Time Value of Money PDF

Part 1 Introduction to Financial Management Part 4 Capital Structure and Dividend Policy (Chapters 1, 2, 3, 4) (Chapters 15, 16) Part 2 Valuation of Financial Assets...

Part 1 Introduction to Financial Management Part 4 Capital Structure and Dividend Policy (Chapters 1, 2, 3, 4) (Chapters 15, 16) Part 2 Valuation of Financial Assets Part 5 Liquidity Management and Special Topics (Chapters 5, 6, 7, 8, 9, 10) in Finance (Chapters 17, 18, 19, 20) Part 3 Capital Budgeting (Chapters 11, 12, 13, 14) 5 C H A P T E R The Time Value of Money The Basics Chapter Outline 5.1 Using Timelines to Visualize Objective 1. Construct cash flow timelines to organize your analysis of problems involving the time value of Cash Flows (pgs. 162–163) money. 5.2 Compounding and Future Objective 2. Understand compounding and calculate the future value of cash flows using mathematical Value (pgs. 164–171) formulas, a financial calculator, and an Excel spreadsheet. 5.3 Discounting and Present Objective 3. Understand discounting and calculate the present value of cash flows using mathematical Value (pgs. 171–176) formulas, a financial calculator, and an Excel spreadsheet. 5.4 Making Interest Rates Objective 4. Understand how interest rates are quoted and know how to make them comparable. Comparable (pgs. 177–181) 160 M05_TITM2189_13_GE_C05.indd 160 17/05/17 11:10 AM  Principle P 1 Applied Chapters 5 and 6 are dedicated to P Principle 1: Money Has future value of money you have today. In Chapter 6, we ex- a Time Value. This basic idea—a dollar received today, other tend our analysis to multiple cash flows spread out over time. things being the same, is worth more than a dollar received a In later chapters, we will use the skills we gain from Chapters 5 year from now—underlies many financial decisions faced in and 6 to analyze bond prices (Chapter 9) and stock prices business. In Chapter 1, we discussed capital budgeting, capi- (Chapter 10), calculate the value of investment opportuni- tal structure, and working capital management decisions— ties (Chapters 11–13), and determine the cost of financing a each of these decisions involves aspects of the time value of firm’s investments (Chapter 14). money. In this chapter, we learn how to calculate the value today of money you will receive in the future as well as the Payday Loans Sometimes marketed to college students as quick relief for urgent expenses, a payday loan is a short-term loan to cover expenses until the next payday. As some borrowers turn to these loans dur- ing times of financial desperation, lenders can charge them ex- tremely high rates of interest. For example, in 2016, one payday lender, Advance America,1 advertised that you could borrow $100 and repay $126.40 in 14 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 effective annual interest rate for this payday loan is a whopping 44,844 percent! (We will examine this later in the chapter in section 5.4, pages 177–181.) The very high rates of interest charged by these lenders have led some states to impose limits on the interest rates payday lend- ers can charge. Even so, the cost of this type of loan can be ex- tremely high. Understanding the time value of money is an essential tool to analyzing the cost of this and other types of financing. 1 https://www.advanceamerica.net/apply-for-a-loan/service/payday-loan accessed June 28, 2016. 161 M05_TITM2189_13_GE_C05.indd 161 17/05/17 11:10 AM  162 PART 2 | Valuation of Financial Assets Regardless of Your Major… A Dollar Saved Is “ Suppose that you and your classmate each receive a gift of $10,000 from grandparents Two Dollars Earned” but choose different ways to invest the new- found money. You immediately invest your gift until retirement, whereas 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 percent per year until you retire, you’ll end up with over $800,000. If your classmate invests his $10,000 for 31 years (remember that he carried his money around for 15 years in his wallet) and earns the same 10 percent per year, he’ll end up with only 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. Your Turn: See Study Questions 5–7 and 5–8. 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 logical decisions. 