OPTIMAL RISKY PORTFOLIOS

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CHAPTER 6

• OPTIMAL RISKY PORTFOLIOS

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• Top-down process
• Capital allocation between risky portfolio and
  risk-free assets
• Asset allocation across broad asset classes
  (bond/ stock)
• Security selection of individual assets within
  each asset class
• Risky portfolio (E/D )
• Risky portfolio and risk-free asset (E/D/Rf)

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6.1 Diversification and Portfolio risk
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Diversification and Portfolio Risk

• Consider a portfolio composed of only one stock,
  what would be the sources of risk? What if add
  one stock?
• Insurance principle
• Market risk
• Risk that remains even after extensive
  diversification, attributable to market wide risk
  sources
• Systematic or non-diversifiable
• Firm-specific risk
• Risk that can be eliminated by diversification
• Diversifiable or nonsystematic

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Figure 7.1 Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
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Figure 7.2 Portfolio Diversification
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6.2 Portfolio of Two Risk Assets
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Portfolios of Two Risky Assets

• Study the efficient diversification
• Construct risky portfolios to provide the lowest
  possible risk for any given level of expected
  return
• Consider a portfolio comprised of two mutual
  funds, a bond portfolio D, and a stock fund E

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Portfolios of Two Risky Assets
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Portfolios of Two Risky Assets

• Rate of return on the portfolio
• Expected rate of return on the portfolio
• Variance of the rate of return on the portfolio

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Two-Security Portfolio Return

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Two-Security Portfolio Risk
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Portfolios of Two Risky Assets

• Covariance
• Another way to express variance of the portfolio

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Table 7.2 Computation of Portfolio Variance From
the Covariance Matrix
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Portfolios of Two Risky Assets

• Variance is reduced
• Correlation coefficient
• Portfolio variance

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Portfolios of Two Risky Assets
Range of values for r D,E -1.0 lt r lt 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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Portfolios of Two Risky Assets

• When Perfect positive correlation
• Portfolio variance
• Portfolio SD is no bigger than the weighted
  average of the individual security SD

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Portfolios of Two Risky Assets

• When perfect positive correlation
• When perfect negative correlation

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Portfolios of Two Risky Assets

• Hedge asset negative correlation with the other
  assets in the portfolio
• Expected return is unaffected by correlation,
  standard deviation is less than the weighted
  average of the component standard deviation
• portfolio of less than perfectly correlated
  assets always offer better risk-return
  opportunities than the individual component
  securities on their own
• The lower the correlation, the greater the gain
  in efficiency.

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Portfolios of Two Risky Assets

• Perfect hedged position
• When
• To solve

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Portfolios of Two Risky Assets

• Experiment with different proportions

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Table 7.3 Expected Return and Standard Deviation
with Various Correlation Coefficients
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Figure 7.3 Portfolio Expected Return as a
Function of Investment Proportions
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Figure 7.4 Portfolio Standard Deviation as a
Function of Investment Proportions
Minimum variance portfolio
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Portfolios of Two Risky Assets

• The Minimum-Variance Portfolio
• In the Example ,what is the minimum
  level
• Diversification effect Minimum-variance
  portfolio has a standard deviation smaller than
  that of either of the individual component assets
• Pass through the two undiversified portfolios of
  WD1,WE1

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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
PORTFOLIO OPPORTUNITY SET For any pair of wD and
wE
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Portfolios of Two Risky Assets
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Portfolios of Two Risky Assets
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Portfolios of Two Risky Assets

• Portfolio Opportunity set
• All combination of portfolio expected return and
  std deviation that can be constructed from the
  two available assets
• The best portfolio will depend on risk aversion

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Correlation Effects

• The relationship depends on the correlation
  coefficient
• -1.0 lt ? lt 1.0
• The smaller the correlation, the greater the risk
  reduction potential
• If r 1.0, no risk reduction is possible
• When r -1.0 , perfect hedging opportunity and
  maximum advantage from diversification

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6.3 Asset Allocation with Stocks, Bonds and Bills
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Extending to Include Riskless Asset

• Asset allocation across the three key asset
  classes
• Stocks, bonds, risk-free assets
• Risk-free T-bills yielding 5
• Two possible CAL, comparing their
  reward-to-variability ratio
• CAL A
• CAL B

