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Discounted Cash Flow Analysis (Time Value of Money)

Discounted Cash Flow Analysis (Time Value of Money). Future value Present value Rates of return. Time lines show timing of cash flows. 0. 1. 2. 3. i%. CF 0. CF 1. CF 2. CF 3.

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Discounted Cash Flow Analysis (Time Value of Money)

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  1. Discounted Cash Flow Analysis(Time Value of Money) • Future value • Present value • Rates of return

  2. Time lines show timing of cash flows. 0 1 2 3 i% CF0 CF1 CF2 CF3 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; and so on.

  3. Time line for a $100 lump sum due at the end of Year 2. 0 1 2 Years i% 100

  4. Time line for an ordinary annuity of $100 for 3 years. 0 1 2 3 i% 100 100 100

  5. 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

  6. 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.

  7. After 1 year FV1 = PV + INT1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years FV2 = FV1(1 + i) = PV(1 + i)2 = $100(1.10)2 = $121.00.

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

  9. 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% PV = ? 100

  10. Solve FVn = PV(1 + i )n for PV: PV = = FVn( ). n FVn (1 + i)n 1 1 + i PV = $100( ) = $100(PVIFi, n) = $100(0.7513) = $75.13. 1 1.10 3

  11. 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 (1.20)n = 2 n ln(1.20) = ln 2 n(0.1823) = 0.6931 n = 0.6931/0.1823 = 3.8 years.

  12. Graphical Illustration: FV 2 3.8 1 Years 0 1 2 3 4

  13. ordinary annuity and annuity due • An ordinary or deferred annuity consists of a series of equal payments made at the end of each period. • An annuity due is an annuity for which the cash flows occur at the beginning of each period. • Annuity due value=ordinary due value x (1+r)

  14. What’s the difference between an ordinary annuity and an annuity due? Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT

  15. 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

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

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

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

  19. What interest rate would cause $100 to grow to $125.97 in 3 years? $100 (1 + i )3 = $125.97.

  20. 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.

  21. 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.

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

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

  24. 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%.

  25. Effective Annual Rate (EAR = EFF%): The annual rate which causes PV to grow to the same FV as under multiperiod compounding. Example: EFF% for 10%, semiannual: FV= (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.

  26. An investment with monthly compounding is different from one with quarterly compounding. Must put on EFF% basis to compare rates of return. • Banks say “interest paid daily.” Same as compounded daily.

  27. An Example: Effective Annual Rates • Suppose you’ve shopped around and come up with the following three rates: • Bank A: 15% compounded daily • Bank B: 15.5% compounded quarterly • Bank C: 16% compounded annually • Which of these is the best if you are thinking of opening a savings account?

  28. Bank A is compounding every day. = 0.15/365 = 0.000411 At this rate, an investment of $1 for 365 periods would grow to ($1x1.000411365)= $1.1618 So, EAR = 16.18% Bank B is paying 0.155/4 = 0.03875 or 3.875% per quarter. At this rate, an investment of $1 of 4 quarters would grow to: ($1x1.038754)= 1.1642 So, EAR = 16.42%

  29. Bank C is paying only 16% annually. In summary, Bank B gives the best offer to savers of 16.42%. The facts are (1) the highest quoted rate is not necessarily the best and (2) the compounding during the year can lead to a significant difference between the quoted rate and the effective rate.

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

  31. 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%.

  32. 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.

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

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

  35. EAR = EFF%: Used to compare returns on investments with different compounding patterns. Also used for calculations if dealing with annuities where payments don’t match interest compounding periods.

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

  37. What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semi-annually? 4 5 0 1 2 3 6 6-mos. periods 5% 100 100 100

  38. Payments occur annually, but compounding occurs each 6 months. • So we can’t use normal annuity valuation techniques.

  39. Compound Each CF 0 1 2 3 4 5 6 5% 100 100.00 100 110.25 121.55 331.80 FVA3 = 100(1.05)4 + 100(1.05)2 + 100 = 331.80.

  40. b. The cash flow stream is an annual annuity whose EFF% = 10.25%.

  41. What’s the PV of this stream? 0 1 2 3 5% 100 100 100 90.70 82.27 74.62 247.59

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

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

  44. 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.

  45. 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.

  46. 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.

  47. $ 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.

  48. 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.

  49. 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.)

  50. 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. Note: % in calculator, decimal in equation.

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