Chapter 5 Introduction to Valuation: The Time Value of Money 5.1 Future Value and Compounding

1. Chapter 5Introduction to Valuation: The Time Value of Money • 5.1 Future Value and Compounding • 5.2 Present Value and Discounting • 5.3 More on Present and Future Values • 5.4 Summary and Conclusions

2. 5.2 Future Value for a Lump Sum • Given r, the interest rate, every $1 today will produce (1+r) of future value (FV). • 1. $110 = $100 x (1 + .10) • 2. $121 = $110 x (1 + .10) = $100 x 1.1 x 1.1 = $100 x 1.12 • 3. $133.10 = $121 x (1 + .10) = $100 x 1.1 x 1.1 x 1.1 = $100 x (1+ .10) 3 • In general, the future value, FVt, of $1 invested today at r% for t periods is FVt= $1 x (1 + r)t • The expression (1 + r)t is the future value interest factor.

3. 5.2 Future Value for a Lump Sum (continued) • 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 x 1.12 6_ = $5000 x 1.9738227 = $9869.1135 At 12%, the simple interest is .12 x $5000 = $600 per year. After 6 years, this is 6 x $600 = $3600 ; the difference between compound and simple interest is thus $4869.11 - $3600 = $1269.11

4. 5.2 Future Value for a Lump Sum (concluded) • Basic Vocabulary: • 1.The expression (1 + r)t is called the future value interest factor or FVIF • 2. The r is usually called the Compound interest rate • 3. The approach is often called compounding

5. 5.3 Example 1. • In 1934, a book was sold for $3.37. By 1996, it was sold for $7,500. What is the annually compounded rate of increase in the value of the book? • Set this up as a future value (FV) problem. Future value = $7,500 Present value = $3.37 t = 1996 - 1934 = 62 years

6. 5.3 Example 1 (concluded) • FV = PV x (1 + r)t $7,500 = $3.37 x (1 + r)62 (1 + r)62 = $7,500/3.37 = 2,225.52 • Solve for r: r = (2,225.52)1/62 - 1 = .1324 = 13.24%

7. 5.4 Example 2-Interest Q. You have just won a $1 million jackpot in the state lottery. You can buy a 10 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 10 year period. Which alternative will provide you with more money at the end of ten years?

8. 5.4 Example 2-Interest cont. 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. The difference is: $191,000.

9. 5.5 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

10. 5.6 Example 3. • Suppose you are currently 21 years old, and can earn 10 percent on your money. How much must you invest today in order to accumulate $1 million by the time you reach age 65? FV = $1 millionr = 10% t = 65 - 21 = 44 years PV = ?

11. 5.6 Example 3. Cont. • Set this up as a future value equation and solve for the present value: $1 million = PV x (1.10)44 PV = $1 million/(1.10)44 = $15,091.

12. 5.7 Example 4. • Q. Suppose you need $20,000 in 3 years to pay your college tuition. If you can earn 8% on your money, how much do you need today? • A. FV = $20,000, r = (8%), t= 3, t = 3 FVt= PV x (1 + r)t $20,000 = PV x (1.08)3 Rearranging: PV = $20,000/(1.08)3 = $15,877 In general, the present value, PV, of a $1 to be received in t periods when the rate is r is: $1 PV = (1 + r )t

13. 5.7 Present Value for a Lump Sum (concluded) • Basic Vocabulary: • 1.The expression 1/(1 + r)t is called the present value interest factor or, more often, the PVIF . • 2. The r is usually called the discount rate . • 3. The approach is often called discounting .

14. 5.8 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

15. 5.9 Example 5. • 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 = $10,000 PV = $ 5,000 t = 10 years PV = FVt/(1 + r)t $5000 = $10,000/(1 + r)10 • Now solve for r: (1 + r)10 = $10,000/$5,000 = 2.00 r = (2.00)1/10 - 1 = .0718 = 7.18%

16. 5.11 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).

17. 5.11 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