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Chapter 6

Chapter 6. Portfolio Selection. 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. Allocating Capital Between Risky & Risk Free Assets.

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Chapter 6

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  1. Chapter 6 Portfolio Selection

  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. Allocating Capital Between Risky & Risk Free Assets • Possible to split investment funds between safe and risky assets • Risk free asset: proxy; T-bills • Risky asset: stock (or a portfolio)

  4. Allocating Capital Between Risky & Risk Free Assets • Examine risk/return tradeoff • Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

  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. Portfolios of One Risky Asset and One Risk-Free Asset • Assume a risky portfolio P defined by : E(rp) = 15% and p = 22% • The available risk-free asset has: • rf = 7% and rf = 0% • And the proportions invested: y% in P and (1-y)% in rf

  7. E(rc) = yE(rp) + (1 - y)rf rc = complete or combined portfolio If, for example, y = .75 E(rc) = .75(.15) + .25(.07) = .13 or 13% Expected Returns for Combinations

  8. = 0, then Since rf *   = y c p * Rule 4 in Chapter 5 Variance on the Possible Combined Portfolios

  9. E(r) E(rp) = 15% P E(rc) = 13% C rf = 7% F  c 0 22% Possible Combinations

  10. If y = .75, then = .75(.22) = .165 or 16.5%  c If y = 1 = 1(.22) = .22 or 22%  c If y = 0 =0(.22) = .00 or 0%  c Combinations Without Leverage

  11. E(r) P E(rp) = 15% E(rp) - rf = 8% ) S = 8/22 rf = 7% F  0 p= 22% CAL (Capital Allocation Line)

  12. Using Leverage with Capital Allocation Line • Borrow at the Risk-Free Rate and invest in stock • Using 50% Leverage rc = (-.5) (.07) + (1.5) (.15) = .19 c = (1.5) (.22) = .33

  13. A = 4 E(r) A = 2 rf=7%  p = 22% Indifference Curves and Risk Aversion Certainty equivalent of portfolio P’s expected return for two different investors P E(rp)=15%

  14. 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

  15. E(r) P Borrower 7% Lender  p = 22% CAL with Risk Preferences

  16. E(r) P ) S = .27 9% 7% ) S = .36  p = 22% CAL with Higher Borrowing Rate

  17. St. Deviation Unique Risk Market Risk Number of Securities Risk Reduction with Diversification

  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. Two-Security Portfolio: Return w1 = proportion of funds in Security 1 w2 = proportion of funds in Security 2 r1 = expected return on Security 1 r2 = expected return on Security 2

  20. Two-Security Portfolio: Risk 12 = variance of Security 1 22 = variance of Security 2 Cov(r1,r2) = covariance of returns for Security 1 and Security 2

  21. Covariance 1,2 = Correlation coefficient of returns 1 = Standard deviation of returns for Security 1 2 = Standard deviation of returns for Security 2

  22. Correlation Coefficients: Possible Values 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

  23. Three-Security Portfolio

  24. Generally, for an n-Security Portfolio:

  25. Returning to the Two-Security Portfolio and , or Question: What happens if we use various securities’ combinations, i.e. if we vary r?

  26. E(r) 13% r = -1 r = .3 r = 1 %8 r = -1 St. Dev 12% 20% Two-Security Portfolios with Different Correlations

  27. Portfolio of 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

  28. Minimum-Variance Combination • Suppose our investment universe comprises the following two securities: • What are the weights of each security in the minimum-variance portfolio?

  29. Minimum-Variance Combination:  = .2 • Solving the minimization problem we get: • Numerically:

  30. Minimum -Variance: Return and Risk with  = .2 • Using the weights wA and wB we determine minimum-variance portfolio’s characteristics:

  31. Minimum -Variance Combination:  = -.3 • Using the same mathematics we obtain: wA = 0.6087 wB = 0.3913 • While the corresponding minimum-variance portfolio’s characteristics are: rP = 11.57% and sP = 10.09%

  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. 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

  34. E(r) Efficient frontier Global minimum variance portfolio Individual assets Minimum variance frontier s The Minimum-Variance Frontier of Risky Assets

  35. Extending to Include A Riskless Asset • 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

  36. E(r) CAL (P) CAL (A) M M P P A CAL (Global minimum variance) A G F  P P&F M Alternative CALs

  37. E(r) U’’’ U’’ U’ Efficient frontier of risky assets S P Less risk-averse investor Q More risk-averse investor s Portfolio Selection & Risk Aversion

  38. CAL E(r) B Q P A rf F s Efficient Frontier with Lending & Borrowing

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