Chapter 6 Efficient Diversification Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Chapter 6 Efficient Diversification 6-2

6.1 Diversification and Portfolio Risk 6.2 Asset Allocation With Two Risky Assets 6-3

Combinations of risky assets market risk,systematic risk or nondiversifiable risk. 2-unique risk, firm- specific risk, nonsystematic risk, or diversifiable risk.

Covariance and correlation The problem with covariance  Covariance does not tell us the intensity of the comovement of the stock returns, only the direction. We can standardize the covariance however and calculate the correlation coefficient which will tell us not only the direction but provides a scale to estimate the degree to which the stocks move together. 6-5

Correlation –Highly correlated investments - moves proportionally in the same direction~ do not diversify away risk(+1)~Poor –Negatively correlated investments : moves in opposition to each other ~provide high degree of risk reduction(-1)~ Excellent –Uncorrelated investments - no predictions about the movement ~provide some overall reduction in risk(0)~ Good

The effects of correlation & covariance on diversification Asset A Asset B Portfolio AB 6-7

6-8 The effects of correlation & covariance on diversification Asset C Portfolio CD

Naïve diversification 6-9 Most of the diversifiable risk eliminated at 25 or so stocks The power of diversification

Covariance calculations Using scenario analysis with probabilities the covariance can be calculated with the following formula: If when r 1 > E[r 1 ], r 2 > E[r 2 ], and when r 1 < E[r 1 ], r 2 < E[r 2 ], then COV will be postive If when r 1 > E[r 1 ], r 2 E[r 2 ], then COV will be negative Which will “average away” more risk? 6-10

Measuring the correlation coefficient Standardized covariance is called the _____________________ For Stock 1 and Stock 2 correlation coefficient or  6-11

= W 1 + W 2 W 1 = Proportion of funds in Security 1 W 2 = Proportion of funds in Security 2 = Expected return on Security 1 = Expected return on Security 2 Two-Security Portfolio: Return r1r1 E( ) rprp r2r2 r1r1 r2r2 6-12

E(r p ) = W 1 r 1 + W 2 r 2 W 1 = W 2 = = Two-Security Portfolio Return E(r p ) = 0.6(9.28%) + 0.4(11.97%) = 10.36% Wi = % of total money invested in security i % 11.97% r1r1 r2r2 6-13

 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 6-14

Problem 3 a.Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio. i. E(r NP ) = w OP E(r OP ) + w ABC E(r ABC ) = ii Cov =    OP   ABC = iii.  NP = [w OP 2  OP 2 + w ABC 2  ABC w OP w ABC (Cov OP, ABC )] 1/2 = [(0.9 2  ) + (0.1 2  ) + (2  0.9  0.1 .00028)] 1/2 = %  2.27% (0.9  0.67) + (0.1  1.25) = 0.728% 0.40 .0237 .0295 = 

 = +1  =.3 E(r) 13% 8% 12%20% St. Dev TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS Stock A Stock B W A = 0% W B = 100% W A = 100% W B = 0%  = 0  = -1 50%A 50%B 6-16

Summary: Portfolio Risk/Return Two Security Portfolio Amount of risk reduction depends critically on _________________________. Adding securities with correlations _____ will result in risk reduction. To sum up the idea: The less correlated the greater the risk reduction possible through diversification correlations or covariances <

Minimum Variance Combinations -1<  < Cov(r 1 r 2 ) W1W1 W1W1 = = Cov(r 1 r 2 ) 2 2 W2W2 W2W2 = (1 - W 1 )  2 Choosing weights to minimize the portfolio variance 6-18

1 1 Minimum Variance Combinations -1<  < E(r 2 ) =.14 =.20 Stk 2 12 =.2 E(r 1 ) =.10 =.15 Stk 1          1  2 Cov(r 1 r 2 ) =    1 

E[r p ] = Minimum Variance: Return and Risk with  =.2 p2 =p2 = p2 =p2 = (.10) (.14) =.1131 or 11.31% W 1 2  W 2 2  W 1 W 2  1,2  1 

Minimum Variance Combination with  =    1  2 Cov(r 1 r 2 ) =    1 

Minimum Variance Combination with  = E[r p ] = p2 =p2 = p2 =p2 = (.10) (.14) =.1157 = 11.57% W 1 2  W 2 2  W 1 W 2  1,2  1  2 1 Notice lower portfolio standard deviation but higher expected return with smaller   12 =.2 E(r p ) = 11.31%  p = 13.08% 6-22

Extending Concepts to All Securities Consider all possible combinations of securities, with all possible different weightings and keep track of combinations that provide more return for less risk or the least risk for a given level of return and graph the result. The set of portfolios that provide the optimal trade-offs are described as the efficient frontier. The efficient frontier portfolios are dominant or the best diversified possible combinations. All investors should want a portfolio on the efficient frontier. … Until we add the riskless asset 6-23

E(r) The minimum-variance frontier of risky assets Efficient Frontier is the best diversified set of investments with the highest returns 6-24Efficientfrontier Individualassets Minimumvariancefrontier St. Dev. Found by forming portfolios of securities with the lowest covariances at a given E(r) level.

