Introduction to Financial Management (Chapters 1, 2, 3, 4) Part 4 Capital Structure and Dividend Policy (Chapters 15,
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Introduction to Financial Management (Chapters 1, 2, 3, 4)
Part 4
Capital Structure and Dividend Policy (Chapters 15, 16)
Part 2
Valuation of Financial Assets (Chapters 5, 6, 7, 8, 9, 10)
Part 5
Liquidity Management and Special Topics in Finance (Chapters 17, 18, 19, 20)
Part 3
Capital Budgeting (Chapters 11, 12, 13, 14)
C H A P T E R
5
Part 1
Time Value of Money The Basics Chapter Outline 5.1
Using Timelines to Visualize Cash Flows
5.2
(pgs. 128–130)
Compounding and Future Value (pgs. 130–137)
5.3
Discounting and Present Value (pgs. 137–143)
5.4
Making Interest Rates Comparable (pgs. 144–147)
Objective 1. Construct cash flow timelines to organize your analysis of time value of money problems.
Objective 2. Understand compounding and calculate the future value of cash flows using mathematical formulas, a financial calculator, and an Excel spreadsheet. Objective 3. Understand discounting and calculate the present value of cash flows using mathematical formulas, a financial calculator, and an Excel spreadsheet. Objective 4. Understand how interest rates are quoted and know how to make them comparable.
ISBN 1256147850 Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
Principle
P
1 Applied
Chapters 5 and 6 are dedicated to P Principle 1: Money Has a Time Value. This basic idea—a dollar received today, other things being the same, is worth more than a dollar received a year from now—underlies many financial decisions faced in business. In Chapter 1 we discussed capital budgeting, capital structure, and working capital management decisions—each of these decisions involves aspects of the time value of money. In this chapter, we learn how to calculate the value today of money
you will receive in the future, as well as the future value of money you have today. In Chapter 6 we extend our analysis to multiple cash flows spread out over time. In later chapters, we will use the skills we gain from Chapters 5 and 6 to analyze bond (Chapter 9) and stock (Chapter 10) prices, analyze the value of investment opportunities (Chapters 11–13), and determine the cost of financing the firm’s investments (Chapter 14).
ISBN 1256147850
Payday Loans Sometimes marketed to college students as quick relief for urgent expenses, a payday loan is a shortterm loan to cover expenses until the next payday. As some borrowers turn to these loans during times of financial desperation, lenders can charge them extremely high rates of interest. For example, in early 2010, one payday lender advertised that you could borrow $500 and repay $626.37 in eight days. This might not sound like a bad deal on the surface, but if we apply some basic rules of finance to analyze this loan, we see quite a different story. The annual interest rate for this payday loan is a whopping 2,916,780%! (will examine this on pages 144–145) The very high rates of interest charged by these lenders have led some states to impose limits on the interest rates payday lenders can charge. Even so, the cost of this type of loan can be extremely high. Understanding the time value of money is an essential tool to analyzing the cost of this and other types of financing.
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
127
128
PART 2

Valuation of Financial Assets
Regardless of Your Major... Suppose that you and your classmate each receive a gift of $10,000 from grandparents, but choose different ways to invest the new found money. You immediately invest your gift until retirement, while your classmate carries around his gift in his wallet in the form of 100 crisp $100 bills. Then, after 15 years of carrying around a fat wallet, your classmate decides to invest his $10,000 for retirement. If you invest your $10,000 for 46 years and earn 10% per year until you retire, you’ll end up with over $800,000. If your classmate invests his $10,000 for 31 years (remember he carried his money around for 15 years in his wallet) and earns the same 10% per year, he’ll only end up with about $192,000. Knowing about the power of the time value of money provided you with an additional $600,000 at retirement. In this chapter we’ll learn more about these kinds of valuation problems. For now, keep in mind that the time value of money is a concept you will want to understand, regardless of your major.
A Dollar Saved Is Two Dollars Earned” “
5.1
Using Timelines to Visualize Cash Flows
To evaluate a new project, a financial manager must be able to compare benefits and costs that occur at different times. We will use the time value of money tools we develop in this chapter to make the benefits and costs comparable, allowing us to make a logical decision. We begin our study of time value analysis by introducing some basic tools. As a first step, we can construct a timeline, a linear representation of the timing of cash flows. A timeline identifies the timing and amount of a stream of payments—both cash received and cash spent—along with the interest rate earned. Timelines are a critical first step that financial analysts use to solve financial problems, and we will refer to timelines throughout this text. To learn how to construct a timeline, consider the following example, where we have annual cash inflows and outflows over the course of four years. The following timeline illustrates these cash inflows and outflows from time Period 0 (the present) until the end of year 4: i = 10% Time Period 0 Cash Flow $100
Beginning of Period 1
1
2
3
4
$30
$20
$10
$50
End of Period 2 & Beginning of Period 3
Years
End of Period 4 & Beginning of Period 5
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
For our purposes, time periods are identified on the top of the timeline. In this example, the time periods are measured in years, indicated on the far right of the timeline. For example, time Period 0 in the above example is the current year. The dollar amount of the cash flow received or spent during each time period is shown below the timeline. Positive values represent cash inflows. Negative values represent cash outflows. For example, in the timeline shown, a $100 cash outflow occurs at the beginning of the first year (at time 0), followed by cash inflows of $30 and $20 in years 1 and 2, a cash outflow (a negative cash flow) of $10 in year 3, and finally a cash inflow of $50 in year 4. Timelines are typically expressed in years, but could be expressed in months, days, or, for that matter, any unit of time. For now, let’s assume we’re looking at cash flows that occur annually, so the distance between 0 and 1 represents the time period between today and the end of the first year. The interest rate, 10% in this example, is listed above the timeline.
CHAPTER 5  Time Value of Money Checkpoint 5.1
Creating a Timeline Suppose you lend a friend $10,000 today to help him finance a new Jimmy John’s Sub Shop franchise and in return he promises to give you $12,155 at the end of the fourth year. How can one represent this as a timeline? Note that the interest rate is 5%. STEP 1: Picture the problem A timeline provides a tool for visualizing cash flows and time:
i = rate of interest 0
1
2
3
4
Cash Flow Year 0
Cash Flow Year 1
Cash Flow Year 2
Cash Flow Year 3
Cash Flow Year 4
Time Period Cash Flow
Years
STEP 2: Decide on a solution strategy To complete the timeline we simply record the cash flows onto the template. STEP 3: Solve We can input the cash flows for this investment on the timeline as shown below. Time period zero (the present) is shown at the left end of the timeline, and future time periods are shown above the timeline, moving from left to right, with the year that each cash flow occurs shown above the timeline. Keep in mind that year 1 represents the end of the first year as well as the beginning of the second year.
i = 5% Time Period
0
1
2
3
Cash Flow $10,000
4
Years
$12,155
STEP 4: Analyze Using timelines to visualize cash flows is useful in financial problem solving. From analyzing the timeline, we can see that there are two cash flows, an initial $10,000 cash outflow, and a $12,155 cash inflow at the end of year 4. STEP 5: Check yourself Draw a timeline for an investment of $40,000 today that returns nothing in one year, $20,000 at the end of year 2, nothing in year 3, and $40,000 at the end of year 4, where the interest rate is 13.17%. ANSWER: i = 13.17% Time Period Cash Flow
0
1
2
3
4
$40,000
$0
$20,000
$0
$40,000
Years
ISBN 1256147850
>> END Checkpoint 5.1
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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Valuation of Financial Assets
Before you move on to 5.2
Concept Check  5.1 1. What is a timeline, and how does it help you solve time value of money problems? 2. Does year 5 represent the end of the fifth year, the beginning of the sixth year, or both?
5.2
Compounding and Future Value
If we assume that an investment will only earn interest on the original principal, we call this simple interest. Suppose that you put $100 in a savings account earning 6% interest annually. How much will your savings grow after one year? If you invest for one year at an interest rate of 6%, you will earn 6% simple interest on your initial deposit of $100, giving you a total of $106 in your account. What if you left your $100 in the bank for two years? In this case, you will earn interest not only on your original $100 deposit but also on the $6 in interest you earned during the first year. This process of accumulating interest on an investment over multiple time periods is called compounding. And, when interest is earned on both the initial principal and the reinvested interest during prior periods, the result is called compound interest. Time value of money calculations are essentially comparisons between what we will refer to as Present Value, what a cash flow would be worth to you today, and Future Value, what a cash flow will be worth in the future. The following is a mathematical formula that shows how these concepts relate to each other when the future value is in 1 year: Future Value in 1 year = Present Value * 11 + interest rate2
(5–1)
In the above example, you began with a $100 investment, so the present value is $100. The future value in one year is then given by the equation $100 * 11 + .062 = $106.00
To see how to calculate the future value in two years, let’s do a timeline and a few calculations: i = 6% Time Period
Cash Flow
0
1
$100
x 1.06 =
2
$106
x 1.06 =
Years
$112.36
During the first year, your $100 deposit earns $6 in interest. Summing the interest and the original deposit gives you a balance of $106 at the end of the first year. In the second year, you earn $6.36 in interest, giving you a future value of $112.36. Why do you earn $0.36 more in interest during the second year than during the first? Because in the second year, you earn an additional 6% on the $6 interest you earned in the first year. This amounts to $0.36 (or $6 .06). Again, this result is an example of compound interest. Anyone who has ever had a savings account or purchased a government savings bond has received compound interest. What happens to the value of your investment at the end of the third year, assuming the same interest rate of 6%? We can follow the same approach to calculate the future value in three years. Using a timeline, we can calculate the future value of your $100 as follows: i = 6% Time Period
$100
1
x1.06
= $106
2
x 1.06
= $112.36
3
x 1.06
Years
= $119.10
Note that every time we extend the analysis for one more period we just multiply the previous balance by (1 interest rate). Consequently, the future value of any amount of money for any number of periods can be expressed with the following equation where n the number of periods during which the compounding occurs: Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
Cash Flow
0
CHAPTER 5  Time Value of Money
Future Present Interest * a1 + b = ValuePeriod n Value 1Deposit2 rate 1i2
131
n
or Number of Years (n) Future Value Annual Present 1 + in year n = a Interest Rate 1i2 b Value 1PV2 1FVn 2
(5–1a)
Important Definitions and Concepts:
• FVn the future value of the investment at the end of n periods. • i the interest (or growth) rate per period. • PV the present value, or original amount invested at the beginning of the first period. We also refer to (1 i)n as the Future Value Interest Factor. To find the future value of a dollar amount, simply multiply that dollar amount by the appropriate Future Value Interest Factor, FVn = PV11 + i2n FVn = PV * Future Value Interest Factor
where, Future Value Interest Factor = 11 + i2n Panel A in Figure 5.1 shows what your investment of $100 would grow to in four years if it continues to earn an annual compound interest rate of 6%. Notice how the amount of interest earned increases each year. In the first year, you earn only $6, but by year 4, you earn $7.15 in interest. Prior to the introduction of inexpensive financial calculators and Excel, future values were commonly calculated using time value of money tables containing Future Value Interest Factors for different combinations of i and n. Table 5.1 provides an abbreviated Future Value Interest Factor Table; you can find the expanded Future Value Interest Factor Tables in Appendix B at the back of the book. So, to find the value of $100 invested for 4 years at 6%, we would simply look at the intersection of the n 4 row and the 6% column, finding a Future Value Interest Factor of 1.262. We would then multiply this value by $100 to find that our investment of $100 at 6% for 4 years would grow to $126.20.
