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Learning Objectives

Learn the concept of time value of money and how to calculate present values, future values, and annuities based on interest rates and time periods. Includes tables and calculator instructions.

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Learning Objectives

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  1. Learning Objectives • Explain the concept of the time value of money. (LO1) • Calculate present values, future values, and annuities based on the number of time periods involved and the going interest rate. (LO2) • Calculate yield based on the time relationships between cash flows. (LO3)

  2. LO2 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. PV FV FV 0 1 2

  3. Figure 9-1Relationship of present value and future value LO2 $ $1,464.10 future value $1,000 present value 10% interest 1 2 3 0 4 Number of periods

  4. Future Value LO2 FV = ? N = 4 PV = $1,000 I = 10% Alternatively, • FV = PV x FVIF • FVIF is the future value interest factor (Appendix A) • FV = $1,000(1.464) = $1,464

  5. Future value of $1 (FVIF) Percent Period1% 2% 3% 4% 6% 8% 10% LO2 1 . . . . 1.010 1.020 1.030 1.040 1.060 1.080 1.100 2 . . . . 1.020 1.040 1.061 1.082 1.124 1.166 1.210 3 . . . . 1.030 1.061 1.093 1.125 1.191 1.260 1.331 4 . . . . 1.041 1.082 1.126 1.170 1.262 1.360 1.464 5 . . . . 1.051 1.104 1.159 1.217 1.338 1.469 1.611 10 . . . . 1.105 1.219 1.344 1.480 1.791 2.159 2.594 20 . . . . 1.220 1.486 1.806 2.191 3.207 4.661 6.727 An expanded table is presented in Appendix A

  6. 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

  7. Present Value LO2 FV = $1,464.10 N= 4 PV = ? I = 10% Alternatively, • PV = FV x PVIF • PVIF is the present value interest factor (Appendix B) • PV = $1,464.10(0.683) = $1,000

  8. Present value of $1 (PVIF) LO2 Percent Period1% 2% 3% 4% 6% 8% 10% 1 . . . . . 0.990 0.980 0.971 0.962 0.943 0.926 0.909 2 . . . . . 0.980 0.961 0.943 0.925 0.890 0.857 0.826 3 . . . . . 0.971 0.942 0.915 0.889 0.840 0.794 0.751 4 . . . . . 0.961 0.924 0.888 0.855 0.792 0.735 0.683 5 . . . . . 0.951 0.906 0.863 0.822 0.747 0.681 0.621 10 . . . . . 0.905 0.820 0.744 0.676 0.558 0.463 0.386 20 . . . . . 0.820 0.673 0.554 0.456 0.312 0.215 0.149 An expanded table is presented in Appendix B

  9. 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

  10. 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

  11. 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

  12. Figure 9-2Compounding process for annuity LO2 $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 Period 0 Period 1 Period 2 Period 3 Period 4 $4,641

  13. Future Value – Annuity LO2 FV = ? A = $1,000 $1,000 $1,000 $1,000 $1,000 0 2 3 4 1 Alternatively, FVA = A x FVIFA FVIFA is the future value interest factor for an annuity (Appendix C) FVA = $1,000(4.641) = $4,641

  14. Future value of an annuity of $1 (FVIFA) LO2 Percent Period1% 2% 3% 4% 6% 8% 10% 1 . . . . 1.000 1.000 1.000 1.000 1.000 1.000 1.000 2 . . . . 2.010 2.020 2.030 2.040 2.060 2.080 2.100 3 . . . . 3.030 3.060 3.091 3.122 3.184 3.246 3.310 4 . . . . 4.060 4.122 4.184 4.246 4.375 4.506 4.641 5 . . . . 5.101 5.204 5.309 5.416 5.637 5.867 6.105 10 . . . . 10.462 10.950 11.464 12.006 13.181 14.487 15.937 20 . . . . 22.019 24.297 26.870 29.778 36.786 45.762 57.275 30 . . . . 34.785 40.588 47.575 56.085 79.058 113.280 164.490 An expanded table is presented in Appendix C

