efficient diversification chapter 6. diversification and portfolio risk market risk –systematic or...
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Efficient Efficient DiversificationDiversificationCHAPTER 6CHAPTER 6CHAPTER 6CHAPTER 6
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Diversification and Portfolio Risk
• Market risk– Systematic or Nondiversifiable
• Firm-specific risk– Diversifiable or nonsystematic
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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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Two Asset Portfolio Return – Stock and Bond
ReturnStock
htStock Weig
Return Bond
WeightBond
Return Portfolio
rwrwr
S
S
B
B
p
rwrwr SSBBp
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Covariance
1,2 = Correlation coefficient of returns
1,2 = Correlation coefficient of returns
Cov(r1r2) = 12Cov(r1r2) = 12
1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2
1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2
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Correlation Coefficients: Possible Values
If If = 1.0, the securities would be = 1.0, the securities would be perfectly positively correlatedperfectly positively correlated
If If = - 1.0, the securities would be = - 1.0, the securities would be perfectly negatively correlatedperfectly negatively correlated
Range of values for 1,2
-1.0 < < 1.0
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Two Asset Portfolio St Dev – Stock and Bond
Deviation Standard Portfolio
Variance Portfolio
2
2
,
22222 2
p
p
SBBSSBSSBBp wwww
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rp = Weighted average of the n securitiesrp = Weighted average of the n securities
p2 = (Consider all pair-wise
covariance measures)p
2 = (Consider all pair-wise covariance measures)
In General, For an n-Security Portfolio:
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Numerical Example: Bond and Stock
ReturnsBond = 6% Stock = 10%
Standard Deviation Bond = 12% Stock = 25%
WeightsBond = .5 Stock = .5
Correlation Coefficient (Bonds and Stock) = 0
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Return and Risk for Example
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
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Figure 6.3 Investment Opportunity Set for Stock and Bonds
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Figure 6.4 Investment Opportunity Set for Stock and Bonds with
Various Correlations
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Figure 6.3 Investment Opportunity Set for Bond and Stock Funds
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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) / E(RP) - Rf) / PE(RA) - Rf) /
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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Single Factor Modelri = 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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Single Index Model
Risk PremRisk Prem Market Risk PremMarket Risk Prem or Index Risk Premor Index Risk Prem
ii= the stock’s expected return if the= the stock’s expected return if the market’s excess return is zeromarket’s excess return is zero
ßßii(r(rmm - r - rff)) = the component of return due to= the component of return due to
movements in the market indexmovements in the market index
(r(rmm - r - rff)) = 0 = 0
eei i = firm specific component, not due to market= firm specific component, not due to market
movementsmovements
errrr ifmiifi
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Let: RLet: Ri i = (r= (rii - r - rff))
RRm m = (r= (rmm - r - rff))Risk premiumRisk premiumformatformat
RRi i = = ii + ß + ßii(R(Rmm)) + e+ eii
Risk Premium Format
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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
i2 = i
2 m2 + 2(ei)
where;
i2 = total variance
i2 m
2 = systematic variance
2(ei) = unsystematic variance
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Total Risk = Systematic Risk + Unsystematic Risk
Systematic Risk/Total Risk = 2
ßi2
m2 / 2 = 2
i2 m
2 / i2 m
2 + 2(ei) = 2
Examining Percentage of Variance
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Advantages of the Single Index Model
• Reduces the number of inputs for diversification
• Easier for security analysts to specialize