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Time Value of Money

Time Value of Money

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Time Value of Money

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  1. Time Value of Money FIL 404 Prepared by Keldon Bauer

  2. Cash Flow Time Lines • You win a contest, and you have the option of taking $1.4 million now or $250,000 per year for five years. • Which should you take? • The answer comes through taking into consideration the time value of money.

  3. 2 3 4 5 0 1 Cash Flow Time Lines • The first step is visualizing the cash flows by drawing a cash flow time line. • Time lines show when cash flows occur. • Time 0 is now.

  4. 2 3 4 5 0 1 8% $250K $250K $250K $250K $250K Cash Flow Time Lines • Outflows are listed as negatives. • Inflows are positive. • State the appropriate “interest rate,” which represents your opportunity costs

  5. Future Value • Future value is higher than today, because if I had the money I would put it to work, it would earn interest. • The interest could then earn interest. • Compounding: allowing interest to earn interest on itself.

  6. 2 3 4 5 0 1 8% Principal -1 Interest 0.08 0.0864 0.0933 0.1008 0.1088 Prev. Interest 0.00 0.0800 0.1664 0.2597 0.3605 Total 1.08 1.1664 1.2597 1.3605 1.4693 Future Value - Example • If you invest $1,000 today at 8% interest per year, how much should you have in five years (in thousands).

  7. Future Value • For one year, the future value can be defined as:

  8. Future Value • The second year, the future value can be stated as follows:

  9. Future Value • Therefore, the general solution to the future value problem is: • The Excel formula is: • =FV(Interest, Term, Payments, [Present Value], [Type])

  10. Future Value - Excel

  11. Future Value • Interest can be seen as the opportunity growth rate of money.

  12. 2 3 4 5 0 1 8% PV=? $500 Present Value • Present value is the value in today’s dollars of a future cash flow. • If we are interested in the present value of $500 delivered in 5 years:

  13. Present Value • The general solution to this problem follows from the solution to the future value problem:

  14. Present Value - Excel • The Excel formula is: • =PV(Interest, Term, Payments, [Future Value], [Type])

  15. Present Value - Excel

  16. Present Value • Since the discount rate is the opportunity cost, the present value represents what I would have to give up now to get the future value specified.

  17. 2 3 4 5 0 1 ?% $100 $500 Interest Rates • If we know the amount we need at time n and the amount we can invest at time zero, then we must only solve for the interest rate.

  18. Interest Rates • Solving for interest rates algebraically:

  19. 2 3 4 5 0 1 ?% $100 $500 Interest Rates - Example In Excel: =RATE(Term, Payment, Present Value, [Future Value], [Type], [Guess])

  20. Interest Rate - Excel

  21. Time Periods • If the present value, future value and interest rate are known, but the number of time periods is not. Then n can be found algebraically:

  22. Time Periods - Example • If we use the last example of investing $100, we want $500 in future, and the current market interest is 8%, n can be found: In Excel: =NPER(Interest, Payment, Present Value, [Future Value], [Type])

  23. Time Periods - Excel

  24. Annuities • Definition: A series of equal payments at a fixed interval. • Two types: • Ordinary annuity: Payments occur at the end of each period. (Default in Excel) • Annuity due: Payments occur at the beginning of each period. (Set the type = 1 in Excel) • In Excel, use the same formulas introduced so far, just specify payment and type.

  25. 2 3 4 5 0 1 8% $100 $100 $100 $100 $100 FV=? What is the future value? Ordinary Annuity • Example is a regular payment of $100 for five years earning 8% interest.

  26. Ordinary Annuity – Future Value • The future value of an ordinary annuity can be found as follows:

  27. 2 3 4 5 0 1 8% $100 $100 $100 $100 $100 Ordinary Annuity - Example

  28. Ordinary Annuity - Example

  29. 2 3 4 5 0 1 8% $100 $100 $100 $100 $100 FV=? What is the future value? Annuity Due • Example is a regular payment of $100 for five years earning 8% interest.

  30. Annuity Due – Future Value • The future value of an annuity due can be found by noticing that the annuity due is the same as an ordinary annuity, with one more compounding period:

  31. 8% $100 $100 $100 $100 $100 2 3 4 5 0 1 Annuity Due - Example

  32. Annuity Due - Excel

  33. 2 3 4 5 0 1 8% $100 $100 $100 $100 $100 PV=? What is the present value? Ordinary Annuity - Present Value • Example is a regular payment of $100 for five years earning 8% interest.

  34. Ordinary Annuity - Present Value • The present value of an ordinary annuity can be found as follows:

  35. 2 3 4 5 0 1 8% $100 $100 $100 $100 $100 Ordinary Annuity - Example

  36. Ordinary Annuity - Excel

  37. 2 3 4 5 0 1 8% $100 $100 $100 $100 $100 PV=? What is the present value? Annuity Due - Present Value • Example is a regular payment of $100 for five years earning 8% interest.

  38. Annuity Due - Present Value • The future value of an annuity due can be found as follows:

  39. 8% $100 $100 $100 $100 $100 2 3 4 5 0 1 Annuity Due - Example

  40. Annuity Due - Excel

  41. Annuities - Finding Interest Rate • Interest rates cannot be solved directly. • Calculators and computers search for the correct answer (there is only one correct answer). • It guesses and then iteratively goes higher or lower.

  42. Perpetuities • What would you have to pay to be paid $2,000 per year forever (given a market rate of 8%)?

  43. Uneven Cash Flow Streams • If payments are irregular or come at irregular intervals, we can still find the PV (or FV). • Take the present value (or future value) of individual payments and sum them together.

  44. $ 92.59 2 3 4 5 0 1 8% $171.47 $238.15 $294.01 $100 $200 $300 $400 $500 $340.29 Uneven Cash Flows - Example $1,136.51 = Present Value

  45. Uneven Cash Flows - Excel • Excel can do this in one argument: • =NPV(Interest, Array of Payments [starting with payment for time 1]). • If you want to include a payment in time zero, add it to the above argument separately.

  46. Uneven Cash Flows - Excel

  47. $432.00 2 3 4 5 0 1 $349.92 8% $251.94 $136.05 $100 $200 $300 $400 $500 $1,136.51 = Present Value Uneven Cash Flow - Example Future Value = $1,669.91

  48. Uneven Cash Flows - Excel

  49. Finding Interest Rate • As with annuities, interest rates for uneven cash flow streams cannot be solved directly. • Calculators search for the correct answer, called an IRR (there may be more than one correct answer). • It guesses and then iteratively goes higher or lower.

  50. 8% 2 3 4 5 0 1 -$100 $146.93 0 1 2 3 4 5 6 7 8 9 10 4% -$100 $148.02 Compounding • The more often one compounds interest, the faster it grows. Annual Semi-Annual