Compound Interest and Time As Panel C of Figure 5.1 shows, the future value of an investment grows with the number of periods we let it compound. For example, after five years, the future value of $100 earning 10% interest each year will be $161.05. However, after 25 years, the future value of that investment will be $1,083.47. Note that although we increased the number of years threefold, the future value increases by more than sixfold ($1,083.47/$161.05 6.7 fold). This illustrates an important point. Future value is not directly proportional to time. Instead, future value grows exponentially. This means it grows by a fixed percentage each year, which means that the dollar value grows by an increasing amount each year.
ISBN 1256147850
Compound Interest and the Interest Rate Panel C of Figure 5.1 illustrates that future value increases dramatically with the level of the rate of interest. For example, the future value of $100 in 25 years, given a 10% interest rate, compounded annually, is $1,083.47. However, if we double the rate of interest to 20%, the future value increases almost ninefold in 25 years to equal $9,539.62. This illustrates another important point. The increase in future value is not directly proportional to the increase in the rate of interest. We doubled the rate of interest, and the future value of the investment increased by 8.80 times. Why did the future value jump by so much? Because there is a lot of time over 25 years for the higher interest rate to result in more interest being earned on interest.
Techniques for Moving Money through Time In this book, we will refer to three methods for solving time value of money problems: mathematical formulas, financial calculators, and spreadsheets. Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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Valuation of Financial Assets
Figure 5.1 Future Value and Compound Interest Illustrated (Panel A)
Interest earned = Beginning Value the interest rate
Calculating Compound Interest
This panel shows how interest compounds annually. During the first year, $100 invested at a 6% interest rate earns only $6. Since we earn 6% on the ending value for year 1 (or $106), in year 2 we earn $6.36 in interest. This increase in the amount of interest results from interest being earned on both the initial deposit of $100 plus the $6.00 in interest earned during year 1. The fact that we earn interest on both principal and interest is why we refer to this as compound interest. Simple interest, on the other hand, would be earning only $6.00 interest each and every year. $400 Future value of $100 compounded at a rate of 8% per year for 15 years = $317.22
$350
$317.22
(Panel B)
Future value
$300 Future value of $100 compounded at a rate of 8% per year for 10 years = $215.89
$250 $200
Year
Beginning Value
Interest Earned
Ending Value
1
$ 100.00
$ 6.00
$ 106.00
2
$ 106.00
$ 6.36
$ 112.36
3
$ 112.36
$ 6.74
$ 119.10
4
$ 119.10
$ 7.15
$ 126.25
The Power of Time
This figure illustrates the importance of time when it comes to compounding. Because interest is earned on past interest, the future value of $100 deposited in an account that earns 8% compounded annually grows over threefold in 15 years. If we were to expand this figure to 45 years (which is about how long you have until you retire, assuming you’re around 20 years old right now), it would grow to over 31 times its initial value.
$215.89
$150 $100 Future value of $100 compounded at a rate of 0% per year equals $100 regardless of how many years it is invested.
$50 $0
0
5 10 Number of periods in years
15 $25,000 $23,737.63 Future value of $100 compounded at a rate of 20% per year for 30 years = $23,737.63
$20,000
The Power of the Rate of Interest
This figure illustrates the importance of the interest rate in the power of compounding. As the interest rate climbs, so does the future value. In fact, when we change the interest rate from 10% to 20%, the future value in 25 years increases by over 8 times, jumping from $1,083.47 to $9,539.62.
Future value
(Panel C)
Future value of $100 compounded at a rate of 20% per year for 25 years = $9,539.62
$15,000
Future value of $100 compounded at a rate of 15% per year for 25 years = $3,291.90
$10,000
$9,539.62 Future value of $100 compounded at a rate of 10% per year for 25 years = $1,083.47 $5,000
$1,083.47 $0
0
5
10
15
20
25
30
Number of periods in years
>> END FIGURE 5.1 Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
$3,291.90
CHAPTER 5  Time Value of Money Table 5.1
133
Future Value Interest Factors
Number of Periods (n)
i 3%
i 6%
i 9%
i 12%
1
1.030
1.060
1.090
1.120
2
1.061
1.124
1.188
1.254
3
1.093
1.191
1.295
1.405
4
1.126
1.262
1.412
1.574
• Do the math. You can use the mathematical formulas just as we have done in this chapter. You simply substitute the values that you know into the appropriate time value of money equation to find the answer. • Use financial calculators. Financial calculators have preprogrammed functions that make time value of money calculations simple. • Use a spreadsheet on your personal computer. Spreadsheet software such as Excel has preprogrammed functions built into it. The same inputs that are used with a financial calculator are also used as inputs to Excel. As a result, if you can correctly set a problem up to solve on your financial calculator, you can easily set it up to solve using Excel. In the business world, Excel is the spreadsheet of choice and is the most common way of moving money through time. In Appendix A at the back of the book, we will show how to solve valuation problems using each of these methods. Because the authors of this book believe that spending enough time solving problems the oldfashioned way—by doing the math—leads to a deeper understanding and better retention of the concepts found in this book, we will first demonstrate how to solve problems using the formulas; but we will also demonstrate, whenever possible, how to derive the solution using a financial calculator and Excel.
Applying Compounding to Things Other Than Money While this chapter focuses on moving money through time at a given interest rate, the concept of compounding applies to almost anything that grows. For example, let’s suppose we’re interested in knowing how big the market for wireless printers will be in five years, and assume the demand for them is expected to grow at a rate of 25% per year over the next five years. We can calculate the future value of the market for printers using the same formula we used to calculate the future value for a sum of money. If the market is currently 25,000 printers per year, then 25,000 would be PV, n would be 5, and i would be 25%. Substituting into Equation (5–1a) we would solve for FV,
ISBN 1256147850
Number of Future Value Years (n) Annual Present in year n = = 25,000(1 + .25)5 = 76,293 a1 + b Value (PV) Interest Rate (i) (FVn )
The power of compounding can also be illustrated through the story of a peasant who wins a chess tournament sponsored by the king. The king then asks him what he would like as his prize. The peasant answers that, for his village, he would like one grain of wheat to be placed on the first square of his chessboard, two pieces on the second square, four on the third square, eight on the fourth square, and so forth, until the board is filled up. The king, thinking he was getting off easy, pledged his word of honor that this would be done. Unfortunately for the king, by the time all 64 squares on the chessboard were filled, there were 18.5 million trillion grains of wheat on the board, because the kernels were compounding at a rate of 100% over the 64 squares of the chessboard. In fact, if the kernels were onequarter inch long they would have stretched, if laid end to end, to the sun and back 391,320 times! Needless to say, no one in the village ever went hungry. What can we conclude from this story? There is incredible power in compounding.
Compound Interest with Shorter Compounding Periods So far, we have assumed that the compounding period is always a year in length. However, this isn’t always the case. For example, banks often offer savings accounts that compound interest Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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PART 2

Valuation of Financial Assets
Checkpoint 5.2
Calculating the Future Value of a Cash Flow You are put in charge of managing your firm’s working capital. Your firm has $100,000 in extra cash on hand and decides to put it in a savings account paying 7% interest compounded annually. How much will you have in your account in 10 years? STEP 1: Picture the problem We can set up a timeline to identify the cash flows from the investment as follows: i = 7% 0
Time Period
1
2
3
4
5
6
7
8
9
Cash Flow $100,000
10
Years
Future Value = ?
STEP 2: Decide on a solution strategy This is a simple future value problem. We can find the future value using Equation 5–1a. STEP 3: Solve Using the Mathematical Formulas. Substituting PV $100,000, i 7%, and n 10 years into Equation (5–1a), we get Number of Future Value Years (n) Present Annual in year n = a1 + b Value (PV) Interest Rate (i) (FVn )
(5–1a)
FVn = $100,000(1 + .07)10 = $100,00011.967152 = $196,715 At the end of ten years, you will have $196,715 in the savings account. Using a Financial Calculator. Enter
10
7.0
–100,000
0
N
I/Y
PV
PMT
Solve for
FV 196,715
Using an Excel Spreadsheet. = FV(rate,nper,pmt,pv) or with values entered = FV(0.07,10,0,–100000) STEP 4: Analyze Notice that you input the present value with a negative sign because present value represents a cash outflow. In effect, the money leaves your firm when it’s first invested. In this problem your firm invests $100,000 at 7% and found that it will grow to $196,715 after 10 years. Put another way, given a 7% compound rate, your $100,000 today will be worth $196,715 in 10 years. STEP 5: Check yourself What is the future value of $10,000 compounded at 12% annually for 20 years?
