1. Chapter 4
Time Value of Money

2. Topics in Chapter
Read Sections 4-1, 4-2, 4-3, 4-4, 4-5, 4-6, 4-15, 4-17.
Future Value and Compounding
Present Value and Discounting
Rates of Return/Interest Rates
Amortization

3. Determinants of Intrinsic Value: The Present Value Equation
Net operating
profit after taxes
Required investments
in operating capital −
Free cash flow
(FCF) =
FCF1
FCF2
FCF∞
...
Value =
(1 + WACC)1
(1 + WACC)2
(1 + WACC)∞
Weighted average
cost of capital
(WACC)
Market interest rates
Firm’s debt/equity mix
Cost of debt
Cost of equity
Market risk aversion
Firm’s business risk

4. Overview
Of all the concepts used in finance, none is more important than the time value of money (TVM), also called discounted cash flow (DCF) analysis.

5. 4-1 Time Lines: show timing of cash flows
4-1 Time Lines: show timing of cash flows. The basic tool for TVM/DCF analysis.
CF0
CF1
CF3
CF2 1 2 3 I%
Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

6. Time line for a $100 lump sum due at the end of Year 2.1
2 Year I%

7. Time line for uneven CFs
100 50 75 1 2 3 I%
-50

8. Cash Inflow (+) Vs. Outflow (-)
Cash Inflow (+): a cash receipt.
Cash Outflow (-): a cash deposit, cost, or amount paid.
When deciding whether a cash flow is inflow or outflow: first take a perspective; second remember that the sign doesn’t represent ownership, but just indicate the direction of money movement. For example, you deposited $100 into a bank account today, and plan to withdraw the money one year later. From your perspective, at time 0, you have a cash outflow (-) of $100, then at time 1, you will have a cash inflow (+) of more than $100 (principal plus interest). From the bank’s perspective, the signs of cash flows are just the opposite.

9. 4-2 Future Values (FV) Example: FV of an initial $100 after 3 years with I = 5%1 2 3 5%
Finding FVs (moving to the right
on a time line) is called compounding.
100

10. 4-2a & 4-2b Step-by-step/Formula/Regular Calculator Approach
After 1 year
FV1 = PV + INT = PV + PV*I
= PV(1 + I)1
= $100*( )
= $105

11. After 2 years FV2 = FV1(1+I) = PV(1+I)(1+I) = PV(1+I)2 = $100(1.05)2
= $110.25

12. After 3 years FV3 = FV2(1+I)=PV(1 + I)2(1+I) = PV(1+I)3 = $100(1.05)3
= $115.76
In general, Equation (4-1) on P143:
FVN = PV(1 + I)N

13. Three Ways to Find FVs
Step-by-step/Formula/Regular calculator approach using time line (as shown in Slides 10-12).
Use a financial calculator.
Use a spreadsheet.

14. 4-3c Financial Calculators: TI BAII+
Set number of decimal places to display: 2ND FORMAT; use the up and down arrows to display DEC=; press the number of decimal places you want to display, e.g. 2/3/4 etc.; press ENTER; press 2ND QUIT.

15. BAII +: Set Time Value Parameters
To set END mode (for cash flows occurring at the end of the period), hit 2ND BGN; 2ND SET will toggle between cash flows at the beginning of the period (BGN) and end of the period (END), finally press 2ND QUIT. Leave it as END.
To set 1 payment per period, press 2ND P/Y, press 1, press ENTER; press 2ND QUIT. Leave it as 1.

16. BAII+ Settings (Continued)
To reset TVM calculations:
2ND CLR TVM (above the FV key).
Box on P145: Hints on Using Financial Calculators

17. Financial Calculator Solution
Financial calculators solve this equation:
PV (1+I)N + FVN = 0
There are 4 variables. If 3 are known, the calculator will solve for the 4th.
(Set PMT = 0. This key is reserved to represent a series of equal payments as in section 4.17 Amortized Loans.)

