Chapter 5 Introduction to Valuation: Time Value of Money

1. Chapter 5Introduction to Valuation: Time Value of Money • Homework: 3, 5, 17, 18, 19, & 20

2. Time Value of Money • Goals • Why Do We Care About This Topic? • “Exchange Rates” between present dollars and futuredollars.

3. Future Value for a Lump Sum • Answer to “Invest $100 in an account paying 10% per year. How much will you have after 1 year?” • FV = 100(1 + 0.10)FV = 100(1.1)FV = 110 Formula for Future Value (Single Period) Future Value = Principal(1+interest rate) FV = $X(1+r)

4. Future Value: Investing for Multiple Periods • Suppose the interest rate is 10% over the second year • => $110 will grow at 10% FV = • Notice that • 1. $110 = $100 (1 + .10) • 2. $121 = $110 (1 + .10) = $100 (1.1)(1.1) = $100 (1.1)2 • 3. $133.10 = $121 (1 + .10) = $100 (1.1) (1.1) (1.1) = $100 ________

5. Investing for More Than One Period: FV for a Lump Sum If you reinvest your money $X for more than one period, you get compounding on your investment FV = X(1+r)(1+r). . . FV = X(1+r)t where t = # of periods

6. Future Value for a Lump Sum: Basic Vocabulary: • 1.The expression (1 + r)t is called the future value interest factor or _____________ . • 2. The r is usually called the _____________ . • 3. The approach is often called _____________ .

7. Future Value of $100 at 10 Percent (Table 5.1) Year Beginning Amount Interest Earned Ending Amount 1 $100.00 $10.00 $110.00 2 110.00 11.00 121.00 3 121.00 12.10 133.10 4 133.10 13.31 146.41 5 146.41 14.64 161.05 Total interest $61.05

8. Interest on Interest Calculation

9. Future Values, Interest Rates and Time r = 20% r=15% r=10% r=5%

10. The Tale of Manhattan • In 1626, Peter Minuit bought all of Manhattan Island from the Indians for $24 in goods and trinkets. Who got the better end of the deal? • 375 years has passed. How much would $24 be worth today if it could grow at 10% per year?

11. Future Value for a Lump Sum (example) • Q. Deposit $5,000 today in an account paying 12%. How much will you have in 6 years? How much is simple interest? How much is compound interest? • A. Multiply the $5000 by the future value interest factor: $5000 (1 + r)t = $5000 ___________ = At 12%, the simple interest is the difference between compound and simple interest is

12. Interest on Interest Illustration Q. You have just won a $1 million jackpot in the lottery. You can buy a ten year certificate of deposit which pays 6% compounded annually. Alternatively, you can give the $1 million to your brother-in-law, who promises to pay you 6% simple interest annually over the ten year period. Which alternative will provide you with more money at the end of ten years? A. FV of the CD: $1 million x (1.06)10 = $1,790,847.70 FV of the investment with your brother-in-law: $1 million + $1 million (.06)(10) = $1,600,000 Thus, compounding (or interest on interest), results in incremental wealth of nearly $191,000. What would the advantage to the CD be (if any) if your brother-in-law had offered 7.5% simple interest annually?

13. Chapter 5 Quick Quiz - Part 1 of 4 • In 1934, the first edition of a book described by many as the “bible” of financial statement analysis was published. Security Analysis has proven so popular among financial analysts that it has never been out of print. According to an item in The Wall Street Journal, a copy of the first edition was sold by a rare book dealer in 1996 for $7,500. The original price of the first edition was $3.37. What is the annually compounded rate of increase in the value of the book?

