1. Chapter 5 The Time Value of Money— The Basics

2. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-2 Slide Contents Learning Objectives Principles Applied in this Chapter –5.1 Using Timelines to Visualize Cash Flows –5.2 Compounding and Future Value –5.3 Discounting and Present Value –5.4 Making Interest Rates Comparable Key Terms

3. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-3 Learning Objectives 1.Construct cash flow timelines to organize your analysis of problems involving the time value of money. 2.Understand compounding and calculate the future value of cash flows using mathematical formulas, a financial calculator, and an Excel spreadsheet.

4. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-4 Learning Objectives (cont.) 3.Understand discounting and calculate the present value of cash flows using mathematical formulas, a financial calculator and an Excel spreadsheet. 4.Understand how interest rates are quoted and know how to make them comparable.

5. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-5 Principles Applied in this Chapter Principle 1: Money Has a Time Value.

6. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-6 5.1 USING TIMELINES TO VISUALIZE CASHFLOWS

7. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-7 Using Timelines to Visualize Cashflows A timeline identifies the timing and amount of a stream of payments – both cash received and cash spent - along with the interest rate earned. A timeline is typically expressed in years, but it could also be expressed as months, days or any other unit of time.

8. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-8 Time Line Example i=10% Years Cash flow-$100$30$20-$10$50 The 4-year timeline illustrates the following: –The interest rate is 10%. –A cash outflow of $100 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 of $10 in year 3 and cash inflow of $50 in year 4. 01234

9. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-9 5.2 COMPOUNDING AND FUTURE VALUE

10. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-10 Compounding and Future Value Time value of money calculations involve 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).

11. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-11 Compound Interest and Time Example: Suppose that you deposited $500 in your savings account that earns 5% annual interest. How much will you have in your account after two years? After five years? FV 2 = PV(1+i) n = 500(1.05) 2 = $551.25

12. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-12 Compound Interest and Time YEARPV or Beginning Value Interest Earned (5%)FV or Ending Value 1$500.00$500*.05 = $25$525 2$525.00$525*.05 = $26.25$551.25 3 $551.25*.05 =$27.56$578.81 4 $578.81*.05=$28.94$607.75 5 $607.75*.05=$30.39$638.14 Using Equation 5-1a: FV = PV(1+i) n = 500(1.05) 5 = $638.14

13. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-13 Figure 5.1 Future Value and Compound Interest Illustrated (Panel A) Calculating Compound Interest

14. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-14 Figure 5.1 Future Value and Compound Interest Illustrated (cont.) (Panel B) The Power of Time

15. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-15 Figure 5.1 Future Value and Compound Interest Illustrated (cont.) (Panel C) The Power of the Rate of Interest

16. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-16 Applying Compounding to Things Other Than Money Example A DVD rental firm is currently renting 8,000 DVDs per year. How many DVDs will the firm be renting in 10 years if the demand for DVD rentals is expected to increase by 7% per year? Using Equation 5-1a, –FV = 8000(1.07) 10 = 15,737.21 DVDs

17. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-17 CHECKPOINT 5.2: CHECK YOURSELF Calculating the FV of a Cash Flow What is the FV of $10,000 compounded at 12% annually for 20 years?

18. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-18 Step 1: Picture the Problem i=12% Years Cash flow-$10,000 01 2 … 20 Future Value=?

19. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-19 Step 2: Decide on a Solution Strategy This is a simple future value problem. We can find the future value using Equation 5-1a.

20. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-20 Step 3: Solve Solve Using a Mathematical Formula FV = $10,000(1.12) 20 = $10,000(9.6463) = $96,462.93

21. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-21 Step 3: Solve (cont.) Solve Using a Financial Calculator N = 20 I/Y = 12% PV = -10,000 PMT = 0 FV = $96,462.93 Solve Using an Excel Spreadsheet =FV(rate,nper,pmt, pv) =FV(0.12,20, 0,-10000) = $96,462.93

22. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-22 Step 4: Analyze If you invest $10,000 at 12%, it will grow to$96,462.93 in 20 years.

23. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-23 Compound Interest with Shorter Compounding Periods Banks frequently offer savings account that compound interest every day, month, or quarter. More frequent compounding will generate higher interest income and lead to higher future values.

24. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-24 Table 5-2 The Value of $100 Compounded at Various Non-Annual Periods and Various Rates

25. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-25 CHECKPOINT 5.3: CHECK YOURSELF Calculating Future Values Using Non-Annual Compounding Periods 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?

26. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-26 Step 1: Picture the Problem i=10% Months Cash flow -$50,000 01 2 … 120 FV of $50,000 Compounded for 120 months @ 10%/12

27. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-27 Step 2: Decide on a Solution Strategy This involves solving for future value of $50,000. Since the interest is compounded monthly, we will use equation 5-1b.

28. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-28 Step 3: Solve Using a Mathematical Formula FV = PV (1+i/12) m*12 = $50,000 (1+0.10/12) 10*12 = $50,000 (2.7070) = $135,352.07 Using a Financial Calculator N = 120 I/Y =.833% PV = -50,000 PMT = 0 FV = $135,352

29. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-29 Step 3: Solve (cont.) Using an Excel Spreadsheet =FV(rate,nper,pmt, pv) =FV(0.00833,120, 0,-50000) = $135,346.71

30. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-30 Step 4: Analyze More frequent compounding leads to a higher FV as you are earning interest more often on interest you have previously earned. If the interest was compounded annually, the FV would have been equal to only $129,687.12 –$50,000 (1.10) 10 = $129,687.12

31. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-31 5.3 DISCOUNTING AND PRESENT VALUE

32. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-32 The Key Question What is value today of cash flow to be received in the future? The answer to this question requires computing the present value (PV) i.e. the value today of a future cash flow, and the process of discounting, determining the present value of an expected future cash flow.

33. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-33 The Mechanics of Discounting Future Cash Flows The term in the bracket is known as the Present Value Interest Factor (PVIF). PV = FV n × PVIF

34. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-34 Figure 5.2 The Present Value of $100 Compounded at Different Rates and for Different Time Periods

35. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-35 CHECKPOINT 5.4: CHECK YOURSELF Solving for the PV of a Future Cash Flow What is the present value of $100,000 to be received at the end of 25 years given a 5% discount rate?

36. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-36 Step 1: Picture the Problem i=5% Years Cash flow$100,000 01 2 … 25 Present Value =?

37. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-37 Step 2: Decide on a Solution Strategy Here we are solving for the present value (PV) of $100,000 to be received at the end of 25 years using a 5% interest rate. We can solve using equation 5-2.

38. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-38 Step 3: Solve Using a Financial Calculator N = 25 I/Y = 5 PMT = 0 FV = 100,000 PV = -$29,530 Using a Mathematical Formula PV = $100,000 [1/(1.05) 25 ) = $100,000 [0.2953] = $29,530

39. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-39 Step 4: Analyze Once you’ve found the present value, it can be compared to other present values. Present value computation makes cash flows that occur in different time periods comparable so that we can make good decisions.

40. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-40 Two Additional Types of Discounting Problems Solving for: (1) Number of Periods; and (2) Rate of Interest (1): How long will it take to accumulate a specific amount in the future? It is easier to solve for “n” using the financial calculator or Excel rather than mathematical formula. (See checkpoint 5.5)

41. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-41 The Rule of 72 It determine the number of years it will take to double the value of your investment. N = 72/interest rate For example, if you are able to generate an annual return of 9%, it will take 8 years (=72/9) to double the value of investment.

42. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-42 CHECKPOINT 5.5: CHECK YOURSELF Solving for Number of Periods, n How many years will it take for $10,000 to grow to $200,000 given a 15% compound growth rate?

43. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-43 Step 1: Picture the Problem i=15% Years Cash flow -$10,000 $200,000 01 2 … N =? We know FV, PV, and i and are solving for N

44. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-44 Step 2: Decide on a Solution Strategy In this problem, we are solving for “n”. We know the interest rate, the present value and the future value. We can calculate “n” using a financial calculator or an Excel spreadsheet.

45. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-45 Step 3: Solve Using a Financial Calculator I/Y = 15 PMT = 0 PV = -10,000 FV = 200,000 N = 21.4 years Using an Excel Spreadsheet N = NPER(rate,pmt,pv,fv) = NPER(.15,0,-10000,200000) = 21.4 years

46. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-46 Step 4: Analyze It will take 21.4 years for $10,000 to grow to $200,000 at an annual interest rate of 15%.

47. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-47 Solving for the Rate of Interest (2): What rate of interest will allow your investment to grow to a desired future value? We can determine the rate of interest using mathematical equation, the financial calculator or the Excel spread sheet.

48. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-48 CHECKPOINT 5.6: CHECK YOURSELF Solving for the Interest Rate, i At what rate will $50,000 have to grow to reach $1,000,000 in 30 years?

49. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-49 Step 1: Picture the Problem i=?% Years Cash flow-$50,000$1,000,000 01 2 … 30 We know FV, PV and N and are Solving for “interest rate”

50. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-50 Step 2: Decide on a Solution Strategy Here we are solving for the interest rate. The number of years, the present value, the future value are known. We can compute the interest rate using mathematical formula, a financial calculator or an Excel spreadsheet.

51. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-51 Step 3: Solve Using a Mathematical Formula I = (FV/PV) 1/n - 1 = (1000000/50000) 1/30 - 1 = (20) 0.0333 - 1 = 1.1050 - 1 =.1050 or 10.50% Using an Excel Spreadsheet =Rate (nper, pmt, pv, fv) =Rate (30,0,-50000,1000000) =10.50%

52. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-52 Step 4: Analyze You will have to earn an annual interest rate of 10.50 percent for 30 years to increase the value of investment from $50,000 to $1,000,000.

53. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-53 5.4 MAKING INTEREST RATES COMPARABLE

54. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-54 Annual Percentage Rate (APR) The annual percentage rate (APR) indicates the interest rate paid or earned in one year without compounding. APR is also known as the nominal or quoted (stated) interest rate.

55. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-55 Calculating the Interest Rate and Converting it to an EAR We cannot compare two loans based on APR if they do not have the same compounding period. To make them comparable, we calculate their equivalent rate using an annual compounding period. We do this by calculating the effective annual rate (EAR)

56. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-56 CHECKPOINT 5.7: CHECK YOURSELF Calculating an EAR What is the EAR on a quoted or stated rate of 13 percent that is compounded monthly?

57. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-57 Step 1: Picture the Problem i= an annual rate of 13% that is compounded monthly Months 01 2 … 12 Compounding periods are expressed in months (i.e. m=12) and we are Solving for EAR

58. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-58 Step 2: Decide on a Solution Strategy Here we need to solve for Effective Annual Rate (EAR). We can compute the EAR by using equation 5-4

59. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-59 Step 3: Solve EAR = [1+.13/12] 12 - 1 = 1.1380 – 1 =.1380 or 13.80%

60. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-60 Step 4: Analyze There is a significant difference between APR and EAR (13.00% versus 13.80%). If the interest rate is not compounded annually, we should compute the EAR to determine the actual interest earned on an investment or the actual interest paid on a loan.

61. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-61 To the Extreme: Continuous Compounding As m (number of compounding period) increases, so does the EAR. When the time intervals between when interest is paid are infinitely small, we can use the following mathematical formula to compute the EAR. EAR = (e quoted rate ) – 1 –Where “e” is the number 2.71828

62. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-62 Continuous Compounding (cont.) Example What is the EAR on your credit card with continuous compounding if the APR is 18%? EAR = e.18 - 1 = 1.1972 – 1 =.1972 or 19.72%

63. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-63 Key Terms Annual Percentage Rate (APR) Compounding Compound Interest Discounting Discount Rate Effective Annual Rate (EAR) Future Value

64. Copyright ©2014 Pearson Education, Inc. All rights reserved.5-64 Key Terms (cont.) Future Value Interest Factor Nominal or Stated Interest Rate Present Value Present Value Interest Factor Simple Interest Timeline