1. © 2003 McGraw-Hill Ryerson Limited 9 9 Chapter The Time Value of Money McGraw-Hill Ryerson©2003 McGraw-Hill Ryerson Limited Prepared by: Terry Fegarty Seneca College

2. © 2003 McGraw-Hill Ryerson Limited Chapter 9 - Outline  Time Value of Money  Future Value and Present Value  Annuities  Time-Value-of-Money Formulas  Adjusting for Non-Annual Compounding  Compound Interest Tables  Summary and Conclusions PPT 9-2

3. © 2003 McGraw-Hill Ryerson Limited Time Value of Money  The basic idea behind the concept of time value of money is:  $1 received today is worth more than $1 in the future OR  $1 received in the future is worth less than $1 today Why?  because interest can be earned on the money  The connecting piece or link between present (today) and future is the interest or discount rate PPT 9-3

4. © 2003 McGraw-Hill Ryerson Limited Future Value and Present Value  Future Value (FV) is what money today will be worth at some point in the future  Present Value (PV) is what money at some point in the future is worth today PPT 9-4

5. © 2003 McGraw-Hill Ryerson Limited Figure 9-1 Relationship of present value and future value $1,000 present value $1,464.10 future value Number of periods 12340 $ 10% interest PPT 9-5

6. © 2003 McGraw-Hill Ryerson Limited 1....1.0101.0201.0301.0401.0601.0801.100 2....1.0201.0401.0611.0821.1241.1661.210 3....1.0301.0611.0931.1251.1911.2601.331 4....1.0411.0821.1261.1701.2621.3601.464 5....1.0511.1041.1591.2171.3381.4691.611 10....1.1051.2191.3441.4801.7912.1592.594 20....1.2201.4861.8062.1913.2074.6616.727 An expanded table is presented in Appendix A Future value of $1 (FV IF ) Periods1%2%3%4%6%8%10% PPT 9-6

7. © 2003 McGraw-Hill Ryerson Limited 1..... 0.9900.9800.9710.9620.9430.9260.909 2.....0.9800.9610.9430.9250.8900.8570.826 3.....0.9710.9420.9150.8890.8400.7940.751 4.....0.9610.9240.8880.8550.7920.7350.683 5.....0.9510.9060.8630.8220.7470.6810.621 10.....0.9050.8200.7440.6760.5580.4630.386 20.....0.8200.6730.5540.4560.3120.2150.149 Present value of $1 (PV IF ) Periods1%2%3%4%6%8%10% An expanded table is presented in Appendix B PPT 9-7

8. © 2003 McGraw-Hill Ryerson Limited 2 Questions to Ask in Time Value of Money Problems 1.Future Value or Present Value?  Future Value: Present (Now)  Future  Present Value: Future  Present (Now) 2.Single amount or Annuity?  Single amount: one-time (or lump) sum  Annuity: equal amount per year for a number of years PPT 9-8

9. © 2003 McGraw-Hill Ryerson Limited Annuities Annuity: a stream or series of equal payments to be received in the future  The payments are assumed to be received at the end of each period (unless stated otherwise)  A good example of an annuity is a lease, where a fixed monthly charge is paid over a number of years PPT 9-9

10. © 2003 McGraw-Hill Ryerson Limited Figure 9-2 Compounding process for annuity Period 0Period 1Period 2Period 3Period 4 $1,000 x 1.000 = $1,000 FV = $1,100 FV = $1,210 FV = $1,331 $4,641 $1,000 for one period—10% $1,000 for two periods—10% $1,000 for three periods—10% PPT 9-10

11. © 2003 McGraw-Hill Ryerson Limited Future value of an annuity of $1 (FV IFA ) An expanded table is presented in Appendix C 1....1.0001.0001.0001.0001.0001.0001.000 2....2.0102.0202.0302.0402.0602.0802.100 3....3.0303.0603.0913.1223.184 3.2463.310 4....4.0604.1224.1844.2464.3754.5064.641 5....5.1015.2045.3095.4165.6375.8676.105 10....10.46210.95011.46412.00613.18114.487 15.937 20....22.01924.29726.87029.77836.78645.76257.275 30....34.78540.58847.57556.08579.058113.280164.490 Periods1%2%3%4%6%8%10% PPT 9-11