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The Optimal Risky Portfolio
B Wd0.7,wE0.3
A minimum variance portfolio wD0.82,wE0.18
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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
Tangency portfolio, yield the CAL with highest
feasible reward-to-volatility ratio, the optimal
risky portfolio to mix with T-bills
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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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The Optimal Risky Portfolio

• Maximize the slope of the CAL
• The tangency portfolio is the optimal portfolio
  to mix with T-bills

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The Optimal Risky Portfolio

• The solution for the optimal risky portfolio
• (see xls optimal risky portfolio)

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The Optimal Complete Portfolio

• The optimal complete portfolio
• Specify the return characteristics of all
  securities (expected return, variance,
  covariance)
• The opportunity set of risky assets
• The optimal risky portfolio The CAL tangent
  with the opportunity set
• Use the investors degree of risk aversion

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The Optimal Complete Portfolio

• Investor with risk aversion A4
• Percentage in bonds0.74390.40.2976
• Percentage in stocks0.74390.60.4463

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The Complete Portfolio
Indifference Curve
Opportunity Set of Risky Assets
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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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6.4 The Markowitz Portfolio Selection Model
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Security Selection

• Generalize the portfolio construction problem to
  many risky assets and a risk-free asset
• To determine the risk-return opportunities
  available (Minimum-variance frontier) from the
  set of risky assets
• Involve the risk-free asset (CAL tangent to the
  efficient frontier) and get the optimal risky
  portfolio P
• Choose the appropriate mix between P and T-bill

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Security Selection

• Minimum-variance frontier
• A graph of the lowest possible variance that can
  be attained for a given portfolio expected return
• Efficient frontier of risky assets
• The part of the frontier that lies above

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Figure 7.10 The Minimum-Variance Frontier of
Risky Assets
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Figure 7.11 The Efficient Frontier of Risky
Assets with the Optimal CAL
search for the CAL with the highest
reward-to-variability ratio
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Figure 7.12 The Efficient Portfolio Set
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Markowitz Portfolio Selection Model

• 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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Markowitz Portfolio Selection Model

• Efficient frontier of risky assets
• Harry Markowitz, 1952
• Principle for any risk level, we are interested
  only in that portfolio with the highest expected
  return, the frontier is the set of portfolios
  that minimizes the variance for any target
  expected return

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Capital Allocation and the Separation Property

• The separation property
• 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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The Power of Diversification

• Remember
• For an equally weighted portfolio
• If define the average variance and average
  covariance of the securities as
• variance of an equally weighted portfolio as

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The Power of Diversification

• variance of an equally weighted portfolio as
• If average covariance is zero (uncorrelated),
  when all risk is firm-specific, portfolio
  variance approaches zero (power of
  diversification) when n gets larger
• Assume all securities same, discuss

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Table 7.4 Risk Reduction of Equally Weighted
Portfolios in Correlated and Uncorrelated
Universes
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6.5 Risk Pooling, Risk Sharing and Risk in the
Long Run
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Risk Pooling, Risk Sharing and Risk in the Long
Run

• Property value is 100,000, payouts on the
  1-year policy as following
• Risk free rate5, up-front charge120,Compute
  the risk premium and SD
• Expected return
• 100000(15)120(15)-0.001100000
• Risk premium0.26
• SD3160.7/1000003.16

Risk premium
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Risk Pooling and the Insurance Principle

• Sell 10000 of policies, uncorrelated, same E(r)
  and SD
• the variance of the portfolio
• SD of the 10000 policies
• selling more policies causes risk to fall

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Risk Pooling and the Insurance Principle Continued

• When we combine n uncorrelated insurance policies
  each with an expected profit of , both
  expected total profit and SD grow in direct
  proportion to n
• Ratio of mean and SD not change when n increases,
  risk-return trade-off not improve with additional
  policies

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Risk Sharing

• What does explain the insurance business?
• Risk sharing or the distribution of a fixed
  amount of risk among many investors
• Risk sharing
• Example insure a fraction of the project risk,
  fixed amount of equity capital
• Underwriter diversifies its risk by allocating
  its investment budget across many projects that
  are not perfectly correlated
• Limit exposure to any single source of risk by
  sharing the risk with other underwriters
• Each one diversifies a largely fixed portfolio
  across many projects
• Risk pooling pooling many sources of risk in a
  portfolio of given size

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6.6 A Spreadsheet Model
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SOLVER
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6.7 Optimal Portfolios with Restrictions on the
Risk-Free Asset
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Optimal Portfolios with Restrictions

