1. 7 - 1 Copyright © 2002 by Harcourt, Inc.All rights reserved. Future value Present value Rates of return Amortization CHAPTER 7 Time Value of Money

2. 7 - 2 Copyright © 2002 by Harcourt, Inc.All rights reserved. Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF 2 0123 i% Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

3. 7 - 3 Copyright © 2002 by Harcourt, Inc.All rights reserved. Time line for a $100 lump sum due at the end of Year 2. 100 012 Year i%

4. 7 - 4 Copyright © 2002 by Harcourt, Inc.All rights reserved. Time line for an ordinary annuity of $100 for 3 years. 100 0123 i%

5. 7 - 5 Copyright © 2002 by Harcourt, Inc.All rights reserved. Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 100 50 75 0123 i% -50

6. 7 - 6 Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? 0123 10% 100 Finding FVs is compounding.

7. 7 - 7 Copyright © 2002 by Harcourt, Inc.All rights reserved. FV 1 = PV + INT 1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years: FV 2 = PV(1 + i) 2 = $100(1.10) 2 = $121.00. After 1 year:

8. 7 - 8 Copyright © 2002 by Harcourt, Inc.All rights reserved. After 3 years: FV 3 = PV(1 + i) 3 = $100(1.10) 3 = $133.10. In general, FV n = PV(1 + i) n.

9. 7 - 9 Copyright © 2002 by Harcourt, Inc.All rights reserved. Four Ways to Find FVs Solve the equation with a regular calculator. Use tables. Use a financial calculator. Use a spreadsheet.

10. 7 - 10 Copyright © 2002 by Harcourt, Inc.All rights reserved. Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4th. FV n = PV(1 + i) n. Financial Calculator Solution

11. 7 - 11 Copyright © 2002 by Harcourt, Inc.All rights reserved. Here’s the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set:P/YR = 1, END INPUTS OUTPUT 3 10 -100 0 NI/YR PV PMT FV 133.10

12. 7 - 12 Copyright © 2002 by Harcourt, Inc.All rights reserved. 10% What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 100 0123 PV = ?

13. 7 - 13 Copyright © 2002 by Harcourt, Inc.All rights reserved. Solve FV n = PV(1 + i ) n for PV:  PV = = FV n. FV n (1 + i) n  1 1 + i PV= $100 = $100(PVIF i,n ) = $100(0.7513) = $75.13.  1 1.10 3 n

14. 7 - 14 Copyright © 2002 by Harcourt, Inc.All rights reserved. Financial Calculator Solution 3 10 0100 N I/YR PV PMTFV -75.13 Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years. INPUTS OUTPUT

15. 7 - 15 Copyright © 2002 by Harcourt, Inc.All rights reserved. If sales grow at 20% per year, how long before sales double? Solve for n: FV n = $1(1 + i) n ; $2 = $1(1.20) n Use calculator to solve, see next slide.

16. 7 - 16 Copyright © 2002 by Harcourt, Inc.All rights reserved. 20 -1 0 2 N I/YR PV PMTFV 3.8 Graphical Illustration: 0 1234 1 2 FV 3.8 Year INPUTS OUTPUT

17. 7 - 17 Copyright © 2002 by Harcourt, Inc.All rights reserved. Ordinary Annuity PMT 0123 i% PMT 0123 i% PMT Annuity Due What’s the difference between an ordinary annuity and an annuity due?

18. 7 - 18 Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the FV of a 3-year ordinary annuity of $100 at 10%? 100 0123 10% 110 121 FV= 331

19. 7 - 19 Copyright © 2002 by Harcourt, Inc.All rights reserved. 310 0 -100 331.00 Financial Calculator Solution Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT I/YRNPMTFVPV

20. 7 - 20 Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the PV of this ordinary annuity? 100 0123 10% 90.91 82.64 75.13 248.68 = PV

21. 7 - 21 Copyright © 2002 by Harcourt, Inc.All rights reserved. Have payments but no lump sum FV, so enter 0 for future value. 3 10 100 0 -248.69 INPUTS OUTPUT NI/YRPVPMTFV

22. 7 - 22 Copyright © 2002 by Harcourt, Inc.All rights reserved. Find the FV and PV if the annuity were an annuity due. 100 0123 10% 100

23. 7 - 23 Copyright © 2002 by Harcourt, Inc.All rights reserved. 3 10 100 0 -273.55 Switch from “End” to “Begin.” Then enter variables to find PVA 3 = $273.55. Then enter PV = 0 and press FV to find FV = $364.10. INPUTS OUTPUT NI/YRPVPMTFV

24. 7 - 24 Copyright © 2002 by Harcourt, Inc.All rights reserved. What is the PV of this uneven cash flow stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

25. 7 - 25 Copyright © 2002 by Harcourt, Inc.All rights reserved. Input in “CFLO” register: CF 0 = 0 CF 1 = 100 CF 2 = 300 CF 3 = 300 CF 4 = -50 Enter I = 10, then press NPV button to get NPV = $530.09. (Here NPV = PV.)

