# Chapter 4. Time Value of Money

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1 Chapter 4 Time Value of Money

2 Learning Goals 1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow. 2. Understand the concept of future value and present value, their calculation for single amounts, and the relationship between them. 3. Find the future value and the present value of both an ordinary annuity and an annuity due, and the present value of a perpetuity. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-2

3 Learning Goals (cont.) 4. Calculate both the future value and the present value of a mixed stream of cash flows. 5. Understand the effect that compounding interest more frequently than annually has on future value and the effective annual rate of interest. 6. Describe the procedures involved in (1) determining deposits needed to accumulate to a future sum, (2) loan amortization, (3) finding interest or growth rates, and (4) finding an unknown number of periods. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-3

4 The Role of Time Value in Finance Most financial decisions involve costs & benefits that are spread out over time. Time value of money allows comparison of cash flows from different periods. Question Would it be better for a company to invest \$100,000 in a product that would return a total of \$200,000 after one year, or one that would return \$220,000 after two years? Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-4

5 The Role of Time Value in Finance (cont.) Most financial decisions involve costs & benefits that are spread out over time. Time value of money allows comparison of cash flows from different periods. Answer It depends on the interest rate! Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-5

6 Basic Concepts Future Value: compounding or growth over time Present Value: discounting to today s value Single cash flows & series of cash flows can be considered Time lines are used to illustrate these relationships Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-6

7 Computational Aids Use the Equations Use the Financial Tables Use Financial Calculators Use Electronic Spreadsheets Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-7

8 Computational Aids (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-8

9 Computational Aids (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-9

10 Computational Aids (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-10

11 Computational Aids (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-11

12 Advantages Electronic Spreadsheets Spreadsheets go far beyond the computational abilities of calculators. Spreadsheets have the ability to program logical decisions. Spreadsheets display not only the calculated values of solutions but also the input conditions on which solutions are based. Spreadsheets encourage teamwork. Spreadsheets enhance learning. Spreadsheets communicate as well as calculate. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-12

13 Basic Patterns of Cash Flow The cash inflows and outflows of a firm can be described by its general pattern. The three basic patterns include a single amount, an annuity, or a mixed stream: Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-13

14 Simple Interest With simple interest, you don t earn interest on interest. Year 1: 5% of \$100 = \$5 + \$100 = \$105 Year 2: 5% of \$100 = \$5 + \$105 = \$110 Year 3: 5% of \$100 = \$5 + \$110 = \$115 Year 4: 5% of \$100 = \$5 + \$115 = \$120 Year 5: 5% of \$100 = \$5 + \$120 = \$125 Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-14

15 Compound Interest With compound interest, a depositor earns interest on interest! Year 1: 5% of \$ = \$ \$ = \$ Year 2: 5% of \$ = \$ \$ = \$ Year 3: 5% of \$ = \$5.51+ \$ = \$ Year 4: 5% of \$ = \$ \$ = \$ Year 5: 5% of \$ = \$ \$ = \$ Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-15

16 Time Value Terms PV 0 = present value or beginning amount i = interest rate FV n = future value at end of n periods n = number of compounding periods A = an annuity (series of equal payments or receipts) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-16

17 Four Basic Models FV n = PV 0 (1+i) n = PV x (FVIF i,n ) PV 0 = FV n [1/(1+i) n ] = FV x (PVIF i,n ) FVA n = A (1+i) n - 1 = A x (FVIFA i,n ) i PVA 0 = A 1 - [1/(1+i) n ] = A x (PVIFA i,n ) i Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-17

18 Future Value of a Single Amount Future Value techniques typically measure cash flows at the end of a project s life. Future value is cash you will receive at a given future date. The future value technique uses compounding to find the future value of each cash flow at the end of an investment s life and then sums these values to find the investment s future value. We speak of compound interest to indicate that the amount of interest earned on a given deposit has become part of the principal at the end of the period. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-18

19 Future Value of a Single Amount: Using FVIF Tables If Fred Moreno places \$100 in a savings account paying 8% interest compounded annually, how much will he have in the account at the end of one year? \$100 x (1.08) 1 = \$100 x FVIF 8%,1 \$100 x 1.08 = \$108 Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-19

20 Future Value of a Single Amount: The Equation for Future Value Jane Farber places \$800 in a savings account paying 6% interest compounded annually. She wants to know how much money will be in the account at the end of five years. FV 5 = \$800 X ( ) 5 = \$800 X (1.338) = \$1, Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-20

21 Future Value of a Single Amount: Using a Financial Calculator Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-21

22 Future Value of a Single Amount: Using Spreadsheets Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-22

23 Future Value of a Single Amount: A Graphical View of Future Value Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-23

