1. 28 - 1 Copyright © 2002 South-Western Time Value of Money Bond Valuation Risk and Return Stock Valuation WEB CHAPTER 28 Basic Financial Tools: A Review

2. 28 - 2 Copyright © 2002 South-Western 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. 28 - 3 Copyright © 2002 South-Western Time line for a $100 lump sum due at the end of Year 2. 100 012 Year i%

4. 28 - 4 Copyright © 2002 South-Western Time line for an ordinary annuity of $100 for 3 years. 100 0123 i%

5. 28 - 5 Copyright © 2002 South-Western What’s the FV of an initial $100 after 1, 2, and 3 years if i = 10%? FV = ? 0123 10% Finding FVs (moving to the right on a time line) is called compounding. 100FV = ?

6. 28 - 6 Copyright © 2002 South-Western After 1 year: 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.

7. 28 - 7 Copyright © 2002 South-Western After 3 years: FV 3 = PV(1 + i) 3 = $100(1.10) 3 = $133.10. In general, FV n = PV(1 + i) n.

8. 28 - 8 Copyright © 2002 South-Western What’s the FV in 3 years of $100 received in Year 2 at 10%? 100 0123 10% 110

9. 28 - 9 Copyright © 2002 South-Western What’s the FV of a 3-year ordinary annuity of $100 at 10%? 100 0123 10% 110 121 FV= 331

10. 28 - 10 Copyright © 2002 South-Western 3 10 0 -100 331.00 NI/YRPVPMTFV Financial Calculator Solution Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT

11. 28 - 11 Copyright © 2002 South-Western 10% What’s the PV of $100 due in 2 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 100 012 PV = ?

12. 28 - 12 Copyright © 2002 South-Western Solve FV n = PV(1 + i ) n for PV:   PV= $100 1 1.10 = $100PVIF = $1000.8264 = $82.64. i,n       2

13. 28 - 13 Copyright © 2002 South-Western What’s the PV of this ordinary annuity? 100 0123 10% 90.91 82.64 75.13 248.69 = PV

14. 28 - 14 Copyright © 2002 South-Western Have payments but no lump sum FV, so enter 0 for future value. 3 10 100 0 NI/YRPVPMTFV -248.69 INPUTS OUTPUT

15. 28 - 15 Copyright © 2002 South-Western How much do you need to save each month for 30 years in order to retire on $145,000 a year for 20 years, i = 10%? 0 36022012 PMT... 119 months before retirementyears after retirement -145k...

16. 28 - 16 Copyright © 2002 South-Western How much must you have in your account on the day you retire if i = 10%? How much do you need on this date? 220... 119 years after retirement -145k... 0

17. 28 - 17 Copyright © 2002 South-Western You need the present value of a 20- year 145k annuity--or $1,234,467. 2010-145000 0 NI/YRPVFV PMT 1,234,467 INPUTS OUTPUT

18. 28 - 18 Copyright © 2002 South-Western How much do you need to save each month for 30 years in order to have the $1,234,467 in your account? You need $1,234,467 on this date. 0 36012 PMT... months before retirement...

19. 28 - 19 Copyright © 2002 South-Western You need a payment such that the future value of a 360-period annuity earning 10%/12 per period is $1,234,467. 360 10/12 0 1234467 NI/YRPVFV PMT 546.11 INPUTS OUTPUT It will take an investment of $546.11 per month to fund your retirement.

20. 28 - 20 Copyright © 2002 South-Western Key Features of a Bond 1.Par value: Face amount; paid at maturity. Assume $1,000. 2.Coupon interest rate: Stated interest rate. Multiply by par value to get dollars of interest. Generally fixed. (More…)

21. 28 - 21 Copyright © 2002 South-Western 3.Maturity: Years until bond must be repaid. Declines. 4.Issue date: Date when bond was issued.

22. 28 - 22 Copyright © 2002 South-Western 1010 100 1000 NI/YR PV PMTFV -1,000 The bond consists of a 10-year, 10% annuity of $100/year plus a $1,000 lump sum at t = 10: $ 614.46 385.54 $1,000.00 PV annuity PV maturity value PV annuity ====== INPUTS OUTPUT

23. 28 - 23 Copyright © 2002 South-Western 1013 100 1000 NI/YR PV PMTFV -837.21 When k d rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount. What would happen if expected inflation rose by 3%, causing k = 13%? INPUTS OUTPUT

24. 28 - 24 Copyright © 2002 South-Western What would happen if inflation fell, and k d declined to 7%? 10 7 100 1000 NI/YR PV PMTFV -1,210.71 If coupon rate > k d, price rises above par, and bond sells at a premium. INPUTS OUTPUT

25. 28 - 25 Copyright © 2002 South-Western The bond was issued 20 years ago and now has 10 years to maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%, or at 7%?

