1. CHAPTER 4 Risk and Return: The Basics Basic return concepts Basic risk concepts Stand-alone risk Portfolio (market) risk Risk and return: CAPM/SML

2. HW CHAPTER 4 ST1, pg 164 B&E 4-3, 4-7, 4-8, 4-9, 4-13 pg 166-167 B&E

3. What are investment returns? Investment returns measure the financial results of an investment. Returns may be historical or prospective (anticipated). Returns can be expressed in: Dollar terms. Percentage terms.

4. What is the return on an investment that costs $1,000 and is sold after 1 year for $1,100? Dollar return: Percentage return: $ Received - $ Invested $1,100 - $1,000 = $100. $ Return/$ Invested $100/$1,000 = 0.10 = 10%.

5. What is investment risk? Typically, investment returns are not known with certainty. Investment risk pertains to the probability of earning a return less than that expected. The greater the chance of a return far below the expected return, the greater the risk.

6. Selected Realized Returns, 1926 – 2001 Average Standard Return Deviation Small-company stocks17.3%33.2% Large-company stocks12.720.2 L-T corporate bonds 6.1 8.6 L-T government bonds 5.7 9.4 U.S. Treasury bills 3.9 3.2 Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2002 Yearbook (Chicago: Ibbotson Associates, 2002), 28.

7. Probability distribution Rate of return (%) 50150-20 Stock X Stock Y Which stock is riskier? Why?

8. Assume the Following Investment Alternatives EconomyProb.T-BillAltaRepoAm F.MP Recession 0.10 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 0.10 8.0 50.0-20.0 30.0 43.0 1.00

9. 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. (nominal)

10. Do the returns of Alta Inds. and Repo Men move with or counter to the economy? Alta Inds. moves with the economy, so it is positively correlated with the economy. This is the typical situation. Repo Men moves counter to the economy. Such negative correlation is unusual.

11. Calculate the expected rate of return on each alternative. r = expected rate of return. r Alta = 0.10(-22%) + 0.20(-2%) + 0.40(20%) + 0.20(35%) + 0.10(50%) = 17.4%. ^ ^

12. Alta has the highest rate of return. Does that make it best? r Alta17.4% Market15.0 Am. Foam13.8 T-bill 8.0 Repo Men 1.7 ^

13. What is the standard deviation of returns for each alternative?

14. Standard Deviation: Another View Why doesn’t this formula have “Probability” in it? All states assumed equally likely. What does the “-1” in the denominator tell us? This is the calculation assuming a sample.

15.  T-bills = 0.0%.  Alta = 20.0%.  Repo =13.4%.  Am Foam =18.8%.  Market =15.3%. Alta Inds:  = ((-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%.

16. Prob. Rate of Return (%) T-bill Am. F. Alta 0813.817.4

17. 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.

18. Comments on standard deviation as a measure of risk Standard deviation (σ i ) measures total, or stand- alone, risk. The larger σ i is, the lower the probability that actual returns will be closer to expected returns. Larger σ i is associated with a wider probability distribution of returns. Difficult to compare standard deviations, because expected return has not been accounted for - comparing two risky propositions, without any idea of the payoff.

19. Expected Return versus Risk Expected Securityreturn Risk,  Alta Inds. 17.4% 20.0% Market 15.0 15.3 Am. Foam 13.8 18.8 T-bills 8.0 0.0 Repo Men 1.7 13.4

20. Standardized Risk Coefficient of Variation is a measure of relative variability. Should you take the investment with the lowest coefficient of variation (small CV is generally better)? Shows risk per unit of return. (pain/gain ratio)

21. Coefficient of Variation: CV = Standard deviation/expected return CV T-BILLS = 0.0%/8.0% = 0.0. CV Alta Inds = 20.0%/17.4%= 1.1. CV Repo Men = 13.4%/1.7%= 7.9. CV Am. Foam = 18.8%/13.8%= 1.4. CV M = 15.3%/15.0%= 1.0. *** sigma(Portfolio) = [w1 sigma1 + w2 sigma2 ], ie weighted average of individual StDevs. If and only If -> ρ12 =1 CV has sigma in the numerator, hence for the reason above, taking a wt. avg. of CV will involve taking a wt. avg. of sigma (or variance) - which is not allowed.