• Unique optimal risky portfolio
• When all investors can borrow and lend at the
  risk-free rate
• Maximize the reward-to-variability ratio
• Without a risk-free asset
• No tangency portfolio
• Superimpose a personal set of indifference curves
  on the efficient frontier

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Portfolio Selection without a Risk-Free Asset
Expected return
More risk-tolerant investor
B
?
S
?
P
Q
?
Std deviation
More risk-averse investor
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Optimal Portfolios with Restrictions

• When risk-free investment available, but cannot
  borrow
• CAL exist but limited to FP
• A net lenders at rate of rf
• B borrowing at risk-free, risk-tolerant
  investors
• Q with restriction on borrowing, B turned to Q

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Portfolio Selection with Risk-Free Lending but No
Borrowing
Expected return
CAL
B
?
?
Q
?
P
A
?
rf
F
Std deviation
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Optimal Portfolios with Restrictions

• Investors wish to borrow to invest in a risky
  portfolio at a rate higher than risk-free rate
• Borrowing rate greater than the lending rate
• CAL1
• FP1, efficient portfolio set for risk-averse
  investors
• P1 as the optimal risky portfolio
• A as the complete portfolio

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Defensive investors with different Lending and
Borrowing rate
Expected return
CAL1
CAL2
B
?
P2
Efficient Frontier
?
rBf
P1
?
A
rf
F
Std deviation
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Optimal Portfolios with Restrictions

• CAL2
• Right of P2, efficient portfolio set for
  risk-tolerant investors, borrow at the higher
  rate rBf to invest
• Left of P2 unavailable, because lending only at
  rf
• P2 as the optimal risky portfolio
• B as the complete portfolio

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Aggressive investors with different Lending and
Borrowing rate
Expected return
CAL1
CAL2
B
?
?
Efficient Frontier
P2
?
rBf
P1
?
A
rf
F
Std deviation
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Optimal Portfolios with Restrictions

• Investors in the middle range
• Choose the risky portfolio from range P1P2,

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Moderately risk-tolerant investors with different
Lending and Borrowing rate
Expected return
CAL1
CAL2
?
P2
?
Efficient Frontier
C
?
rBf
P1
rf
Std deviation
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Review of Portfolio Mathematics

• Rule 1 Expected return of an asset
• Rule 2 Variance of an assets return

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Review of Portfolio Mathematics

• Rule 3 Rate of Return on a Portfolio
• Rule 4 portfolio a risky asset a risk-free
  asset

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Review of Portfolio Mathematics

• To quantify the hedging or diversification
  potential of an asset, use 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 returns on two assets to
  vary

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Review of Portfolio Mathematics

• Covariance
• Correlation Coefficient

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Review of Portfolio Mathematics

• Rule 5 when two risky assets with variances
  and respectively, are combined into a
  portfolio with portfolio weights and ,
  the portfolio variance is given by

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Review of Portfolio Mathematics

• Correlation coefficient
• Range of values for -1.0 lt lt 1.0
• If 1.0, the securities would be perfectly
  positively correlated
• If - 1.0, the securities would be
  perfectly negatively correlated

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Examples

• Humanex portfolio
• 50 T-bill , rate of return 5
• 50 Best Candy stock

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Examples

• Scenario analysis of best candy stock

Normal year Normal year Abnormal year
bullish bearish crisis
probability 0.5 0.3 0.2
Rate of return 25 10 -25

• Expected rate of return of best candy stock

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Examples

• Variance
• Standard deviation

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Examples

• The portfolios expected rate of return
• The portfolios standard deviation

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Examples

• Humanex portfolio
• 50 Best Candy stock
• 50 Sugarcanes stock

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Examples

• Scenario analysis of Sugarcanes stock

Normal year Normal year Abnormal year
bullish bearish crisis
probability 0.5 0.3 0.2
Rate of return 1 -5 35

• Expected rate of return of best candy stock

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Examples

• Scenario analysis of the portfolio

Normal year Normal year Abnormal year
bullish bearish crisis
probability 0.5 0.3 0.2
Rate of return 13 2.5 5

• Expected rate of return of best candy stock

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Examples

• Summarize

Portfolio Expected return Standard deviation
All in best candy 10.5 18.9
Half in T-bills 7.75 9.45
Half in Sugarcane 8.25 4.83
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Examples

• Covariance of the return of best candy and
  sugarcane stock

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Examples

• Portfolio variance