1. © 2009 Cengage Learning/South-Western The Time Value Of Money Chapter 3

2. 2 Time Value of Money Financial managers compare the marginal benefits and marginal cost of investment projects. Projects usually have a long-term horizon: timing of benefits and costs matters. Time-value of money: A dollar received today is worth more than a dollar received in the future.

3. 3 Future Value Future Value: The value of an investment made today measured at a specific future date using compound interest. FV n = PV x (1+r) n Future Value depends on: Interest rate Number of periods Compounding interval

4. 4 Future Value of $200 4 years, 7% interest 0 1 2 3 4 PV = $200 End of Year FV 4 = $262.16 FV 3 = $245.01 FV 2 = $228.98 FV 1 = $214 Compound interest: Interest earned both on the principal amount and on the interest earned in previous periods.

5. 5 Compounding Year 1: FV 1 = $214 Earns 7% interest on initial $200 FV 1 = $200+$14 = $214 Year 2: FV 2 = $228.98 Earn $14 interest again on $200 principal Earns $0.98 on previous year’s interest of $14: $14 x 7% = $0.98 FV 2 = $214+$14+$0.98 = $228.98 Year 3: FV 3 = $245.01 Earn $14 interest again on $200 principal Earns $2.03 on previous years’ interest of $28.98: $28.98 x 7% = $2.03 FV 3 = $228.98+$14+$2.03 = $245.01 Earn $14 interest again on $200 principal Earns $3.15 on previous years’ interest of $45.01: $45.01 x 7% = $3.15 FV 4 = $245.01+$14+$3.15 = $262.16 Year 4: FV 4 = $262.16

6. 6 The Power of Compound Interest Periods 0% 10% 5% 15% 20%

7. 7 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.

8. 8 Present Value of $200 4 Years, 7% Interest 0 1 2 3 4 Discounting PV = $186.92 FV 1 = $200 Discounting: The process of calculating present values. FV 2 = $200 PV = $174.69 FV 3 = $200 PV = $163.26 FV 4 = $200 PV = $152.58 End of Year

9. 9 The Power of Discounting Periods Present Value of One Dollar ($) 0 2 4 6 8 10 12 14 16 18 20 22 24 0.5 0.75 1.00 0.25 10% 5% 15% 20% 0%

10. 10 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.

11. 11 Future and Present Values of An Ordinary Annuity Present Value 0 1 2 3 4 5 $1,000 $1,000 $1,000 $1,000 $1,000 Discounting End of Year Future Value Compounding

12. 12 Future Value of An Ordinary Annuity 5 Years, 5.5% Interest $1,055.00 $1,113.02 $1,174.24 $1,238.82 $1,000.00 0 1 2 3 4 5 $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.

13. 13 Future Value of An Annuity Due 5 Years, 5.5% Interest 0 1 2 3 4 5 $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. $1,113.02 $1,174.24 $1,238.82 $1,306.96 $1,055.00

14. 14 Present Value of Cash Flow Streams Mixed streams Annuities Perpetuities: cash flow streams that continue forever

15. 15 Present Value of An Ordinary Annuity 5 Years, 5.5% Interest $947.87 $898.45 $851.61 $807.22 $1,000 $1,000 $1,000 $1,000 $1,000 End of Year 0 1 2 3 4 5 $765.13

16. 16 Present Value of An Annuity Due 5 Years, 5.5% Interest $947.87 $898.45 $851.61 $807.22 End of Year $1,000 $1,000 $1,000 $1,000 $1,000 0 1 2 3 4 5 $1,000.00

17. 17 Future and Present Values of A Mixed Steam 5 Years, 4% Interest PV $5,271.7 0 1 2 3 4 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

18. 18 Present Value of A Perpetuity For a constant stream of cash flows that continues forever

19. 19 Present Value of A Growing Perpetuity Growing Perpetuity CF 1 = $1,000 r = 7% per year g = 2% per year 0 1 2 3 4 $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

20. 20 Compounding More Frequently Than Annually continuous compounding The more frequent the compound period, the larger the FV! m compounding periods

21. 21 Compounding More Frequently Than Annually FV at end of 2 years of $125,000 at 5% interest Semiannual compounding: Quarterly compounding: Continuous compounding:

22. 22 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.

23. 23 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.

24. 24 Additional Applications of Time Value Deposits needed to accumulate a future sum Loan amortization Implied interest or growth rates Number of compounding periods

25. 25 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.