Your Turn: For more practice, do related Study Problems 5–1, 5–2, 5–4, 5–6, and 5–8 through 5–11 at the end of this chapter.
>> END Checkpoint 5.2
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
ANSWER: $96,462.93
CHAPTER 5  Time Value of Money
135
every day, month, or quarter. Savers prefer more frequent compounding because they earn interest on their interest sooner and more frequently. Fortunately, it’s easy to adjust for different compounding periods, and later in the chapter we will examine how to compare two loans with different compounding periods in more detail. Consider the following example: You invest $100 for five years to earn a rate of 8%, and the investment is compounded semiannually (twice a year). This means that interest is calculated every six months. Essentially, you are investing your money for ten sixmonth periods, and in each period, you will receive 4% interest. In effect, we divide the annual interest rate (i) by the number of compounding periods per year (m), and we multiply the number of years (n) times the number of compounding periods per year (m) to convert the number of years into the number of periods. So, our future value formula found in Equation (5–1a) must be adjusted as follows: m * A Number of Years (n) B
Annual Future Value Interest Rate (i) Present ±1 + ≤ in year n = Value (PV) Compounding (FVn) Periods per Year (m)
(5–1b)
Substituting into Equation (5–1b) gives us the following estimate of the future value in five years: FVn = $100 11 + .08>222 * 5 = $10011.48022 = $148.02 But if the compounding had been annual rather than semiannual, the future value of the investment would have been only $146.93. Although the difference here seems modest, it can be significant when large sums of money are involved and the number of years and the number of compounding periods within those years are large. For example, for your $100 investment, the difference is only $1.09. But if the amount were $50 million (not an unusually large bank balance for a major company), the difference would be $545,810.41. Table 5.2 shows how shorter compounding periods lead to higher future values. For example, if you invested $100 at 15% for one year, and the investment was compounded daily Table 5.2
The Value of $100 Compounded at Various NonAnnual Periods and Various Rates
ISBN 1256147850
Notice the impact of shorter compounding periods is heightened by both higher interest rates and compounding over longer time periods.
For 1 Year at i Percent
i2%
5%
10%
15%
Compounded annually
$102.00
$105.00
$110.00
$115.00
Compounded semiannually
102.01
105.06
110.25
115.56
Compounded quarterly
102.02
105.09
110.38
115.87
Compounded monthly
102.02
105.12
110.47
116.08
Compounded weekly (52)
102.02
105.12
110.51
116.16
Compounded daily (365)
102.02
105.13
110.52
116.18
For 10 Years at i Percent
i2%
5%
10%
15%
Compounded annually
$121.90
$162.89
$259.37
$404.56
Compounded semiannually
122.02
163.86
265.33
424.79
Compounded quarterly
122.08
164.36
268.51
436.04
Compounded monthly
122.12
164.70
270.70
444.02
Compounded weekly (52)
122.14
164.83
271.57
447.20
Compounded daily (365)
122.14
164.87
271.79
448.03
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
x
$1.18
x $43.47
Checkpoint 5.3
Calculating Future Values Using NonAnnual Compounding Periods You have been put in charge of managing your firm’s cash position and noticed that the Plaza National Bank of Portland, Oregon, has recently decided to begin paying interest compounded semiannually instead of annually. If you deposit $1,000 with Plaza National Bank at an interest rate of 12%, what will your account balance be in five years? STEP 1: Picture the problem If you earn a 12% annual rate compounded semiannually for five years, you really earn 6% every six months for 10 sixmonth periods. Expressed as a timeline, this problem would look like the following: i = 12% ÷ 2 = 6% every 6 months 0
Time Period
Cash Flow
1
2
3
4
5
6
7
8
9
10
6Month Periods
Future Value of $1,000 compounded for 10 sixmonth periods at 12%/2 every 6 months
$1,000
STEP 2: Decide on a solution strategy In this instance we are simply solving for the future value of $1,000. The only twist is that interest is calculated on a semiannual basis. Thus, if you earn 12% compounded semiannually for five years, you really earn 6% every six months for 10 sixmonth periods. We can calculate the future value of the $1,000 investment using Equation (5–1b). STEP 3: Solve Using the Mathematical Formulas. Substituting number of years (n) 5, number of compounding periods per year (m) 2, annual interest rate (i) 12%, and PV $1,000 into Equation (5–1b): m * A Number of Years (n) B Annual Interest Rate (i) ≤ Compounding Periods per Year (m)
Future Value Present ±1 + in year n = Value (PV) (FVn )
Future Value in year n (FVn) = $1,000 a1 +
.12 2 * 5 = $1,000 * 1.79085 = $1,790.85 b 2
Using a Financial Calculator. Enter
10
6.0
1,000
0
N
I/Y
PV
PMT
Solve for
FV 1,790.85
You will have $1,790.85 at the end of five years. Using an Excel Spreadsheet. FV(rate,nper,pmt,pv) or with values entered FV(0.06,10,0,1000) STEP 4: Analyze The more often interest is compounded per year—that is, the larger m is, resulting in a larger value of nper— the larger the future value will be. That’s because you are earning interest more often on the interest you’ve previously earned.
If you deposit $50,000 in an account that pays an annual interest rate of 10% compounded monthly, what will your account balance be in 10 years? ANSWER: $135,352.07 Your Turn: For more practice, do related Study Problems 5–5 and 5–7 at the end of this chapter.
>> END Checkpoint 5.3
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
STEP 5: Check yourself
CHAPTER 5  Time Value of Money
Life
ss
e Busin
The Business of Life Saving for Your First House There was a time in the early and mid2000s when you didn’t need to worry about a down payment when you bought a new house. But that all changed as the housing bubble burst and home prices fell. Today you may be able to get away with only putting down around 10%, but the rate on your mortgage will be lower if you can come up with 20%. To buy a median priced home, which was just over $170,000 at the beginning of 2010, you’d have to come up with a 10% down payment of $17,000 or a 20% down payment of $34,000. On top of that, you’ll need to furnish your new home, and that costs money too. Putting into practice what you have learned in this chapter, you know that the sooner you start to save for your first home, the easier it will be. Once you estimate how much you’ll need for that Your Turn: See
137
new house, you can easily calculate how much you’ll need to save annually to reach your goal. All you need to do is look at two variables: n (the number of years you’ll be saving the money) and i (the interest rate at which the savings will grow). You can start saving earlier, which gives you a larger value for n. Or, you can earn more on your investments. That is, invest at a higher value for i. Of course, you always prefer getting a higher i on your savings, but this is not something you can control. First, let’s take a look at a higher value for i, which translates into a higher return. For example, let’s say you’ve just inherited $10,000, and you invest it at 6% annually for 10 years—after which you want to buy your first house. The calculation is easy. At the end of 10 years, you will have accumulated $17,908 on this investment. But suppose you are able to earn 12% annually for 10 years. What would the value of your investment be then? In this case, your investment would be worth $31,058. Needless to say, the rate of interest that you earn plays a major role in determining how quickly your investment will grow. Now consider what happens if you wait five years before investing your $10,000. The value of n drops from 10 to 5, and, as a result, the amount you’ve saved also drops. In fact, if you invested your $10,000 for five years at 6%, you’d end up with $13,382, and even at 12% you’d end up with only $17,623. The bottom line is the earlier you begin saving, the more impact every dollar you save will have.
Study Problem 5–3.
rather than annually, you would end up with $1.18 ($116.18 $115.00) more. However, if the period were extended to 10 years, then the difference grows to $43.47 ($448.03 $404.56).
Before you move on to 5.3
Concept Check  5.2 1. What is compound interest, and how is it calculated? 2. Describe the three basic approaches that can be used to move money through time. 3. How does increasing the number of compounding periods affect the future value of a cash sum?
ISBN 1256147850
5.3
Discounting and Present Value
So far we have only been moving money forward in time; that is, we have taken a known present value of money and determined how much it will be worth at some point in the future. Financial decisions often require calculating the future value of an investment made today. However, there are many instances where we want to look at the reverse question: What is the value today of a sum of money to be received in the future? To answer this question, we now turn our attention to the analysis of a present value—the value today of a future cash flow—and the process of discounting, determining the present value of an expected future cash flow. Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
PART 2

Valuation of Financial Assets
The Mechanics of Discounting Future Cash Flows Discounting is actually the reverse of compounding. We can demonstrate the similarity between compounding and discounting by referring back to the future value formula found in Equation (5–1a): Number of
Future Value Years (n) Annual Present 1 + in year n = a b Interest Rate (i) Value (PV) (FVn )
(5–1a)
To determine the present value of a known future cash flow, we simply take Equation (5–1a) and solve for PV: 1 Future Value Number of Years (n) Annual Present = in year n J a 1 + Interest Rate (i) b K Value (PV) (FVn)
(5–2)
We also refer to the term in the brackets as the Present Value Interest Factor, which is the value that multiples the future value to calculate the present value. Thus, to find the present value of a future cash flow, we multiply the future cash flow by the Present Value Interest Factor.1 Present = Value (PV)
Present Value Future Value in year n * Interest Factor P (PVIF) Q (FVn)
where Present Value Interest Factor 1PVIF2 =
1 11 + i2n
Note that the present value of a future sum of money decreases as we increase the number of periods, n, until the payment is received, or as we increase the interest rate, i. That, of course, only makes sense because the Present Value Interest Factor is the inverse of the Future Value Interest Factor. Graphically, this relationship can be seen in Figure 5.2. Thus, given a discount rate, or interest rate at which money is being brought back to present, of 10%, $100 received in 10 years would be worth only $38.55 today. By contrast, if the discount rate is 5%, the present value would be $61.39. If the discount rate is 10%, but the $100 is received in five years Figure 5.2 The Present Value of $100 Compounded at Different Rates and for Different Time Periods The present value of $100 to be received in the future becomes smaller as both the interest rate and number of years rise. At i 10%, notice that when the number of years goes up from 5 to 10, the present value drops from $62.09 to $38.55. $120.00 Present value (dollars)
138
1 $62.09 = $100 (1 + .10)5
[
]
$100.00 $80.00 $62.09
$60.00 $40.00
$38.55 $20.00 $0.00
1 $38.55 = $100 (1 + .10)10
[
2
4
6
8
]
10
Year
>> END FIGURE 5.2
1
Related tables appear in Appendix C at the end of the book.
