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Time Value of Money

Time Value of Money. Chapter Outcomes. What is the time value of money Interest rate : Simple VS Compound Calculation : Future Value, Present Value, Interest rate, Number of periods Single Cash flow (FV, PV) Annuities (FVA, PVA). $. $. $. $. $. $. $. $. $. $. $. $. $. $. $. $.

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Time Value of Money

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  1. Time Value of Money

  2. Chapter Outcomes • What is the time value of money • Interest rate : Simple VS Compound • Calculation : Future Value, Present Value, Interest rate, Number of periods • Single Cash flow (FV, PV) • Annuities (FVA, PVA)

  3. $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ Time Value of Money • TIME VALUE OF MONEY: Interest is earned over time by saving or investing money

  4. $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ Time Value of Money • PRESENT VALUE (PV) • FUTURE VALUE (FV) • Interest rate (i) • Number of periods (n)

  5. Simple Interest • Interest earned only on the principal • EQUATION: FV = PV + (PV x i x n) • EXAMPLE:You save $100 today and a bank pays you 10% per year. How much will be your savings after 2 years? 0 1 2 $100 FV2 = ??? PV +( PV x i x n ) 100 10% 2 100 = $ 120

  6. Compound Interest • Interest earned on both principal and interest • EQUATION: FV = PV x (1 + i)n • EXAMPLE:You save $100 today and a bank pays you 10% per year. How much will be your savings after 2 years? 0 1 2 2 FV2 = ??? PV x ( 1 + i ) n 100 $100 10% = $ 121

  7. Future Value $100 + $100(0.05) = $105 PV + Interest = FV PV + PV*i = FV PV = Present Value FV = Future Value i = interest rate (as a percentage)

  8. Future Value Future Value in one year FV = PV*(1+i)

  9. Future Value Future Value in two years $100+$100(0.05)+$100(0.05) + $5(0.05) =$110.25 Present Value of the Initial Investment + Interest on the initial investment in the 1st Year + Interest on the initial investment in the 2nd Year+ Interest on the Interest from the 1stYear in the 2nd Year = Future Value in Two Years

  10. Future Value Future Value in n years FVn = PV*(1+i)n

  11. Future Value is the value on some future date of an investment made today. Future Value (FV) Year 0 1 2 3 4 5……….. n 1 ?

  12. Using Interest Factor Tables • FV = PV x FVIF i, nFVIFi,n = look from tableA1 • What is the future value of $100 invested now at 10% interest for 2 years? FV = PV x FVIF i, n FV = PV x FVIF10%, 2 FV = $100 x 1.21 = $121

  13. Using Interest Factor Tables • FV = PV x FVIF i, nFVIFi,n = look from tableA1 • Ann deposits $250 today, how much is her account value after 7 years if the saving rate is 1%? • How much do you have to deposit today in order to have $50,000 in the next 4 years, saving rate is 2%? • Sam deposits $100 today and two years from now, he is going to deposit another $200. With 3% rate, how much is his account value 5 years from now?

  14. Present Value Present Value (PV) is the value today (in the present) of a payment that is promised to be made in the future. OR Present Value is the amount that must be invested today in order to realize a specific amount on a given future date.

  15. Present Value Present Value (PV) is the value today (in the present) of a payment that is promised to be made in the future. Year 0 1 2 3 4 5……….. n 1 ?

  16. Present Value Present Value is the amount that must be invested today in order to realize a specific amount on a given future date. Year 0 1 2 3 4 5……….. n 1 ?

  17. Discounting from FV to PV FV = PV x (1 + i)n FV = PV xFVIF i, n PV = FV ÷ (1 + i)n PV = FV x [1 ÷ (1 + i)n] PV = FV x PVIF i, n Table A1 Table A2 17

  18. Practice : Using Table What is the present value of $1,000 to be received 10 years from now if the interest rate is 8%? PV = FV x PVIF i, n = FV x PVIF 8%, 10 = $1,000 x 0.463 = $463

  19. A series of equal payments that occur at the end of each period Annuities – A level Stream Year 0 1 2 3 4……….. n 1 1 1 1 1

  20. Annuity Example: You save $100 at the end of each year for 3 years 2 3 0 n 1 $ 100 $ 100 $ 100 $$$ $$$ $$$ 20

  21. Compound Annuities • Compound annuity involves depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow. • Perhaps we are saving money for education, a new car or a vacation home. • In any case, we want to know how much savings will have grown by some point in the future.

  22. Future value of the annuity Year 0 1 2 3 4……….. n 1 1 1 1 1 Money deposit at the end of the year

  23. A 5 year $500 annuity compounded at 6 % Year 0 1 2 3 4 5 500 500 500 500 500 500.0 530.0 562.0 595.5 631.0 2,818.50 “Future value of the annuity”

  24. Future Value of an Annuity 0 n PMT PMT PMT EQUATION: • FVA n = PMT x [(1 + i)n - 1] ÷ i • FVA n = PMT x FVIFA i, n FVA n = ?