24 Present Value of a Single Amount Present value is the current dollar value of a future amount of money. It is based on the idea that a dollar today is worth more than a dollar tomorrow. It is the amount today that must be invested at a given rate to reach a future amount. Calculating present value is also known as discounting. The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, or the cost of capital. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-24

25 Present Value of a Single Amount: Using PVIF Tables Paul Shorter has an opportunity to receive \$300 one year from now. If he can earn 6% on his investments, what is the most he should pay now for this opportunity? \$300 x [1/(1.06) 1 ] = \$300 x PVIF 6%,1 \$300 x = \$ Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-25

26 Present Value of a Single Amount: The Equation for Future Value Pam Valenti wishes to find the present value of \$1,700 that will be received 8 years from now. Pam s opportunity cost is 8%. PV = \$1,700/( ) 8 = \$1,700/1.851 = \$ Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-26

27 Present Value of a Single Amount: Using a Financial Calculator Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-27

28 Present Value of a Single Amount: Using Spreadsheets Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-28

29 Present Value of a Single Amount: A Graphical View of Present Value Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-29

30 Annuities Annuities are equally-spaced cash flows of equal size. Annuities can be either inflows or outflows. An ordinary (deferred) annuity has cash flows that occur at the end of each period. An annuity due has cash flows that occur at the beginning of each period. An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-30

31 Types of Annuities Fran Abrams is choosing which of two annuities to receive. Both are 5-year \$1,000 annuities; annuity A is an ordinary annuity, and annuity B is an annuity due. Fran has listed the cash flows for both annuities as shown in Table 4.1 on the following slide. Note that the amount of both annuities total \$5,000. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-31

32 Future Value of an Ordinary Annuity Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-32

33 Future Value of an Ordinary Annuity (cont.) Fran Abrams wishes to determine how much money she will have at the end of 5 years if he chooses annuity A, the ordinary annuity and it earns 7% annually. Annuity a is depicted graphically below: Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-33

34 Future Value of an Ordinary Annuity: Using the FVIFA Tables FVA = \$1,000 (FVIFA,7%,5) = \$1,000 (5.751) = \$5,751 Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-34

35 Future Value of an Ordinary Annuity: Using a Financial Calculator Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-35

36 Future Value of an Ordinary Annuity: Using Spreadsheets Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-36

37 Future Value of an Annuity Due: Using the FVIFA Tables Fran Abrams now wishes to calculate the future value of an annuity due for annuity B in Table 4.1. Recall that annuity B was a 5 period annuity with the first annuity beginning immediately. FVA = \$1,000(FVIFA,7%,5)(1+.07) = \$1,000 (5.751) (1.07) = \$6,154 Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-37

38 Future Value of an Annuity Due: Using a Financial Calculator Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-38

39 Future Value of an Annuity Due: Using Spreadsheets Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-39

40 Present Value of an Ordinary Annuity Braden Company, a small producer of plastic toys, wants to determine the most it should pay to purchase a particular annuity. The annuity consists of cash flows of \$700 at the end of each year for 5 years. The required return is 8%. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-40

41 Present Value of an Ordinary Annuity: The Long Method Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-41

42 Present Value of an Ordinary Annuity: Using PVIFA Tables PVA = \$700 (PVIFA,8%,5) = \$700 (3.993) = \$2, Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-42

43 Present Value of an Ordinary Annuity: Using a Financial Calculator Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-43

44 Present Value of an Ordinary Annuity: Using Spreadsheets Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-44

45 Present Value of an Annuity Due: Using PVIFA Tables In the earlier example, we found that the value of Braden Company s \$700, 5 year ordinary annuity discounted at 8% to be about \$2,795. If we now assume that the cash flows occur at the beginning of the year, we can find the PV of the annuity due. PVA = \$700 (PVIFA,8%,5) (1.08) = \$700 (3.993) (1.08) = \$3, Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-45

46 Present Value of an Annuity Due: Using a Financial Calculator Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-46

47 Present Value of an Annuity Due: Using Spreadsheets Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-47

48 Present Value of a Perpetuity A perpetuity is a special kind of annuity. With a perpetuity, the periodic annuity or cash flow stream continues forever. PV = Annuity/Interest Rate For example, how much would I have to deposit today in order to withdraw \$1,000 each year forever if I can earn 8% on my deposit? PV = \$1,000/.08 = \$12,500 Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-48

49 Future Value of a Mixed Stream Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-49

50 Future Value of a Mixed Stream (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-50

51 Future Value of a Mixed Stream: Using Excel Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-51

52 Present Value of a Mixed Stream Frey Company, a shoe manufacturer, has been offered an opportunity to receive the following mixed stream of cash flows over the next 5 years. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-52

