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# CHAPTER 5 Time Value of Money

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1. CHAPTER 5Time Value of Money • Read Chapter 6 (Ch. 5 in the 4th edition) • Future value • Present value • Rates of return • Amortization

2. Time Value of Money Problems • Use a financial calculator • Bring your calculator to class • Will need on exams • We will not use the tables

3. 0 1 2 3 i% CF0 CF1 CF2 CF3 • Time lines show timing of cash flows. • Tick marksat ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

4. A. (1) a. Time line for a \$100 lump sum due at the end of Year 2. 0 1 2 Year i % 100

5. 0 1 2 3 i% 100 100 100 A. (1) b. Time line for anordinary annuity of \$100 for3 years.

6. 0 1 2 3 i% 100 75 50 A. (1) c. Time line for uneven CFs -\$50 at t=0 and\$100, \$75, and \$50 at the end of Years 1 through 3. -50

7. What’s the FV of an initial\$100 after 3 years if i = 10%? 0 1 2 3 10% 100 FV = ? Finding FVs is Compounding.

8. After 1 year: FV1 = PV + I1 = PV + PV (i) = PV(1 + i) = \$100 (1.10) = \$110.00. After 2 years: FV2 = PV(1 + i)2 = \$100 (1.10)2 = \$121.00.

9. After 3 years: FV3 = PV(1 + i)3 = 100 (1.10)3 = \$133.10. In general, FVn = PV (1 + i)n

10. Three ways to find FVs: 1. ‘Solve’ the Equation with a Scientific Calculator 2. Use Tables (the book describes this but not for use in this class) 3. Use a Financial Calculator 4. Spreadsheet (has built-in formulas) -- won’t work on exams

11. Here’s the setup to find FV: INPUTS 3 10 -100 0 N I/YR PV PMT FV 133.10 OUTPUT Clearing automatically sets everything to 0, but for safety enter PMT = 0. Check your calculator. Set: P/YR = 1 and END (“BEGIN” should not show on the display)

12. 0 1 2 3 10% 100 What’s the PV of \$100 duein 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. PV = ?

13. Financial Calculator Solution: INPUTS 3 10 0 100 N I/YR PV PMT FV -75.13 OUTPUT Either PV or FV must be negative. Here PV = -75.13. Put in \$75.13 today, take out \$100 after 3 years.

14. If sales grow at 20% per year,how long before sales double? Solve for n: FVn = 1(1 + i)n; In our case 2 = (1.20)n . Take the log of both sides: ln(2) = n ln(1.2) n = ln(2)/ln(1.2)=.693…/0.1823.. =3.8017

15. INPUTS 20 -1 0 2 N I/YR PV PMT FV 3.8 OUTPUT Financial calculator solution Graphical Illustration: FV 2 3.8 1 Year 0 1 2 3 4

16. 0 1 2 3 i% PMT PMT PMT 0 1 2 3 i% PMT PMT Ordinary vs. Annuity Due PMT

17. What’s the FV of a 3-yearordinary annuity of \$100 at10%? 0 1 2 3 10% 100 100 100 110 121 FV = 331

18. INPUTS 3 10 0 -100 331.00 N I/YR PV PMT FV OUTPUT Financial Calculator Solution: If you enter PMT of 100, you get FV of -331. Get used to the fact that you have to figure out the sign.

19. 0 1 2 3 10% 100 100 100 What’s the PV of this ordinaryannuity? 90.91 82.64 75.13 248.69 = PV

20. INPUTS 3 10 100 0 -248.69 OUTPUT Financial Calculator Solution: N I/YR PV PMT FV Have payments but no lump sum FV, so enter 0 for future value.

21. Technical Aside: Your calculator really is assuming a NPV equation, with PV as a time zero cash flow as follows: When you use the top row of calculator keys, the calculator assumes NPV=0 and solves for one variable.

22. 0 1 2 3 10% 100 100 Find the FV and PV if theannuity were an annuity due. 100

23. Switch from “End” to “Begin”. Then enter variables to find PVA3 = \$273.55. INPUTS 3 10 100 0 -273.55 N I/YR PV PMT FV OUTPUT Then enter PV = 0 and press FV to find FV = \$364.10.

24. Alternative: • The first payment is in the present and thus has a PV of 100. • The next two payments comprise a two period ordinary annuity -- use the formula with n=2, PMT=100, and i=.10. • Sum the above two for the present value. • If you already have the PV, multiply by To get FV

25. Perpetuities • A perpetuity is a stream of regular payments that goes on forever An infinite annuity • Future value of a perpetuity Makes no sense because there is no end point • Present value of a perpetuity A diminishing series of numbers • Each payment’s present value is smaller than the one before

26. Q: The Longhorn Corporation issues a security that promises to pay its holder \$5 per quarter indefinitely. Money markets are such that investors can earn about 8% compounded quarterly on their money. How much can Longhorn sell this special security for? A: Convert the k to a quarterly k and plug the values into the equation. You may also work this by inputting a large n into your calculator (to simulate infinity), as shown below. Example N 999 I/Y 2 PMT 5 FV 0 PV 250 Answer Perpetuities—Example

27. 1 2 3 100 300 300 What is the PV of this uneven cashflow stream? 4 0 10% -50 90.91 247.93 225.39 -34.15 530.08 = PV

28. Input in “CFLO” register ( CFj ): CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 • Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.)