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
0
CHAPTER 5  Time Value of Money Checkpoint 5.4
Solving for the Present Value of a Future Cash Flow Your firm has just sold a piece of property for $500,000, but under the sales agreement, it won’t receive the $500,000 until ten years from today. What is the present value of $500,000 to be received ten years from today if the discount rate is 6% annually? STEP 1: Picture the problem Expressed as a timeline, this problem would look like the following: i = 6% Time Period
Cash Flow
0
1
2
3
4
5
6
7
8
Present Value = ?
9
10
Years
$500,000
STEP 2: Decide on a solution strategy In this instance we are simply solving for the present value of $500,000 to be received at the end of 10 years. We can calculate the present value of the $500,000 using Equation (5–2). STEP 3: Solve Using the Mathematical Formulas. Substituting FV10 $500,000, n 10, and i 6% into Equation (5–2), we find PV = $500,000 c = $500,000 c
1 (1 + .06)10
d
1 d 1.79085
$500,000 [.558394] $279,197 The present value of the $500,000 to be received in ten years is $279,197. Earlier we noted that discounting is the reverse of compounding. We can easily test this calculation by considering this problem in reverse: What is the future value in ten years of $279,197 today if the rate of interest is 6%? Using our FV Equation (5–1a), we can see that the answer is $500,000. Using a Financial Calculator. Enter
10
6.0
N
I/Y
Solve for
PV
0
500,000
PMT
FV
279,197
Using an Excel Spreadsheet. PV(rate,nper,pmt,fv) or with values entered PV(0.06,10,0,500000) STEP 4: Analyze
ISBN 1256147850
Once you’ve found the present value of any future cash flow, that present value is in today’s dollars and can be compared to other present values. The underlying point of this exercise is to make cash flows that occur in different time periods comparable so that we can make good decisions. Also notice that regardless of which method we use to calculate the future value—computing the formula by hand, with a calculator, or with Excel— we always arrive at the same answer. STEP 5: Check yourself What is the present value of $100,000 to be received at the end of 25 years given a 5% discount rate? ANSWER: $29,530. Your Turn: For more practice, do related Study Problems 5–12, 5–15, and 5–19 at the end of this chapter.
>> END Checkpoint 5.4
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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Valuation of Financial Assets
instead of 10 years, the present value would be $62.09. This concept of present value will play a central role in valuing stocks, bonds, and new proposals. You can easily verify this calculation using any of the discounting methods we describe next.
Two Additional Types of Discounting Problems Time value of money problems do not always involve calculating either the present value or future value of a series of cash flows. There are a number problems that require that you solve for either the number of periods in the future, n, or the rate of interest, i. For example, to answer the following question you will need to calculate the value for n. • How many years will it be before the money I have saved will be enough to buy a second home? • How long will it take to accumulate enough money for a down payment on a new retail outlet? And to answer the following questions you must solve for the interest rate, i. • What rate do I need to earn on my investment to have enough money for my newborn child’s college education (n 18 years)? • If our firm introduces a new product line, what interest rate will this investment earn? Fortunately, with the help of the mathematical formulas, a financial calculator, or an Excel spreadsheet, you can easily solve for i or n in any of the above situations.
Solving for the Number of Periods Suppose you want to know how many years it will take for an investment of $9,330 to grow to $20,000 if it’s invested at 10% annually. Let’s take a look at how to solve this using the mathematical formulas, a financial calculator, and an Excel spreadsheet. Using Mathematical Formulas. Substituting for FV, PV, and i in Equation (5–1a), Number of
Future Value Years (n) Annual Present in year n = a 1 + Interest Rate (i) b Value (PV) (FVn )
(5–1a)
$20,000 = $9,33011.102n Solving for n mathematically is tough. One way is to solve for n using a trialanderror approach. That is, you could substitute different values of n into the equation—either increasing the value of n to make the righthand side of the equation larger, or decreasing the value of n to make it smaller, until the two sides of the equation are equal—but that will be a bit tedious. Using the time value of money features on a financial calculator or in Excel is much easier and faster. Using a Financial Calculator. Using a financial calculator or an Excel spreadsheet, this problem becomes much easier. With a financial calculator, all you do is substitute in the values for i, PV, and FV, and solve for n: Enter N Solve for
10.0
9,330
0
20,000
I/Y
PV
PMT
FV
8.0
Using an Excel Spreadsheet. With Excel, solving for n is straightforward. You simply use the NPER(rate,pmt,pv,fv) or with variables entered, NPER(0.10,0,–9330,20000). Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
You’ll notice that PV is input with a negative sign. In effect the financial calculator is programmed to assume that the $9,330 is a cash outflow (the money leaving your hands), whereas the $20,000 is money that you will receive. If you don’t give one of these values a negative sign, you can’t solve the problem.
CHAPTER 5  Time Value of Money
141
Checkpoint 5.5
Solving for the Number of Periods, n Let’s assume that the Toyota Corporation has guaranteed that the price of a new Prius will always be $20,000, and you’d like to buy one but currently have only $7,752. How many years will it take for your initial investment of $7,752 to grow to $20,000 if it is invested so that it earns 9% compounded annually? STEP 1. Picture the problem In this case we are solving for the number of periods: i =9% 0
Time Period
Cash Flow
1
2
3
4
.... ....
5
$7,752
?
Years
$20,000
STEP 2. Decide on a solution strategy In this problem we know the interest rate, the present value, and the future value, and we want to know how many years it will take for $7,752 to grow to $20,000 at 9% per year. We are solving for n, and we can calculate it using Equation (5–1a). STEP 3. Solve Using a Financial Calculator. Enter N Solve for
9.0
7,752
0
20,000
I/Y
PV
PMT
FV
11.0
Using an Excel Spreadsheet. NPER(rate,pmt,pv,fv) or with values entered NPER(0.09,0,7752,20000) STEP 4. Analyze It will take about 11 years for $7,752 to grow to $20,000 at 9% compound interest. This is the kind of calculation that both individuals and business make in trying to plan for major expenditures. STEP 5: Check yourself How many years will it take for $10,000 to grow to $200,000 given a 15% compound growth rate? ANSWER: 21.4 years. Your Turn: For more practice, do related Study Problems 5–13 and 5–18 at the end of this chapter.
>> END Checkpoint 5.5
ISBN 1256147850
The Rule of 72 Now you know how to determine the future value of any investment. What if all you want to know is how long it will take to double your money in that investment? One simple way to approximate how long it will take for a given sum to double in value is called the Rule of 72. This “rule” states that you can determine how many years it will take for a given sum to double by dividing the investment’s annual growth or interest rate into 72. For example, if an investment grows at an annual rate of 9 percent per year, according to the Rule of 72 it should take 72/9 8 years for that sum to double. Keep in mind that this is not a hard and fast rule, just an approximation—but it’s a pretty good approximation. For example, the Future Value Interest Factor of (1 i)n for 8 years (n 8) at 9 percent (i 9%) is 1.993, which is pretty close to the Rule of 72’s approximation of 2.0. Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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Solving for the Rate of Interest You have just inherited $34,946 and want to use it to fund your retirement in 30 years. If you have estimated that you will need $800,000 to fund your retirement, what rate of interest would you have to earn on your $34,946 investment? Let’s take a look at solving this using the mathematical formulas, a financial calculator, and an Excel spreadsheet to calculate i. Using Mathematical Formulas. If you write this problem using our time value of money formula you get, Number of
Future Value Years (n) Annual Present in year n = a 1 + Interest Rate (i) b Value (PV) (FVn )
(5–1a)
$800,000 = $34,94611+ i230 Once again, you could resort to a trialanderror approach by substituting different values of i into the equation and calculating the value on the righthand side of the equation to see if it is equal to $800,000. However, again, this would be quite cumbersome and unnecessary. Alternatively, you can solve for i directly by dividing both sides of the equation above by $34,946, 11 +i230 = $800,000>$34,946 = 22.8925
and then taking the 30th root of this equation to find the value of (1 i). Since taking the 30th root of something is the same as taking something to the 1/30 (or 0.033333) power, this is a relatively easy process if you have a financial calculator with a “yn” key. In this case, you (1) enter 22.8925, (2) press the “yn” key, (3) enter 0.033333, and (4) press the “” key. The answer should be 1.109999, indicating that (1 i) 1.109999, and i 10.9999% or 11%. As you might expect, it’s faster and easier to use the time value of money functions on a financial calculator or in Excel. Using a Financial Calculator. Using a financial calculator or an Excel spreadsheet, this problem becomes much easier. With a financial calculator, all you do is substitute in the values for n, PV, and FV, and solve for i: Enter
30 N
Solve for
I/Y
34,946
0
800,000
PV
PMT
FV
11.0
Using an Excel Spreadsheet. RATE(nper,pmt,pv,fv) or with values entered RATE(30,0,34946,800000)
Before you move on to 5.4
Concept Check  5.3 1. What does the term discounting mean with respect to the time value of money?