12. © 2003 McGraw-Hill Ryerson Limited Periods1%2%3%4%6%8%10% An expanded table is presented in Appendix D 1....0.9900.9800.9710.9620.9430.9260.909 2....1.9701.9421.9131.8861.8331.7831.736 3....2.9412.8842.8292.7752.6732.5772.487 4....3.9023.8083.7173.6303.4653.3123.170 5....4.8534.7134.5804.4524.2123.9933.791 8....7.6527.3257.0206.7736.2105.7475.335 10....9.4718.9838.5308.1117.3606.7106.145 20....18.04616.35114.87713.59011.4709.8188.514 30....25.80822.39619.60017.29213.76511.2589.427 Present value of an annuity of $1 (PV IFA ) PPT 9-12

13. © 2003 McGraw-Hill Ryerson Limited Table 9-1 Relationship of present value to annuity Annual BeginningInterestAnnualEnding YearBalance(6 percent)WithdrawalBalance 1....$10,000.00$600.00$2,886.00$7,714.00 2....7,714.00462.842,886.005,290.84 3....5,290.84317.452,886.002,722.29 4....2,722.29163.712,886.000 PPT 9-13

14. © 2003 McGraw-Hill Ryerson Limited Table 9-2 Payoff table for loan (amortization table) AnnualRepayment BeginningAnnualInterestonEnding PeriodBalancePayment(8%)PrincipalBalance 1....$40,000$4,074$3,200 $ 874$39,126 2....39,1264,0743,130 94438,182 3....38,1824,0743,055 1,01937,163 PPT 9-14

15. © 2003 McGraw-Hill Ryerson Limited Formula Appendix Future value—–single amount.. (9-1) A Present value—–single amount. (9-3) B Future value—–annuity....... (9-4a) C Future value—–annuity in advance................... (9-4b) – Present value—annuity....... (9-5a) D Determining the Yield on an Investment (a) PPT 9-15

16. © 2003 McGraw-Hill Ryerson Limited Formula Appendix Present value—annuity in advance................ (9-5b) – Annuity equalling a future value.................. (9-6a) C Annuity in advance equalling a future value............ (9-6b) – Annuity equalling a present value.................. (9-7a) D Annuity in advance equalling a present value........... (9-7b) – Determining the Yield on an Investment (b) PPT 9-16

17. © 2003 McGraw-Hill Ryerson Limited Adjusting for Non-Annual Compounding  Interest is often compounded quarterly, monthly, or semiannually in the real world  Since the time value of money tables assume annual compounding, an adjustment must be made:  the number of years is multiplied by the number of compounding periods  the annual interest rate is divided by the number of compounding periods PPT 9-17

18. © 2003 McGraw-Hill Ryerson Limited The present value of a deferred annuity ($1,000 per year to be paid 4 - 8 years in the future) (first step) A 1 A 2 A 3 A 4 A 5 Present $1,000 $1,000 $1,000 $1,000$1,000 value 0 1* 2 3 4 5 6 7 8 *Each number represents the end of the period; that is, 4 represents the end of the fourth period Beginning of fourth period $3,993 PPT 9-18

19. © 2003 McGraw-Hill Ryerson Limited $3,170 $3,993 A 1 A 2 A 3 A 4 A 5 Present (single amount) $1,000$1,000$1,000$1,000$1,000 value 012345678 The present value of a deferred annuity ($1,000 per year to be paid 4 - 8 years in the future) (second step) End of third period—Beginning of fourth period PPT 9-19

20. © 2003 McGraw-Hill Ryerson Limited Summary and Conclusions  The financial manager uses the time value of money approach to value cash flows that occur at different points in time  A dollar invested today at compound interest will grow a larger value in future. That future value, discounted at compound interest, is equated to a present value today  Cash values may be single amounts, or a series of equal amounts (annuity) PPT 9-20