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**Efficient Diversification**

Ch 6 Efficient Diversification

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**Diversification and Portfolio Risk**

Total risk: Market risk Systematic or Nondiversifiable Firm-specific risk Diversifiable or nonsystematic or unique

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**Figure 6.1 Portfolio Risk as a Function of the Number of Stocks**

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**Figure 6.2 Portfolio Risk as a Function of Number of Securities**

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Exercise 42 1. Risk that can be eliminated through diversification is called ______ risk. A) unique B) firm-specific C) diversifiable D) all of the above 2. The risk that can be diversified away is ___________. A) beta B) firm specific risk C) market risk D) systematic risk

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**Two Asset Portfolio Return – Stock and Bond**

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**Covariance Cov(r1r2) = r1,2s1s2 r1,2 = Correlation coefficient of**

returns s1 = Standard deviation of returns for Security 1 s2 = Standard deviation of returns for Security 2

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**Correlation Coefficients: Possible Values**

Range of values for r 1,2 -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

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**Two Asset Portfolio St Dev – Stock and Bond**

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**In General, For an n-Security Portfolio:**

rp = Weighted average of the n securities sp2 = (Consider all pair-wise covariance measures)

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**Numerical Example: Bond and Stock**

Returns Bond = 6% Stock = 10% Standard Deviation Bond = 12% Stock = 25% Weights Bond = .5 Stock = .5 Correlation Coefficient (Bonds and Stock) = 0

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**Return and Risk for Example**

.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

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**Figure 6.3 Investment Opportunity Set for Stock and Bonds**

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**Minimum variance portfolio**

Ws = σB2 - Cov(rS, rB) / (σs2 + σB2 -2Cov(rS, rB))

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**Figure 6.4 Investment Opportunity Set for Stock and Bonds with Various Correlations**

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**Extending to Include Riskless Asset**

The optimal combination becomes linear A single combination of risky and riskless assets will dominate

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**Figure 6.5 Opportunity Set Using Stock and Bonds and Two Capital Allocation Lines**

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**Dominant CAL with a Risk-Free Investment (F)**

CAL(O) dominates other lines -- it has the best risk/return or the largest slope Slope = (E(R) - Rf) / s [ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA] Regardless of risk preferences combinations of O & F dominate

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**Figure 6.6 Optimal Capital Allocation Line for Bonds, Stocks and T-Bills**

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**Figure 6.7 The Complete Portfolio**

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**Figure 6.8 The Complete Portfolio – Solution to the Asset Allocation Problem**

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**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

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**Figure 6.9 Portfolios Constructed from Three Stocks A, B and C**

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**Figure 6.10 The Efficient Frontier of Risky Assets and Individual Assets**

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Exercise 22 1. Adding additional risky assets will generally move the efficient frontier _____ and to the _______. A) up, right B) up, left C) down, right D) down, left 2. Rational risk-averse investors will always prefer portfolios ______________. A) located on the efficient frontier to those located on the capital market line B) located on the capital market line to those located on the efficient frontier C) at or near the minimum variance point on the efficient frontier D) Rational risk-averse investors prefer the risk-free asset to all other asset choices.

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Exercise33 1. The standard deviation of return on investment A is .10 while the standard deviation of return on investment B is If the covariance of returns on A and B is .0030, the correlation coefficient between the returns on A and B is __________. A) .12 B) .36 C) .60 D) .77 2. Consider two perfectly negatively correlated risky securities, A and B. Security A has an expected rate of return of 16% and a standard deviation of return of 20%. B has an expected rate of return 10% and a standard deviation of return of 30%. The weight of security B in the global minimum variance is __________. A) 10% B) 20% C) 40% D) 60%

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Exercise32 1. Which of the following correlations coefficients will produce the least diversification benefit? A) -0.6 B) -1.5 C) 0.0 D) 0.8 2. The expected return of portfolio is 8.9% and the risk free rate is 3.5%. If the portfolio standard deviation is 12.0%, what is the reward to variability ratio of the portfolio? A) 0.0 B) 0.45 C) 0.74 D) 1.35

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**Single Factor Model ri = E(Ri) + ßiF + e**

ßi = index of a securities’ particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor

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**( ) ( ) b a r r r r e Single Index Model - = + - + Risk Prem**

f i m f i i Risk Prem Market Risk Prem or Index Risk Prem a = the stock’s expected return if the market’s excess return is zero i (rm - rf) = 0 ßi(rm - rf) = the component of return due to movements in the market index ei = firm specific component, not due to market movements

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**Risk Premium Format Let: Ri = (ri - rf) Risk premium format**

Rm = (rm - rf) Risk premium format Ri = ai + ßi(Rm) + ei

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**Figure 6.11 Scatter Diagram for Dell**

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**Figure 6.12 Various Scatter Diagrams**

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Components of Risk Market or systematic risk: risk related to the macro economic factor or market index Unsystematic or firm specific risk: risk not related to the macro factor or market index Total risk = Systematic + Unsystematic

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**Measuring Components of Risk**

si2 = bi2 sm2 + s2(ei) where; si2 = total variance bi2 sm2 = systematic variance s2(ei) = unsystematic variance

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**Examining Percentage of Variance**

Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk = r2 ßi2 s m2 / s2 = r2 bi2 sm2 / (bi2 sm2 + s2(ei)) = r2

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