2. How is discounting related to compounding?
ISBN 1256147850 Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
CHAPTER 5  Time Value of Money Checkpoint 5.6
Solving for the Interest Rate, i Let’s go back to that Prius example in Checkpoint 5.5. Recall that the Prius always costs $20,000. In 10 years, you’d really like to have $20,000 to buy a new Prius, but you only have $11,167 now. At what rate must your $11,167 be compounded annually for it to grow to $20,000 in 10 years? STEP 1: Picture the problem We can visualize the problem using a timeline as follows: i = ?% Time Period
0
1
2
3
4
5
6
7
8
Cash Flow $11,167
9
10
Years
$20,000
STEP 2: Decide on a solution strategy Here we know the number of years, the present value, and the future value, and we are solving for the interest rate. We’ll use Equation (5–1a) to solve this problem. STEP 3: Solve Using the Mathematical Formulas. $20,000 $11,167 (1 i )10, or 1.7910= (1 i )10 We then take the 10th root of this equation to find the value of (1 i). Since taking the 10th root of something is the same as taking something to the 1/10 (or 0.10) power, this can be done if you have a financial calculator with a “yn” key. In this case, you (1) enter 1.7910, (2) press the “yn” key, (3) enter 0.10, and (4) press the “” key. The answer should be 1.06, indicating that (1 i ) 1.06, and i 6%. Using a Financial Calculator. Enter
10 N
Solve for
11,167
0
20,000
PV
PMT
FV
I/Y 6.0
Using an Excel Spreadsheet. RATE(nper,pmt,pv,fv) or with values entered RATE(10,0,11167,20000) STEP 4: Analyze You can increase your future value by growing your money at a higher interest rate or by letting your money grow for a longer period of time. For most of you, when it comes to planning for your retirement, a large n is a real positive for you. Also, if you can earn a slightly higher return on your retirement savings, or any savings for that matter, it can make a big difference. STEP 5: Check yourself At what rate will $50,000 have to grow to reach $1,000,000 in 30 years?
ISBN 1256147850
ANSWER: 10.5%. Your Turn: For more practice, do related Study Problems 5–14, 5–16, 5–17, 5–20 to 5–22, 5–26, and 5–27 at the end of this chapter.
>> END Checkpoint 5.6
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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5.4
Making Interest Rates Comparable
Sometimes it’s difficult to determine exactly how much you are paying or earning on a loan. That’s because the loan might not be quoted as compounding annually, but rather as compounding quarterly or daily. To illustrate, let’s look at two loans, one that is quoted as 8.084% compounded annually and another quoted as 7.85% compounded quarterly. Unfortunately, because on one the interest is compounded annually (you pay interest just once a year), but on the other, interest is compounded quarterly (you pay interest four times a year), they are difficult to compare. To allow borrowers to compare rates between different lenders, the U.S. TruthinLending Act requires what is known as the annual percentage rate (APR) to be displayed on all consumer loan documents. The annual percentage rate (APR) indicates the interest rate paid or earned in one year without compounding. We can calculate APR as the interest rate per period (for example, per month or week) multiplied by the number of periods that compounding occurs during the year (m): Annual Percentage Interest Rate per Compounding Rate (APR) = Period (for example * Periods per P per month or week) Q or simple interest year (m)
(5–3)
Thus, if you are paying 2% per month, m, the number of compounding periods per year, would be 12, and the APR would be: APR = 2%>month * 12 months>year = 24% Unfortunately, the APR does not help much when the rates being compared are not compounded for the same number of periods per year. In fact, the APR is also called the nominal or quoted (stated) interest rate because it is the rate that the lender states you are paying. In our example, both 8.084% and 7.85% are the annual percentage rates (APRs), but they aren’t comparable because the loans have different compounding periods. To make them comparable, we calculate their equivalent rate using an annual compounding period. We do this by calculating the effective annual rate (EAR), the annual compounded rate that produces the same return as the nominal, or stated, rate. The EAR can be calculated using the following equation: m
Quoted Annual Rate 1 Effective Annual Rate 1EAR2 = ± 1 + Compounding Periods ≤ per year (m)
(5–4)
We calculate the EAR for the loan that has a 7.85% quoted annual rate of interest compounded quarterly (i.e., m 4 times per year) using Equation (5–4) as follows: EAR = c1 +
0.0785 4 d  1 = .08084 or 8.084% 4
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
So if your banker offers you a loan with a 7.85% rate that is compounded quarterly or an 8.084% rate with annual compounding, which should you prefer? If you didn’t know how the time value of money is affected by compounding, you would have chosen the 7.85% rate because, on the surface, it looked like the loan with the lower cost. However, you should be indifferent since these two offers have the same cost to you—that is to say, they have the same EAR. The key point here is that to compare the two loan terms you need to convert them to the same number of compounding periods (annual in this case). Given the wide variety of compounding periods used by businesses and banks, it is important to know how to make these rates comparable so you can make logical decisions. Now let’s return to that payday loan we introduced at the chapter opening. What is its EAR? In that example, we looked at a payday lender that advertised that you could borrow $500 and repay $626.37 eight days later. On the surface, that looks like you are paying 25.274% ($626.37/$500 1.25274), but that’s really what you are paying every eight days. To find the quoted annual rate we multiply the 8day rate of 25.274% times the number of eightday periods in a year (in effect, you are paying 25.274% every 8 days, or 25.274% 45.625 1,153.13%).
CHAPTER 5  Time Value of Money
145
Checkpoint 5.7
Calculating an EAR or Effective Annual Rate Assume that you just received your first credit card statement and the APR, or annual percentage rate listed on the statement, is 21.7%. When you look closer you notice that the interest is compounded daily. What is the EAR, or effective annual rate, on your credit card? STEP 1: Picture the problem We can visualize the problem using a timeline as follows: If i = an annual rate of 21.7% which is compounded on a daily basis, what is the EAR? Time Period
Cash Flow
0
1
2
3
4
5
6
7
8
9
10
Daily Periods
$ Amount
STEP 2: Decide on a solution strategy We’ll use Equation (5–4) to solve this problem, Effective m Quoted Annual Rate b  1 Annual Rate = a 1 + Compounding Periods per year (m) (EAR)
(5–4)
STEP 3. Solve To calculate the EAR we can use Equation (5–4), where the quoted annual rate is 21.7%, or 0.217, and m is 365. Substituting in these values, we get, EAR = c1 +
0.217 365 1 d 365
EAR = 1.242264  1 = 0.242264 or 24.2264% You were right in thinking the amount of interest you owed seemed high. In reality, the EAR, or effective annual rate, is actually 24.2264%. Recall that whenever interest is compounded more frequently, it accumulates faster. STEP 4: Analyze When you invest in a certificate of deposit, or CD, at a bank, the rate they will quote you is the EAR—that’s because it actually is the rate that you will earn on your money—and it’s also higher than the simple APR. It’s important to make sure when you compare different interest rates that they are truly comparable, and the EAR allows you to make them comparable. For example, if you’re talking about borrowing money at 9% compounded daily, while the APR is 9%, the EAR is actually 9.426%. That’s a pretty big difference when you’re paying the interest. STEP 5: Check yourself What is the EAR on a quoted or stated rate of 13% that is compounded monthly?
ANSWER: 13.80%.
ISBN 1256147850
Your Turn: For more practice, do related Study Problems 5–35 through 5–38 at the end of this chapter.
>> END Checkpoint 5.7
In this case m is 45.625 because there are 45.625 eightday periods in a year (365 days), and the annual rate is 0.25274 45.625 11.5313 (the eightday rate times the number of eightday periods in a year). Substituting into Equation (5–4), we get, EAR = c 1 +
11.5313 45.625 d 1 45.625
EAR = 29,168.80  1 = 29,167.80, or 2,916,780% Needless to say, you’ll want to stay away from payday loans. Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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Calculating the Interest Rate and Converting It to an EAR When you have nonannual compounding and you calculate a value for i using your financial calculator or Excel, you’re calculating the rate per nonannual compounding period, which is referred to as the periodic rate, Periodic Rate =
Quoted Annual Rate Compounding Periods per year 1m2
You can easily convert the periodic rate into an APR by multiplying it by the number of times that compounding occurs per year (m). However, if you’re interested in the EAR, you’ll have to subsequently convert the value you just calculated into an EAR. Let’s look at an example. Suppose that you’ve just taken out a 2year, $100,000 loan with monthly compounding and that at the end of two years you will pay $126,973 to pay the loan off. How can we find the quoted interest rate on this loan and convert it to an EAR? This problem can be solved using either a financial calculator or Excel.2 Because the problem involves monthly compounding, m, the number of compounding periods per year is 12; n, the number of periods, becomes 24 (number of years times m, or 2 times 12); and the solution, i, will be expressed as the monthly rate. Financial Calculator. Substituting in a financial calculator we find, Enter
24 N
Solve for
I/Y
100,000
0
126,973
PV
PMT
FV
1.0
To determine the APR you’re paying on this loan, you need to multiply the value you just calculated for i times 12, thus the APR on this loan is 12%, but that is not the loan’s EAR. It’s merely the APR. To convert the APR to an EAR, we can use Equation (5–4), Quoted Annual Rate ≤ Effective Annual Rate 1EAR2 = ± 1 + Compounding Periods per year (m)
m
1
(5–4)
where the quoted annual rate is 0.12, and m is 12. Substituting these values into the above equation we get, EAR = c1 +
0.12 12 d 1 12
EAR = 1.1268  1 = 0.126825 or 12.6825% In reality, the EAR, or effective annual rate, is actually 12.6825%. In effect, if you took out a twoyear loan for $100,000 at 12.6825% compounded annually, your payment at the end of two years would be $126,973, the same payment you had when you borrowed $100,000 at 12% compounded monthly.
