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Chapter 9 The Time Value of Money

Chapter 9 The Time Value of Money. Notes:. Although it is easiest to use your financial calculator to solve time value problems, you MUST understand what you are doing. This will require a lot of practice to eliminate mistakes.

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Chapter 9 The Time Value of Money

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  1. Chapter 9 The Time Value of Money

  2. Notes: • Although it is easiest to use your financial calculator to solve time value problems, you MUST understand what you are doing. This will require a lot of practice to eliminate mistakes. • Understanding the concept of Time Value of Money NOW is extremely important because all the remaining chapters will require TVM concept application.

  3. Notes: • In your Test, you will be REQUIRED to show both your financial calculator solution and Mathematical solution (either using the formula or the Financial Tables) • Only in multiple choice questions will you not be required to show your calculation. Therefore, you can use the financial calculator alone. You must however, make sure that you know how to use your calculator properly; otherwise you will easily make mistakes with the use of a financial calculator.

  4. Using your Financial Calculators (TI BAII Plus and Sharp EL-733A) • For the first time that you are using your calculator, perform the following: • TI BAII Plus users, Set Payment Frequency and Compounding Frequency to 1 (press 2nd, press P/Y, press 1, press ENTER, press down arrow, press 1, press ENTER, press 2nd, press QUIT) • Sharp EL-733Ausers, Set to FIN Mode if not yet in FIN Mode ( press2nd F, press Mode). You should see FIN on the Display. • Set Decimal to 4 places • TI BAII Plus users, press 2nd, press Format, press 4,press ENTER , press 2nd, press QUIT. • Sharp EL-733Ausers, press 2ndF, press TAB, press 4.

  5. Using your Financial Calculators (TI BAII Plus and Sharp EL-733A) • To start each calculation, • TI BAII Plus users, press CE/C,press 2nd, press CLR TVM, press 2nd,press CLR Work. BAII Plus has a continuous memory. Turning-off the calculator does not erase what was previously stored in its memory, although turning it on again resets the display to zero. Therefore, it is extremely important to clear memory before each calculation. • Sharp EL-733Ausers, press 2ndF, press CA. • To erase the previously entered number, • TI BAII Plus users, simply press CE/C. • Sharp EL-733Ausers, simply press C CE. • Enter Outflow Value as negative. To enter it as negative enter the value/s, press +/-. DO NOT use the minus sign key.

  6. Using your Financial Calculators (TI BAII Plus and Sharp EL-733A) • The order in which data (PV, n, I, etc) are entered does not matter. • To compute for the result, press CPT forTI BAII Plus users, press COMP for Sharp EL-733Ausers. Then press whatever variable you are computing for (PV. FV, etc) • To perform calculations involving annuity dues, payment must be set to the begin mode. • TI BAII Plus users, press 2nd, press BGN, press 2nd,press SET, press 2nd, press QUIT. • Sharp EL-733Ausers, press BGN. • It is important to reset the mode back to END. Most payment problems are made at the end of each year (ordinary annuities).

  7. Today Future We know that receiving $1 today is worth more than $1 in the future. This is dueto thetime value of money. The cost of receiving $1 in the future is theinterestwe could have earned if we had received the $1 sooner.

  8. If we can MEASURE this interest cost, we can: Today Future • Translate $1 today into its equivalent in the future(COMPOUNDING). ?

  9. If we can MEASURE this interest cost, we can: Today Future Today Future • Translate $1 today into its equivalent in the future(COMPOUNDING). • Translate $1 in the future into its equivalent today(DISCOUNTING). ? ?

  10. Future Value – single sum

  11. Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Calculator Solution: I/Y = i = 6 N = n = 5 PV = -100 FV = $133.82 PV = -100 FV = 0 5

  12. Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Calculator Solution: I/Y = i = 6 N = n = 5 PV = -100 FV = $133.82 PV = -100 FV = 133.82 0 5

  13. Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Mathematical Solution: FV = PV (FVIF i, n) FV = 100 (FVIF .06, 5) (use FVIF table, or) FV = PV (1 + i)n FV = 100 (1.06)5 = $133.82 PV = -100 FV = 133.82 0 5

  14. Future Value - single sumsIf you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? Calculator Solution: I/Y = i = 1.5 N = n = 20 PV = -100 FV = $134.68 PV = -100 FV = 0 20

  15. Future Value - single sumsIf you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? Calculator Solution: I/Y = i = 1.5 N = n = 20 PV = -100 FV = $134.68 PV = -100 FV = 134.68 0 20

  16. Future Value - single sumsIf you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? Mathematical Solution: FV = PV (FVIF i, n) FV = 100 (FVIF .015, 20) (can’t use FVIF table) FV = PV (1 + i/m) m x n FV = 100 (1.015)20 = $134.68 PV = -100 FV = 134.68 0 20

  17. Present Value – single sum

  18. Present Value - single sumsIf you will receive $100 5 years from now, what is the PV of that $100 if the interest rate is 6%? Calculator Solution: I/Y = i =6 N = n = 5 FV = 100 PV = -74.73 PV = FV = 100 0 5

  19. Present Value - single sumsIf you will receive $100 5 years from now, what is the PV of that $100 if the interest rate is 6%? Mathematical Solution: PV = FV (PVIF i, n) PV = 100 (PVIF .06, 5) (use PVIF table, or) PV = FV / (1 + i)n PV = 100 / (1.06)5 = $74.73 PV = -74.73 FV = 100 0 5

