Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 1 Chapter 6 Risk and Return: Past and Prologue

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 2 Rates of Return: Single Period HPR = Holding Period Return P 1 = Ending price P 0 = Beginning price D 1 = Dividend during period one

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 3 Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( )/ ( 20) = 25%

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 4 Data from Text Example p Assets(Beg.) HPR (.20).25 TA (Before Net Flows Net Flows (0.8) 0.0 End Assets

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 5 Returns Using Arithmetic and Geometric Averaging Arithmetic r a = (r 1 + r 2 + r r n ) / n r a = ( ) / 4 =.10 or 10% Geometric r g = {[(1+r 1 ) (1+r 2 ).... (1+r n )]} 1/n - 1 r g = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 =.0829 = 8.29%

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 6 Dollar Weighted Returns Internal Rate of Return (IRR) - the discount rate that results present value of the future cash flows being equal to the investment amount Considers changes in investment Initial Investment is an outflow Ending value is considered as an inflow Additional investment is a negative flow Reduced investment is a positive flow

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 7 Dollar Weighted Average Using Text Example Net CFs $ (mil) Solving for IRR 1.0 = -.1/(1+r) /(1+r) 2 +.8/(1+r) /(1+r) 4 r =.0417 or 4.17%

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 8 Quoting Conventions APR = annual percentage rate (periods in year) X (rate for period) EAR = effective annual rate ( 1+ rate for period) Periods per yr - 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01) = 12.68%

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 9 Characteristics of Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 10 r r Symmetric distribution Normal Distribution s.d.

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 11 r r NegativePositive Skewed Distribution: Large Negative Returns Possible Median

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 12 r rNegativePositive Skewed Distribution: Large Positive Returns Possible Median

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 13 Subjective returns p(s) = probability of a state r(s) = return if a state occurs 1 to s states p(s) = probability of a state r(s) = return if a state occurs 1 to s states Measuring Mean: Scenario or Subjective Returns E(r) = p(s) r(s)  s

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 14 Numerical Example: Subjective or Scenario Distributions StateProb. of Stater in State E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) E(r) =.15

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 15 Standard deviation = [variance] 1/2 Measuring Variance or Dispersion of Returns Subjective or Scenario Variance=  s p(s) [r s - E(r)] 2 Var =[(.1)( ) 2 +(.2)( ) ( ) 2 ] Var= S.D.= [.01199] 1/2 =.1095 Using Our Example:

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 16 Real vs. Nominal Rates Fisher effect: Approximation nominal rate = real rate + inflation premium R = r + i or r = R - i Example r = 3%, i = 6% R = 9% = 3% + 6% or 3% = 9% - 6% Fisher effect: Exact r = (R - i) / (1 + i) 2.83% = (9%-6%) / (1.06)

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 17 Annual Holding Period Returns From Figure 6.1 of Text GeomArithStan. SeriesMean%Mean%Dev.% Lg Stk Sm Stk LT Gov T-Bills Inflation

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 18 Annual Holding Period Risk Premiums and Real Returns Risk Real SeriesPremiums%Returns% Lg Stk Sm Stk LT Gov T-Bills Inflation

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 19 Possible to split investment funds between safe and risky assets Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio) Allocating Capital Between Risky & Risk-Free Assets

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 20 Allocating Capital Between Risky & Risk-Free Assets (cont.) Issues –Examine risk/ return tradeoff –Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 21 r f = 7%  rf = 0% E(r p ) = 15%  p = 22% y = % in p (1-y) = % in r f Example Using the Numbers in Chapter 6 (pp )

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 22 E(r c ) = yE(r p ) + (1 - y)r f r c = complete or combined portfolio For example, y =.75 E(r c ) =.75(.15) +.25(.07) =.13 or 13% Expected Returns for Combinations

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 23 E(r) E(r p ) = 15% r f = 7% 22% 0 P F Possible Combinations 

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 24 p p c c = = Since rfrf rfrf y y Variance on the Possible Combined Portfolios = 0, then      

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 25 c c =.75(.22) =.165 or 16.5% If y =.75, then c c = 1(.22) =.22 or 22% If y = 1 c c = 0(.22) =.00 or 0% If y = 0 Combinations Without Leverage      

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 26 Using Leverage with Capital Allocation Line Borrow at the Risk-Free Rate and invest in stock Using 50% Leverage r c = (-.5) (.07) + (1.5) (.15) =.19  c = (1.5) (.22) =.33

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 27 E(r) E(r p ) = 15% r f = 7% = 22% = 22% 0 P F F P P ) S = 8/22 ) S = 8/22 E(r p ) - r f = 8% CAL(CapitalAllocationLine) 

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 28 Risk Aversion and Allocation Greater levels of risk aversion lead to larger proportions of the risk free rate Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 29 Quantifying Risk Aversion E(r p ) = Expected return on portfolio p r f = the risk free rate.005 = Scale factor A x  p = Proportional risk premium The larger A is, the larger will be the added return required for risk