To the Extreme: Continuous Compounding
2
Using either a TI BAIIPlus or an HP 10BII calculator, there is a shortcut key that allows you to enter the number of compounding periods and the nominal rate to calculate the EAR. Those keystrokes are shown in Appendix A in the back of the book.
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
As m (the number of compounding periods per year) increases, so does the EAR. That only makes sense because the greater the number of compounding periods, the more often interest is earned on interest. As you just saw, we can easily compute the EAR when interest is compounded daily (m 365). We can just as easily calculate EAR if the interest, i, is compounded hourly (m 8,760), compounded every minute (m 525,600), or every second
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(m 31,536,000). We can even calculate the EAR when interest is continuously compounded, that is, when the time intervals between interest payments are infinitely small, as EAR = 1equoted annual rate2  1
(5–5)
where e is the number 2.71828, with the corresponding calculator key generally appearing as “ex.” This number e is an irrational number that is used in applications that involve things that grow continuously over time. It is similar to the number Π in geometry.3 Let’s take another look at the credit card example we looked at in Checkpoint 5.7, but with continuous compounding. Again, the APR, or annual percentage rate, is listed at 21.7%. With continuous compounding, what’s the EAR, or effective annual rate, on your credit card? EAR = e.217  1 = 1.2423  1 = 0.2423 or 24.23%
Finance in a Flat World Financial Access at Birth
Russia Norway Finland England Sweden Denmark Germany France Italy Spain Turkey Morocco North Africa
USA
Mexico
China
Japan South Korea Philippines
Thailand Malaysia Brazil Australia Argentina
South Africa
Approximately half the world’s population has no access to financial services such as savings, credit, and insurance. Inspired by the OneLaptopPerChild campaign, UCLA finance professor Bhagwan Chowdhry has a plan to take this number to zero by 2030, the Financial Access at Birth (FAB) Campaign. This is how FAB could work. Each child would have an online bank account opened at birth with an initial deposit of $100. The bank account would be opened together with the child’s birth
registration, and the deposit plus interest could be withdrawn when the child reaches 16 years of age. If the program were launched in 2011, in just 20 short years every child and young adult in the world would have access to financial services. Assuming a 5% annual rate of interest on the deposit, the $100 deposit would grow to about $218 when the child reaches 16. If we wait until the child reaches 21 before turning over the account, it will have grown to about $279. In many parts of the world this would be a princely sum of money. Moreover, the recipient would have a bank account! So what’s the cost of implementing FAB? Currently there are about 134 million children born annually and assuming that a quarter of these children would not need the service, this leaves 100 million children that otherwise would not have access to a bank account. The cost of the program would then be just $10 billion per year, which is less than the amount spent per week on military expenditures around the world. If 100 million individuals would contribute just $100 per year the dream of the FAB could become a reality. Every person in the world would have access to financial services in just 20 years! Want to learn more? Go to http://tr.im/fabcam.
Before you begin end of chapter material
Concept Check  5.4 1. How does an EAR differ from an APR? ISBN 1256147850
2. What is the effect of having multiple compounding periods within a year on future values?
Like the number Π, it goes on forever. In fact, if you’re interested, you can find the first 5 million digits of e at http://antwrp.gsfc.nasa.gov/htmltest/gifcity/e.5mil
3
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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C H A P T E R
5
Applying the Principles of Finance to Chapter 5 P Principle 1: Money Has a Time Value This chapter begins our study of the time value of money—a dollar received today, other things being the same, is worth more than a dollar received a year from now. The concept
of time value of money underlies many financial decisions faced in business. In this chapter, we learn how to calculate the value today of a sum of money received in the future and the future value of a present sum.
Chapter Summary 5.1
Construct cash flow timelines to organize your analysis of time value of money problems. (pgs. 128–130) SUMMARY: Timelines can help you visualize and then solve time value of money problems. Time periods—with 0 representing today, 1 the end of Period 1 and so forth—are listed on top of the timeline. Note that Period 1 represents the end of Period 1 and the beginning of Period 2. The periods can consist of years, months, days, or any unit of time. However, in general when people analyze cash flows, they are looking at yearly periods. The cash flows appear below the timeline. Cash inflows are labeled with positive signs. Cash outflows are labeled with negative signs.
Concept Check  5.1 1. What is a timeline, and how does it help you solve time value of money problems? 2. Does year 5 represent the end of the fifth year, the beginning of the sixth year, or both?
5.2
KEY TERM
Timeline, page 128 A linear representation of the timing of cash flows.
Understand compounding and calculate the future value of cash flows using mathematical formulas, a financial calculator, and an Excel spreadsheet. (pgs. 130–137) SUMMARY: Compounding begins when the interest earned on an investment during a past period begins earning interest in the current period. Financial managers must compare the costs and benefits of alternatives that do not occur during the same time period. Calculating the time value of money makes all dollar values comparable; because money has a time value, it moves all dollar flows either back to the present or out to a common future date. All time value formulas presented in this chapter actually stem from the single compounding formula FVn PV(1 i)n. The formulas are used to deal simply with common financial situations, for example, discounting single flows or moving single flows out into the future. Financial calculators are a handy and inexpensive alternative to doing the math. However, most professionals today use spreadsheet software, such as Excel. KEY TERMS
Compounding, page 130 The process of determining the future value of a payment or series of payments when applying the concept of compound interest.
Future value interest factor, page 131 The value (1 i)n used as a multiplier to calculate an amount’s future value.
Present value, page 130 The value in today’s dollars of a future payment discounted back to the present at the required rate of return.
Simple interest, page 130 The interest earned on the principal.
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
Compound interest, page 130 The situation in which interest paid on the investment during the first period is added to the principal and, during the second period, interest is earned on the original principal plus the interest earned during the first period.
Future value, page 130 What a cash flow will be worth in the future.
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KEY EQUATIONS Future Value Number of Years Annual Present b a1 + in year n = Interest Rate (i) Value (PV) (FVn)
Concept Check  5.2 1. What is compound interest, and how is it calculated?
Annual Future Value Interest Rate (i) Present ±1 + ≤ in year n = Value (PV) Compounding (FVn) Periods per year (m)
2. Describe the three basic approaches that can be used to move money through time. 3. How does increasing the number of compounding periods affect the future value of a cash sum?
(5–1a)
Number of m * a Years (n) b
(5–1b)
Where m is the number of compounding periods per year.
5.3
Understand discounting and calculate the present value of cash flows using mathematical formulas, a financial calculator, and an Excel spreadsheet. (pgs. 137–143) SUMMARY: Previously we were solving for the future value (FVn) of the present value (PV) of a sum of money. When we are solving for the present value, we are simply doing the reverse of solving for the future value. We can find the present value by solving for PV,
PV = FVn c
1 d 11 + i2n
In addition, increasing the number of compounding periods within the year, while holding the rate of interest constant, will magnify the effects of compounding. That is, even though the rate of interest does not change, increasing the number of compounding periods means that interest gets compounded sooner than it would otherwise. This magnifies the effects of compounding. KEY TERMS Discount rate, page 138 The interest rate used in the discounting process.
Discounting, page 137 The inverse of compounding. This process is used to determine the present value of a future cash flow.
Rule of 72, page 141 A method for estimating the time it takes for an amount to double in value. To determine the approximate time it takes for an amount to double in value 72 is divided by the annual interest rate.
Present value interest factor, page 138 The value [1/(1 i)n] used as a multiplier to calculate a future payment’s present value.
KEY EQUATIONS
Present = Value (PV)
Future Value in year n (FVn) Number of Annual Years (n) a 1 + Interest Rate (i) b
ISBN 1256147850
1
Concept Check  5.3 1. What does the term discounting mean with respect to the time value of money? 2. How is discounting related to compounding?
Present = Value (PV)
Future Value in year n F ±1 + (FVn)
Annual Interest Rate (i) ≤ Compounding Periods per Year (m)
m* a
Number of b Years (n)
Where m is the number of compounding periods per year.
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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(5–2)
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5.4
Understand how interest rates are quoted and know how to make them comparable. (pgs. 144–147) SUMMARY: One way to compare different interest rates is to use the annual percentage rate (APR), which indicates the amount of interest earned in one year without compounding. The APR is the simple interest rate and is calculated as the interest rate per period multiplied by the number of periods in the year: APR = Interest Rate per Period * Periods per Year
(5–3)
The problem with the APR is that if compounding occurs more than once a year—for example, if the interest you owe is calculated every month, then in the second month, and from then on, you will end up paying interest from the first month. The end result of this is that the actual interest rate you are paying is greater than the APR. To find out the actual amount of interest we would pay over the course of one time period, we must convert the quoted APR rate to an effective annual rate (EAR). The EAR is the annual compounded rate that produces the same cash flow as the nominal interest rate: EAR = a1 +
Quoted Annual Rate m b 1 m
(5–4)
Where EAR is the effective annual rate, and m is the number of compounding periods within a year. KEY TERMS
Annual percentage rate (APR), page 144 The interest rate paid or earned in one year without compounding. It is calculated as the interest rate per period (for example, per month or week) multiplied by the number of periods that compounding occurs during the year (m).
Effective annual rate (EAR), page 144 The annual compounded rate that produces the same return as the nominal, or stated, rate. Nominal or quoted (stated) interest rate, page 144 The interest rate paid on debt securities without an adjustment for any loss in purchasing power.
KEY EQUATIONS Interest Annual Percentage Compounding Rate (APR) = Rate per * Periods per P Period Q or simple interest Year
Concept Check  5.4
Quoted Effective Annual Rate ≤ Annual Rate = ± 1 + Compounding Periods (EAR) per year (m)
1. How does an EAR differ from an APR? 2. What is the effect of having multiple compounding periods within a year on future values?
