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Chapter 5

Chapter 5. The Time Value of Money. Time Value. The process of expressing the present in the future (compounding) the future in the present (discounting). Time Value. Payments are either a single payment a series of equal payments (an annuity). Time Value.

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Chapter 5

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  1. Chapter 5 The Time Value of Money

  2. Time Value • The process of expressing • the present in the future (compounding) • the future in the present (discounting)

  3. Time Value • Payments are either • a single payment • a series of equal payments (an annuity)

  4. Time Value • Time value of money problems may be solved by using • Interest tables • Financial calculators • Software

  5. Variables for Time Value of Money Problems • PV = present value • FV = future value • PMT = annual payment • N = number of time periods • I = interest rate per period

  6. Financial Calculators • Express the cash inputs (PV, FV, and PMT) as cash inflows and cash outflows • At least one of the cash variables must be • an inflow (+) • an outflow (-)

  7. Future Value • The future value of $1 takes a single payment in the present into the future • The general equation for the future value of $1: P0(1 + i)n = Pn

  8. Future Value Illustrated • PV = -100 • I = 8 • N = 20 • PMT = 0 • FV = ? • = 466.10

  9. Greater Terminal Values • Higher interest rates • Longer time periods • Result in greater terminal values

  10. Greater Terminal Values

  11. Present Value • The present value of $1 brings a single payment in the future back to the present • The general equation for the present value of $1:P0 = Pn (1+i)n

  12. Present Value Illustrated • FV = 100 • I = 6 • N = 5 • PMT = 0 • PV = ? • = -74.73

  13. Lower Present Values • Higher interest rates • Longer time periods • Result in lower present values

  14. Lower Present Values

  15. Annuities - Future Sum • The future sum of annuity takes a series of payments into the future • Payments may be made • at the end of each time period (ordinary annuity) • at the beginning of each time period (annuity due)

  16. FV Time Lines • Ordinary annuity Year Payment - $100 100 100 0 1 2 3

  17. FV Time Lines • Annuity due Year Payment$100 100 100 - 0 1 2 3

  18. Future Value of an Ordinary Annuity Illustrated • PV = 0 • PMT = -100 • I = 5 • N = 3 • FV = ? • = 315.25

  19. Greater Terminal Values • Higher interest rates • Longer time periods • Result in greater terminal values

  20. Greater Terminal Values

  21. Annuities - Present Value • The present value of an annuity brings a series of payments in the future back to the present

  22. Present Value of an Ordinary Annuity Illustrated • FV = 0 • PMT = 100 • I = 6 • N = 3 • PV = ? • = -267.30

  23. Annuities - Present Value • Higher interest rates result in lower present values • But longer time periods increases the present value (because more payments are received)

  24. Annuities - Present Value

  25. Additional Time Value Illustrations • The following is a series of problems or questions that use the time value of money.

  26. Illustration 1 • You deposit $1,000 in an account at the end of each year for twenty years. What is the total amount in the account if you earn 6 percent annually?

  27. Future Value of an Ordinary Annuity • The unknown: FV • The givens: • PV = 0 • PMT = -1,000 • N = 20 • I = 6 The answer:$36,786

  28. Interpretation • For an annual cash payment of $1,000, you will have $36,786 after twenty years • Of the $36,786 • $20,000 is the total cash outflow • $16,786 is the earned interest

  29. Illustration 2 • What is the present value of (or required cash outflow to purchase) an ordinary annuity of $1,000 for twenty years, if the rate of interest is 6 percent?

  30. Present Value of an Annuity • The unknown: PV • The givens: • FV = 0 • PMT = 1,000 • N = 20 • I = 6 The answer:$11,470

  31. Interpretation • For a present payment of $11,470, the individual will annually receive $1,000 for the next twenty years • The $11,470 is an immediate cash outflow • The $1,000 annual payment to be received is a cash inflow

  32. Illustration 3 • What is the future value after ten years of a stock that cost $10 and appreciates at 9 percent annually?

  33. Future Value of $1 • The unknown: FV • The givens: • PV = 10 • PMT = 0 • N = 10 • I = 9 The answer: $23.67

  34. Interpretation • A $10 stock will be worth $23.67 after 10 year if its price grows 9% annually.

  35. Illustration 4 • What is the cost of a stock that was sold for $23.67, held for 10 years and whose value appreciated 9 percent annually?

  36. Present Value of $1 • The unknown: PV • The givens: • FV = 23.67 • PMT = 0 • N = 10 • I = 9 The answer:$10

  37. Interpretation • $23.67 received after ten years is worth $10 today if the return rate is 9 percent.

  38. Interpretation of Future and Present Values These two problems are the same: • In the first case the $10 is compounded into its future value ($23.67) • In the second case the future value ($23.67) is discounted back to its present value ($10)

  39. Illustration 5 • A stock was purchased for $10 and sold for $23.67 after 10 years. What was the return?

  40. Future Value of I • The unknown: I • The givens: • PV = 10 • PMT = 0 • N = 0 • FV = 23.67 The answer: 9%

  41. Interpretation • The yield on a $10 investment that was sold after 10 years for $23.67 is 9%.

  42. Illustration 6 • If an investment pay $50 a year for 10 years and repays $1,000 after 10 years, what is this investment worth today if you can earn 6 percent?

  43. Determination of Present Value • The unknown: PV • The givens: • FV = 1,000 • PMT = 50 • I = 6 • N = 10 The answer: $926

  44. Interpretation • If you collect $50 a year for 10 years and receive $1,000 after 10 years, those cash inflows are currently worth $926 at 6 percent.

  45. Illustration 7 • Time value is used to determine a loan repayment schedule such as a mortgage.

  46. Loan Repayment Schedule • Amount borrowed (PV) = $80,000 • Interest rate (I) = 8% • Term of the loan (N) = 25 years • No future value since loan is repaid • Amount of the annual payment = $7,494.30

  47. Loan Repayment Schedule

  48. Illustration 8 • You have $115,000 and spend $24,000 a year. If you earn 8% annually, how long will your funds last?

  49. Determination of Number of Years • The unknown: N • The givens: • PV = 115,000 • I = 8 • FV = 0 • PMT = -24,000 The answer: 6.3 years

  50. Interpretation • If you have $115,000 and earn 8 percent annually, you can spend $24,000 per year for approximately 6 years and 4 months.

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