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Chapter 7

Chapter 7. CHAPTER SEVEN Capital Allocation Between The Risky And The Risk-Free Asset. CHAPTER OVERVIEW

gregory-vaughn
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Chapter 7

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  1. Chapter 7 CHAPTER SEVEN Capital Allocation Between The Risky And The Risk-Free Asset

  2. CHAPTER OVERVIEW • This chapter develops the concept of allocating investment funds between risk-free and riskless assets. Combining risk free assets (U. S. Treasury bills or money market mutual funds) with risky mutual fund portfolios operationalizes the concepts of producing portfolios of acceptable risk/return trade-offs for various investors.

  3. LEARNING OBJECTIVES • The student should be able to construct portfolios of different risk levels, given information about risk free rates and returns on risky assets. The student should be able to calculate the expected return and standard deviation of these portfolios.

  4. 7.1 Capital allocation across risky and risk-free portfolios

  5. Allocating Capital: Risky & Risk Free Assets • It’s possible to split investment funds between safe and risky assets. • Risk free asset: T-bills;money market instruments(CD,CP) • Risky asset: stock (or a risky portfolio) • Asset allocation is the most important part of portfolio construction.

  6. Capital allocation across risky and risk-free portfolios • Example:assume that the total market value of an initial portfolio is $300,000,of which $90,000 is invested in T-bills,a risk-free asset for practice.The remaining $210,000 is invested in risky securities-$113,400 in Equity(E)and $96,600 in long-term bonds(B).The Equities and long-term bonds holdings comprise the risky portfolio,54% in E and 46% in B.

  7. Analyze example: complete portfolio • Equity(E):$113,400 T-bills:$90,000 • Bonds(B):$96,600 • $210,000 • E in risky-portfolio=113400/210000=54% • B in risky-portfolio=96000/210000=46% Risky portfolio Risk-free asset

  8. Risky assets y=210000/300000=70% • Risk-free assets 1-y=90000/300000=30% E in completeportfolio=113400/300000=0.378 B in complete portfolio=96600/300000=0.322 Risky-asset y=0.7

  9. If reduce the allocation to the risky portfolio from y=0.7 to y=0.56.the risky portfolio would then total only 300000×0.56=168000. • the proportions of each asset in risky-portfolio are unchanged: • E=168000×54%=90720 • B=168000×46%=77280

  10. Exercise • 150000×54%=81000, E in complete portfolio=81000/300000=27% 50%risk-free Equity(54%) 300000×50% =150000 Bond(46%)

  11. 7.2 Portfolios of one risky asset and one risk-free asset

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

  13. rf = 7% rf = 0% E(rp) = 15% p = 22% y = % in p (1-y) = % in rf Example Using Chapter 7.3 Numbers

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

  15. 由y份风险资产与1-y份无风险资产组成的整个资产组合，记为C，其收益率记为rc,有由y份风险资产与1-y份无风险资产组成的整个资产组合，记为C，其收益率记为rc,有 • 对资产组合的收益率取期望，有 • 任意资产组合的基本收益率是无风险资产收益率。资产组合期望获得一个风险溢价。

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

  17. Investment Opportunity Set with a Risk-Free Investment 投资机会集（investment opportunity set）， 即由不同投资于风险资产组合P的比例y值所产生的所有资产组合的可能期望 收益与标准方差配对的集合。其图形是由rf点引出，穿过P点的直线，该 条直线叫做资本配置线（capital allocation line,CAL）,它表示投资者的所有可行 的风险收益组合。它的斜率S，可称为报酬与波动性比率（reward-to-variability ratio）

  18. Since = 0, then rf = y c p Variance For Possible Combined Portfolios *   * Rule 4 in Chapter 6

  19. 风险资产与无风险资产组合的风险（Portfolio Risk with Risk-Free Asset） 规则4：当一个风险资产与一个无风险资产相组合时，资产组合的标准差等于风险资产的标准差乘以该资产组合投资于这部分资产上的比例。

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

  21. Capital Allocation Line with Leverage 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

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

  23. Borrowing rate 9%,lending rate 7%,if use Leverage to construct portfolio.then S0=0.36 SL=0.27 σp=22% E(rp) = 15%

  24. CAL with Higher Borrowing Rate E(r) S=(15%-9%)/22%=6/22=0.27 P ) S = .27 9% 7% ) S = .36 S=(15%-7%)/22%=8/22=0.36  p = 22%

  25. If you invest a proportion,y,in a risky fund with expected return E(rp)and standard deviation σp,and the remainder,1-y,in a risk-free asset with a sure rate rf,Then the • Portfolio’s expected return and standard deviation are: • E(rc)=rf+y[E(rp)-rf] • σc,=y σp,

  26. 7.3Risk Aversion and asset 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.

