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Principles of Business Finance Fin 510

Principles of Business Finance Fin 510. Dr. Lawrence P. Shao Marshall University Spring 2002. CHAPTER 6 Time Value of Money. Future value Present value Rates of return Amortization. Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 0. 1. 2.

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Principles of Business Finance Fin 510

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  1. Principles of Business FinanceFin 510 Dr. Lawrence P. Shao Marshall University Spring 2002

  2. CHAPTER 6Time Value of Money • Future value • Present value • Rates of return • Amortization

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

  4. 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.

  5. After 1 year: FV1 = PV + INT1 = 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.

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

  7. Three Ways to Find FVs • Use a financial calculator. • Use future value table. • Use future value formula. FVn = PV(1 + i)n. There are 4 variables. If 3 are known, you can solve for the 4th.

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

  9. Solve FVn = PV(1 + i )n for PV: 3 1 ö æ ( ) ÷ PV = $100 = $100 PVIF ç ø è i, n 1.10 ( ) = $100 0.7513 = $75.13.

  10. If sales grow at 20% per year, how long before sales double? Solve for n: FVn = 1(1 + i)n; 2 = 1(1.20)n Use FVIF formula to solve. Solution: 3.8 years

  11. What’s the difference between an ordinaryannuity and an annuitydue? Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT

  12. What’s the FV of a 3-year ordinary annuity of $100 at 10%? 0 1 2 3 10% 100 100 100 110 121 FV = 331

  13. What’s the PV of this ordinary annuity? 0 1 2 3 10% 100 100 100 90.91 82.64 75.13 248.68 = PV

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

  15. 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.

  16. 0 1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10. 0 1 2 3 4 5 6 0 1 2 3 5% 100 134.01 Semiannually: FV6 = 100(1.05)6 = 134.01.

  17. We will deal with 3 different rates: iNom = nominal, or stated, or quoted, rate per year. iPer = periodic rate. EAR = EFF% = . effective annual rate

  18. iNom is stated in contracts. Periods per year (m) must also be given. • Examples: • 8%; Quarterly • 8%, Daily interest (365 days)

  19. Periodic rate = iPer = iNom/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: iPer = 8%/4 = 2%. 8% daily (365): iPer = 8%/365 = 0.021918%.

  20. Effective Annual Rate (EAR = EFF%): The annual rate which causes PV to grow to the same FV as under multi-period compounding. Example: EFF% for 10%, semiannual: FV = (1 + iNom/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.

  21. How do we find EFF% for a nominal rate of 10%, compounded semiannually?

  22. 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(360) = (1 + 0.10/360)360 - 1 = 10.52%.

  23. 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.

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

  25. iPer: Used in calculations, shown on time lines. If iNom has annual compounding, then iPer = iNom/1 = iNom.

  26. FV of $100 after 3 years under 10% semiannual compounding? Quarterly? mn i æ ö Nom FV = PV 1 . + ç ÷ è ø n m 2x3 0.10 æ ö FV = $100 1 + ç ÷ è ø 3S 2 = $100(1.05)6 = $134.01. FV3Q = $100(1.025)12 = $134.49.

  27. Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

  28. Step 1: Find the required payments. 0 1 2 3 10% -1,000 PMT PMT PMT 3 10 -1000 0 INPUTS N I/YR PV FV PMT OUTPUT 402.11

  29. Step 2: Find interest charge for Year 1. INTt = Beg balt (i) INT1 = $1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT = $402.11 - $100 = $302.11.

  30. Step 4: Find ending balance after Year 1. End bal = Beg bal - Repmt = $1,000 - $302.11 = $697.89. Repeat these steps for Years 2 and 3 to complete the amortization table.

  31. BEG PRIN END YR BAL PMT INT PMT BAL 1 $1,000 $402 $100 $302 $698 2 698 402 70 332 366 3 366 402 37 366 0 TOT 1,206.34 206.34 1,000 Interest declines. Tax implications.

  32. $ 402.11 Interest 302.11 Principal Payments 0 1 2 3 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling.

  33. 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.

  34. On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.)

  35. iPer = 10.0% / 365 = 0.027397% per day. 0 1 2 273 0.027397% ... FV = ? -100 273 ( ) FV = $100 1.00027397 273 ( ) = $100 1.07765 = $107.77.

  36. Now suppose you leave your money in the bank for 21 months, which is 1.75 years or 273 + 365 = 638 days. How much will be in your account at maturity?

  37. iPer = 0.027397% per day. 0 365 638 days ... ... -100 FV = 119.10 FV = $100(1 + .10/365)638 = $100(1.00027397)638 = $100(1.1910) = $119.10.

  38. You are offered a note which pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank which pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of 0.019178% and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless. Should you buy it?

  39. iPer = 0.019178% per day. 0 365 456 days ... ... -850 1,000 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV

  40. 1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1,000. FVBank=$850(1.00019178)456 =$927.67 in bank. Buy the note: $1,000 > $927.67.

  41. 2. Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV = $1,000/(1.00019178)456 = $916.27. PV of note is greater than its $850 cost, so buy the note.

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