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 an 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 or Year 0 (the present) until the end of Year 4: i = 10% Time Period 0 1 2 3 4 Years Cash Flow -$100 $30 $20 -$10 $50 Beginning of Period 1 End of Period 2 or End of Period 4 or beginning of Period 3 beginning of Period 5 For our purposes, time periods are identified above 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 this 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 (a negative cash flow) occurs at the beginning of the first year (at Time Period 0), fol- lowed by cash inflows (positive cash flows) of $30 and $20 in Years 1 and 2, a cash outflow of $10 in Year 3, and, finally, a cash inflow of $50 in Year 4. Timelines are typically expressed in years, but they 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 percent, in this example—is listed above the timeline. M05_TITM2189_13_GE_C05.indd 162 19/07/17 2:51 PM  CHAPTER 5 | The Time Value of Money 163 Checkpoint 5.1 MyLab Finance Video Creating a Timeline Suppose you lend a friend $10,000 today to help him finance a new Jimmy John’s Sandwiches 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 percent. STEP 1: Picture the problem A timeline provides a tool for visualizing cash flows and time: i = rate of interest Time Period 0 1 2 3 4 Years Cash Flow Cash Flow Cash Flow Cash Flow Cash Flow Cash Flow Year 0 Year 1 Year 2 Year 3 Year 4 STEP 2: Decide on a solution strategy To complete the timeline, we simply record the cash flows on the template. STEP 3: Solve We can input the cash flows for this investment on the timeline as shown below. Time Period 0 (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; each cash flow is listed below the timeline at the appropriate time period. 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 4 Years Cash Flow -$10,000 $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 Year 1, $20,000 at the end of Year 2, nothing in Year 3, and $40,000 at the end of Year 4; the interest rate is 13.17 percent. ANSWER: i = 13.17% Time Period 0 1 2 3 4 Years Cash Flow -$40,000 $0 $20,000 $0 $40,000 >> END Checkpoint 5.1 Before you move on to 5.2 Concept Check | 5.1 1. What is a timeline, and how does it help you solve problems involving the time value of money? 2. Does Year 5 represent the end of the fifth year, the beginning of the sixth year, or both? M05_TITM2189_13_GE_C05.indd 163 25/07/17 4:31 PM  164 PART 2 | Valuation of Financial Assets 5.2 Compounding and Future Value If we assume that an investment will earn interest only on the original principal, we call this simple interest. Suppose that you put $100 in a savings account earning 6 percent interest annually. How much will your savings grow after one year? If you invest for one year at an interest rate of 6 percent, you will earn 6 percent simple interest on your initial deposit of $100, giving you a total of $106 in your account. What if you leave 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 re- fer to as present value, what a cash flow is worth to you today, and future value, what a cash flow will be worth to you in the future. The following is a mathematical formula that shows how these concepts relate to each other when the future value is in one year: Future Value in 1 Year = Present Value * (1 + Interest Rate) (5–1) In the savings account 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 × (1 +.06) = $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 0 1 2 Years Cash Flow $100 x 1.06 = $106 x 1.06 = $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 percent on the $6 in 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 percent? 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 0 1 2 3 Years Cash Flow $100 x1.06 = $106 x 1.06 = $112.36 x 1.06 = $119.10 Note that every time we extend the analysis for one more period, we just multiply the previous balance by (1 1 Interest Rate). Consequently, we can use the following equation to express the future value of any amount of money for any number of periods (where n 5 the number of periods during which the compounding occurs): Future Present Interest n = a1 + b Value Period n Value (Deposit) Rate (i) or Future Value Number of Years (n) Present Annual in Year n = a1 + b (5–1a) Value (PV) Interest Rate (i) (FVn) M05_TITM2189_13_GE_C05.indd 164 17/05/17 11:10 AM  CHAPTER 5 | The Time Value of Money 165 Important Definitions and Concepts: FVn 5 the future value of the investment at the end of n periods. i 5 the interest (or growth) rate per period. PV 5 the present value, or original amount invested at the beginning of the first period. We also refer to (1 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 = PV (1 + i)n = PV * Future Value Interest Factor where Future Value Interest Factor = (1 + i)n. Panel A in Figure 5.1 shows what your investment of $100 will grow to in four years if it continues to earn an annual compound interest rate of 6 percent. Notice how the amount of interest earned increases each year. In the first year, you earn only $6 in interest, but by Year 4, you earn $7.15. 