  20. PV = -5,000 FV = 11,933 0 5 Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return? Calculator Solution: N = n = 5 PV = -5,000 FV = 11,933 I/Y = i =19%

  21. Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return? Mathematical Solution: PV = FV (PVIF i, n ) 5,000 = 11,933 (PVIF ?, 5 ) PV = FV / (1 + i)n 5,000 = 11,933 / (1+ i)5 .419 = ((1/ (1+i)5) 2.3866 = (1+i)5 (2.3866)1/5 = (1+i) i = .19

  22. PV = FV = 0 Present Value - single sumsSuppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500?

  23. PV = -100 FV = 500 0 ? Present Value - single sumsSuppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Calculator Solution: • FV = 500 • I/Y = i = 0.8 PV = -100 • N = n = 202 months

  24. Present Value - single sumsSuppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Mathematical Solution: PV = FV / (1 + i)n 100 = 500 / (1+ .008)N 5 = (1.008)N ln 5 = ln (1.008)N ln 5 = N ln (1.008) 1.60944 = .007968 N N = 202 months

  25. Hint for single sum problems: • In every single sum future value and present value problem, there are 4 variables: • FV, PV, i, and n • When doing problems, you will be given 3 of these variables and asked to solve for the 4th variable. • Keeping this in mind makes “time value” problems much easier!

  26. The Time Value of Money 0 1 2 3 4 Compounding and Discounting Cash Flow Streams

  27. Annuities 0 1 2 3 4 • Annuity: a sequence of equal cash flows, occurring at the end of each period.

  28. Examples of Annuities: • If you buy a bond, you will receive equal coupon interest payments over the life of the bond. • If you borrow money to buy a house or a car, you will pay a stream of equal payments.

  29. 0 1 2 3 Future Value - annuityIf you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?

  30. 0 1 2 3 Future Value - annuityIf you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? Calculator Solution: I/Y= i = 8 N = n = 3 PMT = -1,000 FV = $3,246.40 1000 1000 1000

  31. 0 1 2 3 Future Value - annuityIf you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? Calculator Solution: I/Y= i = 8 N = n = 3 PMT = -1,000 FV = $3,246.40 1000 1000 1000

  32. Future Value - annuityIf you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? Mathematical Solution: FV = PMT (FVIFA i, n) FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or)

  33. Future Value - annuityIf you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? Mathematical Solution: FV = PMT (FVIFA i, n) FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or) FV = PMT (1 + i)n - 1 i

  34. Future Value - annuityIf you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? Mathematical Solution: FV = PMT (FVIFA i, n) FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or) FV = PMT (1 + i)n - 1 i FV = 1,000 (1.08)3 - 1 = $3246.40 .08

  35. 0 1 2 3 Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the discount rate is 8%?

  36. 0 1 2 3 Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the discount rate is 8%? Calculator Solution: I/Y = i = 8 N = n = 3 PMT = -1,000 PV = $2,577.10 1000 1000 1000

  37. 0 1 2 3 Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the discount rate is 8%? Calculator Solution: I/Y = i = 8 N = n = 3 PMT = -1,000 PV = $2,577.10 1000 1000 1000

  38. Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the discount rate is 8%? Mathematical Solution: PV = PMT (PVIFA i, n) PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or)

  39. Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the discount rate is 8%? Mathematical Solution: PV = PMT (PVIFA i, n) PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or) 1 PV = PMT 1 - (1 + i)n i

  40. Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the discount rate is 8%? Mathematical Solution: PV = PMT (PVIFA i, n) PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or) 1 PV = PMT 1 - (1 + i)n i 1 PV = 1000 1 - (1.08 )3 = $2,577.10 .08

  41. Ordinary Annuity vs. Annuity Due

  42. Earlier, we examined this “ordinary” annuity: 0 1 2 3 1000 1000 1000 Using an interest rate of 8%, we find that: • The Future Value (at 3) is $3,246.40. • The Present Value (at 0) is $2,577.10.

  43. What about this annuity? 0 1 2 3 1000 1000 1000 • Same 3-year time line, • Same 3 $1000 cash flows, but • The cash flows occur at the beginning of each year, rather than at the end of each year. • This is an “annuity due.”

  44. 0 1 2 3 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?

  45. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Calculator Solution: Mode = BEGIN I/Y = i = 8 N = n = 3 PMT = -1,000 FV = $3,506.11 -1000 -1000 -1000 0 1 2 3

  46. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Calculator Solution: Mode = BEGIN I/Y = i = 8 N = n = 3 PMT = -1,000 FV = $3,506.11 -1000 -1000 -1000 0 1 2 3

  47. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution:Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n) (1 + i) FV = 1,000 (FVIFA .08, 3) (1.08) (use FVIFA table, or)

  48. (1 + i) Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution:Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n) (1 + i) FV = 1,000 (FVIFA .08, 3) (1.08) (use FVIFA table, or) FV = PMT (1 + i)n - 1 i

  49. (1 + i) (1.08) Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution:Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n) (1 + i) FV = 1,000 (FVIFA .08, 3) (1.08) (use FVIFA table, or) FV = PMT (1 + i)n - 1 i FV = 1,000 (1.08)3 - 1 = $3,506.11 .08

  50. 0 1 2 3 Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if the discount rate is 8%?

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