©2009 McGraw-Hill Ryerson Limited 1 of 37 ©2009 McGraw-Hill Ryerson Limited 9 9 The Time Value of Money ©2009 McGraw-Hill Ryerson Limited Prepared by: Michel Paquet SAIT Polytechnic

©2009 McGraw-Hill Ryerson Limited 2 of 37 Chapter 9 - Outline Basic Idea of Time Value of Money Application to Capital Budgeting Decision and Cost of Capital Future Value and Present Value Nominal and Effective Annual Interest Rates Annuities Summary and Conclusions

©2009 McGraw-Hill Ryerson Limited 3 of 37 Learning Objectives 1.Explain the concept of the time value of money. (LO1) 2.Calculate present values, future values, and annuities based on the number of time periods involved and the going interest rate. (LO2) 3.Calculate the yield based on the time relationships between cash flows. (LO3)

©2009 McGraw-Hill Ryerson Limited 4 of 37 Time Value of 0123 $1 > >> 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 Basic Idea: Money has different values at different points in time. LO1

©2009 McGraw-Hill Ryerson Limited 5 of 37 Capital Budgeting Decision and Cost of Capital Capital budgeting decision is about comparing current outlays with future benefits: Cost of capital is the discount rate used in this comparison process. -$1,000$60$100$ LO1

©2009 McGraw-Hill Ryerson Limited 6 of 37 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. 012 PVFV LO2

©2009 McGraw-Hill Ryerson Limited 7 of 37 Figure 9-1 Relationship of present value and future value $1,000 present value $ $1, future value 10% interest Number of periods LO2

©2009 McGraw-Hill Ryerson Limited 8 of 37 Future Value FV = ? n = 4 PV = $1,000 i = 10% Alternatively, FV = PV x FV IF FV IF is the future value interest factor (Appendix A) FV = $1,000(1.464) = $1,464 LO2

©2009 McGraw-Hill Ryerson Limited 9 of 37 Future value of $1 (FV IF ) An expanded table is presented in Appendix A Percent Period1%2%3%4%6%8%10% LO2

©2009 McGraw-Hill Ryerson Limited 10 of 37 Calculator (Texas Instruments BA-II Plus) The first step to use a business calculator is to clear the registers. Then set P/Y to PV 10 I/Y 4 N 0 PMT CPT FV = LO2 Future Value

©2009 McGraw-Hill Ryerson Limited 11 of 37 Present Value FV = $1, n = 4 PV = ? i = 10% Alternatively, PV = FV x PV IF PV IF is the present value interest factor (Appendix B) PV = $1,464.10(0.683) = $1,000 LO2

©2009 McGraw-Hill Ryerson Limited 12 of 37 Present value of $1 (PV IF ) An expanded table is presented in Appendix B LO2 Percent Period1%2%3%4%6%8%10 %

©2009 McGraw-Hill Ryerson Limited 13 of 37 Calculator (Texas Instruments BA-II Plus) FV 0 PMT 10 I/Y 4 N CPT PV = LO2 Present Value

©2009 McGraw-Hill Ryerson Limited 14 of 37 Nominal and Effective Annual Interest Rates Interest is often compounded quarterly, monthly, or semiannually in the real world. A nominal annual interest rate assumes annual compounding. It must be adjusted to reflect the more frequent than annual compounding. Effective Annual Interest Rate = (1 + i/m) m – 1 where i = nominal annual interest rate m = number of compounding periods per year LO2

©2009 McGraw-Hill Ryerson Limited 15 of 37 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. LO2

©2009 McGraw-Hill Ryerson Limited 16 of 37 Figure 9-2 Compounding process for annuity Period 0Period 1Period 2Period 3Period 4 $1,000 for three periods—10% FV = $1,331 $1,000 for two periods—10% FV = $1,210 $1,000 for one period—10% FV = $1,100 $1,000 x = $1,000 $4,641 LO2

©2009 McGraw-Hill Ryerson Limited 17 of 37 Future Value – Annuity $1,000 A = $1,000 FV = ? Alternatively, FV A = A x FV IFA FV IFA is the future value interest factor for an annuity (Appendix C) FV A = $1,000(4.641) = $4,641 LO2

©2009 McGraw-Hill Ryerson Limited 18 of 37 Future value of an annuity of $1 (FV IFA ) An expanded table is presented in Appendix C LO2 Percent Period1%2%3%4%6%8%10 %

©2009 McGraw-Hill Ryerson Limited 19 of 37 Calculator (Texas Instruments BA-II Plus) 1000 PMT 0 PV 10 I/Y 4 N CPT FV = LO2 Future Value – Annuity

