate table: 5 FV (FVIF n 5 4 ) r 5 2% FV 5 (FVIF n 5 ) r 5 0% Look up table values and substitute, 5 (FVIF n 5 )(FVIF n 5 4 ) r 5 0% r 5 0% 5 $,000(.0)(.0824) 5 $,90.64 Calculator Finding FV : Clear registers. 000 [PV] 0 [PMT] 0 [I%] [N] [FV] Answer: $,00 Finding : Clear registers. 00 [PV] 0 [PMT] 2 [I%] 4 [N] [FV] Answer: $, An electronics shop advertises that it will sell the top-of-the-line big-screen TV for $2,000. The shop offers an interest rate of 2% per month for 24 months. What payments must be made in this amortized loan? solution Illustrate problem with a time line. n = 0 n = n = 2 n = 3 n = 23 n = 24 PMT PMT PMT PMT PMT =? = $2,000 2%

28 26 Chapter 4 Time Is Money Algebra Solution Set up the problem as the present value of an ordinary annuity. 5 (PMT) c 2 3>s rdn 4 d r $2,000 5 (PMT) c 2 3>s.02d24 4 d r $2,000 5 PMT [8.939] Table Solution $ payment 5 (PMT)(PVIFA n 5 24 ) $2,000 5 PMT(8.939) $2,000 5 PMT $ payment Calculator Clear registers [PV] 0 [FV] 2 [I%] 24 [N] [PMT] Answer: $05.74 Problems $2, PMT r 5 2%. Find the present value of $,000 to be received in one year for each of the following annual discount rates. a. 0% per year compounded annually b. 0% per year compounded monthly c. 0% per year compounded continuously 2. Find the present value of $2,500 to be received in 2 years for each of the following discount rates. a. 2% per year compounded annually b. 2% per year compounded semiannually c. 2% per year compounded quarterly d. 2% per year compounded monthly e. 2% per year compounded continuously 3. Find the future value of $,000 deposited for year in each of three banks, which pay the following annual rates of interest. a. 2% compounded annually b. 2% compounded monthly c. 2% compounded continuously