  25. Future Value of an Annuity 0 1 2 n 3 i = 8% PMT PMT $1,000 PMT $1,000 $1,000 • EXAMPLE:You plan to invest $1,000 at the end of each year for 3 years at an 8% compound interest rate. What will be the future value of the investment? FVA n = ? 3

  26. Future Value of an Annuity 0 1 2 n 3 i = 8% $$$ $$$ $1,000 $1,000 $1,000 $$$ FVA 3 = PMT x FVIFA i, n = 1,000 x FVIFA 8%, 3 = 1,000 x 3.2464 = 3,246.40 FVA n = ? 3

  27. Future Value of an Annuity 0 1 2 n 3 i = 8% $$$ $$$ $1,000 $1,000 $1,000 $$$ FVA 3 = PMT x [(1 + i)n - 1] ÷ i = 1,000 x [(1 + 0.08)3 - 1] ÷ 0.08 = 1,000 x [(1.08)3 - 1] ÷ 0.08 = 1,000 x 3.2464 = 3,246.40 FVA n = ? 3

  28. Future Value of an Annuity 0 1 2 3 100 100 100 FVA 3 = PMT x FVIFA i, n = 100 x FVIFA 8%, 3 = 100 x 3.2464 = 324.64

  29. Find future value at year 5 i = 8% 4 5 0 1 2 3 FVA 3 = PMT x FVIFA i, n = 100 x FVIFA 8%, 3 = 100 x 3.2464 = 324.64 100 100 100 324.64 FV5 = FVA3 x FVIF i, n = 324.64 x FVIF 8%,2 = 324.64 x 1.1664 = 378.66

  30. Present value of the annuity Money deposit at the end of the year Year 0 1 2 3 4……….. n 1 1 1 1 1

  31. A 5 year $500 annuity compounded discounted back to the present at 6 % Money deposit at the end of the year Year 0 1 2 3 4 5 500 500 500 500 500 “Present value of the annuity” 471.5 445.0 420.0 396.0 373.5 2,106.0

  32. Present Value of an Annuity 0 n PMT PMT PMT EQUATION: • PVA 0 = PMT x [1 - (1÷(1 +i)n)] ÷ i • PVA 0 = PMT x PVIFA i, n PVA 0 = ?

  33. Present Value of an Annuity 0 1 2 3 n i = 8% PMT PMT PMT $1,000 $1,000 $1,000 • EXAMPLE:You plan to invest $1,000 at the end of each year for 3 years at an 8% compound interest rate. What is the present value of the investment? PVA 0 = ?

  34. Present Value of an Annuity 0 1 2 3 n i = 8% $$$ $$$ $$$ PVA 0 = PMT x PVIFA i, n = 1,000 x PVIFA 8%, 3 = 1,000 x 2.5771 = 2,577.10 $1,000 $1,000 $1,000 PVA 0 = ?

  35. Present Value of an Annuity 0 1 2 n 3 i = 8% $$$ $$$ $$$ $1,000 $1,000 $1,000 PVA 0 = ? PVA 0 = PMT x [1 - (1÷(1 +i)n)] ÷ i =1,000x [1 - (1÷(1.08)3)] ÷ 0.08 =1,000x [1 – 0.7938] ÷ 0.08 =1,000x 2.5771 = 2,577.10

  36. Find present value (period 0) 0 1 2 n 3 i = 8% 100 100 100 PVA 0 = PMT x PVIFA i, n = 100 x PVIFA 8%, 3 = 100 x 2.5771 = 257.71

  37. Find present value (period 0) i = 8% n 4 0 1 2 n 3 PVA 1 = PMT x PVIFA i, n = 100 x PVIFA 8%, 3 = 100 x 2.5771 = 257.71 100 100 100 257.71 PV0 = PVA1 x PVIF i, n = 257.71 x PVIF 8%,1 = 257.71 x 0.9259 = 238.61

  38. Find present value (period 0) i = 8% 2 3 n 4 n 5 0 1 n PVA 2 = PMT x PVIFA i, n = 100 x PVIFA 8%, 3 = 100 x 2.5771 = 257.71 100 100 100 257.71 PV0 = PVA2 x PVIF i, n = 257.71 x PVIF 8%,2 = 257.71 x 0.8573 = 220.93

  39. More than 1 period per year • Interest rate per period → i ÷ m • Number of periods → n x m • Annually → m = 1 • Semiannually → m = 2 • Quarterly → m = 4 • Monthly → m = 12 • Daily → m = 365 * m = number of periods / year

  40. Conclusion A1: FV = PV x FVIF i/m, nxm A2: PV = FV x PVIF i/m, nxm A3: FVA = PMT x FVIFA i/m, nxm A4: PVA = PMT x PVIFA i/m, nxm * i /m = Interest rate per period ** n x m = Number of periods

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