26. 28 - 26 Copyright © 2002 South-Western M Bond Value ($) Years remaining to Maturity 1,372 1,211 1,000 837 775 3025 20 15 10 5 0 k d = 7%. k d = 13%. k d = 10%.

27. 28 - 27 Copyright © 2002 South-Western At maturity, the value of any bond must equal its par value. The value of a premium bond would decrease to $1,000. The value of a discount bond would increase to $1,000. A par bond stays at $1,000 if k d remains constant.

28. 28 - 28 Copyright © 2002 South-Western Assume the Following Investment Alternatives Economy Prob.T-BillHTCollUSRMP Recession0.10 8.0%-22.0% 28.0% 10.0%-13.0% Below avg. 0.20 8.0 -2.0 14.7 -10.0 1.0 Average 0.40 8.0 20.0 0.0 7.0 15.0 Above avg. 0.20 8.0 35.0 -10.0 45.0 29.0 Boom 0.108.0 50.0 -20.0 30.0 43.0 1.00

29. 28 - 29 Copyright © 2002 South-Western What is unique about the T-bill return? The T-bill will return 8% regardless of the state of the economy. Is the T-bill riskless? Explain.

30. 28 - 30 Copyright © 2002 South-Western Do the returns of HT and Collections move with or counter to the economy? HT moves with the economy, so it is positively correlated with the economy. This is the typical situation. Collections moves counter to the economy. Such negative correlation is unusual.

31. 28 - 31 Copyright © 2002 South-Western Calculate the expected rate of return on each alternative. k = expected rate of return. k HT = 0.10(-22%) + 0.20(-2%) + 0.40(20%) + 0.20(35%) + 0.10(50%) = 17.4%. ^ ^

32. 28 - 32 Copyright © 2002 South-Western k HT17.40% Market15.00 USR13.80 T-bill8.00 Collections1.74 ^ HT has the highest rate of return. Does that make it best?

33. 28 - 33 Copyright © 2002 South-Western What is the standard deviation of returns for each alternative? = Standard deviation..

34. 28 - 34 Copyright © 2002 South-Western  T-bills = 0.0%.  HT = 20.0%.  Coll =13.4%.  USR =18.8%.  M =15.3%.. = ((-22 - 17.4) 2 0.10 + (-2 - 17.4) 2 0.20 + (20 - 17.4) 2 0.40 + (35 - 17.4) 2 0.20 + (50 - 17.4) 2 0.10) 1/2 = 20.0%. HT:

35. 28 - 35 Copyright © 2002 South-Western The coefficient of variation (CV) is calculated as follows: CV HT = 20.0%/17.4% = 1.15  1.2. CV T-bills = 0.0%/8.0% = 0. CV Coll = 13.4%/1.74% = 7.7. CV USR = 18.8%/13.8% = 1.36  1.4. CV M = 15.3%/15.0% = 1.0.  /k. ^

36. 28 - 36 Copyright © 2002 South-Western Prob. Rate of Return (%) T-bill US R HT 0813.817.4

37. 28 - 37 Copyright © 2002 South-Western Standard deviation measures the stand-alone risk of an investment. The larger the standard deviation, the higher the probability that returns will be far below the expected return. Coefficient of variation is an alternative measure of stand-alone risk.

38. 28 - 38 Copyright © 2002 South-Western Expected Return versus Risk Security Expected return Risk,  HT 17.4% 20.0%1.2 Market15.015.3 1.0 USR13.8 18.81.4 T-bills8.00.00.0 Collections1.74 13.4 7.7 Which alternative is best? CV

39. 28 - 39 Copyright © 2002 South-Western Portfolio Risk and Return Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections. Calculate k p and  p. ^

40. 28 - 40 Copyright © 2002 South-Western Portfolio Return, k p k p is a weighted average: k p = 0.5(17.4%) + 0.5(1.74%) = 9.6%. k p is between k HT and k Coll. ^ ^ ^ ^ ^^ ^^ k p =   w i k i  n i = 1

41. 28 - 41 Copyright © 2002 South-Western Alternative Method k p = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40 + (12.5%)0.20 + (15.0%)0.10 = 9.6%. ^ Estimated Return EconomyProb.HTColl.Port. Recession 0.10-22.0% 28.0% 3.0% Below avg. 0.20 -2.0 14.7 6.4 Average 0.40 20.0 0.0 10.0 Above avg. 0.20 35.0 -10.0 12.5 Boom 0.10 50.0 -20.0 15.0 (More...)