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 in MyLab Finance. So to find the value of $100 invested for four years at 6 percent, we would simply look at the intersection of the n 5 4 row and the 6% column, which is the future value interest factor of 1.262. We would then multiply this value by $100 to find that our investment of $100 at 6 percent for four years would grow to $126.20. Compound Interest and Time As Panel B 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 percent 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 more than sixfold ($1,083.47/$161.05 5 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. 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 percent interest rate compounded annually, is $1,083.47. However, if we double the rate of inter- est to 20 percent, the future value increases almost ninefold in 25 years to $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 8.8 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 problems involving the time value of money: mathematical formulas, financial calculators, and spreadsheets. Do the math. You can use the mathematical formulas just as we have done in this chap- ter. You simply substitute the values that you know into the appropriate time-value-of- money equation to find the answer. Use a financial calculator. Financial calculators have preprogrammed functions that make time-value-of-money calculations simple. M05_TITM2189_13_GE_C05.indd 165 25/07/17 4:32 PM  166 PART 2 | Valuation of Financial Assets Figure 5.1 Future Value and Compound Interest Illustrated (Panel A) Calculating Compound Interest This panel shows how interest compounds annually. During Interest Earned 5 Beginning Value 3 Interest Rate the first year, $100 invested at a 6% interest rate earns only $6. Because we earn 6% on the ending value for Year 1 (or $106), Year Beginning Value Interest Earned Ending Value we earn $6.36 in interest in Year 2. This increase in the amount 1 $ 100.00 $ 6.00 $ 106.00 of interest results from interest being earned on both the initial 2 $ 106.00 $ 6.36 $ 112.36 deposit of $100 and the $6.00 in interest earned during Year 1. The fact that we earn interest on both principal and interest is 3 $ 112.36 $ 6.74 $ 119.10 why we refer to this as compound interest. Simple interest, on 4 $ 119.10 $ 7.15 $ 126.25 the other hand, would be earning only $6.00 in interest each and every year. $400 Future value of $100 $350 compounded at a rate of 8% per (Panel B) The Power of Time $317.22 year for 15 years = $317.22 This figure illustrates the importance of time when it comes to compounding. $300 Because interest is earned on past interest, the future value of $100 depos- Future value of $100 ited in an account that earns 8% interest compounded annually grows over $250 compounded at a rate threefold in 15 years. If we were to expand this figure to 45 years (which is of 8% per year for Future value about how long you have until you retire, assuming you’re around 20 years 10 years = $215.89 $215.89 $200 old right now), the account would grow to over 31 times its initial value. $150 $100 Future value of $100 compounded at a rate of 0% per year equals $50 $100, regardless of how many years it is invested. $25,000 $0 0 5 10 15 $23,737.63 Number of periods in years Future value of $100 compounded at a rate of 20% per year for 30 years = $23,737.63 $20,000 Future value of $100 compounded at a rate of 20% per year for 25 years = $9,539.62 $15,000 Future value Future value of $100 compounded at a rate of 15% per (Panel C) The Power of the Rate of Interest year for 25 years = $3,291.90 $10,000 This figure illustrates the importance of the interest $9,539.62 rate in the power of compounding. As the interest rate Future value of $100 climbs, so does the future value. In fact, when we compounded at a rate of 10% per change the interest rate from 10% to 20%, the future year for 25 years = $1,083.47 $5,000 value in 25 years increases by 8.8 times, jumping from $1,083.47 to $9,539.62. $3,291.90 $1,083.47 $0 0 5 10 15 20 25 30 Number of periods in years >> END FIGURE 5.1 M05_TITM2189_13_GE_C05.indd 166 17/05/17 11:10 AM  CHAPTER 5 | The Time Value of Money 167 Table 5.1 Future Value Interest Factors Number of Periods (n) i 5 3% i 5 6% i 5 9% i 5 