Portfolio Selection

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Portfolio Selection

• Chapter 8
• Charles P. Jones, Investments Analysis and
  Management,
• Tenth Edition, John Wiley Sons
• Prepared by
• Elshahat, Ahmed, Florida International University

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Building a Portfolio Using Markowitz Principles

• Identify optimal risk-return combinations
  available from the set of risky assets being
  considered.
• Tool Markowitz efficient frontier analysis.
• Input expected returns, variances, covariances
  
• Select the optimal portfolio from among those in
  the efficient set.
• Criteria investor's preferences

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1 Identifying Optimal Risk-Return Combinations

• Assumptions
• A single investment period
• Liquidity of positions (no transaction cost).
• Investor preferences is based only on portfolio's
  expected return and risk.
• Steps
• The Attainable Set of Portfolios
• Efficient Portfolios
• The Efficient Set

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The Attainable Set of Portfolios

1. Determine the risk-return opportunities
   available. A large number of possible portfolios
   exist.

• The attainable (opportunity) set is the entire
  set of all portfolios that could be found from a
  group of n securities.
• Risk-averse investors should be interested only
  in those portfolios with the lowest possible risk
  for any given level of return.

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Efficient Portfolios

• Portfolio that has the smallest portfolio risk
  for a given level of expected return or the
  largest expected return for a given level of risk

• Given the minimum-variance portfolios, we can
  plot the minimum-variance frontier as shown in
  this Figure.
• X dominates Y.

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The Efficient Set (Frontier) (AB)

• The efficient set is determined by the principle
  of dominance portfolio X dominates portfolio Y
  if it has the same level of risk but a larger
  expected return, or the same expected return but
  a lower risk.

• The solution to the Markowitz model revolves
  around the portfolio weights

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Understanding the Markowitz Solution  

• Think of efficient portfolios as being derived in
  the following manner
• Inputs Determine the Required E(R), 10.
• Create all portfolios that can yield 10.
• Choose the one with the lowest risk.
• Determine anther Required E(R), 11.
• Continue the process.

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2 Selecting an Optimal Portfolio of Risky
Assets

• Markowitz model does not specify one optimum
  portfolio. Rather, it generates the efficient set
  of portfolios, all of which, by definition, are
  optimal portfolios.

• Indifference Curves describe investor
  preferences for risk and return.
• Each indifference curve represents the
  combinations of risk and expected return that are
  equally desirable to a particular investor.

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Indifference Curves Properties

1. Indifference curves cannot intersect
2. Investors have an infinite number of indifference
   curves
3. The curves for all risk-averse investors will be
   upward-sloping, but the shapes of the curves can
   vary depending on risk preferences.
4. Higher indifference curves are more desirable
   than lower indifference curves.
5. The greater the slope of the indifference curves,
   the greater the risk aversion of investors.
6. The farther an indifference curve is from the
   horizontal axis, the greater the utility.

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Selecting the Optimal Portfolio  

• The optimal portfolio for a risk-averse investor
  is the one on the efficient frontier that is
  tangent to an investor's indifference curve that
  is highest in return-risk space.

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SOME IMPORTANT CONCLUSIONS ABOUT THE MARKOWITZ
MODEL

1. Markowitz portfolio theory is referred to as a
   two-parameter model.
2. The Markowitz analysis generates an entire set of
   efficient portfolios, all are equally good.
3. The Markowitz model does not address the issue of
   investors using borrowed money
4. Different investors will estimate the inputs to
   the Markowitz model differently.
5. The Markowitz model remains cumbersome to work
   with because of the large variance-covariance
   matrix

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Alternative Methods of Obtaining the Efficient
Frontier SIM MIM

• THE SINGLE-INDEX MODEL (SIM) relates returns on
  each security to the returns on a common index.

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THE SINGLE-INDEX MODEL

• The single-index model divides a security's
  return into two components
• Micro event A unique part, represented by ai,
  affecting an individual company but not all
  companies in general.
• Macro event A market-related part represented by
  ßiRM, broad-based and affects all (or most)
  firms.
• Error term

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Critical Assumption of the SIM  

• Securities are related only in their common
  response to the return on the market (the
  residual errors for security i are uncorrelated
  with those of security j).
• It implies that stocks covary together only
  because of their common relationship to the
  market index.
• There are no influences on stocks beyond the
  market, such as industry effects.
• Thus, covariance depends only on market risk.

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Splitting Risk into Two Parts  
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The Asset Allocation Decision

• The allocation of portfolio assets to broad asset
  markets.
• How much of the portfolio's funds are to be
  invested in stocks, how much in bonds, money
  market assets, and so forth.
• Some Major Asset Classes
• International Investing
• Bonds
• Treasury Inflation-Indexed Securities.
• Real Estate

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COMBINING ASSET CLASSES
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The Impact of Diversification on Risk
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How Many Securities Are Enough to Diversify
Properly?

• As few as 20 stocks could be adequate