1. Ch. 4: Introduction to Valuation: The Time Value of Money
Also known as Discounted Cash Flow or Present Value Analysis
Future Value and Compounding
Present Value and Discounting
Time Lines
Show timing of cash flows (CF). Tick marks are at ends of periods. Time 0 is now, Time 1 is the end of period 1 and also the beginning of period 2. Example: Time line for a $100 lump sum due at the end of Year 2 1

2. What’s the Future Value of an initial $100 after 3 years if r = 10%?1 2 3
100
FV = ?
“$100 compounded for 3 years at 10% annually.“ 5

3. After 1 year: FV1 = PV(1 + r) = PV + r(PV) = $100(1.10) = $110.00
= $100(1.10) = $110.00
After 2 years:
FV2 = PV(1 + r)2
= $100(1.10)2 = $121.00
After 3 years:
FV3 = PV(1 + r)3
= $100(1.10)3 = $133.10
In general: 6

4. Notes: Note the role of 1 in (1+r) 4 variables imply 4 problem types
There are 4 ways to solve for the unknown variable, given the other 3 variables:
do the math
use a financial calculator
use a spreadsheet
use tables (text Appendix A) 7

5. HP-10B Calculator setup to find FV:
N I/YR PV PMT FV
133.10
INPUTS
OUTPUT
Start with Clear All. For these problems, be sure
P/YR = 1 and “BEGIN” is off (so cash flows are at
end of each period). Set display to 4 decimal places. Either PV or FV must be negative.
Spreadsheet: Excel Function Wizard: FV (similar)
Table: see Table A.1: “Future value of $1 at the end of t periods” The table shows Future Value Interest Factors. 8

6. 2nd Type of Problem: Find PV of $100 due in 3 years if r = 10%
Finding the Present Value is discounting,
the reverse of compounding. In PV problems, r is called “the discount rate.”
100 1 2 3
PV = ? 9

7. HP 10B Calculator Solution:
PV = 100(0.7513) = $75.13
The Present Value Interest Factor (PVIFr,t) is from Table A.2: “Present value of $1 to be received after t periods.”
HP 10B Calculator Solution:
-75.13
INPUTS
OUTPUT N
I/YR PV
PMT FV 10

8. 3rd Type of Problem: Solving for t
If sales grow at 7% per year, how long before sales double?
FVt = PV(1 + r)t
2 = 1(1.07)t
2 = (1.07)t
ln 2 = t * ln(1.07)
t = ln2/ln1.07 = 10.2 12

9. HP 10B Calculator Solution:
N I/YR PV PMT FV
10.2
INPUTS
OUTPUT
Excel Function Wizard: NPER
Table: Table A.1 implies a little over 10 years (less precision)
“Rule of 72”: Divide 72 by interest rate (or
growth rate) to approximate doubling time 13

10. 4th Type of Problem: Solving for r
Find r so $100 grows to $120 in 3 years:
$100 (1 + r)3 = $120
INPUTS
OUTPUT N
I/YR PV FV
PMT
6.27%
Excel: RATE; Table A.1 less precise 14

11. Examples Suppose you just invested $1,000 in a mutual fund.
If it earns 12% per year, how much will your shares be worth in 5 years?
How much will the shares be worth in 10 years?
If it earns 20% per year, how much will the shares be worth in 10 years?
In 1958 the average tuition for one year at an Ivy League school was $1,800. Thirty years later, in 1988, the average cost was $13,700. What was the (average annual) growth rate in tuition over the 30-year period? (Old Exam Question) 15

12. Recommended Practice GMAC Questions 6-10, p. 105
PowerBall Lottery example, p. 86
Recommended Practice
Self-Test Problems , p. 104
Problems on pp : 1, 3, 5, 7, 9, 13, 17, 25 (answers are on pp ) 15