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 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 in MyLab Finance, we show you how to solve valuation problems using each of these methods. Because we, the authors of this book, believe that spending enough time solving problems the old-fashioned way—by doing the math—leads to a deeper under- standing and better retention of the concepts found in this book, we will first demonstrate how to solve problems using the formulas. However, we will also demonstrate, whenever possible, how to derive solutions using a financial calculator and Excel. Applying Compounding to Things Other Than Money Although this chapter focuses on moving money through time at a given interest rate, the con- cept 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 we assume the demand for them will grow at a rate of 25 percent per year over those 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 percent. Substituting into Equation (5–1a), we would solve for FV: Number of Future Value Present Annual Years (n) in Year n = a1 + b = 25,000 (1 +.25)5 = 76,293 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 percent over the 64 squares. In fact, if the kernels were one-quarter 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 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 provide more details on how to compare two loans with different compounding periods. M05_TITM2189_13_GE_C05.indd 167 25/07/17 4:34 PM  168 PART 2 | Valuation of Financial Assets Checkpoint 5.2 MyLab Finance Video 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 percent interest compounded annually. How much will your firm have in its savings 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% Time Period 0 1 2 3 4 5 6 7 8 9 10 Years Cash Flow -$100,000 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 5 $100,000, i 5 7%, and n 5 10 years into Equation (5–1a), we get Future Value Number of Present Annual Years (n) in Year n = a1 + b (5–1a) Value (PV) Interest Rate (i) (FVn) FVn = $100,000(1 +.07)10 = $100,000 (1.96715) = $196,715 At the end of 10 years, the firm will have $196,715 in its savings account. Using a Financial Calculator. Enter 10 7.0 –100,000 0 N I/Y PV PMT FV Solve for 196,715 Using an Excel Spreadsheet. 5 FV(rate,nper,pmt,pv) or, with values entered, 5 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 invested $100,000 at 7 percent and found that it will grow to $196,715 after 10 years. Put another way, given a 7 percent 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 percent annually for 20 years? ANSWER: $96,462.93. 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 M05_TITM2189_13_GE_C05.indd 168 25/07/17 4:35 PM  CHAPTER 5 | The Time Value of Money 169 Consider the following example: You invest $100 for five years at an interest rate of 8 percent, 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 10 six- month periods, and in each period, you will receive 4 percent 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 * (Number of Years (n)) Annual Future Value Present Interest Rate (i) in Year n = ±1 + ≤ (5–1b) Value (PV) Compounding (FVn) Periods per Year (m) Substituting into Equation (5–1b) gives us the following estimate of the future value in five years: * 5 FVn = $100 (1 +.08>2)2 = $100 (1.4802) = $148.02 If the compounding had been annual rather than semiannual, the future value of the invest- ment 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 both large. For example, for your $100 invest- ment, the difference is only $1.09. But if the amount was $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 percent for one year and the investment was com- pounded daily rather than annually, you would end up with $1.18 ($116.18 2 $115.00) more. However, if the period was extended to 10 years, then the difference would grow to $43.47 ($448.03 2 $404.56). Table 5.2 The Value of $100 Compounded at Various Non-annual Periods and Various Rates Notice that 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 i 5 2% 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 $1.18 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 i 5 2% 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 $43.47 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 M05_TITM2189_13_GE_C05.indd 169 17/05/17 11:10 AM  170 PART 2 | Valuation of Financial Assets Checkpoint 5.3 Calculating Future Values Using Non-annual Compounding Periods You have been put in charge of managing your firm’s cash position and have 