1. Financial Management: Principles & Applications
Thirteenth Edition
Chapter 5
The Time Value of Money—The Basics
Copyright © 2018, 2014, 2011 Pearson Education, Inc. All Rights Reserved

2. Learning Objectives (1 of 2)
Construct cash flow timelines to organize your analysis of problems involving the time value of money.
Understand compounding and calculate the future value of cash flows using mathematical formulas, a financial calculator, and an Excel spreadsheet.

3. Learning Objectives (2 of 2)
Understand discounting and calculate the present value of cash flows using mathematical formulas, a financial calculator and an Excel spreadsheet.
Understand how interest rates are quoted and know how to make them comparable.

4. Principles Applied in this Chapter
Principle 1: Money Has a Time Value.

5. 5.1 USING TIMELINES TO VISUALIZE CASHFLOWS

6. 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. Timelines are a critical first step that financial analysts use to solve financial problems.
A timeline is typically expressed in years, but it could also be expressed as months, days or any other unit of time.

7. Time Line Example The 4-year timeline illustrates the following:
Years 1 2 3 4
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.

8. 5.2 COMPOUNDING AND FUTURE VALUE

9. 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).

10. Compound Interest and Time (1 of 2)
The Future value of an investment grows with the number of periods we let it compound.
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?
FV2 = PV(1+i)n = 500(1.05)2 = $551.25
FV5 = PV(1+i)n = 500(1.05)5 = $638.14

11. Compound Interest and Time (2 of 2)
YEAR
PV 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 = $27.5
$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

12. Figure 5.1 Future Value and Compound Interest Illustrated (Panel A) Calculating Compound Interest
Year
Beginning Value
Interest Earned
Ending Value 1 $
$ 6.00 $ 2
$ 6.36 $ 3
$ 6.74 $ 4
$ 7.15 $

13. Figure 5.1 Future Value and Compound Interest Illustrated (Panel B) The Power of Time

14. Figure 5.1 Future Value and Compound Interest Illustrated (Panel C) The Power of the Rate of Interest

15. Applying Compounding to Things Other Than Money
Example A car rental firm is currently renting 8,000 cars per year. How many cars will the firm be renting in 10 years if the demand for car rentals is expected to increase by 7% per year?
Using Equation 5-1a,
FV = 8000(1.07)10 = 15, Cars

16. 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?

17. Step 1: Picture the Problem
Years 1
2 … 20
Cash flow
−$10,000
Blank
Future Value = ?

18. Step 2: Decide on a Solution Strategy
This is a simple future value problem. We can find the future value using Equation 5-1a.

19. Step 3: Solve (1 of 2) Solve Using a Mathematical Formula
FV = $10,000(1.12)20 = $10,000(9.6463) = $96,462.93

20. Step 3: Solve (2 of 2)
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

21. Step 4: Analyze
If you invest $10,000 at 12%, it will grow to $96, in 20 years.

22. 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.
(5-1b)

23. Table 5-2 The Value of $100 Compounded at Various Non—Annual Periods and Various Rates (1 of 2)
For 1 Year at i Percent
i = 2% 5%
10%
15%
Blank
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
105.12
110.47
116.08
Compounded weekly (52)
110.51
116.16
Compounded daily (365)
105.13
110.52
116.18

24. Table 5-2 The Value of $100 Compounded at Various Non—Annual Periods and Various Rates (2 of 2)
For 10 Years at i Percent
i = 2% 5%
10%
15%
Blank
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)
164.87
271.79
448.03

25. Table 5-2 Observations
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 compounded daily rather than annually, you would end up with $1.18 more ($ $115.00). However, if the period was extended to 10 years, then the difference would grow to $43.47 ($ $404.56)

26. 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?

27. Step 1: Picture the Problem
Months 1
2 …
120
Cash flow
−$50,000
Blank
FV of $50,000 Compounded for %/12

28. 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.

29. Step 3: Solve (1 of 2)
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

30. Step 3: Solve (2 of 2)
Using an Excel Spreadsheet = FV(rate, nper, pmt, pv) = FV( ,120, 0,−50000) = $135,346.71

31. 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

32. 5.3 DISCOUNTING AND PRESENT VALUE

33. Present Value: 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) - the value today of a future cash flow - and the process of discounting, determining the present value of an expected future cash flow.

34. The Mechanics of Discounting Future Cash Flows
The term in the bracket is known as the Present Value Interest Factor (PVIF).
PV = FVn × PVIF

35. Figure 5.2 The Present Value of $100 Compounded at Different Rates and for Different Time Periods

36. 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?

37. Step 1: Picture the Problem
Years 1
2 … 25
Cash flow
Blank
$100,000
Present Value = ?

38. 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.

39. Step 3: Solve
Using a Mathematical Formula PV = $100,000 [1/(1.05)25) = $100,000 [0.2953] = $29,530
Using a Financial Calculator N = 25 I/Y = 5 PMT = 0 FV = 100,000 PV = −$29,530

40. 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.

41. Two Additional Types of Discounting Problems
Solving for: (1) Number of Periods; and (2) Rate of Interest

42. Solving for the Number of Periods
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)

43. 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.

44. 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?

45. Step 1: Picture the Problem
Years 1
2 …
N = ?
Cash flow
Blank
−$10,000
$200,000
We know FV, PV, and i and are solving for N

46. 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.

47. Step 3: Solve Using a Financial Calculator Using an Excel Spreadsheet
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

48. Step 4: Analyze
It will take 21.4 years for $10,000 to grow to $200,000 at an annual interest rate of 15%.

49. Solving for the Rate of Interest
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.

50. 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?

51. Step 1: Picture the Problem
Years 1
2 … 30
Cash flow
Blank
−$50,000
$1,000,000
We know FV, PV and N and are Solving for “interest rate”

52. 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.

53. Step 3: Solve
Using a Mathematical Formula I = (FV/PV)1/n −1 = ( /50000)1/30 −1 = (20) −1 = −1 = or 10.50%
Using an Excel Spreadsheet = Rate (nper, pmt, pv, fv) = Rate(30,0, −50000, ) = 10.50%

54. Step 4: Analyze
You will have to earn an annual interest rate of percent for 30 years to increase the value of investment from $50,000 to $1,000,000.

55. 5.4 MAKING INTEREST RATES COMPARABLE

56. 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.

57. 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)

58. 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?

59. Step 1: Picture the Problem
I = an annual rate of 13% that is compounded monthly
Months 1
2 … 12
Compounding periods are expressed in months (i.e. m = 12) and we are Solving for EAR

60. 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

61. Step 3: Solve
EAR = [1+.13/12]12 −1 = − 1 = or 13.80%

62. 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.

63. 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

64. EAR with Continuous Compounding
Example What is the EAR on your credit card with continuous compounding if the APR is 18%?
EAR = e.18 −1
= −1
= or 19.72%

65. Key Terms (1 of 2) Annual Percentage Rate (APR) Compounding
Compound Interest
Discounting
Discount Rate
Effective Annual Rate (EAR)
Future Value

66. Key Terms (2 of 2) Future Value Interest Factor
Nominal or quoted (stated) Interest Rate
Present Value
Present Value Interest Factor
Rule of 72
Simple Interest
Timeline

67. Practice Questions

76. Copyright