Class Notes: Set 2 The Time Value of Money

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3. 3 Time Value of Money A dollar today is better than a dollar in the future. Just as rent is a landlord�s compensation for the use of an apartment, investors need to be compensated for the use of their money. We call the �rent� for the use of money interest. For now we will delay the discussion of how this rent is established, and focus on the impact of interest on financial decisions.

4. 4 Interest Rates If a bank pays 6% interest on deposits, how much will a deposit of $100 earn, in interest, in one year? $100 x .06 = $6 How much money will you have accumulated in your account after one year? Balance=Deposit + Interest = $100 + $6 = $106 How much money will you have accumulated in your account after two years? Yr 1 Deposit + Interest = $100 + $6 = $106 Yr 2 Balance + Interest = $106 + $6.36 = $112.36

5. 5 Compounding of Interest The future value relationship assumes that interest is compounded. (Interest is earned on interest) Year One $100 x 1.06 = $106 Year Two $106 x 1.06 = $112.36 Year Three $112.36 x 1.06 = $119.10 Year Four $119.10 x 1.06 = $126.25 Year Five $126.25 x 1.06 = $133.82 This compounding is expressed exponentially Future Value

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7. 7 The Impact Of Compounding Interest Simple Interest - Interest paid only on the original investment. (No interest on interest) Using the same 6% example: Year One $100 x 1.06 = $106 Year Two $106+(100x.06)= $112 Year Three $112+(100x.06)= $118 Year Four $118+(100x.06)= $124 Year Five $124+(100x.06)= $130 The simple interest method results in $3.82 less in accumulated value after five years.

8. 8 Compound Interest The longer the investment period the greater the impact of compound interest. $1,000 invested at 6% for 30 years annually compounded is ? $1,000x(1.06)30 = $ 5,743.49 Simple interest would result in ? $1,000 x (1+.06 x30) = $2,800.00

9. 9 Compound Interest The higher the interest rate the greater the impact of compound interest. $1,000 invested at 10% for 30 years annually compounded is: $1,000x(1.10)30=$17,449.40 Simple interest would result in ? $1,000 x (1+.10 x30) = $4,000.00

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11. 11 Future Value of an Investment What is the future value of $3,000,000 invested for 5 years at 7%? What is the future value of $3,000,000 invested for 30 years at 8%? Calculate the Future Value Factors (1+rate)# of periods from above.

12. 12 Using Your Calculator�s Financial Functions What is the future value of $3,000,000 invested for 5 years at 7%? PMT=0, PV = 3,000,000, N = 5, I/Y = 7% : FV=4,207,655.192 What is the future value of $3,000,000 invested for 30 years at 8%? PMT=0, PV = 3,000,000, N = 30, I/Y = 8% : FV = 30,187,970.67 Future Value Factor is the FV of $1.00 PV = 1, N = 5, I/Y = 7% : FV Factor = 1.402552 PV = 1, N = 30, I/Y = 8% : FV = 10.062657

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16. 16 Finding Present Values Future value formula: FV=PV(1+r)n . We can divide both sides by (1+r)n: . FV/(1+r)n=PV(1+r)n/(1+r)n . We get FV/(1+r)n=PV

17. 17 Present Value If we know the value in the future, and want to calculate the value today we simply reverse the process: Alternatively: Where: PV= present value of original investment FV = future value of investment n = number of periods r = interest rate per period

18. 18 Calculating Present Value Calculate the present value of $133.82 received in 5 years at a 6% interest rate. $133.82 x (1/(1.06)5) = $100.00 1/(1+.06)5 =0.74726 In this case the discount factor is .74726 The interest rate used in a present value calculation is called the discount rate. Restated, the present value of one dollar received five years from now at a 6% discount rate is $0.74726

19. 19 Present Value of an Investment What is the present value or $1,000 discounted for 10 years at 6%? What is the present value or $1,000 discounted for 30 years at 10%? Calculate the Discount Factors 1/(1+rate)# of periods from above.

20. 20 Using Your Calculator�s Financial Functions What is the present value or $1,000 discounted for 10 years at 6%. FV =1,000, N = 10, I/Y = 6% : PV = What is the present value or $1,000 discounted for 30 years at 10%. FV =1,000, N = 30, I/Y = 10% : PV = Present Value (Discount) Factor is the PV of $1.00 FV =1, N = 10, I/Y = 6% : Discount Factor = .61391 FV =1, N = 30, I/Y = 10% : Discount Factor = .05731

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22. 22 Effect of Interest Rates and Time on Present Value The longer the investment period the smaller the present value. $1,000 discounted for 10 years at 6% is $1,000x(1/1.0610) = $558.39 Discount Factor = 1/1.0610 = .55839 $1,000 discounted for 30 years at 6% is $1,000x(1/1.0630) = $174.11 Discount Factor = 1/1.0630= .17411

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24. 24 Present Value of $1

25. 25 Testing Our Intuition Future Value Increase (Decrease) in Interest Rate Increase (Decrease) in Investment Horizon Present Value Increase (Decrease) in Interest Rate Increase (Decrease) in Investment Horizon

26. 26 Solving for the Discount Rate If you know the required current investment (present value) and the future investment payoff (future value) you can solve for the implied discount rate.