22. Expected Return versus Coefficient of Variation ExpectedRisk: Securityreturn  CV Alta Inds 17.4% 20.0%1.1 Market 15.0 15.31.0 Am. Foam 13.8 18.81.4 T-bills 8.0 0.0 Repo Men 1.7 13.47.9

23. Return vs. Risk (Std. Dev.): Which investment is best?

24. Portfolio Risk and Return Assume a two-stock portfolio with $50,000 in Alta Inds. and $50,000 in Repo Men. Calculate r p and  p. ^

25. Portfolio Return, r p r p is a weighted average: r p = 0.5(17.4%) + 0.5(1.7%) = 9.6%. r p is between r Alta and r Repo. ^ ^ ^ ^ ^^ ^^ r p =   w i r i  n i = 1

26. Alternative Method r 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 (More...) EconomyProb.AltaRepoPort. 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

27.  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 Alta and Repo (16.7%). The portfolio provides average return but much lower risk. The key here is negative (or less than perfect) correlation. Var(P) = w1 ^2 Var1+w2 ^2 Var2+2w1w2*Cov(1,2) And Cov(1,2) = corr(1,2)*(Var1Var2)^ 1/2 If corr(1,2) = 1 then Var(P) = [w 1  1 + w 2  2 ]^ 2

28. StDev(P) = [w 1  1 + w 2  2 ], ie weighted average of individual StDevs. IFF -> ρ12 =1 σ = w1 σ1 + w2 σ2  IFF -> ρ12 = σ12/σ1σ2 = 1 If the correlation coefficient is -1 then portfolio standard deviation is equal σ = w1 σ1 - w2 σ2 and it is possible to achieve the zero portfolio standard deviation by varying the proportion of assets weights w1 and ;w2 in the portfolio. Practically impossible since very few assets are perfectly negatively correlated.

29. Correlation Coefficient Correlation coefficients (  ) range from … -1 to +1  = -1 implies perfectly negative correlation  = +1 implies perfectly positive correlation  = 0 implies variables are not related Do most stocks have positive, negative, or zero correlations with each other? Positive, but not perfectly so What is correlation of any security with riskless asset (T-bill – is it riskless?)? Zero

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

31. General comments about risk Most stocks are positively correlated with the market (ρ k,m  0.65). σ  35% for an average stock. (what is range in 2 of 3 years? E® = 12% from market) Combining stocks in a portfolio generally lowers risk.

32. What would happen to the risk of an average 1-stock portfolio as more randomly selected stocks were added?  p would decrease because the added stocks would not be perfectly correlated, but r p would remain relatively constant. ^

33. Large 0 15 Prob. 2 1  1  35% ;  Large  20%. Return

34. # Stocks in Portfolio 102030 40 2,000+ Company Specific (Diversifiable) Risk Market Risk 20 0 Stand-Alone Risk,  p  p (%) 35

35. 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 = +.

36. Two Components of Risk Company-specific (Diversifiable) risk Unique to specific firms. Results from random or uncontrollable events. What are some examples? Natural disasters, accidents, strikes, lawsuits, death of CEO, etc. Market (systematic) risk Relates to forces affecting all investments. What are some examples? Inflation, recession, war, yield inversion etc.

37. 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.

38. No Stand-alone risk is not important to a well- diversified investor – because it vaporizes Rational, risk-averse investors are concerned with σ p, which is based upon market risk. There can be only one price (the market return) for a given security. No compensation should be earned for holding unnecessary, diversifiable risk. Can an investor holding one stock earn a return commensurate with its risk?

39. 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. For stock i, its beta is: b i = cov[i,m]/var[m] =  i  m  iM /  m ^2 = (  iM  i ) /  M How is market risk measured for individual securities?

40. Betas….? In addition to measuring a stock’s contribution of risk to a portfolio, beta also measures the stock’s volatility relative to the market. Shows how the price of a security responds to changes in the overall stock market (not just its variance) Stock’s Beta is the only relevant measure of risk from a portfolio standpoint: in a well diversified portfolio there is no USR so beta gives the stock’s response to a change in the market (syst. risk)

41. Using a Regression to Estimate Beta 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.

42. Calculation of Beta... kiki _ kMkM _ - 505101520 20 15 10 5 -5 -10 Regression line: k i = -2.59 + 1.44 k M ^^ Yeark M k i 115% 18% 2 -5-10 312 16

43. Use the historical stock returns to calculate the beta for PQU. YearMarketPQU 1 25.7% 40.0% 2 8.0%-15.0% 3-11.0%-15.0% 4 15.0% 35.0% 5 32.5% 10.0% 6 13.7% 30.0% 7 40.0% 42.0% 8 10.0%-10.0% 9-10.8%-25.0% 10-13.1% 25.0%

44. Calculating Beta for PQU r PQU = 0.83r M + 0.03 R 2 = 0.36 -40% -20% 0% 20% 40% -40%-20%0%20%40% r M r pqu Show in Excel, and how to get return series on stocks, market

45. What is beta for PQU? The regression line, and hence beta, can be found using a calculator with a regression function or a spreadsheet program. In this example, b = 0.83.

