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**Optimal Risky Portfolios**

CHAPTER 7

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**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 way returns two assets vary

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**Two-Security Portfolio: Return**

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**Two-Security Portfolio: Risk**

= Variance of Security D = Variance of Security E = Covariance of returns for Security D and Security E

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**Two-Security Portfolio: Risk Continued**

Another way to express variance of the portfolio:

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Covariance Cov(rD,rE) = DEDE D,E = Correlation coefficient of returns D = Standard deviation of returns for Security D E = Standard deviation of returns for Security E

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**Correlation Coefficients: Possible Values**

Range of values for 1,2 > 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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**Table 7.1 Descriptive Statistics for Two Mutual Funds**

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**Three-Security Portfolio**

2p = w1212 + w2212 + w3232 + 2w1w2 Cov(r1,r2) Cov(r1,r3) + 2w1w3 + 2w2w3 Cov(r2,r3)

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**Table 7.2 Computation of Portfolio Variance From the Covariance Matrix**

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**Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients**

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**Minimum Variance Portfolio as Depicted in Figure 7.4**

Standard deviation is smaller than that of either of the individual component assets Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk

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**Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation**

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Correlation Effects The relationship depends on the correlation coefficient -1.0 < < +1.0 The smaller the correlation, the greater the risk reduction potential If r = +1.0, no risk reduction is possible

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**Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs**

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The Sharpe Ratio Maximize the slope of the CAL for any possible portfolio, p The objective function is the slope:

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Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio

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**Figure 7.8 Determination of the Optimal Overall Portfolio**

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**Figure 7.9 The Proportions of the Optimal Overall Portfolio**

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**Markowitz Portfolio Selection Model**

Security Selection First step is to determine the risk-return opportunities available All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations

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**Figure 7.10 The Minimum-Variance Frontier of Risky Assets**

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**Markowitz Portfolio Selection Model Continued**

We now search for the CAL with the highest reward-to-variability ratio

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**Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL**

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**Markowitz Portfolio Selection Model Continued**

Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

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**Figure 7.12 The Efficient Portfolio Set**

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**Capital Allocation and the Separation Property**

The separation property tells us that the portfolio choice problem may be separated into two independent tasks Determination of the optimal risky portfolio is purely technical Allocation of the complete portfolio to T-bills versus the risky portfolio depends on personal preference

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**Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set**

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**Diversification and Portfolio Risk**

Market risk Systematic or nondiversifiable Firm-specific risk Diversifiable or nonsystematic

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**The Power of Diversification**

Remember: If we define the average variance and average covariance of the securities as: We can then express portfolio variance as:

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**Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes**

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**Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio**

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