©2009 McGraw-Hill Ryerson Limited 20 of 37 Present Value – Annuity PV = ? $1,000 A = $1,000 Alternatively, PV A = A x PV IFA PV IFA is the present value interest factor for an annuity (Appendix D) PV A = $1,000(3.170) = $3,170 LO2

©2009 McGraw-Hill Ryerson Limited 21 of 37 Present value of an annuity of $1 (PV IFA ) An expanded table is presented in Appendix D LO2 Percent Period1%2%3%4%6%8%10 %

©2009 McGraw-Hill Ryerson Limited 22 of 37 Calculator (Texas Instruments BA-II Plus) 1000 PMT 0 FV 10 I/Y 4 N CPT PV = LO2 Present Value – Annuity

©2009 McGraw-Hill Ryerson Limited 23 of 37 Determining The Annuity Value Formula Approach: Table Approach: A = FV A /FV IFA (Appendix C) A = PV A /PV IFA (Appendix D) Calculator: FV (or PV), I/Y, N, CPT PMT LO2

©2009 McGraw-Hill Ryerson Limited 24 of 37 Your uncle gives you $10,000 now. If you are able to earn 6% on these funds, how much can you withdraw at the end of each year for next 4 year? PV = $10,000A = ? A = PV A /PV IFA = $10,000/3.465 = $2,886 Alternatively, 4 N 6 I/Y PV 0 FV CPT PMT = LO2 Determining The Annuity Value

©2009 McGraw-Hill Ryerson Limited 25 of 37 Table 9-1 Relationship of present value to annuity 1....$10,000.00$600.00$2,886.00$7, , , , , , , , , Year Beginning Balance Annual Interest (6%) + - Annual Withdrawal = Ending Balance LO2

©2009 McGraw-Hill Ryerson Limited 26 of 37 Table 9-2 Payoff table for loan (amortization table) 1....$40,000$4,074$3,200 $ 874$39, ,1264,0743, , ,1824,0743,055 1,01937,163 Period Beginning Balance Annual Payment Annual Interest (8%) Repayment on Principal Ending Balance LO2

©2009 McGraw-Hill Ryerson Limited 27 of 37 Determining Yield on an Investment A $10,000 investment will generate $1,490 a year for the next 10 years, what is the yield on the investment? $1,490 PV = $10, Using a calculator 10 N PV 1490 PMT 0 FV CPT I/Y = 7.996(%) i = ? LO3

©2009 McGraw-Hill Ryerson Limited 28 of 37 Formula Summary Formula Appendix Future value—–single amount.. (9-1) FV = PV(1 + i) n 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 LO2

©2009 McGraw-Hill Ryerson Limited 29 of 37 Formula Summary 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) LO2

©2009 McGraw-Hill Ryerson Limited 30 of 37 Two Questions to Ask in Time Value of Money Problems First, Future Value or Present Value? Future Value: Present (Now)  Future Present Value: Future  Present (Now) Second, Single amount or Annuity? Single amount: one-time (or lump) sum Annuity: same amount per year for a number of years LO2

©2009 McGraw-Hill Ryerson Limited 31 of 37 Finding Present Value (first part) A 1 A 2 A 3 A 4 A 5 $1,000$1,000$1,000$1,000$1,000 PV = ? LO2

©2009 McGraw-Hill Ryerson Limited 32 of 37 Finding Present Value (second part) A 1 A 2 A 3 A 4 A 5 $1,000$1,000$1,000$1,000$1,000 PV = ? Each number represents the end of the period; that is, 4 represents the end of the fourth period. End of third period—Beginning of fourth period $3,993 LO2

©2009 McGraw-Hill Ryerson Limited 33 of 37 Finding Present Value (final part) $3,170 $3,993A 1 A 2 A 3 A 4 A 5 Present (single amount) $1,000$1,000$1,000$1,000$1,000 value End of third period—Beginning of fourth period LO2

©2009 McGraw-Hill Ryerson Limited 34 of 37 Perpetuities Perpetuity Growing perpetuity Growing annuity (with end date) LO2

©2009 McGraw-Hill Ryerson Limited 35 of 37 Canadian Mortgages: An Integrated Application A 20-year, $80,000 mortgage carries an annual interest rate of 8%. How much is the monthly payment? The first step is to calculate the effectively monthly interest rate from the semi-annual compounded annual interest rate FV = 1.04 PV = -1 Using a calculator, 6 N -1 PV 0 PMT 1.04 FV CPT I/Y = (%) LO3

©2009 McGraw-Hill Ryerson Limited 36 of 37 The second step is to calculate the monthly payment PV = $80,000 A = ? Using a calculator, 240 N PV I/Y 0 FV CPT PMT = Canadian Mortgages: An Integrated Application LO2

©2009 McGraw-Hill Ryerson Limited 37 of 37 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 to 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). Five variables are involved in a time value of money problem. The fifth variable can be determined if given the other four.