18. Here’s the setup to find FV
N I/YR PV PMT FV
115.76
INPUTS
OUTPUT
Clearing automatically sets everything to 0, but for safety enter PMT = 0.
Financial calculators are programmed to assume that rates are stated as percentages: e.g. enter 5 not 0.05 to I/YR.
When using financial calculators or financial functions in Excel, it’s important to identify cash inflows versus cash outflows.

19. 4-2d Spreadsheet Solution
Use the FV function: Insert, Function, Financial, FV
= FV(I, N, PMT, PV)
= FV(0.05, 3, 0, -100) =

20. 4-2f Graphic View of the Compounding Process Note that time value concepts can be applied to anything that grows: population, sales, EPS, or your future salary.
Figure 4-2 Growth of $100 at Various Interest Rates and Time Periods

21. 4-2g Simple Interest vs. Compound Interest
Compound Interest: interest is earned not only on the original principal, but also on the interest earned in prior periods.
Simple Interest: interest is earned only on the principal.
In our previous example, the total simple interest earned over 3 years would be: PV*I*N = 100*0.05*3 = $15 compared with the total compound interest of $15.76.
Most applications in finance are based on compound interest, but you should be aware that simple interest is still specified in some legal documents.
In our course, by default, we use compound interest for all applications except where the use of simple interest is explicitly specified.

22. 4-3 Present Values (PV) What’s the PV of $115
4-3 Present Values (PV) What’s the PV of $ due in 3 years if I/YR = 5%? 5%
Finding PVs is called discounting, and it’s the reverse of compounding.
115.76 1 2 3
PV = ?

23. 4-3a Discounting a FV to find the PV
Equation (4-3) on P150: N
FVN 1
PV =
= FVN
(1+I)N
1 + I 3 1 PV =
$115.76
1.05
= $115.76*(0.8638) = $100

24. Financial Calculator Solution
N I/YR PV PMT FV
-100
INPUTS
OUTPUT
Either PV or FV must be negative. Here
PV = From your perspective, put in
$100 today, take out $ after 3 years.

25. Spreadsheet Solution
Use the PV function in Excel: Insert, Function, Financial, PV:
= PV(I, N, PMT, FV)
= PV(0.05, 3, 0, ) = -100

26. 4-3b Graphic View of the Discounting Process There is an inverse relationship between present value and discount rate: ceteris paribus, the higher the interest rate, the faster the present value falls. So, the timing of cash flows matters: Ceteris paribus, the same amount due later is less valuable (smaller PV) than due sooner.
Figure 4-4 Present value of $100 at various Interest Rates and Time Periods

27. 4-4 Finding the Interest Rate I A financial security has a cost of $100 and it will return $150 after 10 years. What is the (annual) rate of return you will earn if you buy the security? ?%
150 1
--- 10
-100
FV = PV(1 + I)N
$150 = $100(1 + I)10
(1.5)(1/10) = (1 + I)
= (1 + I)
I = = 4.14%

28. Financial Calculator Solution The output is 4
Financial Calculator Solution The output is 4.14, so the answer should be 4.14%. Recall that financial calculators assume rates are stated as percentages.
N I/YR PV PMT FV
4.14
INPUTS
OUTPUT

29. Spreadsheet Solution Use the RATE function: = RATE(N, PMT, PV, FV)

30. 4-5 Finding the Number of Periods N Suppose you have $500,000 now and the interest rate is 4.5% per year. How long will it be before you have $1 million?
4.5%
1,000,000 1 2 ?
-500,000
FV = PV(1 + I)N
(Continued on next slide)

31. Time to Double (Continued)
N*LN(1.045)= LN(2)
N = LN(2)/LN(1.045)
N = / = 15.75

32. Financial Calculator Solution
, ,000,000
N I/YR PV PMT FV
15.75
INPUTS
OUTPUT

33. Spreadsheet Solution Use the NPER function: = NPER(I, PMT, PV, FV)

34. 4-6 Perpetuities
A perpetuity (consol) is a series of equal payment continued forever.
Even it’s perpetual, it has a finite present value and the PV is calculated as:
(Equation 4-4) on P155 PV of a Perpetuity =
Example: A consol with a face value of $1,000 has a coupon rate of 2.5% per year. What price would it be if the current market interest rate for similar risk securities is 2%?
The coupon rate in the original contract determines the perpetual annual coupon payment (interest payment): $1,000 * = $25
The current price = PV = PMT/I = $25/0.02 = $1,250 (different from face value if the going interest rate is different from the coupon rate)