29. 0 1 2 3 10% 10 60 80 What’s Project L’s NPV? Project L: -100.00 9.09 49.59 60.11 18.79 = NPVL

30. Calculator Solution: Enter in CFLO for L: -100 10 60 80 10 CF0 CF1 CF2 CF3 i NPV = 18.78 = NPVL

31. TI Calculators • BA-35 doesn’t appear to do uneven cash flows (NPV and IRR) BA II PLUS CF CF0=-100 Enter  C01= 10 Enter  F01= 1.00  C02= 60 Enter  F02= 1.00  C03= 80 Enter  F03= 1.00 NPV I=10 Enter  CPT NPV= 18.78 IRR CPT IRR= 18.13

32. The Sinking Fund Problem • Companies borrow money by issuing bonds for lengthy time periods No repayment of principal is made during the bonds’ lives • Principal is repaid at maturity in a lump sum • A sinking fund provides cash to pay off a bond’s principal at maturity • Problem is to determine the periodic deposit to have the needed amount at the bond’s maturity—a future value of an annuity problem

33. Q: The Greenville Company issued bonds totaling \$15 million for 30 years. The bond agreement specifies that a sinking fund must be maintained after 10 years, which will retire the bonds at maturity. Although no one can accurately predict interest rates, Greenville’s bank has estimated that a yield of 6% on deposited funds is realistic for long-term planning. How much should Greenville plan to deposit each year to be able to retire the bonds with the money put aside? A: The time period of the annuity is the last 20 years of the bond issue’s life. Input the following keystrokes into your calculator. N 20 Example I/Y 6 FV 15,000,000 0 PV 407,768.35 PMT Answer The Sinking Fund Problem –Example

34. INPUTS 3 -100 0 125.97 N I/YR PV FV PMT OUTPUT What interest rate wouldcause \$100 to grow to \$125.97 in 3 years? \$100 (1 + i )3 = \$125.97. 8%

35. Will the FV of a lump sum belarger or smaller if wecompound more often, holdingthe stated i% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semi-annually, quarterly, or daily--interest is earned on interest more often.

36. 0 1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10. Semi-annually: 0 1 2 3 4 5 6 0 1 2 3 5% 100 134.01 FV6/2 = 100(1.05)6 = 134.01.

37. We will deal with 3 different rates: iNom = nominal, or stated, or quoted, rate per year. iPer = periodic rate. The literal rate applied each period EAR = EFF% = effective annual rate.

38. iNom is stated in contracts. Periods per year (m) must also be given. Sometimes (incorrectly) referred to as the “simple” interest rate. • Examples: • 8%, Daily interest (365 days) • 8%; Quarterly

39. Periodic rate = iPer = iNom/m, where m is periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. • Examples: 8% quarterly: iper = 8/4 = 2% 8% daily (365): iper = 8/365 = 0.021918%

40. Effective Annual Rate (EAR = EFF%): The annual rate which cause PV to grow to the same FV as under multiperiod compounding. Example: EFF% for 10%, semiannual: FV = (1 + inom/m)m = (1.05)2 = 1.1025. Any PV would grow to same FV at 10.25% annually or 10% semiannually: (1.1025)1 = 1.1025 (1.05)2 = 1.1025

41. Comparing Financial Investments • 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.

42. How do we find EFF% for a nominal rate of 10%, compounded semi-annually?

43. EAR = EFF% of 10% EARAnnual = 10%. EARQ = (1 + 0.10/4)4 - 1 = 10.38%. EARM = (1 + 0.10/12)12 - 1 = 10.47%. EARD = (1 + 0.10/360)360 - 1= 10.5155572%.

44. Can the effective rate ever beequal 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.

45. When is each rate used? Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. inom:

46. Used in calculations, shown on time lines. iper: If inom has annual compounding, then iper = inom/1 = inom.

47. EAR = EFF%: Used to compare returns on investments with different payments per year and in advertising of deposit interest rates. (Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.)

48. FV of \$100 after 3 yearsunder 10% semi-annualcompounding? Quarterly? = \$100(1.05)6 = \$134.01 FV3Q = \$100(1.025)12 = \$134.49

49. 4 5 6 0 1 2 3 5% 100 100 100 What’s the value at the endof Year 3 of the following CF stream if the quoted interestrate is 10%, compoundedsemi-annually? 6-month periods