42. 28 - 42 Copyright © 2002 South-Western  p = ((3.0 - 9.6) 2 0.10 + (6.4 - 9.6) 2 0.20 + (10.0 - 9.6) 2 0.40 + (12.5 - 9.6) 2 0.20 + (15.0 - 9.6) 2 0.10) 1/2 = 3.3%.  p is much lower than: either stock (20% and 13.4%). average of HT and Coll (16.7%). The portfolio provides average return but much lower risk. The key here is negative correlation.

43. 28 - 43 Copyright © 2002 South-Western Portfolio standard deviation in general  p = Portfolio standard deviation. Where w 1 and w 2 are portfolio weights and r 1,2 is the correlation coefficient between stock 1 and 2.

44. 28 - 44 Copyright © 2002 South-Western Two-Stock Portfolios Two stocks can be combined to form a riskless portfolio if r = -1.0. Risk is not reduced at all if the two stocks have r = +1.0. In general, stocks have r  0.65, so risk is lowered but not eliminated. Investors typically hold many stocks. What happens when r = 0?

45. 28 - 45 Copyright © 2002 South-Western Portfolio beta b p = Portfolio beta Where w 1 and w 2 are portfolio weights, and b 1 and b 2 are stock betas. For our portfolio of 50% HT and 50% Collections, b p = 0.5(1.30) + 0.5(-0.87) = 0.215  0.22. b p = w 1 b 1 + w 2 b 2

46. 28 - 46 Copyright © 2002 South-Western What would happen to the riskiness of an average portfolio as more randomly picked stocks were added?  p would decrease because the added stocks would not be perfectly correlated, but k p would remain relatively constant. ^

47. 28 - 47 Copyright © 2002 South-Western Large 0 15 Prob. 2 1  1  35% ;  Large  20%. Return

48. 28 - 48 Copyright © 2002 South-Western # Stocks in Portfolio 102030 40 2,000+ Company-Specific (Diversifiable) Risk Market Risk 20 0 Stand-Alone Risk,  p  p (%) 35

49. 28 - 49 Copyright © 2002 South-Western Stand-alone Market Diversifiable Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification. Firm-specific, or diversifiable, risk is that part of a security’s stand-alone risk that can be eliminated by diversification. risk risk risk = +.

50. 28 - 50 Copyright © 2002 South-Western Conclusions As more stocks are added, each new stock has a smaller risk-reducing impact on the portfolio.  p falls very slowly after about 40 stocks are included. The lower limit for  p is about 20% =  M. By forming well-diversified portfolios, investors can eliminate about half the riskiness of owning a single stock.

51. 28 - 51 Copyright © 2002 South-Western No. Rational investors will minimize risk by holding portfolios. They bear only market risk, so prices and returns reflect this lower risk. The one-stock investor bears higher (stand-alone) risk, so the return is less than that required by the risk. Can an investor holding one stock earn a return commensurate with its risk?

52. 28 - 52 Copyright © 2002 South-Western Market risk, which is relevant for stocks held in well-diversified portfolios, is defined as the contribution of a security to the overall riskiness of the portfolio. It is measured by a stock’s beta coefficient, which measures the stock’s volatility relative to the market. What is the relevant risk for a stock held in isolation? How is market risk measured for individual securities?

53. 28 - 53 Copyright © 2002 South-Western How are betas calculated? Run a regression with returns on the stock in question plotted on the Y- axis and returns on the market portfolio plotted on the X-axis. The slope of the regression line, which measures relative volatility, is defined as the stock’s beta coefficient, or b.

54. 28 - 54 Copyright © 2002 South-Western Yeark M k i 115% 18% 2 -5-10 312 16... kiki _ kMkM _ - 505101520 20 15 10 5 -5 -10 Illustration of beta calculation: Regression line: k i = -2.59 + 1.44 k M. ^^ Beta Illustration

55. 28 - 55 Copyright © 2002 South-Western How is beta calculated? The regression line, and hence beta, can be found using a calculator with a regression function or a spreadsheet program. In this example, b = 1.44. Analysts typically use five years’ of monthly returns to establish the regression line.

56. 28 - 56 Copyright © 2002 South-Western If b = 1.0, stock has average risk. If b > 1.0, stock is riskier than average. If b < 1.0, stock is less risky than average. Most stocks have betas in the range of 0.5 to 1.5. Can a stock have a negative beta? How is beta interpreted?