  27. Utility Function U = E ( r ) - .005 A s2 Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion s2 = variance of returns

  28. CAL with Risk Preferences E(r) The lender has a larger A when compared to the borrower P Borrower 7% Lender  p = 22%

  29. 校级策略（ passive strategy ）描述了这样一种资产组合决策，该决策不做任何或者间接的证券分析。 • We call the capital allocation line provided by 1-month T-bills and a broad index of commom stocks the capital market line(CML).资本市场线代表了生成投资机会集的一个消极策略。 • A passive strategy generates an investment opportunity set that is represented by the CML.

  30. Costs and Benefits of Passive Investing • Active strategy entails costs • Free-rider benefit • Involves investment in two passive portfolios • Short-term T-bills • Fund of common stocks that mimics a broad market index

  31. Chapter 8 Optimal Risky Portfolios

  32. CHAPTER OVERVIEW • In this chapter, the concept of portfolio formation moves beyond the risky and risk-free asset combinations of the previous chapter to include combinations of two risky assets and of many risky assets. The concept of risk reduction by combining securities with different return patterns is introduced. By combining securities with differing return patterns, efficient portfolios (maximum return for a given level of risk) may be created. Finally, the risky portfolio is expanded to include all risky assets (i. e., the market); the investor may invest in the market (or in an indexed mutual fund) combined with the appropriate investment in risk-free instruments to create the portfolio of the desired risk level.

  33. Learning Objectives • Students should be able to calculate standard deviation and return for two security portfolios and be able to find the minimum variance combinations of two securities. Upon completion of this chapter the student should have a full understanding of systematic and firm-specific risk, and of how one can reduce the amount of firm-specific risk in the portfolio by combining securities with differing patterns of returns. The student should be able to quantify this risk-reduction concept by being able to calculate and interpret covariance and correlation coefficients.

  34. 1 . DIVERSIFICATION AND PORTFOLIO RISK

  35. Diversification and Portfolio Risk • Market risk • Systematic or Nondiversifiable • Firm-specific risk • Diversifiable or nonsystematic

  36. Figure 6.1 Portfolio Risk as a Function of the Number of Stocks

  37. Figure 6.2 Portfolio Risk as a Function of Number of Securities

  38. 2.ASSET ALLOCATION WITH TWO RISKY ASSETS

  39. Covariance and Correlation • Portfolio risk depends on the correlation between the returns of the assets in the portfolio • Covariance and the correlation coefficient provide a measure of the returns on two assets to vary

  40. Two Asset Portfolio Return – Stock and Bond

  41. Covariance and Correlation Coefficient • Covariance: • Correlation Coefficient: 注：相关系数越低，分散化就越有效，资产组合风险就越低。

  42. Correlation Coefficients: Possible Values Range of values for r1,2 -1.0 <r < 1.0 If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated

  43. Two Asset Portfolio St Dev –Stock and Bond

  44. In General, For an n-Security Portfolio: rp = Weighted average of the n securities sp2 = (Consider all pair-wise covariance measures)

  45. Three Rules of Two-Risky-Asset Portfolios • Rate of return on the portfolio: • Expected rate of return on the portfolio:

  46. Three Rules of Two-Risky-Asset Portfolios • Variance of the rate of return on the portfolio:

  47. Numerical Text Example: Bond and StockReturns Returns Bond = 6% Stock = 10% Standard Deviation Bond = 12% Stock = 25% Weights Bond = .5 Stock = .5 Correlation Coefficient (Bonds and Stock) = 0

  48. Numerical Text Example: Bond and Stock Return = 8% .5(6) + .5 (10) Standard Deviation = 13.87% [(.5)2 (12)2 + (.5)2 (25)2 + … 2 (.5) (.5) (12) (25) (0)] ½ [192.25] ½ = 13.87

  49. Figure 6.3 Investment Opportunity Set for Stocks and Bonds

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