Portfolio Selection

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Portfolio Selection

• By
• Gunjan Agrawal

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Portfolio SelectionbyHarry MarkowitzJournal
of Finance (1952)
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Portfolio Selection Efficient Diversification of
Investments byHarry MarkowitzNew York
Wiley, 1959
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Portfolio OptimizationUsing Linear
ProgrammingbyMark BroadieColumbia Business
School
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Historical facts byCampbell HarveyDuke
University
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Portfolio Selection

• What is Portfolio?
• What is Portfolio Selection?
• Efficient Asset Allocation.

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Stages of Process

• Observations and Experience
• Beliefs about the future performance
• Relevant beliefs about future
• Choice of portfolio

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First Stage

• Inputs Historical Data
• Outputs Possible Scenarios
• Scenario Returns ( )
• Scenario Probabilities ( )

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Example

• Discrete Time Stochastic Process

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Second Stage

• Inputs Outputs of First Stage
• Scenario Returns and Probabilities
• Assumptions Short sales are not Permitted
• Output Weight of securities in portfolio

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Objective Functions

• Maximum Discounted Expected Returns
• Maximize Expected Returns
• and
• Minimize Variance of Returns
• (E-V Rule)

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Maximum Discounted Expected Return
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Variables

• Expected return from ith security at time t ( )
• Discount factor for ith security at time t ( )
• Discounted return of the ith security
• Relative amount invested in ith security ( )
• Discounted expected return of the portfolio

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Optimization Problem 1
Objective Maximize Subject to
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Optimized Results

• Solution
• if Ri is maximum
• Otherwise
• Conclusion
• Diversified portfolio (R) All Non-Diversified
  portfolios (R)

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E-V Rule
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Assumption and Variables

• Assumption
• Static Models
• Variables
• Flow of returns from the ith security ( )
• Relative amount invested in ith security ( )
• Total Return from the portfolio

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Mean and Variance

• The mean of the return is
• where
• The variance of the return is
• where

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Optimization Problem 2

• Objective
• Maximize and Minimize
• Subject to

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Graphical Solution

• Attainable Combination
• Set of Feasible Solution
• Efficient Combination
• Minimum V for given E
• Maximum E for given V

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Three Securities Case

• Reduce in two dimension.
• Define
• X3 in terms of X2 and X1.
• Iso-mean Curve
• Iso-Variance Curve

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Feasible Region

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Iso-Mean Curve
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Efficient Portfolio

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Case II
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ExtensionChanging Definition of Risk
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Return

• Return in Scenario s
• Total Expected Return

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Risk

• Downside Risk
• if
• otherwise
• Average Downside Risk

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Optimization Problem

• Objective
• Maximize and Minimize ADR
• Subject to

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Linearization

• Objective
• Minimize
• Subject to

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Input Data

• Scenario Returns and Probabilities

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Results

• Max Return Vs. Min ADR

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Results

• Max Return Min ADR vs. Min ADR Max Return

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Insights

• What is Diversification?
• Is it Holding of Securities?
• Is it Holding Securities in right proportion?

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Example

• U.S. Small Stock and SP 500
• SP 500 and U.S. LT Corp. Bonds

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Values of Different Assets
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Returns from Different Assets
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Conclusion

• First Stage was not considered
• Formation of relevant beliefs
• Considered Second Stage
• Relevant beliefs about the assets involved
• Selection of portfolio
• Third Moment M3 is not included

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E-V Hypothesis

• Diversification
• Right Kind of Diversification
• Right Reason
• Theoretical Analysis
• Change in beliefs
• Change in preferences of E vs. V
• Change in and
• Practical Analysis
• Actual Selection