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 30 Quantifying Risk Aversion Rearranging the equation and solving for A Many studies have concluded that investors’ average risk aversion is between 2 and 4

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 31 Chapter 7 Efficient Diversification

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 32 r p = W 1 r 1 + W 2 r 2 W 1 = Proportion of funds in Security 1 W 2 = Proportion of funds in Security 2 r 1 = Expected return on Security 1 r 2 = Expected return on Security 2 Two-Security Portfolio: Return W i  i=1n = 1

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 33  p 2 = w 1 2  w 2 2  W 1 W 2 Cov(r 1 r 2 )  1 2 = Variance of Security 1  2 2 = Variance of Security 2 Cov(r 1 r 2 ) = Covariance of returns for Security 1 and Security 2 Cov(r 1 r 2 ) = Covariance of returns for Security 1 and Security 2 Two-Security Portfolio: Risk

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 34 Covariance  1,2 = Correlation coefficient of returns  1,2 = Correlation coefficient of returns Cov(r 1 r 2 ) =    1  2  1 = Standard deviation of returns for Security 1  2 = Standard deviation of returns for Security 2  1 = Standard deviation of returns for Security 1  2 = Standard deviation of returns for Security 2

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 35 Correlation Coefficients: Possible Values If  = 1.0, the securities would be perfectly positively correlated If  = - 1.0, the securities would be perfectly negatively correlated Range of values for  1, <  < 1.0

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 36  2 p = W 1 2  W 2 2    + 2W 1 W 2 r p = W 1 r 1 + W 2 r 2 + W 3 r 3 Cov(r 1 r 2 ) + W 3 2  3 2 Cov(r 1 r 3 ) + 2W 1 W 3 Cov(r 2 r 3 ) + 2W 2 W 3 Three-Security Portfolio

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 37 r p = Weighted average of the n securities r p = Weighted average of the n securities  p 2 = (Consider all pair-wise covariance measures)  p 2 = (Consider all pair-wise covariance measures) In General, For an n-Security Portfolio:

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 38 E(r p ) = W 1 r 1 + W 2 r 2 Two-Security Portfolio  p 2 = w 1 2  w 2 2  W 1 W 2 Cov(r 1 r 2 )  p = [w 1 2  w 2 2  W 1 W 2 Cov(r 1 r 2 )] 1/2

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 39  = 0 E(r)  = 1  = -1  =.3 13% 8% 12%20% St. Dev TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 40 Portfolio Risk/Return Two Securities: Correlation Effects Relationship depends on correlation coefficient -1.0 <  < +1.0 The smaller the correlation, the greater the risk reduction potential If  = +1.0, no risk reduction is possible

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus Cov(r 1 r 2 ) W1W1 W1W1 = = Cov(r 1 r 2 ) 2 2 W2W2 W2W2 = (1 - W 1 ) Minimum Variance Combination  E(r 2 ) =.14 =.20 Sec 2 12 =.2 E(r 1 ) =.10 =.15 Sec 1        2

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 42 W1W1W1W1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(.2) W1W1W1W1 =.6733 W2W2W2W2 = ( ) =.3267 Minimum Variance Combination:  =.2

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 43 r p =.6733(.10) (.14) =.1131 p = [(.6733) 2 (.15) 2 + (.3267) 2 (.2) 2 + 2(.6733)(.3267)(.2)(.15)(.2)] 1/2 p = [.0171] 1/2 =.1308 Minimum Variance: Return and Risk with  =.2    

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 44 W1W1W1W1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(-.3) W1W1W1W1 =.6087 W2W2W2W2 = ( ) =.3913 Minimum Variance Combination:  = -.3

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 45 r p =.6087(.10) (.14) =.1157 p = [(.6087) 2 (.15) 2 + (.3913) 2 (.2) 2 + 2(.6087)(.3913)(.2)(.15)(-.3)] 1/2 p = [.0102] 1/2 =.1009 Minimum Variance: Return and Risk with  = -.3    

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 46 Extending Concepts to All Securities The optimal combinations result in lowest level of risk for a given return The optimal trade-off is described as the efficient frontier These portfolios are dominant

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 47 E(r) The minimum-variance frontier of risky assets Efficientfrontier Globalminimumvarianceportfolio Minimumvariancefrontier Individualassets St. Dev.

