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

2. Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF 2 012 3 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; and so on.

3. Time line for a $100 lump sum due at the end of Year 2. 100 01 2 Years i%

4. Time line for an ordinary annuity of $100 for 3 years. 100 0123 i%

5. 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. What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? 0123 10% 100 Finding FVs is compounding.

7. After 1 year FV 1 = PV + INT 1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years FV 2 = FV 1 (1 + i) = PV(1 + i) 2 = $100(1.10) 2 = $121.00.

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

9. 6 - 9 FVIF i,n = (1+i) n EX. i=10%, n=7 = (1+.1) 7 =1.94871 PVIF i,n = 1/(1+i) n PVIF i,n and FVIF i,n

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 10% PV = ?

11. PV = = FV n ( ). Solve FV n = PV(1 + i ) n for PV: PV = $100 ( ) = $100(PVIF i, n ) = $100(0.7513) = $75.13. FV n (1 + i) n 1 1 + i n 1 1.10 3

12. 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 (1.20) n = 2 n ln (1.20)= ln 2 n(0.1823)= 0.6931 n= 0.6931/0.1823 = 3.8 years.

13. Graphical Illustration: 0 1234 1 2 FV 3.8 Years

14. 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. ordinary annuity and annuity due Annuity due value=ordinary due value x (1+r)

15. What’s the difference between an ordinary annuity and an annuity due? PMT 0123 i% PMT 0123 i% PMT Annuity Due Ordinary Annuity

16. What’s the FV of a 3-year ordinary annuity of $100 at 10%? 100 0123 10% 110 121 FV = 331

17. What’s the PV of this ordinary annuity? 100 0123 10% 90.91 82.64 75.13 248.69 = PV

18. 6 - 18 PVIFA i,n and FVIFA i,n PVIFA i,n = [1 - (1+i) -n ] / i Ex. i=10%, n=3 = [1 - (1+0.1) -3 ] / 0.1 = 2.48685 FVIFA i,n = [(1+i) n – 1] / i

19. 6 - 19 Ordinary Annuity and Annuity Due Ordinary Annuity Annuity Due PVA n (OA)= PMT(PVIFA i,n ) FVA n (OA)= PMT(FVIFA i,n ) PVA n (AD)= PMT(PVIFA i,n )(1+i) FVA n (AD)= PMT(FVIFA i,n )(1+i)

20. Find the FV and PV if the annuity were an annuity due. 100 0123 10% 100

21. 300 3 What is the PV of this uneven cash flow stream? 0 100 1 300 2 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

22. What interest rate would cause $100 to grow to $125.97 in 3 years? $100 (1 + i ) 3 = $125.97. (1+i) 3 = 1.2597 (1+i) = 1.07999 i = 8% approx.

23. 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.

24. 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.

25. 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.

26. i Nom is stated in contracts. Periods per year (m) must also be given. Examples: 8%, Quarterly interest 8%, Daily interest

27. 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%.

28. 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.

29. 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.

30. 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?

31. Bank A is compounding every day. = 0.15/365 = 0.000411 At this rate, an investment of $1 for 365 periods would grow to ($1x1.000411 365 )= $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: ($1x1.03875 4 )= 1.1642 So, EAR = 16.42%

32. 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.

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

34. 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(360) =(1 + 0.10/360) 360 - 1= 10.52%.

35. 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.

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

37. Used in calculations, shown on time lines. If i Nom has annual compounding, then i Per = i Nom /1 = i Nom. i Per :

38. 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.

39. 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

40. 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? 01 100 23 5% 45 6 6-mos. periods 100

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

42. 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.

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

44. What’s the PV of this stream? 0 100 1 5% 23 100 90.70 82.27 74.62 247.59

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

46. Step 1: Find the required payments. PMT 0123 10% -1000 3 10 -1000 0 INPUTS OUTPUT NI/YRPVFV PMT 402.11

47. 6 - 47 PVA n (OA)= PMT(PVIFA i,n ) =1000/2.4869 = 402.107 PMT=PVA 3 /(PVIFA 10%,3 ) Step 1: Find the required payments.

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

49. 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.

50. Interest declines. Tax implications. BEGPRINEND YRBALPMTINTPMTBAL 1$1,000$402$100$302$698 269840270332366 3366402373660 TOT1,206.34206.341,000

51. $ 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

52. 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.

53. 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.)

54. i Per = 10.0% / 365 = 0.027397% per day. FV=? 012273 0.027397% -100 Note: % in calculator, decimal in equation.   FV = $100 1.00027397 = $100 1.07765 = $107.77. 273

55. 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?

56. 2 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV i Per =0.019178% per day. 1,000 0365456 days -850

57. 1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1000. FV Bank =$850(1.00019178) 456 =$927.67 in bank. Buy the note: $1000 > $927.67.

58. 2. Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV=$1000(1.00019178) 456 =$916.27.