E(r) The EF and asset allocation Efficientfrontier St. Dev. 20% Stocks 80% Bonds 100% Stocks EF including international & alternative investments Ex-Post % Stocks 20% Bonds 60% Stocks 40% Bonds 40% Stocks 60% Bonds 100% Stocks 6-25

6.3The Optimal Risky Portfolio With A Risk-Free Asset 6.4 Efficient Diversification With Many Risky Assets 6-26

Including Riskless Investments The optimal combination becomes linear A single combination of risky and riskless assets will dominate 6-27

E(r)  The Capital Market Line or CML P E(r P&F ) F Risk Free  P&F Efficient Frontier E(r P ) PP  P&F E(r P&F ) CAL (P) = CML oThe optimal CAL is called the Capital Market Line or CML oThe CML dominates the EF 6-28

Dominant CAL with a Risk-Free Investment (F) CAL(P) = Capital Market Line or CML dominates other lines because it has the the largest slope Slope = (E(r p ) - rf) /  p (CML maximizes the slope or the return per unit of risk or it equivalently maximizes the Sharpe ratio) Regardless of risk preferences some combinations of P & F dominate 6-29

6.5 A Single Index Asset Market 6-30

Individual securities We have learned that investors should diversify. Individual securities will be held in a portfolio. What do we call the risk that cannot be diversified away, i.e., the risk that remains when the stock is put into a portfolio? How do we measure a stock’s systematic risk? Systematic risk Consequently, the relevant risk of an individual security is the risk that remains when the security is placed in a portfolio. 6-31

Systematic risk Systematic risk arises from events that effect the entire economy such as a change in interest rates or GDP or a financial crisis such as occurred in 2007and If a well diversified portfolio has no unsystematic risk then any risk that remains must be systematic. That is, the variation in returns of a well diversified portfolio must be due to changes in systematic factors. 6-32

Single Index Model Parameter Estimation Risk Prem Market Risk Prem or Index Risk Prem = the stock’s expected excess return if the market’s excess return is zero, i.e., (r m - r f ) = 0 ß i (r m - r f ) = the component of excess return due to movements in the market index e i = firm specific component of excess return that is not due to market movements αiαi 6-33

Let: R i = (r i - r f ) R m = (r m - r f ) Risk premium format R i =  i + ß i (R m ) + e i Risk Premium Format The Model: 6-34

Estimating the Index Model Excess Returns (i) Security Characteristic Line Excess returns on market index R i =  i + ß i R m + e i Slope of SCL = beta y-intercept = alpha Scatter Plot 6-35

Components of Risk Market or systematic risk: Unsystematic or firm specific risk: Total risk = Systematic + Unsystematic risk related to the systematic or macro economic factor in this case the market index risk not related to the macro factor or market index ß i M + e i  i 2 = Systematic risk + Unsystematic Risk 6-36

Comparing Security Characteristic Lines Describe    e for each. 6-37

Measuring Components of Risk  i 2 = where;  i 2  m 2 +  2 (e i )  i 2 = total variance  i 2  m 2 = systematic variance  2 (e i ) = unsystematic variance 6-38

Total Risk = Systematic Risk / Total Risk = Examining Percentage of Variance Systematic Risk + Unsystematic Risk ß i 2  m 2 /  i 2 =  2  i 2  m 2 / (  i 2  m 2 +  2 (e i )) =  2 22 6-39

Sharpe Ratios and alphas – When ranking portfolios and security performance we must consider both return & risk “Well performing” diversified portfolios provide high Sharpe ratios: Sharpe = (r p – r f ) /  p You can also use the Sharpe ratio to evaluate an individual stock if the investor does not diversify 6-40

Sharpe Ratios and alphas “Well performing” individual stocks held in diversified portfolios can be evaluated by the stock’s alpha in relation to the stock’s unsystematic risk. 6-41