26. 7 - 26 Copyright © 2002 by Harcourt, Inc.All rights reserved. What interest rate would cause $100 to grow to $125.97 in 3 years? 3-100 0 125.97 8% $100 (1 + i ) 3 = $125.97. INPUTS OUTPUT NI/YRPVPMTFV

27. 7 - 27 Copyright © 2002 by Harcourt, Inc.All rights reserved. A 20-year old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095) in an online stock account. The stock account has an expected annual return of 12%. The Power of Compound Interest

28. 7 - 28 Copyright © 2002 by Harcourt, Inc.All rights reserved. How much money by the age of 65? 45 12 0-1095 1,487,261.89 INPUTS OUTPUT NI/YRPVPMTFV If she begins saving today, and sticks to her plan, she will have $1,487,261.89 by the age of 65.

29. 7 - 29 Copyright © 2002 by Harcourt, Inc.All rights reserved. How much would a 40-year old investor accumulate by this method? 25 12 0 -1095 146,000.59 INPUTS OUTPUT NI/YRPVPMTFV Waiting until 40, the investor will only have $146,000.59, which is over $1.3 million less than if saving began at 20. So it pays to get started early.

30. 7 - 30 Copyright © 2002 by Harcourt, Inc.All rights reserved. How much would the 40-year old investor need to save to accumulate as much as the 20-year old? 25 12 0 1487261.89 -11,154.42 INPUTS OUTPUT NI/YRPVPMTFV The 40-year old investor would have to save $11,154.42 every year, or $30.56 per day to have as much as the investor beginning at the age of 20.

31. 7 - 31 Copyright © 2002 by Harcourt, Inc.All rights reserved. Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

32. 7 - 32 Copyright © 2002 by Harcourt, Inc.All rights reserved. 0123 10% 0123 5% 456 134.01 100133.10 123 0 100 Annually: FV 3 = $100(1.10) 3 = $133.10. Semiannually: FV 6 = $100(1.05) 6 = $134.01.

33. 7 - 33 Copyright © 2002 by Harcourt, Inc.All rights reserved. We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year. i Per = periodic rate. EAR= EFF% =. effective annual rate

34. 7 - 34 Copyright © 2002 by Harcourt, Inc.All rights reserved. i Nom is stated in contracts. Periods per year (m) must also be given. Examples: l 8%; Quarterly l 8%, Daily interest (365 days)

35. 7 - 35 Copyright © 2002 by Harcourt, Inc.All rights reserved. Periodic rate = i Per = i Nom /m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Examples: 8% quarterly: i Per = 8%/4 = 2%. 8% daily (365): i Per = 8%/365 = 0.021918%.

36. 7 - 36 Copyright © 2002 by Harcourt, Inc.All rights reserved. Effective Annual Rate (EAR = EFF%): The annual rate that causes PV to grow to the same FV as under multi-period compounding. Example: EFF% for 10%, semiannual: FV = (1 + i Nom /m) m = (1.05) 2 = 1.1025. EFF% = 10.25% because (1.1025) 1 = 1.1025. Any PV would grow to same FV at 10.25% annually or 10% semiannually.

37. 7 - 37 Copyright © 2002 by Harcourt, Inc.All rights reserved. An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons. Banks say “interest paid daily.” Same as compounded daily.

38. 7 - 38 Copyright © 2002 by Harcourt, Inc.All rights reserved. How do we find EFF% for a nominal rate of 10%, compounded semiannually? EFF= – 1 Or use a financial calculator. = – 1.0 = (1.05) 2 – 1.0 = 0.1025 = 10.25%.  1 +  i Nom m  1 +  0.10 2 2 m

39. 7 - 39 Copyright © 2002 by Harcourt, Inc.All rights reserved. EAR = EFF% of 10% EAR Annual = 10%. EAR Q =(1 + 0.10/4) 4 – 1= 10.38%. EAR M =(1 + 0.10/12) 12 – 1= 10.47%. EAR D(365) =(1 + 0.10/365) 365 – 1= 10.52%.

