© 2009 Cengage Learning/South-Western The Time Value Of Money Chapter 3

Time Value of Money Given two choices, would you prefer $1,000 today or $1,000 one year from now? Why? (1) Possibility of inflation (2) Have the use of the money for one year 2

Future Value Invest $1,000 for one year in a certificate of deposit at 5% interest rate per year. $1,000 x 1.05 = $1,050 Present Value Future Value Invest $1,000 for two years in a CD at 5% interest per year, compounded annually $1,000 x 1.05 = $1,050. $1,050 x 1.05 = $1, PV FV 3

Another way to express it: $1,000 x (1.05)² = $1, PV FV $1,000 invested at 5% for three years: $1,000 x (1.05)³ = $1, Twenty years: $1,000 x (1.05)²º = $2,

5 General Formula for Future Value Future Value: The value of an investment made today measured at a specific future date using compound interest. PV x (1+r) n = FV Future Value depends on: Interest rate Number of periods Compounding interval

6 Future Value of $200 4 years, 7% interest PV = $200 End of Year FV 4 = $ FV 3 = $ FV 2 = $ FV 1 = $214 Compound interest: Interest earned both on the principal amount and on the interest earned in previous periods.

Present Value What is the present value of $1,000 to be received one year from now, discounted back at 5% interest? 7

What is the present value of $1,000 to be received two years from now, discounted back at 5% interest? 8

What is the present value of $1,000 to be received twenty years from now, discounted back at 5% interest? 9

10 Present Value Present value: The value today of a cash flow to be received at a specific date in the future, assuming an opportunity to earn interest at a specified rate.

11 Present Value of $200 4 Years, 7% Interest Discounting PV = $ FV 1 = $200 Discounting: The process of calculating present values. FV 2 = $200 PV = $ FV 3 = $200 PV = $ FV 4 = $200 PV = $ End of Year

What about a stream of cash flows? Examples: Dividends to be received from owning some company’s stock Principal and interest to be received from a bond Revenues a company will receive from a new investment How much a consumer needs to save to buy something in the future 12

13 Future Value of Cash Flow Streams Mixed stream A series of unequal cash flows reflecting no particular pattern. Annuity A stream of equal periodic cash flows.

14 Future and Present Values of An Ordinary Annuity Present Value $1,000 $1,000 $1,000 $1,000 $1,000 Discounting End of Year Future Value Compounding

15 Future Value of An Ordinary Annuity 5 Years, 5.5% Interest $1, $1, $1, $1, $1, $1,000 $1,000 $1,000 $1,000 $1,000 End of Year Ordinary annuity: An annuity for which the payments occur at the end of each period.

16 Future Value of An Annuity Due 5 Years, 5.5% Interest $1,000 $1,000 $1,000 $1,000 $1,000 End of Year Annuity due: An annuity for which the payments occur at the beginning of each period. $1, $1, $1, $1, $1,055.00

17 Present Value of Cash Flow Streams Mixed streams Annuities Perpetuities: cash flow streams that continue forever

18 Present Value of An Ordinary Annuity 5 Years, 5.5% Interest $ $ $ $ $1,000 $1,000 $1,000 $1,000 $1,000 End of Year $765.13

19 Present Value of An Annuity Due 5 Years, 5.5% Interest $ $ $ $ End of Year $1,000 $1,000 $1,000 $1,000 $1, $1,000.00

20 Future and Present Values of A Mixed Steam 5 Years, 4% Interest PV $5, $10,000 $3,000 $5,000 $4,000 $3,000 $2,000.0 Discounting End of Year FV $6,413.8 Compounding - $12,166.5 $3,509.6 $5,624.3 $4,326.4 $3,120.0 $4,622.8 $3,556.0 $2,564.4 $1,643.9 $2,884.6

21 Present Value of A Perpetuity For a constant stream of cash flows that continues forever

Example of a Perpetuity Preferred Stock of DuPont pays $4.50 per share Assume investors require a 6.4% return 22

23 Present Value of A Growing Perpetuity Growing Perpetuity CF 1 = $1,000 r = 7% per year g = 2% per year $1,000 $1,000(1+0.02) 1 $1,000(1+0.02) 2 $1,000(1+0.02) 3 … $1,000 $1,020 $1,040.4 $1,061.2

Example of a Growing Perpetuity Microsoft common stock pays $0.52 per share per year. Assume investors require a 12% return on their investment Assume investors expect Microsoft’s dividends to grow at a rate of 9.1% per year 24

25 Compounding More Frequently Than Annually continuous compounding The more frequent the compound period, the larger the FV! m compounding periods

26 Compounding More Frequently Than Annually FV at end of 2 years of $125,000 at 5% interest Semiannual compounding: Quarterly compounding: Continuous compounding:

The number e, used in continuous compounding 27 e is approximately equal to Often used in calculus FV = PV x e rn e is the solution to the following limit:

28 Stated Versus Effective Annual Interest Rates Stated annual rate The contractual annual rate of interest charged by a lender or promised by a borrower. Effective annual rate The annual rate of interest actually paid or earned, reflecting the impact of compounding frequency.

29 Stated Versus Effective Annual Interest Rates Annual percentage rate (APR) The stated annual rate calculated by multiplying the periodic rate by the number of periods in one year. Annual percentage yield (APY) The annual rate of interest actually paid or earned, reflecting the impact of compounding frequency. The same as the effective annual rate.

30 The Time Value of Money Much of finance involves finding future and present values. The time value of money is central to all financial valuation techniques.