35. Bond Price and Interest Rate
When interest rates change, the prices of outstanding bonds also change, but inversely to the change in rates: bond prices decline if rates rise, and prices increase if rates fall.
This relationship holds for all bonds, both consols and those with finite maturities in chapter 5 Bond Valuation.

36. 4-15 Semiannual and Other Compounding Periods
In our previous examples, we assumed that interest is compounded once a year, or annually. In practice, virtually all bonds pay interest semiannually; most stocks pay dividends quarterly; most mortgages, student loans, and auto loans involve monthly payments; and most money market fund accounts pay interest daily. Therefore, it is essential that you understand how to deal with non-annual compounding.

37. 4-15a Types of Interest Rates
Nominal/Quoted Annual Rate or APR
Stated in contracts, and quoted by banks and brokers.
Not used in calculations or shown on time lines: used to calculate periodic rate.
Number of Compounding Periods per year (M) must be given.
Examples:
8%; Quarterly compounding (M=4)
8%, Daily compounding (M=365)

38. Periodic rate (IPER )
IPER = INOM/M, where M is number of compounding periods per year. M = 4 for quarterly, 12 for monthly, and 365 (or sometimes 360) for daily compounding.
Used in calculations, shown on time lines.
Examples:
8% quarterly: IPER = 8%/4 = 2%.
8% daily (365): IPER = 8%/365 = %.

39. FV Formula with Different Compounding Periods
Equation (4-13) on P172:
INOM
FVN = PV M
M*N

40. INOM FVN = PV 1 + M 0.12 FV8Q = $100 1 + 4 = $100(1.03)8 = $126.68
Example: What is the FV of $100 at a 12% nominal rate with quarterly compounding for 2 years?
INOM
FVN = PV M
M*N
0.12
FV8Q = $ 4
4*2
= $100(1.03)8 = $126.68

41. Example Continued: Financial Calculator and Excel
Inputs: N= 4*2 = 8; I = 12/4 = 3, PV = -100, PMT = 0
Output: CPT FV =
Excel financial function:
= FV(I, N, PMT, PV)
= FV(0.03, 8, 0, -100) =

42. Effective Annual Rate (EAR or EFF%)
The EAR is the annual rate that causes PV to grow to the same FV as compounding at the periodic rate for M times per year.
Equation (4-14) on P172:

43. Effective Annual Rate Example
Example: borrow $1 for one year at 12%, compounded quarterly:
FV = PV(1 + INOM/M)M
FV = $1 ( /4)4 = $
EFF% = 12.55%, because $1 borrowed for one year at 12% quarterly compounding would grow to the same amount as $1 borrowed for one year at 12.55% annual compounding.

44. Another Example: EFF% for a credit card loan with APR of 12% (credit card: monthly compounding)
INOM M
= − 1
0.12 12
= (1.01)
= = 12.68%.

45. Finding EFF% with TI BAII+
Previous Example: credit card APR 12%
Press 2ND, Press ICONV;
Type in the nominal annual rate (12), then press ENTER;
Press up arrow, you will see C/Y on display, type in the compounding frequency: 12, then press Enter
Press up arrow again, you will see EFF on display, press CPT, you get 12.68
Then your final answer: 12.68%

46. Comparing Rates with EFF%
The EFF% rate is rarely used in TMV calculations: Use Periodic Rate (& the correct number of Periods) for TMV/DCF calculations.
However, EAR/EFF% must be used to compare the effective costs of different loans or rates of return on different investments when payment periods (compounding frequency) differ.

47. Summary: when is each rate used?
INOM:
Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.
How often interest is compounded should also be given (M).