27. 27 Solving for the Discount Rate Example: the implied return based on receiving $133.82 after five years on an initial investment of $100 is as follows: Example: the implied return based on receiving $2000 after ten years on an initial investment of $1000 is as follows:

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29. 29 !!! IMPORTANT !!! You should never compare cash flows occurring at different times without first discounting or compounding them to a common date.

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42. 42 Example Which of the following interest rates offers is the best if you are thinking of opening a savings account? Note: Most institutions report APRs (like banks) Bank A: 7% compounded annually Bank B: 7% compounded quarterly Bank C: 7% compounded daily

43. 43 Example Which of the following interest rates offers is the best if you are thinking of opening a savings account? Bank A: 7 % compounded annually Bank B: 6.9 % compounded quarterly Bank C: 6.8 % compounded daily

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49. 49 Present Value of Perpetuity It follows that the value of a perpetuity is the amount you would be willing to pay for the right to receive a constant payment forever. Therefore the formula for the present value of a perpetuity is:

50. 50 Valuing Perpetuities What is the value of $75 received annually forever based on a 7.5% discount rate? $1,000= $75/.075 What is the value of $75 received annually forever based on a 5% discount rate? $1,500=$75/.05

51. 51 Formula for Present Value of Perpetuity

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54. 54 What if the first payment is four years from today? A perpetuity might make its first payment after n+1 periods. This is called a delayed (by n periods) perpetuity.

55. 55 Delayed Perpetuity Again, you wish to endow a chair at your old university. But now the first perpetuity payment will not be received until four years from today, how much money needs to be set aside today?

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59. 59 Annuity Annuity - a stream of equal payments received at regular intervals for a specified number of periods. Examples of annuities: -48 equal monthly payments on a car loan -20 equal annual payments from winning the lottery

60. 60 Intuitive Development of the Valuation of an Annuity

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62. 62 Intuitive Development of the Valuation of an Annuity Value of Annuity=Perpetuity - Delayed Perpetuity This formula is called the Annuity Factor

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64. 64 Present Value of an Annuity What is the present value of a 10 year annuity of $1,000 a year at a discount rate of 9%?

65. 65 Using Your Calculator�s Financial Functions What is the present value or $1,000 received annually for 10 years at 9%. PMT =1,000, N = 10, I/Y = 9% : PV = Present Value Annuity Factor (PVAF) is the PV of $1.00 Payments PMT =1, N = 10, I/Y = 9% : PVAF=

66. 66 Alternative Interpretation of the Annuity FactorSpreadsheet Calculation

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68. 68 Example At what interest rate is the present value of an annuity of $1,000 per year for 10 years equal to $6,418? (The present value is for one year before the first payment.)

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71. 71 Loan Types Pure Discount Loans Interest-Only Loans Amortized Loans

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73. 73 Note on Annuities Due The annuities we have discussed thus far are called Ordinary Annuities because the first payment is received one period in the future. If the first payment is received immediately, the annuity is called an Annuity Due. This distinction is unnecessary from a valuation standpoint since every n period Annuity Due is equivalent to the sum of the payment amount and an n-1 period Ordinary Annuity.

74. 74 Future Value of an Annuity We may be interested in calculating the accumulated value of a stream of annuity payments invested at a constant rate of interest. Example � You intend to make monthly contributions to a college savings plan for your newborn niece and you want to estimate the value of this plan in the future. Retirement planning makes extensive use of this methodology

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76. 76 Future Value of an Annuity What is the future value of a 10 year annuity of $1,000 a year at an interest rate of 9%?

77. 77 Using Your Calculator�s Financial Functions What is the Future value or $1,000 invested annually for 10 years at 9%. PMT =1,000, N = 10, I/Y = 9% : FV = Future Value Annuity Factor (FVAF) is the FV of $1.00 Payments PMT =1, N = 10, I/Y = 9% : FVAF=

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84. 84 Rates of Return and Inflation The value of a nominal return to an individual depends on the rate at which prices of goods and services are increasing (Inflation). CPI (Consumer Price Index) is a proxy for inflation. The percentage increases in CPI from one year to the next measures the annual rate of inflation Rate of Inflation= (Year End CPI � Year beginning CPI)/ Year Beginning CPI A rate of return adjusted for inflation is called a �real� rate of return.

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87. 87 Example CPI at the beginning of 2000: 744 CPI at the beginning of 2001: 756 Therefore, the inflation rate during 2000 was 1.61%

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