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

• 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

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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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Markowitz Problem

• Objective
• Maximize and Minimize
• Subject to

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Portfolio with Parametric Weights

• Model Assets weight
• Function of assets Characteristics
• Firms Market capitalization
• Book to Market ratio
• Lagged Return (etc.)
• Estimate coefficients
• Estimating portfolios return
• Optimizing expected utility of the return over a
  sample period

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• Our Approach
• Computationally Simple
• Easily extended
• Produces sensible weights
• Robust Performance

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Parametric Weight Problem
Objective Where
(CRRA)
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Constant Relative Risk Aversion

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CRRA

• In this graph y is an outcome from a
  state-contingent output combination
• The certainty equivalent is the distance between
  point y and point y in state-space.

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Summary Statistics

• Cross-Sectional Means of Assets Characteristics
• ME (Log of market equity)
• BTM (Log of 1 ratio of book equity by market
  equity)
• MOM ( Lagged 12 month return)
• Computation
• For each firm
• For each month

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Market Equity
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Book To Market Ratio
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Lagged 12 Month Return
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Testing Asset Pricing Models

• Equilibrium Asset Pricing Model
• CAPM
• EAPT
• Testing
• Market Capitalization Weight
• Base Weight as MCW
• H0 µ 0

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Base Case
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Portfolio Characteristics Over Time
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Commodity Futures Market

• Market Capitalization is zero
• Momentum
• Term Structure

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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.

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Values of Different Assets
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Returns from Different Assets