(5–3)
m
 1
(5–4)
Where EAR the effective annual rate
Study Questions What is the time value of money? Give three examples of how the time value of money might take on importance in business decisions.
5–2.
The processes of discounting and compounding are related. Explain this relationship.
5–3.
What is the relationship between the number of times interest is compounded per year on an investment and the future value of that investment? What is the relationship between the number of times compounding occurs per year and the EAR?
5–4.
How would an increase in the interest rate (i) or a decrease in the number of periods (n) affect the future value (FVn) of a sum of money?
5–5.
How would an increase in the interest rate (i) or a decrease in the number of periods until the payment is received (n) affect the present value (PV) of a sum of money?
5–6.
Compare some of the different financial calculators that are available on the Internet. Look at Kiplinger Online calculators (www.kiplinger.com/tools/index.html) which include saving and investing, mutual funds, bonds, stocks, home, auto, credit cards,
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
5–1.
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and budgeting online calculators. Also go to www.dinkytown.net, www.bankrate.com/ calculators.aspx, and www.interest.com and click on the “Calculators” link. Which financial calculators do you find to be the most useful? Why? 5–7.
(Related to Chapter Introduction: Payday Loans on page 127) The introduction to this
chapter examined payday loans. Recently, Congress passed legislation limiting the interest rate charged to active military to 36%. Go to the Predatory Lending Association website www.predatorylendingassociation.com and find the military base closest to you and identify the payday lenders that surround that base. Also, identify any payday lenders near you. 5–8.
(Related to Chapter Introduction: Payday Loans on page 127) In the introduction to
this chapter, payday loans were examined. Go to the Responsible Lending Organization website www.responsiblelending.org/paydaylending/. How does the “debt trap” (www.responsiblelending.org/paydaylending/toolsresources/debttrap .html) associated with payday loans work?
SelfTest Problems Problem ST.1 (NonAnnual Compounding) Your local credit union is offering threeyear certificates of deposit that pay 7% compounded quarterly. If you put $10,000 into one of these certificates of deposit, what will it be worth at the end of three years? Solution ST.1 STEP 1: Picture the problem Expressed as a timeline, this problem would look like the following: i = (7%/4 or 1.75%) Time Period
0
1
2
3
4
5
6
7
8
Cash Flow $10,000
9
10
11
12
3Month Periods
Future Value of $10,000 compounded for 12 threemonth periods at 7%/4 every 3 months
STEP 2: Decide on a solution strategy In this instance we are simply solving for the future value of $10,000 where the interest is calculated on a quarterly basis, or four times a year using Equation (5–1b). So, if you are earning 7% compounded quarterly, for 3 years, you are really earning 7%/4 1.75% every three months for 12 threemonth periods. STEP 3: Solve Using the Mathematical Formulas. FVn = PV a1 +
annual interest rate mn b m
= $10,000a1 +
.07 4 * 3 b 4
= $10,00011.2314392 = $12,314 ISBN 1256147850
Using a Financial Calculator. Enter
Solve for
12
7/4
10,000
0
N
I/Y
PV
PMT
FV 12,314.39 (SOLUTION ST.1 CONTINUED >> ON NEXT PAGE)
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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Using an Excel Spreadsheet. FV(rate,nper,pmt,pv) or with values entered FV(0.0175,12,0,–10000) STEP 4: Analyze The more often interest is compounded per year, the larger the future value will be. That’s because you are earning interest on interest you’ve previously earned. >> END Solution ST.1
Problem ST.2 (Solving for the Interest Rate, i) Your Grandmother, who has a beachfront house on Hilton Head Island, is planning on moving back to a retirement community just outside of Atlanta in ten years. She has offered to sell you her beachfront house in 12 years for $600,000. Right now you have $200,000 that you have set aside for the purchase. At what rate must your $200,000 be compounded annually for it to grow to $600,000 at the end of 12 years?
Solution ST.2 STEP 1: Picture the problem In this problem we know the cash flows and know the number of years, 12 years, and are solving for i. We can visualize the problem using a timeline as follows: i = ?% Time Period
0
1
2
3
4
5
6
7
8
Cash Flow $200,000
9
10
11
12
Years
$600,000
STEP 2: Decide on a solution strategy Here we know the number of years, the present value, and the future value, and we are solving for the interest rate using equation (5–1a)—that is, at what rate must $200,000 grow in order to reach $600,000 at the end of 12 years? STEP 3: Solve Using the Mathematical Formulas. $600,000 $200,000(1 i )12, or 3.0 (1 i )12 Solving, (1 i) 1.0959, and i 9.59%. Using a Financial Calculator. Enter
12.0 N
Solve for
I/Y
200,000
0
600,000
PV
PMT
FV
9.59
Using an Excel Spreadsheet. RATE(nper,pmt,pv,fv) or with values entered RATE(12,0,–200000,600000)
Compound interest is powerful. In this case, your money tripled in value in 12 years while growing at less than 10% a year. >> END Solution ST.2
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
STEP 4: Analyze
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Study Problems Compound Interest Go to www.myfinancelab.com to complete these exercises online and get instant feedback.
5–1.
(Related to Checkpoint 5.2 on page 134) (Future value) To what amount will the following investments accumulate? a. $5,000 invested for 10 years at 10% compounded annually b. $8,000 invested for 7 years at 8% compounded annually c. $775 invested for 12 years at 12% compounded annually d. $21,000 invested for 5 years at 5% compounded annually
5–2.
(Future value) Leslie Mosallam, who recently sold her Porsche, placed $10,000 in a savings account paying annual compound interest of 6%. a. Calculate the amount of money that will accumulate if Leslie leaves the money in the bank for 1, 5, and 15 years. b. Suppose Leslie moves her money into an account that pays 8% or one that pays 10%. Rework part (a) using 8% and 10%. c. What conclusions can you draw about the relationship between interest rates, time, and future sums from the calculations you just did?
5–3.
(Related to The Business of Life: Saving for Your First House on page 137) (Future value)
You are hoping to buy a house in the future and recently received an inheritance of $20,000. You intend to use your inheritance as a down payment on your house. a. If you put your inheritance in an account that earns 7% interest rate compounded annually, how many years will it be before your inheritance grows to $30,000? b. If you let your money grow for 10.25 years at 7%, how much will you have? c. How long will it take your money to grow to $30,000 if you move it into an account that pays 3% compounded annually? How long will it take your money to grow to $30,000 if you move it into an account that pays 11%? d. What does all this tell you about the relationship among interest rates, time, and future sums? 5–4.
(Related to Checkpoint 5.2 on page 134) (Future value) Bob Terwilliger received $12,345 for his services as financial consultant to the mayor’s office of his hometown of Springfield. Bob says that his consulting work was his civic duty and that he should not receive any compensation. So, he has invested his paycheck into an account paying 3.98% annual interest and left the account in his will to the city of Springfield on the condition that the city could not collect any money from the account for 200 years. How much money will the city receive from Bob’s generosity in 200 years?
5–5.
(Related to Checkpoint 5.3 on page 136) (Compound interest with nonannual periods) Calculate the amount of money that will be in each of the following accounts at the end of the given deposit period:
Account Holder
ISBN 1256147850
Theodore Logan III Vernell Coles Tina Elliott Wayne Robinson Eunice Chung Kelly Cravens
5–6.
Amount Deposited
Annual Interest Rate
Compounding Periods Per Year (M)
Compounding Periods (Years)
$
10% 12 12 8 10 12
1 12 6 4 2 3
10 1 2 2 4 3
1,000 95,000 8,000 120,000 30,000 15,000
(Related to Checkpoint 5.2 on page 134) (Compound interest with nonannual periods)
You just received a $5,000 bonus. a. Calculate the future value of $5,000, given that it will be held in the bank for five years and earn an annual interest rate of 6%. b. Recalculate part (a) using a compounding period that is (1) semiannual and (2) bimonthly. c. Recalculate parts (a) and (b) using a 12% annual interest rate. Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
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d. Recalculate part (a) using a time horizon of 12 years at a 6% interest rate. e. What conclusions can you draw when you compare the answers in parts (c) and (d) with the answers in parts (a) and (b)? 5–7.
(Related to Checkpoint 5.3 on page 136) (Compound interest with nonannual periods) Your grandmother just gave you $6,000. You’d like to see what it might grow to if you invest it. a. Calculate the future value of $6,000, given that it will be invested for five years at an annual interest rate of 6%. b. Recalculate part (a) using a compounding period that is (1) semiannual and (2) bimonthly. c. Now let’s look at what might happen if you can invest the money at a 12% rate rather than 6% rate; recalculate parts (a) and (b) for a 12% annual interest rate. d. Now let’s see what might happen if you invest the money for 12 years rather than 5 years; recalculate part (a) using a time horizon of 12 years (annual interest rate is still 6%). e. With respect to the changes in the stated interest rate and length of time the money is invested in parts (c) and (d), what conclusions can you draw?
5–8.
(Related to Checkpoint 5.2 on page 134) (Future value) A new finance book sold 15,000 copies following the first year of its release, and was expected to increase by 20% per year. What sales are expected during years two, three, and four? Graph this sales trend and explain.
5–9.
(Future value) You have just introduced “must have” headphones for the iPod. Sales of the new product are expected to be 10,000 units this year and are expected to increase by 15% per year in the future. What are expected sales during each of the next three years? Graph this sales trend and explain why the number of additional units sold increases every year.
5–10. (Future value) If you deposit $3,500 today into an account earning an 11% annual rate of return, what would your account be worth in 35 years (assuming no further deposits)? In 40 years? 5–11. (Simple and compound interest) If you deposit $10,000 today into an account earning an 11% annual rate of return, in the third year how much interest would be earned? How much of the total is simple interest and how much results from compounding of interest?