48. When is each rate used? (Continued)
IPER:
Used in calculations, shown on time lines.
Then the correct number of periods used in calculation is M*N.
If INOM has annual compounding,
then IPER = INOM/1 = INOM

49. When is each rate used? (Continued)
EAR (or EFF%): used to compare returns on investments or effective costs of loans with different payments per year.

50. 4-15b The Result of Frequent Compounding
Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant?
Why?

51. The Impact of Compounding (Answer)
LARGER!
If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily --- interest is earned on interest more often.

52. Example: FV of $100 at a 12% nominal rate for 1 year with different compounding frequencies
FV(Ann.)
= $100(1.12)1
= $112
FV(Semi.)
= $100(1.06)2
= $112.36
FV(Quar.)
= $100(1.03)4
= $112.55
FV(Mon.)
= $100(1.01)12
= $112.68
FV(Daily)
= $100(1+(0.12/365))(365)
= $112.75

53. EAR Annual = Nominal Rate (APR) = 12%.
Example Continued: EAR (or EFF%) for a Nominal Rate of 12% with different compounding frequencies
EAR Annual = Nominal Rate (APR) = 12%.
EARS = ( /2)2 - 1 = %.
EARQ = ( /4)4 - 1 = %.
EARM = ( /12) = %.
EARD(365) = ( /365) = %.

54. Can the effective rate ever be equal to the nominal rate?
Yes, but only if annual compounding is used, i.e., if M = 1.
If M > 1, EFF% will always be greater than the nominal rate.

55. 4-17 Amortized Loans
An amortized loan is a loan that is repaid in equal amounts on a monthly, quarterly, or annual basis etc.
Examples: mortgage loans, auto loans, student loans, and many business loans.

56. 4-17a & 4-17b Payments & Amortization Schedules: an Example
Construct an amortization schedule for a $100,000, 6% annual rate loan with 5 equal payments at the end of the next 5 years.

57. Step 1: Find the required payments
PMT 1
--- 5 6%
100,000 ,
INPUTS
OUTPUT N
I/YR PV FV
-23,739.64

58. Step 2: Find interest charge for Period 1.
INTt = Beg balt*(I)
INT1 = $100,000(0.06) = $6,000

59. Step 3: Find repayment of principal in Period 1.
Repmt = PMT - INT
= $ $6,000
= $17,739.64

60. Step 4: Find ending balance after Period 1.
End bal = Beg bal - Repmt
= $100,000 - $17, = $82,260.36
Repeat these steps for Periods 2 through 5
to complete the amortization table.

61. Amortization Table: Figure 4-11 on P177
Period
BEG BAL (1)
PMT (2)
INT (3)=(1)*I
PRIN RePay (4)=(2)-(3)
END BAL (5)=(1)-(4) 1
$100,000
$23,739.64
$6,000
$17,739.64
$82,260.36 2
82,260.36
63,456.34 3
3,807.38
19,932.26
43,524.08 4
2,611.44
21,128.20
22,395.89 5
1,343.75
TOTAL
$118,698.20
$18,698.19

62. Amortization Schedules
Amortization schedules are widely used--for home mortgages, auto loans, business loans, retirement plans, and more. They are very important!
Spreadsheets are great for setting up amortization schedules.

63. 4-17c Mortgage Interest Payments: another example in Excel
Question: Consider a 30-year home mortgage of $250,000 at an annual rate (APR) of 6%. How much interest will the borrower pay over the life of the loan? How much in the first year?
The Amortization Schedule in Excel for this question is provided in course materials.
(Please also see the discussion in the handout on how to use Excel to develop a model for sensitivity analysis, e.g. with different loan amounts and APRs.)

64. Homework Problems: 1: e, g, h, i, j; 4, 5
Questions on P183:
1: e, g, h, i, j; 4, 5
Problems on P184:
1, 2, 3, 4, 16, 17, 19, 27, 29
(You can use formula/financial calculator/spreadsheet: for each question one approach is enough. You may use multiple approaches to double check.)
Develop an Amortization Schedule in Excel:
5-year Auto Loan of $30,000 with APR 3%
(use the home mortgage loan example in course materials as a template)