Optimal Risky Portfolios

1
Optimal Risky Portfolios

• Finance 7320
• Lecture 9

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Optimal Risky Portfolios

• Diversification Portfolio Risk
• Diversification
• Systematic and Firm Specific Risk
• Portfolios of 2 Risky Assets
• Perfectly Positively Correlated
• Perfectly Negatively Correlated
• Less than perfectly Correlated
• Combination of Risky Portfolio with Rf

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Security Risk

• Systematic Risk - market risk, or risk which is
  attributable to market-wide sources
• Non-systematic risk - also firm specific or
  unique risk, which can be virtually eliminated
  through diversification (exposure to any one
  specific risk is reduced)

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Diversification

• If risk sources are independent, then exposure to
  any specific risk is negligible.
• Diversification strategies
• Naïve - just randomly add more stocks
• Efficient - solve optimization problem

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Example w1w2.5
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Example, cont.

• E (R) (.5 x 11) (.5 x 7) 9
• Portfolio expected return is a weighted average
  of the two individual expected returns.
• ? (Rp) 3.1 lt (.5 x 14.3) (.5 x 8.2) 11.2
• Portfolio S.D. is less than a weighted average of
  individual S.D.s
• ? Diversification has reduced risk

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Portfolios of Two Risky Assets

• We want to examine two risky assets
• Consider two mutual funds
• Bond fund
• Stock fund
• We want to examine the technological
  possibilities for these two risky assets

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Portfolios of Two Risky Assets

• Let
• Rp portfolio return
• w1 invested in security 1
• w2 invested in security 2
• R1 return on security 1
• R2 return on security 2
• Rp w1R1 w2R2
• E(Rp) w1E(R1) w2E(R2) (weighted avg)

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Portfolios of Two Risky Assets

• ?2(Rp) w12 ?12 w22 ?22 2w1w2Cov(1,2)
• Since Cov(1,2) ?12 ?1 ?2
• Where ?12 correlation of asset 1 and 2
• ?(Rp) w12 ?12 w22 ?22 2w1w2?12 ?1 ?2
• We will want to consider three values for ?12

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Two Risky Assets ?12 1

• ?2(Rp) (w1?1 w2 ?2)2
• ?(Rp) (w1?1 w2 ?2)
• Perfect positive correlation gt standard
  deviation of portfolio is weighted average of
  component standard deviations
• For any ?12 lt 1, ?p is less than weighted average

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E(R)
? (R)
?1
?2
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Two Risky Assets ?12 -1

• ?2(Rp) (w1?1 - w2 ?2)2
• Can be set 0
• ? w1 ?1 /(?1 ?2)
• Perfect negative correlation gt standard
  deviation can be set 0.

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E(R)
? (R)
14
-1 lt ?12 lt1

• Here, the equation is that of a rectangular
  hyperbola
• Graph

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E(R)
r -1
r.3
r1
r 0
? (R)
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Example

• Suppose w1w2 .5

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Example

• Suppose w1 .75, w2 .25

?2p 10.96
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E(R)
r -1
r.3
r1
r 0
? (R)
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Portfolio Opportunity Set

• Previous slide is Figure 8.5 in text
• Note No gain from diversification when p1
• As Correlation moves from 1 to 1, opportunity
  set is pushed to northwest

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Combine Risk-Free Asset w/ Risky

• Suppose ?12 lt 1
• Hyperbola is relevant feasible set of risky
  assets
• Now combine with a risk-free asset
• Can combine risk-free with any of possible risky
  portfolios
• CAL from before more risky choices

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.
E(R)
P
.
A
? (R)
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Combination w/ Risk-Free

• Risky Portfolio A is feasible, but
• Recall Slope of Line
• E(R) Rf/ ? (R)
• Risk Premium per unit of risk
• P is also feasible
• P maximizes the premium per unit of risk ?
  optimal risky portfolio

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Summary, So Far

• Determine characteristics of securities
• Means, variances and covariances
• Determine opportunity set (hyperbola)
• Combine with Rf and determine optimal risky
  portfolio, P (determine wis)
• Combine with Preferences ? determine complete
  portfolio, C (wf and wp)

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.
E(R)
P
.
C
Optimal Risky Portfolio
Optimal Complete Portfolio
? (R)
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Example

• Suppose you have 1,000,000 to invest
• Suppose P is 20 Asset 1 80 asset 2
• Suppose wf .3 wp .7
• Then, 300,000 in T-bills 700,000 in P
• 700,000 x .2 140,000 (w1 14)
• 700,000 x .8 560,000 (w2 56)

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Many Risky Securities

• Suppose instead of 2 we have n risky assets
• Determine all risk-return combinations
• Result is a bullet shape, where individual
  securities lie within the bullet
• Shell is minimum variance frontier
• Minimum variance for each level of return
• Top half dominates bottom half

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Efficient Frontier
E(R)
. Individual Assets
.
Global minimum variance portfolio
? (R)
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Efficient Frontier

• No other portfolio has
• Higher return for the same variance
• Lower variance for the same return
• Found by
• Maximizing return subject to a given risk level
• Minimizing risk subject to a given return level

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Summary, Again

• Step 1 Markowitz portfolio selection
• Given n securities
• Define efficient frontier
• Maxize E(R) s.t. ?
• Minimize ? s.t. E(R)
• Delete dominated portfolios

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Summary, cont.

• Step 2 Combine with Rf to determine optimal
  risky portfolio, P
• Portfolio P is the tangency portfolio
• Separation Property determination of optimal
  risky portfolio can be separated from investor
  preferences
• Fund manager would put all clients in same risky
  portfolio
• Construction of optimal portfolio, P, is a
  technical problem (E(R), s.d., cov)

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Summary, cont.

• Step 3 Combine with preferences to determine wf
  and wp
• Note A portfolio manager will put all clients in
  the same risky portfolio.
• Best risky portfolio is same for all investors,
  regardless of risk aversion!

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Implications/ Practice

• Different investment managers will have different
  inputs
• Inputs are a function of security analysis
• Different inputs ? different efficient frontier
• Managers may have different constraints
• Taxes
• Dividends
• Other preferences (age)

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Risk-Free Restrictions

• No risk-free asset available for
  borrowing/lending
• Optimal Risky portfolio depends on preferences
• Will still choose from efficient frontier
• Cannot borrow
• Unaffected if did not borrow anyway
• Risk tolerant choose inferior portfolio

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No Risk-Free Asset
E(R)
Preferences
Efficient frontier
? (R)
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Cannot Borrow
.
E(R)
.
? (R)
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Risk-Free Restrictions

• 3) Different borrowing lending rates
• Three-part CAL
• Part 1 corresponds to lending rate
• Part 2 corresponds to borrowing
• Part 3 100 in risky portfolio no T-bills

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Different Borrowing Lending Rates
.
.
E(R)
.
Part 2
Part 3
Part 1
? (R)
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Worth Repeating

• Portfolio return is weighted average of
  individual expected returns
• Portfolio ? is weighted average of individual ?s
  and covariances
• Portfolio ? lt weighted average of individual ?s
  as long as not perfectly positively correlated

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Worth Repeating

• Efficient frontier is set of optimized portfolios
• Each managers frontier a function of her inputs
• Inputs a function of security analysis
• Risk-free asset ? all investors will choose SAME
  risky portfolio