Efficient Diversification

1
Chapter 6

• Efficient Diversification

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Sources of Uncertainty

• General Economic Conditions Business cycle,
  inflation, interest rates are macro-economic
  conditions that are unknown.
• Firm Specific Conditions RD success,
  Management, sector, etc.

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Benefits of Diversification

• Investing in several stocks reduces portfolio
  risk by reducing exposure to firm specific risks.
  Ex. Invest in Enron and IBM. Management
  decisions independent of each other.
• Example Invest in Crystal Farms and Hershey.

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Diversification
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Diversification cont.

• Expected return from investing in Crystal Farm
  .5(15) .5(-5) 5
• Expected return from investing in Hershey
  .5(-5) .5(15) 5
• VAR of Crystal Farms returns 100
• VAR of Hersheys returns 100
• Why invest in both companies?

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Two-Security Portfolio Return
rp W1r1 W2r2 W1 Proportion of funds in
Security 1 W2 Proportion of funds in Security
2 r1 Expected return on Security 1 r2
Expected return on Security 2
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Two-Security Portfolio Risk
sp2 w12s12 w22s22 2W1W2 Cov(r1r2)
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Combined Portfolio

• Invest ½ our money in Crystal and ½ in Hersheys
• Expected Return of combined portfolio ERcomb
  .5 ERc .5 ERH 5

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Covariance?

• The covariance is a measure of the linear
  association between two variables.
• A positive covariance indicates a positive
  relation
• A negative covariance indicates a negative
  relation

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Covariance Calculation

• VAR(Rcomb) .52 (100) .52100
  2.5.5(-100) 0
• ERcomb 5
• VARRcomb 0
• By diversifying we were able to earn the same
  expected return and reduce risk to zero.

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Correlation Coefficient

• Allows us to interpret the strength of a linear
  relationship between two variables.
• Related to the covariance
• Ranges between -1 and 1

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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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Example

• Rho Cov(Rc, Rh)/(std(Rc) std(Rh))
• Rho -100/(10)(10) -1
• Returns on crystal farm and hersheys are
  perfectly negatively correlated, which allows us
  to reduce the firm specific risk to zero.
• 20 stocks equals a diversified portfolio.

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Risk-Return Trade-Off with Two-Security Portfolio
E(rp) W1r1 W2r2
sp2 w12s12 w22s22 2W1W2 Cov(r1r2)
sp w12s12 w22s22 2W1W2 Cov(r1r2)1/2
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Portfolio Risk/Return Two Securities Correlation
Effects

• Relationship depends on correlation coefficient
• -1.0 lt r lt 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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Investment Opportunity Set

• Combinations of risk/return offered by different
  portfolios.
• Excel file ch6.2.xls displays this set for 2
  risky assets.
• Mean-Variance Criterion Portfolio A dominates
  portfolio B if E(rA)gtE(rB) and sA lt sB

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Minimum Variance Combination
1
s 2
- Cov(r1r2)
2

W1
s 2
s 2
- 2Cov(r1r2)

2
1
(1 - W1)
W2
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TWO-SECURITY PORTFOLIOS WITH DIFFERENT
CORRELATIONS
r 0
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Minimum Variance Combination r .2
(.2)2 - (.2)(.15)(.2)

W1
(.15)2 (.2)2 - 2(.2)(.15)(.2)
W1
.6733
W2
(1 - .6733) .3267
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Minimum Variance Return and Risk with r .2
rp .6733(.10) .3267(.14) .1131
s
(.6733)2(.15)2 (.3267)2(.2)2
p
1/2
2(.6733)(.3267)(.2)(.15)(.2)
1/2
.0171
.1308
s
p
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Minimum Variance Combination r -.3
(.2)2 - (.2)(.15)(.2)

W1
(.15)2 (.2)2 - 2(.2)(.15)(-.3)
W1
.6087
W2
(1 - .6087) .3913
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Minimum Variance Return and Risk with r -.3
rp .6087(.10) .3913(.14) .1157
s
(.6087)2(.15)2 (.3913)2(.2)2
p
1/2
2(.6087)(.3913)(.2)(.15)(-.3)
1/2
.0102
.1009
s
p
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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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ALTERNATIVE CALS
CAL (P)
CAL (A)
E(r)
M
M
P
P
CAL (Global minimum variance)
A
A
G
F
s
P
PF
AF
M
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Dominant CAL with a Risk-Free Investment (F)

• CAL(P) dominates other lines -- it has the best
  risk/return or the largest slope
• Slope (E(R) - Rf) / s
• E(RP) - Rf) / s P gt E(RA) - Rf) / sA
• All investors will select the same risky
  portfolio because it offers the highest return
  per unit of risk.

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Investment Decision

• Return/Risk using optimal CAL
• If we want to earn 12 returns then we must
  invest 1.9 in the optimal portfolio and -.9 in
  T-Bills
• Std. Deviation 34.14

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Efficient Diversification

• Determine the most efficient investment
  opportunity set from the risky assets considered.
• Find the optimal portfolio of risky assets by
  finding the portfolio with the steepest CAL.
• Find the right mix of Rf and risky portfolio
  based on risk preferences of investor.

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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
• Calculated in Ch6.4.xls

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The minimum-variance frontier of risky assets
E(r)
Efficient frontier
Individual assets
Global minimum variance portfolio
Minimum variance frontier
St. Dev.
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Single Factor Model

• Ri E(Ri) ßiM e
• Ri ri - rf
• ßi sensitivity of a securities particular
  return to factor/macro events.
• M some macro factor in this case M is
  unanticipated movement M is expected to be zero.
  

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Single Index Model
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b
a
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r
r
r
r

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f
m
f
i
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i
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Risk Prem
Market Risk Prem
or Index Risk Prem
a
the stocks expected return if the markets
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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Estimating the Index Model
Excess Returns (i)
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Security Characteristic Line
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Excess returns on market index
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Ri a i ßiRm ei
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Regression Analysis

• Intercept Expected excess return.
• Beta Measures the responsiveness of security
  excess returns to market factors. Risk measure
• Beta gt 1 has above average risk aggressive
  investment.

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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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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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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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Advantages of the Single Index Model

• Reduces the number of inputs for diversification
• Easier for security analysts to specialize