Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-1 Chapter 6

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-2 Chapter Summary  Objective:To present the basics of modern portfolio selection process Capital allocation decision Two-security portfolios and extensions The Markowitz portfolio selection model

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-3  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

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-4  Examine risk/return tradeoff  Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets Allocating Capital Between Risky & Risk Free Assets

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-5 The Risk-Free Asset  Perfectly price-indexed bond – the only risk free asset in real terms;  T-bills are commonly viewed as “the” risk-free asset;  Money market funds - the most accessible risk-free asset for most investors.

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-6 Portfolios of One Risky Asset and One Risk-Free Asset  Assume a risky portfolio P defined by : E(r p ) = 15% and  p = 22%  The available risk-free asset has: r f = 7% and  rf = 0%  And the proportions invested: y% in P and (1-y)% in r f

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-7 E(r c ) = yE(r p ) + (1 - y)r f r c = complete or combined portfolio If, for example, y =.75 E(r c ) =.75(.15) +.25(.07) =.13 or 13% Expected Returns for Combinations

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-8 rfrf p c = Since y = 0, then    * Rule 4 in Chapter 5 * Variance on the Possible Combined Portfolios

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-9 Possible Combinations E(r) E (r p ) = 15% r f = 7% 22% 0 P F  cc E(r c ) = 13% C

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-10 c =.75(.22) =.165 or 16.5% If y =.75, then c = 1(.22) =.22 or 22% If y = 1 c =0(.22) =.00 or 0% If y = 0    Combinations Without Leverage

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-11 CAL (Capital Allocation Line) E(r) E(r p ) = 15% r f = 7%  p = 22% 0 P F ) S = 8/22 E(r p ) - r f = 8% 

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-12  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 Using Leverage with Capital Allocation Line

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-13 Indifference Curves and Risk Aversion Certainty equivalent of portfolio P’s expected return for two different investors P E(r)  r f =7% A = 4 A = 2  p = 22% E(r p )=15%

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-14  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 Risk Aversion and Allocation

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-15 CAL with Risk Preferences P E(r)  7% Lender Borrower  p = 22%

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-16 CAL with Higher Borrowing Rate E(r)  9% 7% ) S =.36 ) S =.27 P  p = 22%

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-17 Risk Reduction with Diversification Number of Securities St. Deviation Market Risk Unique Risk

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-18 Summary Reminder  Objective:To present the basics of modern portfolio selection process Capital allocation decision Two-security portfolios and extensions The Markowitz portfolio selection model

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-19 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

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-20  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 Two-Security Portfolio: Risk

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-21  1,2 = Correlation coefficient of returns  1 = Standard deviation of returns for Security 1  2 = Standard deviation of returns for Security 2 Covariance

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-22 Range of values for  1, >   >  -1.0 If   = 1.0, the securities would be perfectly positively correlated If  = - 1.0, the securities would be perfectly negatively correlated Correlation Coefficients: Possible Values

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-23 Three-Security Portfolio

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-24 Generally, for an n-Security Portfolio:

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-25 Returning to the Two-Security Portfolio and, or Question: What happens if we use various securities’ combinations, i.e. if we vary  ?

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-26 Two-Security Portfolios with Different Correlations  = 1 13% %8 E(r) St. Dev 12% 20%  =.3  = -1

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-27  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 Portfolio of Two Securities: Correlation Effects

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-28 Minimum-Variance Combination  Suppose our investment universe comprises the following two securities: AB  A,B E(r)10%14% 0.2  15%20%  What are the weights of each security in the minimum-variance portfolio?

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-29 Minimum-Variance Combination:  =.2  Solving the minimization problem we get:  Numerically:

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-30 Minimum -Variance: Return and Risk with  =.2  Using the weights w A and w B we determine minimum-variance portfolio’s characteristics:

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-31 Minimum -Variance Combination:  = -.3  Using the same mathematics we obtain: w A = w B =  While the corresponding minimum- variance portfolio’s characteristics are: r P = 11.57% and s P = 10.09%

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-32 Summary Reminder  Objective:To present the basics of modern portfolio selection process Capital allocation decision Two-security portfolios and extensions The Markowitz portfolio selection model

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-33  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 Extending Concepts to All Securities

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-34 The Minimum-Variance Frontier of Risky Assets E(r) Efficient frontier Global minimum variance portfolio Minimum variance frontier Individual assets 

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-35  The set of opportunities again described by the CAL  The choice of the optimal portfolio depends on the client’s risk aversion  A single combination of risky and riskless assets will dominate Extending to Include A Riskless Asset

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-36 Alternative CALs M E(r) CAL (Global minimum variance) CAL (A) CAL (P) P A F PP&F M A G P M 

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-37 Portfolio Selection & Risk Aversion E(r) Efficient frontier of risky assets More risk-averse investor U’’’U’’U’ Q P S  Less risk-averse investor

Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-38 Efficient Frontier with Lending & Borrowing F P E(r) rfrf A Q B CAL s