40. 7 - 40 Copyright © 2002 by Harcourt, Inc.All rights reserved. Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if m = 1. If m > 1, EFF% will always be greater than the nominal rate.

41. 7 - 41 Copyright © 2002 by Harcourt, Inc.All rights reserved. When is each rate used? i Nom :Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

42. 7 - 42 Copyright © 2002 by Harcourt, Inc.All rights reserved. i Per :Used in calculations, shown on time lines. If i Nom has annual compounding, then i Per = i Nom /1 = i Nom.

43. 7 - 43 Copyright © 2002 by Harcourt, Inc.All rights reserved. (Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.) EAR = EFF%: Used to compare returns on investments with different payments per year.

44. 7 - 44 Copyright © 2002 by Harcourt, Inc.All rights reserved. FV of $100 after 3 years under 10% semiannual compounding? Quarterly? = $100(1.05) 6 = $134.01. FV 3Q = $100(1.025) 12 = $134.49. FV = PV1.+ i m n Nom mn       FV = $1001+ 0.10 2 3S 2x3      

45. 7 - 45 Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 01 100 23 5% 45 6-mos. periods 100 6

46. 7 - 46 Copyright © 2002 by Harcourt, Inc.All rights reserved. Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

47. 7 - 47 Copyright © 2002 by Harcourt, Inc.All rights reserved. 1st Method: Compound Each CF 01 100 23 5% 456 100100.00 110.25 121.55 331.80 FVA 3 = $100(1.05) 4 + $100(1.05) 2 + $100 = $331.80.

48. 7 - 48 Copyright © 2002 by Harcourt, Inc.All rights reserved. Could you find FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: 2nd Method: Treat as an Annuity EAR = ( 1 + ) – 1 = 10.25%. 0.10 2 2

49. 7 - 49 Copyright © 2002 by Harcourt, Inc.All rights reserved. Or, to find EAR with a calculator: NOM% = 10. P/YR = 2. EFF% = 10.25.

50. 7 - 50 Copyright © 2002 by Harcourt, Inc.All rights reserved. EFF% = 10.25 P/YR = 1 NOM% = 10.25 3 10.25 0 -100 INPUTS OUTPUT NI/YRPVFVPMT 331.80 b. The cash flow stream is an annual annuity. Find k Nom (annual) whose EFF% = 10.25%. In calculator, c.

51. 7 - 51 Copyright © 2002 by Harcourt, Inc.All rights reserved. What’s the PV of this stream? 0 100 1 5% 23 100 90.70 82.27 74.62 247.59

52. 7 - 52 Copyright © 2002 by Harcourt, Inc.All rights reserved. Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

53. 7 - 53 Copyright © 2002 by Harcourt, Inc.All rights reserved. Step 1:Find the required annual payments. PMT 0123 10% -1,000 3 10 -1000 0 INPUTS OUTPUT NI/YRPVFV 402.11 PMT

54. 7 - 54 Copyright © 2002 by Harcourt, Inc.All rights reserved. INT t = Beg bal t (i) INT 1 = $1,000(0.10) = $100. Repmt = PMT – INT = $402.11 – $100 = $302.11. Step 3:Find repayment of principal in Year 1. Step 2:Find the interest paid in Year 1.

55. 7 - 55 Copyright © 2002 by Harcourt, Inc.All rights reserved. Step 4: Find ending balance after Year 1. End bal = Beg bal – Repmt = $1,000 – $302.11 = $697.89. Repeat steps 2-4 for Years 2 and 3 to complete the amortization table.

56. 7 - 56 Copyright © 2002 by Harcourt, Inc.All rights reserved. Interest declines. Tax implications. BEGPRINEND YRBALPMTINTPMTBAL 1$1,000$402$100$302$698 2 698 402 70 332 366 3 366 402 37 366 0 TOT1,206.34206.341,000

57. 7 - 57 Copyright © 2002 by Harcourt, Inc.All rights reserved. $ 0123 402.11 Interest 302.11 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. Principal Payments

58. 7 - 58 Copyright © 2002 by Harcourt, Inc.All rights reserved. Amortization tables are widely used-- for home mortgages, auto loans, business loans, retirement plans, etc. They are very important! Financial calculators (and spreadsheets) are great for setting up amortization tables.