Ch. 4: Introduction to Valuation: The Time Value of Money

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

2. What’s the Future Value of an initial $100 after 3 years if r = 10%? 0 1 2 3 100 FV = ? “$100 compounded for 3 years at 10% annually.“

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

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)

5. HP-10B Calculator setup to find FV: 3 10 -100 0 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.

6. 0 1 2 3 100 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.” PV = ?

7. INPUTS 3 10 0 100 -75.13 OUTPUT 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: FV N I/YR PV PMT

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

9. INPUTS 7 -1 0 2 N I/YR PV PMT FV 10.2 OUTPUT HP 10B Calculator Solution: 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

10. INPUTS 3 -100 0 120 N I/YR PV FV PMT OUTPUT 4th Type of Problem: Solving for r Find r so $100 grows to $120 in 3 years: $100 (1 + r)3 = $120 6.27% Excel: RATE; Table A.1 less precise

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)

12. GMAC Questions 6-10, p. 105 PowerBall Lottery example, p. 86 Recommended Practice • Self-Test Problems 4.1 - 4.3, p. 104 • Problems on pp. 105-7: 1, 3, 5, 7, 9, 13, 17, 25 (answers are on pp. 547-8)