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©2012 McGraw-Hill Ryerson Limited 1 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 yield based on the time relationships between cash flows. (LO3)

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©2012 McGraw-Hill Ryerson Limited 2 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

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©2012 McGraw-Hill Ryerson Limited 3 of 37 Figure 9-1 Relationship of present value and future value $1,000 present value $ 0 1 23 4 $1,464.10 future value 10% interest Number of periods LO2

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©2012 McGraw-Hill Ryerson Limited 4 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

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©2012 McGraw-Hill Ryerson Limited 5 of 37 Future value of $1 (FV IF ) 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 Percent Period1%2%3%4%6%8%10% LO2

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©2012 McGraw-Hill Ryerson Limited 6 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 1. Then PV = 1000 Then I/Y = 10 Then N = 4 Then PMT = 0 Finally CPT FV = -1464.10 LO2 Future Value

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©2012 McGraw-Hill Ryerson Limited 7 of 37 Present Value FV = $1,464.10 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

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©2012 McGraw-Hill Ryerson Limited 8 of 37 Present value of $1 (PV IF ) 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 An expanded table is presented in Appendix B LO2 Percent Period1%2%3%4%6%8%10 %

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©2012 McGraw-Hill Ryerson Limited 9 of 37 Calculator (Texas Instruments BA-II Plus) Set FV = 1464.1 Then PMT = 0 Then I/Y = 10 Then N = 4 Finally CPT PV = -1000.00 LO2 Present Value

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©2012 McGraw-Hill Ryerson Limited 10 of 37 Nominal and Effective Annual Interest Rates Interest is often compounded quarterly, monthly, or semiannually. Effective interest rate takes into account compounding. Effective Annual Interest Rate = (1 + i/m) m – 1 where i = nominal annual interest rate m = number of compounding periods per year LO2

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©2012 McGraw-Hill Ryerson Limited 11 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

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©2012 McGraw-Hill Ryerson Limited 12 of 37 Figure 9-2 Compounding process for annuity Period 0 Period 1 Period 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 = $1,000 $4,641 LO2

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©2012 McGraw-Hill Ryerson Limited 13 of 37 Future Value – Annuity 0 1 234 $1,000 A = $1,000 FV = ? 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 Alternatively,

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©2012 McGraw-Hill Ryerson Limited 14 of 37 Future value of an annuity of $1 (FV IFA ) 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 An expanded table is presented in Appendix C LO2 Percent Period1%2%3%4%6%8%10 %

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©2012 McGraw-Hill Ryerson Limited 15 of 37 Calculator (Texas Instruments BA-II Plus) Set PMT = 1000 Then PV = 0 Then I/Y = 10 Then N = 4 Finally CPT FV = -4641.00 LO2 Future Value – Annuity

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©2012 McGraw-Hill Ryerson Limited 16 of 37 Present Value – Annuity PV = ? 0 1 234 $1,000 A = $1,000 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 Alternatively,

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©2012 McGraw-Hill Ryerson Limited 17 of 37 Present value of an annuity of $1 (PV IFA ) 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 An expanded table is presented in Appendix D LO2 Percent Period1%2%3%4%6%8%10 %

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©2012 McGraw-Hill Ryerson Limited 18 of 37 Calculator (Texas Instruments BA-II Plus) Set PMT = 1000 Then FV = 0 Then I/Y = 10 Then N = 4 Finally CPT PV = -3169.87 LO2 Present Value – Annuity

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©2012 McGraw-Hill Ryerson Limited 19 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

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©2012 McGraw-Hill Ryerson Limited 20 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 = -$10,000PMT = ? 01 2 34 A = PV A /PV IFA = $10,000/3.465 = $2,886 Alternatively, N = 4 I/Y = 6 PV = -10,000 FV = 0 CPT PMT = $2,885.91 LO2 Determining The Annuity Value PMT = ?

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©2012 McGraw-Hill Ryerson Limited 21 of 37 Table 9-1 Relationship of present value to annuity 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 Year Beginning Balance Annual Interest (6%) + - Annual Withdrawal = Ending Balance LO2

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©2012 McGraw-Hill Ryerson Limited 22 of 37 Table 9-2 Payoff table for loan (amortization table) 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 Period Beginning Balance Annual Payment Annual Interest (8%) Repayment on Principal Ending Balance LO2

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©2012 McGraw-Hill Ryerson Limited 23 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

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©2012 McGraw-Hill Ryerson Limited 24 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

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©2012 McGraw-Hill Ryerson Limited 25 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

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©2012 McGraw-Hill Ryerson Limited 26 of 37 Special Considerations in Time Value of Money – Deferred Annuity A 1 A 2 A 3 A 4 A 5 $1,000$1,000$1,000$1,000$1,000 PV = ? 012345678 LO2 e.g.: Find the PV of an Annuity of $1,000 being paid for 5 years beginning 4 years in the future

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©2012 McGraw-Hill Ryerson Limited 27 of 37 Deferred Annuity A 1 A 2 A 3 A 4 A 5 $1,000$1,000$1,000$1,000$1,000 PV = ? 012345678 First: Find the PV of an annuity of $1,000 to be received at the end of the year for 5 years with I= 8% N = 5, I/Y = 8%, PMT = $1,000, FV = 0 CPT PV = $3,993 End of third period—Beginning of fourth period $3,993 LO2

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©2012 McGraw-Hill Ryerson Limited 28 of 37 Second: Find the PV of $3,993 to be received at the end of year 3 discounted back to the present (with I = 8%). N = 3, I/Y = 8%, PMT = 0, FV = $3,993, CPT PV = $3,170 Deferred Annuity (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 012345678 End of third period—Beginning of fourth period LO2

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©2012 McGraw-Hill Ryerson Limited 29 of 37 Perpetuities Equal Payments Growing Payments Growing annuity (with End Date) LO2

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©2012 McGraw-Hill Ryerson Limited 30 of 37 The second step is to calculate the monthly payment. 0 1240 PV = $80,000 PMT = ? FV = 0 Canadian Mortgages LO2 Using a calculator N = 240 PV = -80,000 I/Y = 0.65582 FV = 0 CPT PMT = $662.69 N = 240 (20 yrs. x 12) I/Y =.65582

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© 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.

© 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.

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