46. Calculating Beta in Practice Many analysts use the S&P 500 to find the market return. Analysts typically use four or five years’ of monthly returns to establish the regression line. Some analysts use 52 weeks of weekly returns.

47. 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? Beta of.5If … Market goes up 1%, … stock only goes up.5%. But if market goes down 1%, stock drops just.5%. Beta of 0 No correlation with market How is beta interpreted?

48. Finding Beta Estimates on the Web http://finance.yahoo.com/q/ks?s=C http://www.investor.reuters.com/StockEntry.aspx http://new.quote.com/stocks/company.action?sym=C [change the ticker in the URL Address bar, directly] http://finance.google.com/finance?q=C ** Changes in estimation window length, return frequency, choice of market and statistical adjustments will result in varying betas from different sources. It may be best to calculate your own – you can fix the variables and parameters.

49. Expected Return versus Market Risk Which of the alternatives is best? Expected SecurityreturnRisk, b Alta 17.4% 1.29 Market 15.0 1.00 Am. Foam 13.8 0.68 T-bills 8.0 0.00 Repo Men 1.7 -0.86

50. Use the SML to calculate each alternative’s required return. The Security Market Line (SML) is part of the Capital Asset Pricing Model (CAPM). How do we find a security’s required return k i ? SML: r i = r RF + (r M - r RF )b i. Assume r RF = 8%; r M = r M = 15%. RP M = (r M - r RF ) = 15% - 8% = 7%. Higher beta means higher market risk and thus higher expected or required [not realized] return.

51. Security Market Line What is the proxy for the risk-free rate? U.S. Treasury Bills What is the proxy for the return on the market? Typically use the S&P500 Index [ticker is spx or ^spx or spy]

52. Required Rates of Return r Alta = 8.0% + (7%)(1.29) = 8.0% + 9.0%= 17.0%. r M = 8.0% + (7%)(1.00)= 15.0%. r Am. F. = 8.0% + (7%)(0.68)= 12.8%. r T-bill = 8.0% + (7%)(0.00)= 8.0%. r Repo = 8.0% + (7%)(-0.86)= 2.0%.

53. Expected versus Required Returns ^ r r Alta 17.4% 17.0% Undervalued Market 15.0 Fairly valued Am. F. 13.8 12.8 Undervalued T-bills 8.0 Fairly valued Repo 1.7 2.0 Overvalued

54. .. Repo. Alta T-bills r M = 15 r RF = 8 -1 0 1 2. SML: r i = r RF + (RP M ) b i r i = 8% + (7%) b i r i (%) Risk, b i SML and Investment Alternatives Market Buy Alta Sell Repo

55. Portfolio Beta Portfolio beta is just a weighted average of the security betas b 1 =.8, b 2 =1, b 3 =1.2 w i  {.2,.3,.5} b  w =.16+.30+.60=1.06 Note that =  w i = 1

56. Calculate beta for a portfolio with 50% Alta and 50% Repo b p = Weighted average = 0.5(b Alta ) + 0.5(b Repo ) = 0.5(1.29) + 0.5(-0.86) = 0.22.

57. What is the required rate of return on the Alta/Repo portfolio? r p = Weighted average r = 0.5(17%) + 0.5(2%) = 9.5%. Or use SML: r p = r RF + (RP M ) b p = 8.0% + 7%(0.22) = 9.5%.

58. SML 1 Original situation Required Rate of Return r (%) SML 2 00.51.01.52.0 18 15 11 8 New SML  I = 3% Impact of Inflation Change on SML What if investors raise inflation expectations by 3%? (reqd return on all risky assets increases by 3 %, Prices drop) If Expected Return increases ->Current Price drops If Realized Return increases -> Final Price rises

59. r M = 18% r M = 15% SML 1 Original situation Required Rate of Return (%) SML 2 After increase in risk aversion Risk, b i 18 15 8 1.0  RP M = 3% Impact of Risk Aversion Change What if investors’ risk aversion increased, causing the market risk premium to increase by 3% - rise/run=3/1? (km rises from 15 to 18%, so ki for b=.5 rises by 1.5%, and for b=2 rises by 6%)

60. Capital Asset Pricing Model & Security Market Line Does a higher required return mean that the actual return you get will be higher? No, may lose all of your money on the stock. If the expected return on a risky asset is 0 have a positive expected return. So what would happen when the CAPM is empirically tested?

61. CAPM Issues Investors’ required returns are based on future risk, but betas are calculated with historical data. Will a company’s beta be the same this year and next year? No – Non-Stationarity Problem Assumes markets are efficient There have been studies that both support and dispute the CAPM Still used in practice [provides a conceptual framework useful for linking risk and return in financial decisions]

62. Next Chapter (5):Portfolio Risk