57. 28 - 57 Copyright © 2002 South-Western HT T-Bills b = 0 kiki _ kMkM _ - 2002040 40 20 -20 b = 1.30 Collections b = -0.87 Regression Lines of Three Alternatives

58. 28 - 58 Copyright © 2002 South-Western Expected Return versus Market Risk Security Expected return Risk,  b HT17.4%1.30 Market15.01.00 USR13.80.89 T-bills 8.00.00 Collections1.74 -0.87 Which of the alternatives is best?

59. 28 - 59 Copyright © 2002 South-Western Use the SML to calculate each alternative’s required return. The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM). SML: k i = k RF + (k M - k RF )b i. Assume k RF = 8%; k M = k M = 15%. RP M = k M - k RF = 15% - 8% = 7%. ^

60. 28 - 60 Copyright © 2002 South-Western Required Rates of Return k HT = 8.0% + (15.0% - 8.0%)(1.30) = 8.0% + (7%)(1.30) = 8.0% + 9.1%= 17.1%. k M = 8.0% + (7%)(1.00)= 15.0%. k USR = 8.0% + (7%)(0.89)= 14.2%. k T-bill = 8.0% + (7%)(0.00)= 8.0%. k Coll = 8.0% + (7%)(-0.87)= 1.9%.

61. 28 - 61 Copyright © 2002 South-Western Expected versus Required Returns ^ k k HT 17.4% 17.1% Undervalued Market 15.0 15.0 Fairly valued USR 13.8 14.2 Overvalued T-bills 8.0 8.0 Fairly valued Coll 1.74 1.9 Overvalued

62. 28 - 62 Copyright © 2002 South-Western.. Coll.. HT T-bills. USR k M = 15 k RF = 8 -1 0 1 2. SML: k i = 8% + (15% - 8%) b i. k i (%) Risk, b i SML and Investment Alternatives Market

63. 28 - 63 Copyright © 2002 South-Western What is the required rate of return on the HT/Collections portfolio? k p = Weighted average k = 0.5(17%) + 0.5(2%) = 9.5%. Or use SML: b p = 0.22 (Slide 2-45) k p = k RF + (k M - k RF ) b p = 8.0% + (15.0% - 8.0%)(0.22) = 8.0% + 7%(0.22) = 9.5%.

64. 28 - 64 Copyright © 2002 South-Western One whose dividends are expected to grow forever at a constant rate, g. Stock Value = PV of Dividends What is a constant growth stock?.

65. 28 - 65 Copyright © 2002 South-Western For a constant growth stock, If g is constant, then:....

66. 28 - 66 Copyright © 2002 South-Western $ 0.25 Years (t) 0 If g > k, P 0 = negative

67. 28 - 67 Copyright © 2002 South-Western What happens if g > k s ? If k s < g, get negative stock price, which is nonsense. We can’t use model unless (1) g  k s and (2) g is expected to be constant forever. Because g must be a long- term growth rate, it cannot be  k s.

68. 28 - 68 Copyright © 2002 South-Western Assume beta = 1.2, k RF = 7%, and k M = 12%. What is the required rate of return on the firm’s stock? k s = k RF + (k M - k RF )b Firm = 7% + (12% - 7%) (1.2) = 13%. Use the SML to calculate k s :

69. 28 - 69 Copyright © 2002 South-Western D 0 was $2.00 and g is a constant 6%. Find the expected dividends for the next 3 years, and their PVs. k s = 13%. 01 2.2472 2 2.3820 3 g = 6% 4 1.8761 1.7599 1.6508 D 0 = 2.00 13 % 2.12

70. 28 - 70 Copyright © 2002 South-Western What’s the stock’s market value? D 0 = 2.00, k s = 13%, g = 6%. Constant growth model: = = = $30.29. 0.13 - 0.06 $2.12 0.07 = D 0 (1 + g) k s - g D 1 k s - g

71. 28 - 71 Copyright © 2002 South-Western Rearrange model to rate of return form: Then, k s = $2.12/$30.29 + 0.06 = 0.07 + 0.06 = 13%. ^

72. 28 - 72 Copyright © 2002 South-Western If we have supernormal growth of 30% for 3 yrs, then a long-run constant g = 6%, what is P 0 ? k s is still 13%. Can no longer use constant growth model. However, growth becomes constant after 3 years. ^

73. 28 - 73 Copyright © 2002 South-Western Nonconstant growth followed by constant growth: 0 2.3009 2.6470 3.0453 46.1140 1234 k s =13% 54.1072 = P 0 g = 30% g = 6% D 0 = 2.00 2.603.38 4.394 4.6576 ^