53 Present Value of a Mixed Stream If the firm must earn at least 9% on its investments, what is the most it should pay for this opportunity? This situation is depicted on the following time line. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-53

54 Present Value of a Mixed Stream Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-54

55 Present Value of a Mixed Stream: Using Excel Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-55

56 Compounding Interest More Frequently Than Annually Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently. As a result, the effective interest rate is greater than the nominal (annual) interest rate. Furthermore, the effective rate of interest will increase the more frequently interest is compounded. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-56

57 Compounding Interest More Frequently Than Annually (cont.) Fred Moreno has found an institution that will pay him 8% annual interest, compounded quarterly. If he leaves the money in the account for 24 months (2 years), he will be paid 2% interest compounded over eight periods. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-57

58 Compounding Interest More Frequently Than Annually (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-58

59 Compounding Interest More Frequently Than Annually (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-59

60 Compounding Interest More Frequently Than Annually (cont.) A General Equation for Compounding More Frequently than Annually Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-60

61 Compounding Interest More Frequently Than Annually (cont.) A General Equation for Compounding More Frequently than Annually Recalculate the example for the Fred Moreno example assuming (1) semiannual compounding and (2) quarterly compounding. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-61

62 Compounding Interest More Frequently Than Annually: Using a Financial Calculator Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-62

63 Compounding Interest More Frequently Than Annually: Using a Spreadsheet Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-63

64 Continuous Compounding With continuous compounding the number of compounding periods per year approaches infinity. Through the use of calculus, the equation thus becomes: FV n (continuous compounding) = PV x (e kxn ) where e has a value of Continuing with the previous example, find the Future value of the \$100 deposit after 5 years if interest is compounded continuously. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-64

65 Continuous Compounding (cont.) With continuous compounding the number of compounding periods per year approaches infinity. Through the use of calculus, the equation thus becomes: FV n (continuous compounding) = PV x (e kxn ) where e has a value of FVn = 100 x (2.7183).08x2 = \$ Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-65

66 Continuous Compounding: Using a Financial Calculator Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-66

67 Continuous Compounding: Using a Spreadsheet Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-67

68 Nominal & Effective Annual Rates of Interest The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower. The effective interest rate is the rate actually paid or earned. In general, the effective rate > nominal rate whenever compounding occurs more than once per year EAR = (1 + i/m) m - 1 Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-68

69 Nominal & Effective Annual Rates of Interest (cont.) Fred Moreno wishes to find the effective annual rate associated with an 8% nominal annual rate (I =.08) when interest is compounded (1) annually (m=1); (2) semiannually (m=2); and (3) quarterly (m=4). Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-69

70 Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-70

71 Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum (cont.) Suppose you want to buy a house 5 years from now and you estimate that the down payment needed will be \$30,000. How much would you need to deposit at the end of each year for the next 5 years to accumulate \$30,000 if you can earn 6% on your deposits? PMT = \$30,000/5.637 = \$5, Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-71

72 Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-72

73 Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-73

74 Special Applications of Time Value: Loan Amortization Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-74

75 Special Applications of Time Value: Loan Amortization (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-75

76 Special Applications of Time Value: Interest or Growth Rates At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows. Ray Noble wishes to find the rate of interest or growth reflected in the stream of cash flows he received from a real estate investment over the period from 2002 through 2006 as shown in the table on the following slide. Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-76

77 Special Applications of Time Value: Interest or Growth Rates (cont.) PVIF i,5yrs = PV/FV = (\$1,250/\$1,520) = PVIF i,5yrs = approximately 5% Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-77

78 Special Applications of Time Value: Interest or Growth Rates (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-78

79 Special Applications of Time Value: Interest or Growth Rates (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-79

80 Special Applications of Time Value: Finding an Unknown Number of Periods At times, it may be desirable to determine the number of time periods needed to generate a given amount of cash flow from an initial amount. Ann Bates wishes to determine the number of years it will take for her initial \$1,000 deposit, earning 8% annual interest, to grow to equal \$2,500. Simply stated, at an 8% annual rate of interest, how many years, n, will it take for Ann s \$1,000 (PV n ) to grow to \$2,500 (FV n )? Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-80

81 Special Applications of Time Value: Finding an Unknown Number of Periods (cont.) PVIF 8%,n = PV/FV = (\$1,000/\$2,500) =.400 PVIF 8%,n = approximately 12 years Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-81

82 Special Applications of Time Value: Finding an Unknown Number of Periods (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-82

83 Special Applications of Time Value: Finding an Unknown Number of Periods (cont.) Copyright 2006 Pearson Addison-Wesley. All rights reserved. 4-83

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