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

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Presentation on theme: "Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-1 Chapter 6."— Presentation transcript:

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

2 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

3 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

4 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

5 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.

6 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

7 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

8 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

9 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

10 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

11 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% 

12 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

13 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%

14 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

15 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%

16 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%

17 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

18 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

19 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

20 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

21 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

22 Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 6-22 Range of values for  1,2 + 1.0 >   >  -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

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

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

25 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  ?

26 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

27 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

28 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?

29 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:

30 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:

31 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 = 0.6087 w B = 0.3913  While the corresponding minimum- variance portfolio’s characteristics are: r P = 11.57% and s P = 10.09%

32 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

33 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

34 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 

35 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

36 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 

37 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

38 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

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