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 percent, what will your firm’s account balance be in five years? STEP 1: Picture the problem If you earn a 12 percent annual interest rate compounded semiannually for five years, you really earn 6 percent every six months for 10 six-month periods. Expressed as a timeline, this problem would look like the following: i = 12% ÷ 2 = 6% every 6 months Time Period 0 1 2 3 4 5 6 7 8 9 10 6-Month Periods Cash Flow -$1,000 Future value of $1,000 compounded for 10 six-month periods at 12%/2 every 6 months 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 percent interest compounded semiannually for five years, you really earn 6 percent every six months for 10 six-month 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 5, number of compounding periods per year (m) 5 2, annual interest rate (i) 5 12%, and PV 5 $1,000 into Equation (5–1b): Annual m * (Νumber of Years (n)) Future Value Present Interest Rate(i) in Year n = ±1 + ≤ Value (PV) Compounding (FVn) Periods per Year (m).12 2 * 5 FVn = $1,000 a1 + b = $1,000 * 1.79085 = $1,790.85 2 Using a Financial Calculator. Enter 10 6.0 -1,000 0 N I/Y PV PMT FV Solve for 1,790.85 You will have $1,790.85 at the end of five years. Using an Excel Spreadsheet. 5 FV(rate,nper,pmt,pv) or, with values entered, 5 FV(0.06,10,0,−1000) STEP 4: Analyze The more often interest is compounded per year—that is, the larger m is—the larger the future value will be. That’s because you are earning interest more often on the interest you’ve previously earned. STEP 5: Check yourself If you deposit $50,000 in an account that pays an annual interest rate of 10 percent 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 M05_TITM2189_13_GE_C05.indd 170 17/05/17 11:10 AM  CHAPTER 5 | The Time Value of Money 171 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 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 your 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 Finance for Life 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 percent annually for 10 years—after Saving for Your First House 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 There was a time in the early and mid-2000s when you didn’t investment. But suppose you are able to earn 12 percent annually need to worry about a down payment when you bought a new for 10 years. What will the value of your investment be then? In this house. But that all changed as the housing bubble burst and case, your investment will be worth $31,058. Needless to say, the home prices fell. Today, you may be able to get away with put- rate of interest that you earn plays a major role in determining how ting down only around 10 percent, but the rate on your mort- quickly your investment will grow. gage will be lower if you can come up with 20 percent. To buy Now consider what happens if you wait five years before in- a median-priced home, which was just over $180,000 at the vesting your $10,000. The value of n drops from 10 to 5, and, as beginning of 2016 (new houses were considerably more than a result, the amount you save also drops. In fact, if you invest your that), you’d have to come up with a 10 percent down payment $10,000 for five years at 6 percent, you end up with $13,382, and of $18,000 or a 20 percent down payment of $36,000. On top even at 12 percent, you end up with only $17,623. of that, you would need to furnish your new home, and that The bottom line is this: The earlier you begin saving, the more costs money, too. impact every dollar you save will have. Your Turn: See Study Problem 5.3. 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? 5.3 Discounting and Present Value So far we have been moving money forward in time; that is, we have taken a known pres- ent 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 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. M05_TITM2189_13_GE_C05.indd 171 17/05/17 11:10 AM  172 PART 2 | Valuation of Financial Assets 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 the number of years rise. At i 5 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 [ 1 $62.09 = $100 (1 +.10)5 ] Present value (dollars) $100.00 $80.00 $62.09 $60.00 $40.00 $38.55 $20.00 $0.00 [ 1 $38.55 = $100 (1 +.10)10] 0 2 4 6 8 10 Year >> END FIGURE 5.2 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): Future Value Number of Present Annual Years (n) in Year n = a1 + b (5–1a) Value (PV) Interest Rate (i) (FVn) To determine the present value of a known future cash flow, we simply take Equation (5–1a) and solve for PV: Future Value