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 48 Extending to Include Riskless Asset The optimal combination becomes linear A single combination of risky and riskless assets will dominate

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 49 E(r) CAL (Global minimum variance) CAL (A) CAL (P) M P A F PP&FA&F M A G P M  ALTERNATIVE CALS

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 50 Dominant CAL with a Risk-Free Investment (F) CAL(P) dominates other lines -- it has the best risk/return or the largest slope Slope = (E(R) - Rf) /   E(R P ) - R f ) /  P   E(R A ) - R f ) /    Regardless of risk preferences combinations of P & F dominate

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 51 Single Factor Model r i = E(R i ) + ß i F + e ß i = index of a securities’ particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 52 Single Index Model Risk Prem Market Risk Prem or Index Risk Prem or Index Risk Prem i = the stock’s expected return if the market’s excess return is zero market’s excess return is zero ß i (r m - r f ) = the component of return due to movements in the market index movements in the market index (r m - r f ) = 0 e i = firm specific component, not due to market movements movements 

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 53 Let: R i = (r i - r f ) R m = (r m - r f ) R m = (r m - r f ) Risk premium format R i =  i + ß i (R m ) + e i Risk Premium Format

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 54 Estimating the Index Model Excess Returns (i) SecurityCharacteristicLine Excess returns on market index R i =  i + ß i R m + e i

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 55 Components of Risk Market or systematic risk: risk related to the macro economic factor or market index Unsystematic or firm specific risk: risk not related to the macro factor or market index Total risk = Systematic + Unsystematic

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 56 Measuring Components of Risk  i 2 =  i 2  m 2 +  2 (e i ) where;  i 2 = total variance  i 2  m 2 = systematic variance  2 (e i ) = unsystematic variance

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 57 Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk =  2 ß i 2  m 2 /  2 =  2  i 2  m 2 /  i 2  m 2 +  2 (e i ) =  2 Examining Percentage of Variance

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 58 Advantages of the Single Index Model Reduces the number of inputs for diversification Easier for security analysts to specialize

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 59 Chapter 8 Capital Asset Pricing and Arbitrage Pricing Theory

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 60 Capital Asset Pricing Model (CAPM) Equilibrium model that underlies all modern financial theory Derived using principles of diversification with simplified assumptions Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 61 Assumptions Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets No taxes, and transaction costs

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 62 Assumptions (cont.) Information is costless and available to all investors Investors are rational mean-variance optimizers Homogeneous expectations

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 63 Resulting Equilibrium Conditions All investors will hold the same portfolio for risky assets – market portfolio Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 64 Risk premium on the market depends on the average risk aversion of all market participants Risk premium on an individual security is a function of its covariance with the market Resulting Equilibrium Conditions (cont.)

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 65 E(r) E(r M ) rfrfrfrf M CML mmmm Capital Market Line 

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 66 M=Market portfolio r f =Risk free rate E(r M ) - r f =Market risk premium E(r M ) - r f =Market price of risk =Slope of the CAPM Slope and Market Risk Premium M 

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 67 Expected Return and Risk on Individual Securities The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 68 E(r) E(r M ) rfrfrfrf SML M ß ß = 1.0 Security Market Line

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 69 SML Relationships  = [COV(r i,r m )] /  m 2 Slope SML =E(r m ) - r f =market risk premium SML = r f +  [E(r m ) - r f ]

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 70 Sample Calculations for SML E(r m ) - r f =.08r f =.03  x = 1.25 E(r x ) = (.08) =.13 or 13%  y =.6 e(r y ) = (.08) =.078 or 7.8%

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 71 E(r) R x =13% SML m m ß ß 1.0 R m =11% R y =7.8% 3% x x ß 1.25 y y ß.6.08 Graph of Sample Calculations

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 72 E(r) 15% SML ß 1.0 R m =11% r f =3% 1.25 Disequilibrium Example

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 73 Disequilibrium Example Suppose a security with a  of 1.25 is offering expected return of 15% According to SML, it should be 13% Underpriced: offering too high of a rate of return for its level of risk

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 74 Security Characteristic Line Excess Returns (i) SCL Excess returns on market index R i =  i + ß i R m + e i

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 75 Using the Text Example p. 245, Table 8.5: Jan.Feb...DecMean Std Dev Excess Mkt. Ret. Excess GM Ret.

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 76 Estimated coefficient Std error of estimate Variance of residuals = Std dev of residuals = R-SQR = ß ß (1.547)1.1357(0.309) r GM - r f = + ß(r m - r f ) Regression Results:  

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 77 Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit Since no investment is required, an investor can create large positions to secure large levels of profit In efficient markets, profitable arbitrage opportunities will quickly disappear

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 78 Arbitrage Example from Text pp Current ExpectedStandard Stock Price$ Return% Dev.% A B C D

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 79 Arbitrage Portfolio Mean Stan. Correlation ReturnDev. Of Returns Portfolio A,B,C D

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 80 Arbitrage Action and Returns E. Ret. St.Dev. * P * D Short 3 shares of D and buy 1 of A, B & C to form P You earn a higher rate on the investment than you pay on the short sale

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 81 APT and CAPM Compared APT applies to well diversified portfolios and not necessarily to individual stocks With APT it is possible for some individual stocks to be mispriced - not lie on the SML APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio APT can be extended to multifactor models