  15. 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

  16. Present Value – Annuity LO2 A = $1,000 PV = ? $1,000 $1,000 $1,000 $1,000 0 2 3 4 1 Alternatively, PVA = A x PVIFAPVIFA is the present value interest factor for an annuity (Appendix D) PVA = $1,000(3.170) = $3,170

  17. Present value of an annuity of $1 (PVIFA) LO2 Percent Period1% 2% 3% 4% 6% 8% 10% 1 . . . . 0.990 0.980 0.971 0.962 0.943 0.926 0.909 2 . . . . 1.970 1.942 1.913 1.886 1.833 1.783 1.736 3 . . . . 2.941 2.884 2.829 2.775 2.673 2.577 2.487 4 . . . . 3.902 3.808 3.717 3.630 3.465 3.312 3.170 5 . . . . 4.853 4.713 4.580 4.452 4.212 3.993 3.791 8 . . . . 7.652 7.325 7.020 6.773 6.210 5.747 5.335 10 . . . . 9.471 8.983 8.530 8.111 7.360 6.710 6.145 20 . . . . 18.046 16.351 14.877 13.590 11.470 9.818 8.514 30 . . . . 25.808 22.396 19.600 17.292 13.765 11.258 9.427 An expanded table is presented in Appendix D

  18. 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

  19. Determining The Annuity Value Formula Approach: Table Approach: A = FVA/FVIFA (Appendix C) A = PVA/PVIFA (Appendix D) Calculator: FV (or PV), I/Y, N, CPT PMT LO2

  20. 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? Determining The Annuity Value LO2 PMT = ? PMT = ? PMT = ? PV = -$10,000 PMT = ? 0 1 3 4 2 A = PVA/PVIFA = $10,000/3.465 = $2,886 Alternatively, N= 4 I/Y= 6 PV= -10,000 FV= 0 CPT PMT= $2,885.91

  21. Table 9-1Relationship of present value to annuity LO2 Beginning Balance Annual Interest (6%) Annual Withdrawal Ending Balance - = + Year 1 . . . . $10,000.00 $600.00 $2,886.00 $7,714.00 2 . . . . 7,714.00 462.84 2,886.00 5,290.84 3 . . . . 5,290.84 317.45 2,886.00 2,722.29 4 . . . . 2,722.29 163.71 2,886.00 0

  22. Table 9-2Payoff table for loan (amortization table) LO2 Annual Interest (8%) Repayment on Principal Ending Balance Beginning Balance Annual Payment Period 1 . . . . $40,000 $4,074 $3,200 $ 874 $39,126 2 . . . . 39,126 4,074 3,130 944 38,182 3 . . . . 38,182 4,074 3,055 1,019 37,163

  23. Formula Summary FormulaAppendix Future value—–single amount . . (9-1) FV = PV(1 + i)nA 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

  24. 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

  25. 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

  26. A1 A2 A3 A4 A5 $1,000 $1,000 $1,000 $1,000 $1,000 PV = ? 0 1 2 3 4 5 6 7 8 LO2 Special Considerations in Time Value of Money – Deferred Annuity e.g.: Find the PV of an Annuity of $1,000 being paid for 5 years beginning 4 years in the future

  27. A1 A2 A3 A4 A5 $1,000 $1,000 $1,000 $1,000 $1,000 PV = ? 0 1 2 3 4 5 6 7 8 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 LO2 Deferred Annuity End of third period—Beginning of fourth period $3,993

  28. $3,170 $3,993 A1 A2 A3 A4 A5 Present (single amount) $1,000 $1,000 $1,000 $1,000 $1,000 value 0 1 2 3 4 5 6 7 8 LO2 Deferred Annuity (final part) 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 End of third period—Beginning of fourth period

  29. Perpetuities Equal Payments Growing Payments Growing annuity (with End Date) LO2

  30. The second step is to calculate the monthly payment. Canadian Mortgages LO2 FV = 0 PMT= ? 1 240 0 PV = $80,000 N = 240 (20 yrs. x 12) Using a calculator N= 240 PV= -80,000 I/Y= 0.65582 FV= 0 CPT PMT = $662.69 I/Y = .65582

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