Present 1 = in Year n Numbers of Years (n) § (5–2) Value (PV) £ Annual (FVn) a1 + b Interest Rate (i) We refer to the term in the brackets as the present value interest factor, which is the value by which we multiply the future value to calculate the present value. Thus, to find the pres- ent value of a future cash flow, we multiply the future cash flow by the present value interest factor:2 Future Value Present Value Present = in year n * ° Interest Factor ¢ Value (PV ) (FVn) (PVIF ) 1 where Present Value Interest Factor (PVIF) =. (1 + i)n 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 pres- ent, of 10 percent, $100 received in 10 years will be worth only $38.55 today. By contrast, if the discount rate is 5 percent, the present value will be $61.39. If the discount rate is 2 Related tables appear in Appendix C in MyLab Finance. M05_TITM2189_13_GE_C05.indd 172 25/07/17 4:36 PM  CHAPTER 5 | The Time Value of Money 173 Checkpoint 5.4 MyLab Finance Video 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 10 years from today. What is the present value of $500,000 to be received 10 years from today if the discount rate is 6 percent annually? STEP 1: Picture the problem Expressed as a timeline, this problem would look like the following: i = 6% Time Period 0 1 2 3 4 5 6 7 8 9 10 Years Cash Flow Present Value = ? $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 5 $500,000, n 5 10, and i 5 6% into Equation (5–2), we find 1 PV = $500,000 c d (1 +.06)10 1 = $500,000 c d 1.79085 = $500,000 3.5583944 = $279,197 The present value of the $500,000 to be received in 10 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 10 years of $279,197 today if the rate of interest is 6 percent? Using our FV equation, Equation (5–1a), we can see that the answer is $500,000. Using a Financial Calculator. Enter 10 6.0 0 500,000 N I/Y PV PMT FV Solve for -279,197 Using an Excel Spreadsheet. 5 PV(rate,nper,pmt,fv) or, with values entered, 5 PV(0.06,10,0,500000) STEP 4: Analyze 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 differ- ent 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 percent discount rate? ANSWER: $29,530. Your Turn: For more practice, do related Study Problems 5–12, 5–15, 5–19, and 5–28 at the end of this chapter. >> END Checkpoint 5.4 M05_TITM2189_13_GE_C05.indd 173 25/07/17 4:37 PM  174 PART 2 | Valuation of Financial Assets 10 percent but the $100 is received in 5 years instead of 10 years, the present value will be $62.09. This concept of present value plays a central role in the valuation of 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 the future value of a series of cash flows. There are a number of problems that require you to solve for either the number of periods in the future, n, or the rate of interest, i. For example, to answer the following questions, you will need to calculate the number of periods in the future, 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 5 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 Ex- cel spreadsheet, you can easily solve for i or n in any of these or similar 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 percent annually. Let’s take a look at how to solve this using the mathematical formulas, a financial calculator, and an Excel spreadsheet. Using the Mathematical Formulas. Substituting for FV, PV, and i in Equation (5–1a), Future Value Number of Present Annual Years (n) in Year n = a1 + b (5–1a) Value (PV) Interest Rate (i) (FVn) $20,000 = $9,330 (1.10)n Solving for n mathematically is tough. One way is to solve for n using a trial-and-error ap- proach. That is, you could substitute different values of n into the equation—either increasing the value of n to make the right-hand 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 10.0 -9,330 0 20,000 N I/Y PV PMT FV Solve for 8.0 You’ll notice that PV is input with a negative sign. In effect, the financial calculator is pro- grammed 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. Using an Excel Spreadsheet. With Excel, solving for n is straightforward. You simply use 5 NPER(rate,pmt,pv,fv) or, with variables entered, 5 NPER(0.10,0,−9330,20000). M05_TITM2189_13_GE_C05.indd 174 17/05/17 11:10 AM  CHAPTER 5 | The Time Value of Money 175 Checkpoint 5.5 MyLab Finance Video 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 that 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 percent compounded annually? STEP 1: Picture the problem In this case, we are solving for the number of periods: i = 9% Time Period 0 1 2 3 4 5.... ? Years.... Cash Flow -$7,752 $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 percent interest per year. We are solving for n, and we can calculate it using Equation (5–1a). STEP 3: Solve Using a Financial Calculator. Enter 9.0 -7,752 0 20,000 N I/Y PV PMT FV Solve for 11.0 Using an Excel Spreadsheet. 