Fi8000 Capital Allocation and Efficient Portfolios

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Fi8000Capital Allocation andEfficient Portfolios

• Milind Shrikhande

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Today

• Portfolio Theory
• The Mean-Variance Criterion
• The Normal Distribution
• Capital Allocation
• The Mathematics of Portfolio Theory

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The Mean-Variance Criterion(M-V or µ-s criterion)

• Let A and B be two (risky) assets. All
  risk-averse investors prefer asset A to B if
• µA µB and sA lt sB
• or if
• µA gt µB and sA sB
• Note that these rules apply only when we assume
  that the distribution of returns is normal.

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The Mean-Variance Criterion(M-V or µ-s criterion)
?
E(R) µR
?
STD(R) sR
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The Normal Distribution of Returns
Pr(R)
68
95
µ
µ s
µ 2s
µ - s
µ - 2s
R
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The Normal Distribution of Returns
Pr(Return)
sR Risk
µR Reward
0
RReturn
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The Normal DistributionHigher Reward (Expected
Return)
Pr(Return)
µA
µB
RReturn
lt
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The Normal DistributionLower Risk (Standard
Deviation)
Pr(Return)
A
sA lt sB
B
µA µB
RReturn
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Capital Allocation - Outline

• n mutually exclusive assets (i.e. one can only
  invest in one asset but not in a portfolio)
• One risky asset and one risk-free asset
• n risky assets and one risk-free asset (the risky
  investments are mutually exclusive)
• Two risky assets
• n risky assets
• n risky assets and one risk-free asset

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Capital Allocation - Data

• There are three (risky) assets and one risk-free
  asset in the market. The risk-free rate is rf
  1, and the distribution of the returns of the
  risky assets is normal with the following
  parameters

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Capital Allocation n mutually exclusive assets

• State all the possible investments.
• Assuming you can use the Mean-Variance (M-V)
  rule, which investments are M-V efficient (i.e.
  which assets can not be thrown out of the set of
  desirable investments by a risk-averse investor
  who uses the M-V rule)?
• Present your results on the µ-s (mean
  standard-deviation) plane.

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The Expected Return andthe STD of the Return
(µ-s plane)
A
B
C
rf
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The Mean-Variance Criterion(M-V or µ-s criterion)
?
E(R)
?
STD(R)
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Capital Allocation n mutually exclusive assets

• The investment opportunity set
• rf, A, B, C
• The Mean-Variance (M-V or µ-s ) efficient
  investment set
• rf, A, C
• Note that investment B is not in the efficient
  set since investment A dominates it (one dominant
  investment is enough).

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Capital AllocationOne Risky Asset (A) and One
Risk-free Asset

• State all the possible investments how many
  possible investments are there?
• Assuming you can use the Mean-Variance (M-V)
  rule, which investments are M-V efficient?
• Present your results on the µ-s (mean
  standard-deviation) plane.

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The Expected Return and STD of the Return of the
Portfolio

• a the proportion invested in the risky asset A
• p the portfolio with a invested in the risky
  asset A
• and (1- a) invested in the risk-free asset
  rf
• Rp the return of portfolio p
• µp the expected return of portfolio p
• s p the standard deviation of the return of
  portfolio p
• Rp aRA (1-a)rf
• µp E aRA (1-a)rf aµA (1-a)rf
• s2p V aRA (1-a)rf (asA)2 Or
  sp asA

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Capital AllocationOne Risky Asset and One
Risk-free Asset

• The investment opportunity set
• all portfolios with proportion a invested in A
  and (1-a) invested in the risk-free asset rf
• The Mean-Variance (M-V or µ-s ) efficient
  investment set
• all the portfolios in the opportunity set

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The Capital Allocation Line
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The Expected Return andthe STD of the Return
(µ-s plane)
A
A
B
C
rf
rf
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The Capital Allocation Line (CAL)Four Basic
Investment Strategies
A
P2
B
A
P1
C
rf
rf
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Portfolios on the CAL
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Capital Allocation n Mutually Exclusive Risky
Asset and One Risk-free Asset

• State all the possible investments how many
  possible investments are there?
• Assuming you can use the Mean-Variance (M-V)
  rule, which investments are M-V efficient?
• Present your results on the µ-s (mean
  standard-deviation) plane.

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The Expected Return andthe STD of the Return
(µ-s plane)
A
B
C
rf
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Capital AllocationOne Risky Asset and One
Risk-free Asset

• The investment opportunity set
• all the portfolios with proportion a invested in
  the risky asset j and (1-a) invested in the
  risk-free asset, (j A or B or C)
• The Mean-Variance (M-V or µ-s ) efficient
  investment set
• all the portfolios with proportion a invested in
  the risky asset A and (1-a) invested in the
  risk-free asset (why A?)

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Capital AllocationTwo Risky Assets

• State all the possible investments how many
  possible investments are there?
• Assuming you can use the Mean-Variance (M-V)
  rule, which investments are M-V efficient?
• Present your results on the µ-s (mean
  standard-deviation) plane.

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The Expected Return and STD of the Return of the
Portfolio

• wA the proportion invested in the risky asset A
• wB (1-wA) the proportion invested in the
  risky asset B
• p the portfolio with wA invested in the risky
  asset A and
• (1-wA) invested in the risky asset B
• Rp the return of portfolio p
• µp the expected return of portfolio p
• s p the standard deviation of the return of
  portfolio p
• Rp wARA (1-wA)RB
• µp E wARA (1-wA)RB
• s2p V wARA (1-wA)RB

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Two Risky AssetsThe Investment Opportunity Set
E(Rp)
A
B
STD(Rp)
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Two Risky AssetsThe M-V Efficient Set (Frontier)
E(Rp)
A
B
STD(Rp)
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Two Mutually Exclusive Risky Assets The M-V
Efficient Set
E(R)
A
B
STD(R)
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Two Risky AssetsThe M-V Efficient Set (Frontier)
E(R)
A
P
B
STD(R)
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Capital Allocation Two Risky Assets

• The investment opportunity set
• all the portfolios on the frontier with
  proportion wA invested in the risky asset A and
  (1-wA) invested in the risky asset B
• The Mean-Variance (M-V or µ-s ) efficient
  investment set
• all the portfolios on the efficient frontier

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Two Risky AssetsThe M-V Efficient Set (Frontier)
E(R)
P1
A
P2
Pmin
B
P3
STD(R)
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Portfolios on the Efficient Frontier

• wA the proportion invested in the risky asset A
• wB (1-wA) the proportion invested in the
  risky asset B
• What is the value of wA for each on of the
  portfolios indicated on the graph? - Assume that
• µA10 µB5 sA12 s B6 ?AB(-0.5).
• What is the investment strategy that each
  portfolio represents?
• How can you find the minimum variance portfolio?
  What is the expected return and the std of return
  of that portfolio?

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Portfolios on the Frontier
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The Minimum Variance Portfolio
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The Minimum Variance Portfolio
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Investment Strategies

• Lending vs. Borrowing (bonds)
• Long vs. Short position (stocks)
• Passive risk reduction
• Diversification
• The number of risky assets in the portfolio
• The correlation between the returns of the assets
• A perfect hedge

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Practice Problems

• BKM Ch. 7 1-6, 8, 9, 13, 20, 22, 23
• BKM Ch. 8 1-7
• Mathematics of Portfolio Theory
• Read and practice parts 4-10.