14. Chapter 5 Quick Quiz - Part 1 of 4 ((solution) • Set this up as a future value (FV) problem. Future value = Present value = t = • FV = PV x (1 + r)t so, • Solve for r:

15. Present Value for a Lump Sum • Q. Suppose you need $20,000 in three years to pay your MBA tuition. If you can earn 8% on your money, how much do you need today? • A. Here we know the future value is $20,000, the rate (8%), and the number of periods (3). What is the unknown present amount (called the present value)? From before: FVt = PV x (1 + r)t $20,000 = PV x __________ Rearranging: PV = $20,000/(1.08)3 = $_____________

16. Present Value for a Lump Sum: Formula • The Present Value of a Future Amount $X given the interest rate r is: Future Amount ($X) (1 + r)t PV = where t is the number of periods

17. Present Value of $1 for Different Periods and Rates (Figure 5.3) Presentvalueof $1 ($) 1.00 .90 .80 .70 .60 .50 .40 .30 .20 .10 r = 0% r = 5% r = 10% r = 15% r = 20% Time(years) 1 2 3 4 5 6 7 8 9 10

18. Present Value for a Lump Sum: Basic Vocabulary • 1.The expression 1/(1 + r)t is called the present value interest factor or, more often, the ____________ . • 2. The r is usually called the ______________ . • 3. The approach is often called ____________ .

19. Summary of Time Value Calculations (Table 5.4) I. Symbols: PV = Present value, what future cash flows are worth today FVt = Future value, what cash flows are worth in the future r = Interest rate, rate of return, or discount rate per period t = number of periods C = cash amount II. Future value of C dollars invested at r percent per period for t periods: FVt = C  (1 + r)t The term (1 + r)t is called the future value interest factor and often abbreviated FVIFr,t or FVIF(r,t).

20. Summary of Time Value Calculations (concluded) III. Present value of C dollars to be received in t periods at r percent per period: PV = C/(1 + r)t The term 1/(1 + r)t is called thepresent value interest factor and is often abbreviated PVIFr,t or PVIF(r,t). IV. The basic present equation giving the relationship between present and future value is: PV = FVt/(1 + r)t

21. Solution to Problem 5.10 • Imprudential, Inc. has an unfunded pension liability of $425 million that must be paid in 23 years. If the relevant discount rate is 7.5 percent, what is the present value of this liability? • Future value = FV = $425 million • t = 23 • r = 7.5 percent • Present value = ? • Solution: Set this up as a present value problem. • PV = $425 million x PVIF(7.5,23) • PV =

22. Present Values and Future Values FV (1+r)t and PV = FV = PV(1+r)t If PV and FV are known, we can solve for r or t. r = (FV/PV)1/t - 1

23. Chapter 5 Quick Quiz - Part 2 of 4 • Want to be a millionaire? No problem! Suppose you are currently 21 years old, and can earn 10 percent on your money (about what the typical common stock has averaged over the last six decades - but more on that later). How much must you invest today in order to accumulate $1 million by the time you reach age 65?

24. Chapter 5 Quick Quiz - Part 2 of 4 (solution) • Once again, we first define the variables: FV = r = t = • Set this up as a future value equation and solve for the present value:

25. Chapter 5 Quick Quiz - Part 3 of 4 • Suppose you deposit $5000 today in an account paying r percent per year. If you will get $10,000 in 10 years, what rate of return are you being offered? • Set this up as present value equation: FV = PV = t = PV = FVt/(1 + r)t • Now solve for r:

26. Chapter 5 Quick Quiz -- Part 4 of 4 • Now let’s see what we remember! 1. Which of the following statements is/are true? • Given r and t greater than zero, future value interest factors (FVIFr,t) are always greater than 1.00. • Given r and t greater than zero, present value interest factors (PVIFr,t) are always less than 1.00. 2. True or False: For given levels of r and t, PVIFr,t is the reciprocal of FVIFr,t. 3. All else equal, the higher the discount rate, the (lower/higher) the present value of a set of cash flows.

27. Chapter 5 Quick Quiz -- Part 4 of 4 (solution) 1. 2. 3.

28. One last example. • True story: An automobile dealer gave the following offer; Deposit $50,000 today and you will receive a brand new Mercedes Benz with a value of $25,000. In five years time you get your money $50,000 back and get to keep the car. It was 1980 and interest rates were 20%. What do you think of this deal?