Portfolio Analysis and Theory

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Portfolio Analysis and Theory

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Portfolio Analysis

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Definitions

• A portfolio is the collection of securities an
  investor holds
• A portfolio weight is the proportion of total
  wealth allocated to a given security

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Risk and Return for a Portfolio

• The expected portfolio return is the weighted
  average of the expected returns on component
  securities
• The risk of a portfolio is measured by the
  standard deviation of return

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Diversification

• Portfolio standard deviation depends upon the
  correlation of the returns of component stocks
• If stocks are not perfectly correlated, the
  portfolio standard deviation is less than the sum
  of the component standard deviations
• Therefore, diversification reduces risk

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Systematic and Unsystematic Risk

• The risk of a portfolio can be divided into
  systematic and unsystematic risk
• Systematic risk can be explained by factors that
  are common to all securities
• Unsystematic risk can be explained by factors
  that are unique to a given security
• Systematic means part of the system

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Diversification and Risk

• Unsystematic risk can be reduced through
  diversification
• Systematic risk cannot be reduced through
  diversification
• Systematic risk is measured by Beta

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Beta and the Market Model

• Beta is the slope of the regression of the
  historical stock return on the return of a market
  index, like the SP 500, containing a large
  number of stocks
• Unsystematic risk is the variation in the stocks
  return that cannot be explained by the variation
  in the market index

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Market Model (continued)

• The systematic risk of a portfolio is the
  weighted sum of the systematic risk of each
  component
• You cannot reduce systematic risk through
  diversification
• You can only obtain low systematic risk by
  choosing securities with low systematic risk for
  your portfolio

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Systematic Risk

• For very large portfolios unsystematic risk can
  be almost eliminated
• In this situation the risk each security
  contributes to the portfolio is approximately
  equal to its systematic risk or Beta
• Therefore, the relevant risk for an individual
  security held within a well-diversified portfolio
  is its Beta
• Remember, that for a portfolio the relevant risk
  is the standard deviation

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Example

• Suppose you put all your wealth in a General
  Motors stock. Then the relevant risk is the
  standard deviation because your portfolio
  consists of a single security
• On the other hand, suppose your holdings of
  General Motors is a small fraction of a large
  diversified portfolio. Then the relevant risk is
  the Beta

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Portfolio Theory

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Risk Aversion

• Risk averse investors require compensation, in
  the form of extra return, for assuming financial
  risk
• The risk-free asset has zero risk and is usually
  assumed to be the one-year U. S. treasury bill
• A risk averse investor will hold a risky
  portfolio only if its expected return is greater
  than the risk-free rate

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Risk Aversion (continued)

• Investors are only compensated for bearing
  systematic risk
• Investors are not compensated for bearing
  unsystematic risk because it can be eliminated by
  diversification

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Example

• Suppose investors A and B choose portfolios by
  throwing darts at the Wall Street Journal. Assume
  that the average return and average standard
  deviation of stock in the paper is 10 and 14,
  respectively. If investor A throws one dart, then
  his expected return and risk will be 10 and 14.
  If investor B throws ten darts, her expected
  return will still be 10 but her portfolio
  standard deviation will be (probably) less
  because of the effect of diversification.
• Should A be compensated for assuming more risk?

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Risk-Return Relationship

• There should be a positive relationship between
  expected return and the Beta measure of
  systematic risk.
• There should be no relationship between the
  expected return and unsystematic risk.
• Formal economic model is the Capital Asset
  Pricing Model (CAPM)

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The Capital Asset Pricing Model Assumptions

• risk aversion
• rational behavior
• investors choose a portfolio on the basis of
  expected returns and standard deviation
• investors have same expectations about the future
  expected returns, standard deviations and
  correlations among stocks
• existence of risk-free security
• perfect markets no taxes, transaction costs, or
  restrictions on short sales

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The security market line (SML)

• SML is the relationship between the expected
  return on an asset and its Beta measure of
  systematic risk
• Under the CAPM, the relationship is linear

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Beta

• The Beta of the risk-free asset is zero and of
  course its return is the risk-free rate
• The Beta of the market portfolio is one
• Any stock or portfolio with a Beta 1 has the
  same expected return as the market
• Any stock with a Beta 0 returns the risk-free
  rate

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Beta (continued)

• A stock with a Beta gt 1 has more systematic risk
  than the market and has an expected return that
  is greater than the market
• A stock with a Beta lt 1 has less systematic risk
  than the market and an expected return less than
  the market.

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Beta (continued)

• A stock with high systematic risk (Beta gt 1) will
  on average (not always) go up by a greater
  percentage than the market index when the market
  goes up.
• And of course will on average go down by greater
  percentage when the market goes down.
• This is what systematic risk means.
• So if you thought the market were going down, you
  would buy stocks with low Betas

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The Price of Risk

• If systematic risk is priced, than high beta
  stocks should have an average return, over the
  ups and downs in the market, higher than the
  market index.
• The high Beta stock has greater systematic risk
  because when the market goes down the high Beta
  stock will go down even more.
• However, in theory, when you average over the ups
  and downs the risk averse investor will earn a
  higher average return

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Required Return

• An investor demands a positive expected return
  for two reasons time value of money and a risk
  aversion.
• The return to compensate for the time value of
  money is the risk-free rate.
• The extra return to compensate a risk averse for
  bearing risk is called the risk premium.
• The required return equals the risk free rate
  plus the risk premium.

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Security Market Line

• The market risk premium (the premium on the
  market index) by definition equals the expected
  market return minus the risk-free rate
• Under the CAPM, the risk premium on any security
  equals the Beta multiplied by market risk premium

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Security Market Line (continued)

• The security market line relates the required
  return on a security to its Beta systematic risk.
  The required return equals the risk-free rate
  plus the Beta times the market risk premium

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The benefits of diversification and portfolio
analysis are well established. The conclusions
drawn from portfolio theory are tentative and
debatable
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Alternative Explanation

• A relatively small number of stocks have high
  performance
• Most of the return on a diverisified portfolio
  can be explained by a small number of high
  performing stocks in the portfolio
• If you fail to diversify, then you run the risk
  of not holding any high performing stocks
• Your portfolio will have below average return

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Alternative Explanation

• A relatively small number of stocks will perform
  badly
• Some will go under
• If you fail to diversify, then you might end up
  with a large fraction of wealth in a failed stock
• Your return distribution will have fat tails