5 NPER(rate,pmt,pv,fv) or, with values entered, 5 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 percent 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 percent 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 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 5 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 1 i)n for 8 years (n 5 8) at 9 percent (i 5 9%) is 1.993, which is pretty close to the Rule of 72’s approximation of 2.0. 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 will M05_TITM2189_13_GE_C05.indd 175 25/07/17 4:38 PM  176 PART 2 | Valuation of Financial Assets you have to earn on your $34,946 investment? Let’s take a look at solving this using the math- ematical formulas, a financial calculator, and an Excel spreadsheet to calculate i. Using the Mathematical Formulas. If you write this problem using our time-value-of- money formula, you get Future Value Number of Present Annual Years (n) in Year n = a1 + b (5–1a) Value (PV) Interest Rate (i) (FVn) $800,000 = $34,946 (1 + i)30 Once again, you could resort to a trial-and-error approach by substituting different values of i into the equation and calculating the value on the right-hand side of the equation to see if it is equal to $800,000. However, again, this would be quite cumbersome and unnecessary. Alternatively, you could solve for i directly by dividing both sides of the equation above by $34,946 (1 + i)30 = $800,000>$34,946 = 22.8925 and then taking the 30th root of this equation to find the value of (1 1 i). Because 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 “5” key. The answer should be 1.109999, indicating that (1 1 i) 5 1.109999 and i 5 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 -34,946 0 800,000 N I/Y PV PMT FV Solve for 11.0 Using an Excel Spreadsheet. With Excel, you use 5 RATE(nper,pmt,pv,fv) or, with values entered, 5 RATE(30,0,−34946,8000000). 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? 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 have only $11,167 now. At what rate must your $11,167 be com- pounded 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: M05_TITM2189_13_GE_C05.indd 176 17/05/17 11:11 AM  CHAPTER 5 | The Time Value of Money 177 i = ?% Time Period 0 1 2 3 4 5 6 7 8 9 10 Years Cash Flow -$11,167 $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 5 $11,167 (1 1 i)10 1.7910 5 (1 1 i)10 We then take the 10th root of this equation to find the value of (1 1 i). Because 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 “5” key. The answer should be 1.06, indicating that (1 1 i) 5 1.06 and i 5 6%. Using a Financial Calculator. Enter 10 -11,167 0 20,000 N I/Y PV PMT FV Solve for 6.0 = 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? ANSWER: 10.5 percent. 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 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 be quoted not as compounding annually but rather as com- pounding quarterly or daily. To illustrate, let’s look at two loans, one that is quoted as 8.084 percent compounded annually and one that is quoted as 7.85 percent compounded quarterly. Unfortunately, they are difficult to compare because the interest on one is compounded annu- ally (you pay interest just once a year) but the interest on the other is compounded quarterly (you pay interest four times a year). To allow borrowers to compare rates between different lenders, the U.S. Truth-in-Lending 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 M05_TITM2189_13_GE_C05.indd 177 17/05/17 11:11 AM  178 PART 2 | Valuation of Financial Assets APR as the interest rate per period (for example, per month or week) multiplied by the number of periods during which compounding occurs during the year (m): Annual Percentage Interest Rate per Compounding Rate (APR) = ° Period (for example, ¢ * Periods per (5–3) or Simple Interest per month or week) Year (m) Thus, if you are paying 2 percent per month, the number of compounding periods per year (m) 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 com- pounded 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.3 In our example, both 8.084 percent and 7.85 percent are the APRs, but they aren’t comparable because the loans have different compounding periods.