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Efficient Diversification
Bodie, Kane and MarcusEssentials of Investments 9th Global Edition
6
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6.1 DIVERSIFICATION AND PORTFOLIO RISK Market/Systematic/Non diversifiable Risk
Risk factors common to whole economy Unique/Firm-Specific/Nonsystematic/ Diversifiable
Risk Risk that can be eliminated by diversification
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FIGURE 6.1 RISK AS FUNCTION OF NUMBER OF STOCKS IN PORTFOLIO
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FIGURE 6.2 RISK VERSUS DIVERSIFICATION
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6.2 ASSET ALLOCATION WITH TWO RISKY ASSETS
Covariance and CorrelationPortfolio risk depends on covariance between returns
of assetsExpected return on two-security portfolio
2211)( rWrWrE p
2security on return Expected1security on return Expected
2security in funds of Proportion1security in funds of Proportion
2
1
2
1
rrWW
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6.2 ASSET ALLOCATION WITH TWO RISKY ASSETS
Covariance Calculations
Correlation Coefficient
])()()][()()[(),Cov(1
BS
S
iBBSS rEirrEiriprr
BS
BSSB
rrσσ
),Cov(ρ
BSSBBS rr σσρ),Cov(
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SPREADSHEET 6.1 CAPITAL MARKET EXPECTATIONS
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SPREADSHEET 6.2 VARIANCE OF RETURNS
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SPREADSHEET 6.3 PORTFOLIO PERFORMANCE
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SPREADSHEET 6.4 RETURN COVARIANCE
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6.2 ASSET ALLOCATION WITH TWO RISKY ASSETS
Using Historical Data Variability/covariability change slowly over time Use realized returns to estimate
Cannot estimate averages precisely Focus for risk on deviations of returns from average
value
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6.2 ASSET ALLOCATION WITH TWO RISKY ASSETS
Three RulesRoR: Weighted average of returns on
components, with investment proportions as weights
ERR: Weighted average of expected returns on components, with portfolio proportions as weights
Variance of RoR:
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6.2 ASSET ALLOCATION WITH TWO RISKY ASSETS Risk-Return Trade-Off
Investment opportunity set Available portfolio risk-return combinations
Mean-Variance Criterion If E(rA) ≥ E(rB) and σA ≤ σB
Portfolio A dominates portfolio B
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SPREADSHEET 6.5 INVESTMENT OPPORTUNITY SET
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FIGURE 6.3 INVESTMENT OPPORTUNITY SET
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FIGURE 6.4 OPPORTUNITY SETS: VARIOUS CORRELATION COEFFICIENTS
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6.3 THE OPTIMAL RISKY PORTFOLIO WITH A RISK-FREE ASSET
Slope of CAL is Sharpe Ratio of Risky Portfolio
Optimal Risky Portfolio Best combination of risky and safe
assets to form portfolio
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FIGURE 6.5 TWO CAPITAL ALLOCATION LINES
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FIGURE 6.6 BOND, STOCK AND T-BILL OPTIMAL ALLOCATION
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6.3 THE OPTIMAL RISKY PORTFOLIO WITH A RISK-FREE ASSET
Calculating Optimal Risky Portfolio Two risky assets
BSSBfsfBBfsSfB
BSSBfsSfBB rrErrErrErrE
rrErrEw
])()([])([])([])([])([
22
2
BS ww 1
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FIGURE 6.7 THE COMPLETE PORTFOLIO
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FIGURE 6.8 PORTFOLIO COMPOSITION: ASSET ALLOCATION SOLUTION
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6.4 EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS
Efficient Frontier of Risky Assets Graph representing set of portfolios that maximizes
expected return at each level of portfolio risk Three methods
Maximize risk premium for any level standard deviation Minimize standard deviation for any level risk premium Maximize Sharpe ratio for any standard deviation or risk
premium
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FIGURE 6.9 PORTFOLIOS CONSTRUCTED WITH THREE STOCKS
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FIGURE 6.10 EFFICIENT FRONTIER: RISKY AND INDIVIDUAL ASSETS
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6.4 EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS
Choosing Optimal Risky Portfolio Optimal portfolio CAL tangent to efficient frontier
Preferred Complete Portfolio and Separation Property Separation property: implies portfolio choice, separated
into two tasks Determination of optimal risky portfolio Personal choice of best mix of risky portfolio and risk-free
asset
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6.4 EFFICIENT DIVERSIFICATION WITH MANY RISKY ASSETS
Optimal Risky Portfolio: Illustration Efficiently diversified global portfolio using stock
market indices of six countries Standard deviation and correlation estimated from
historical data Risk premium forecast generated from fundamental
analysis
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FIGURE 6.11 EFFICIENT FRONTIERS/CAL: TABLE 6.1
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6.5 A SINGLE-INDEX STOCK MARKET Index model
Relates stock returns to returns on broad market index/firm-specific factors
Excess return RoR in excess of risk-free rate
Beta Sensitivity of security’s returns to market factor
Firm-specific or residual risk Component of return variance independent of market factor
Alpha Stock’s expected return beyond that induced by market index
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6.5 A SINGLE-INDEX STOCK MARKET Excess Return
riskspecific -firm zero is return excessindex -market i.e. neutral, is
factor market if return excess expected sstock' market to nessresponsive ssecurity'
market overall in movements from return
:
::
:
i
i
i
Mi
iiMii
e
ReRR
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FIGURE 6.12 SCATTER DIAGRAM FOR DELL
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6.5 A SINGLE-INDEX STOCK MARKET Statistical and Graphical Representation of Single-
Index Model Security Characteristic Line (SCL)
Plot of security’s predicted excess return from excess return of market
Algebraic representation of regression line
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6.5 A SINGLE-INDEX STOCK MARKET Statistical and Graphical Representation of Single-
Index Model Ratio of systematic variance to total variance
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6.5 A SINGLE-INDEX STOCK MARKET Diversification in Single-Index Security Market
In portfolio of n securities with weights In securities with nonsystematic risk Nonsystematic portion of portfolio return
Portfolio nonsystematic variance
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6.5 A SINGLE-INDEX STOCK MARKET Using Security Analysis with Index Model
Information ratio Ratio of alpha to standard deviation of residual
Active portfolio Portfolio formed by optimally combining analyzed stocks
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5) The standard deviation of the market-index portfolio is 15%. Stock A has a beta of 2.2 and a residual standard deviation of 25%.A. What would make for a larger increase in the stock’s
variance: an increase of .2 in its beta or an increase of 3.84% (from 30% to 33%) in its residual standard deviation?
B. An investor who currently holds the market-index portfolio decides to reduce the portfolio allocation to the market index to 90% and to invest 10% in stock A. Which of the changes in (a) will have a greater impact on the portfolio’s standard deviation?
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20) Investors expect the market rate of return this year to be 10.5%. The expected rate of return on a stock with a beta of 1.3 is currently 13.65%. If the market return this year turns out to be 9%, how would you revise your expectation of the rate of return on the stock?
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21. The following figure shows plots of monthly rates of return and the stock market for two stocks.A. Which stock is riskier to an investor currently
holding her portfolio in a diversified portfolio of common stock?
B. Which stock is riskier to an undiversified investor who puts all of his funds in only one of these stocks?
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