Discounting and Present Value 5–12. (Related to Checkpoint 5.4 on page 139) (Present value) Sarah Wiggum would like to make a single investment and have $2 million at the time of her retirement in 35 years. She has found a mutual fund that will earn 4% annually. How much will Sarah have to invest today? What if Sarah were a finance major and learned how to earn a 14% annual return? How soon could she then retire? 5–13. (Related to Checkpoint 5.5 on page 141) (Solving for n) How many years will the following take? a. $500 to grow to $1,039.50 if it’s invested at 5% compounded annually b. $35 to grow to $53.87 if it’s invested at 9% compounded annually c. $100 to grow to $298.60 if it’s invested at 20% compounded annually d. $53 to grow to $78.76 if it’s invested at 2% compounded annually
5–15. (Related to Checkpoint 5.4 on page 139) (Present value) What is the present value of the following future amounts? a. $800 to be received 10 years from now discounted back to the present at 10% b. $300 to be received 5 years from now discounted back to the present at 5% Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
5–14. (Related to Checkpoint 5.6 on page 143) (Solving for i) At what annual interest rate would the following have to be invested? a. $500 to grow to $1,948.00 in 12 years b. $300 to grow to $422.10 in 7 years c. $50 to grow to $280.20 in 20 years d. $200 to grow to $497.60 in 5 years
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c. $1,000 to be received 8 years from now discounted back to the present at 3% d. $1,000 to be received 8 years from now discounted back to the present at 20% 5–16. (Related to Checkpoint 5.6 on page 143) (Solving for i) Kirk Van Houten, who has been married for 23 years, would like to buy his wife an expensive diamond ring with a platinum setting on their 30year wedding anniversary. Assume that the cost of the ring will be $12,000 in 7 years. Kirk currently has $4,510 to invest. What annual rate of return must Kirk earn on his investment to accumulate enough money to pay for the ring? 5–17. (Solving for i) You are considering investing in a security that will pay you $1,000 in 30 years. a. If the appropriate discount rate is 10%, what is the present value of this investment? b. Assume these investments sell for $365 in return for which you receive $1,000 in 30 years, what is the rate of return investors earn on this investment if they buy it for $365? 5–18. (Related to Checkpoint 5.5 on page 141) (Solving for n) Jack asked Jill to marry him, and she has accepted under one condition: Jack must buy her a new $330,000 RollsRoyce Phantom. Jack currently has $45,530 that he may invest. He has found a mutual fund that pays 4.5% annual interest in which he will place the money. How long will it take Jack to win Jill’s hand in marriage? 5–19. (Related to Checkpoint 5.4 on page 139) (Present value) Ronen Consulting has just realized an accounting error that has resulted in an unfunded liability of $398,930 due in 28 years. In other words, they will need $398,930 in 28 years. Toni Flanders, the company’s CEO, is scrambling to discount the liability to the present to assist in valuing the firm’s stock. If the appropriate discount rate is 7%, what is the present value of the liability? 5–20. (Related to Checkpoint 5.6 on page 143) (Solving for i) Seven years ago, Lance Murdock purchased a wooden statue of a Conquistador for $7,600 to put in his home office. Lance has recently married, and his home office is being converted into a sewing room. His new wife, who has far better taste than Lance, thinks the Conquistador is hideous and must go immediately. Lance decided to sell it on eBay and only received $5,200 for it, and so he took a loss on the investment. What was his rate of return, that is, the value of i? 5–21. (Solving for i) Springfield Learning sold zero coupon bonds (bonds that don’t pay any interest, instead the bondholder gets just one payment, coming when the bond matures, from the issuer) and received $900 for each bond that will pay $20,000 when it matures in 30 years. a. At what rate is Springfield Learning borrowing the money from investors? b. If Nancy Muntz purchased a bond at the offering for $900 and sold it ten years later for the market price of $3,500, what annual rate of return did she earn? c. If Barney Gumble purchased Muntz’s bond at the market price of $3,500 and held it 20 years until maturity, what annual rate of return would he have earned? 5–22. (Solving for i) If you were offered $1,079.50 ten years from now in return for an investment of $500 currently, what annual rate of interest would you earn if you took the offer?
ISBN 1256147850
5–23. (Solving for i) An insurance agent just offered you a new insurance product that will provide you with $2,376.50 ten years from now if you invest $700 today. What annual rate of interest would you earn if you invested in this product? 5–24. (Solving for n with nonannual periods) Approximately how many years would it take for an investment to grow fourfold if it were invested at 16% compounded semiannually? 5–25. (Solving for n with nonannual periods) Approximately how many years would it take for an investment to grow by sevenfold if it were invested at 10% compounded semiannually?
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PART 2

Valuation of Financial Assets
5–26. (Related to Checkpoint 5.6 on page 143) (Solving for i) You lend a friend $10,000, for which your friend will repay you $27,027 at the end of five years. What interest rate are you charging your “friend”? 5–27. (Solving for i) You’ve run out of money for college, and your college roommate has an idea for you. He offers to lend you $15,000, for which you will repay him $37,313 at the end of five years. If you took this loan, what interest rate would you be paying on it? 5–28. (Related to Checkpoint 5.4 on page 139) (Presentvalue comparison) You are offered $100,000 today or $300,000 in 13 years. Assuming that you can earn 11% on your money, which should you choose? 5–29. (Present value comparison) Much to your surprise, you were selected to appear on the TV show “The Price Is Right.” As a result of your prowess in identifying how many rolls of toilet paper a typical American family keeps on hand, you win the opportunity to choose one of the following: $1,000 today, $10,000 in 12 years, or $25,000 in 25 years. Assuming that you can earn 11% on your money, which should you choose? 5–30. (Related to Checkpoint 5.6 on page 143) (Solving for i—financial calculator needed) In September 1963, the first issue of the comic book XMEN was issued. The original price for the issue was 12 cents. By September 2006, 43 years later, the value of this comic book had risen to $9,500. What annual rate of interest would you have earned if you had bought the comic in 1963 and sold it in 2006? 5–31. (Solving for i—financial calculator needed) In March 1963, Ironman was first introduced in issue number 39 of Tales of Suspense. The original price for that issue was 12 cents. By March of 2010, 47 years later, the value of this comic book had risen to $9,000. What annual rate of interest would you have earned if you had bought the comic in 1963 and sold it in 2010? 5–32. (Solving for i) A financial planner just offered you a new investment product that would require an initial investment on your part of $35,000, and then 25 years from now will be worth $250,000. What annual rate of interest would you earn if you invested in this product? 5–33. (Spreadsheet problem) If you invest $900 in a bank where it will earn 8% compounded annually, how much will it be worth at the end of seven years? Use a spreadsheet to calculate your answer. 5–34. (Spreadsheet problem) In 20 years, you would like to have $250,000 to buy a vacation home. If you have only $30,000, at what rate must it be compounded annually for it to grow to $250,000 in 20 years? Use a spreadsheet to calculate your answer.
Making Interest Rates Comparable 5–35. (Related to Checkpoint 5.7 on page 145) (Calculating an EAR) After examining the various personal loan rates available to you, you find that you can borrow funds from a finance company at 12% compounded monthly or from a bank at 13% compounded annually. Which alternative is the most attractive? 5–36. (Calculating an EAR) You have a choice of borrowing money from a finance company at 24% compounded monthly or borrowing money from a bank at 26% compounded annually. Which alternative is the most attractive?
5–38. (Calculating an EAR) Based on effective interest rates, would you prefer to deposit your money into Springfield National Bank, which pays 8.0% interest compounded annually, or into Burns National Bank, which pays 7.8% compounded monthly? (Hint: Calculate the EAR on each account.) Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.
ISBN 1256147850
5–37. (Calculating an EAR) Your grandmother asks for your help in choosing a certificate of deposit (CD) from a bank with a oneyear maturity and a fixed interest rate. The first certificate of deposit, CD #1, pays 4.95% APR compounded daily, while the second certificate of deposit, CD #2, pays 5.0% APR compounded monthly. What is the effective annual rate (the EAR) of each CD, and which CD do you recommend to your grandmother?
CHAPTER 5  Time Value of Money
157
MiniCase Emily Dao, 27, just received a promotion at work that increased her annual salary to $37,000. She is eligible to participate in her employer’s 401(k) retirement plan to which the employer matches dollarfordollar workers’ contributions up to 5% of salary. However, Emily wants to buy a new $25,000 car in three years, and she wants to have enough money to make a $7,000 down payment on the car and finance the balance. Fortunately, she expects a sizable bonus this year that she hopes will cover that down payment in three years.
A wedding is also in her plans. Emily and her boyfriend, Paul, have set a wedding date two years in the future, after he finishes medical school. In addition, Emily and Paul want to buy a home of their own as soon as possible. This might be possible because at age 30, Emily will be eligible to access a $50,000 trust fund left to her as an inheritance by her late grandfather. Her trust fund is invested in 7% government bonds.
Questions 1. Justify Emily’s participation in her employer’s 401(k) plan using the time value of money concepts by explaining how much an investment of $10,000 will grow to in 40 years if it earns 10%. 2. Calculate the amount of money that Emily needs to set aside from her bonus this year to cover the down payment on a new car, assuming she can earn 6% on her savings. What if she could earn 10% on her savings?
of the $50,000 trust fund) at age 30 for a house down payment, and leaves the other half of the money untouched where it is currently invested? 4. What is the relationship between discounting and compounding? 5. List at least two actions that Emily and Paul could take to accumulate more for their retirement (think about i and n).
ISBN 1256147850
3. What will be the value of Emily’s trust fund at age 60, assuming she takes possession of half of the money ($25,000
Financial Management: Principles and Applications, Eleventh Edition, by Sheridan Titman, John D. Martin, and Arthur J. Keown. Published by Prentice Hall. Copyright © 2011 by Pearson Education, Inc.