Index Models and the Arbitrage Pricing Theory

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Chapter 9
Index Models and the Arbitrage Pricing Theory
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Chapter Summary

• Objective To discuss the nature and illustrate
  the use of arbitrage. To introduce the index
  model and the APT.
• The Single Index Model
• The Arbitrage Pricing Theory

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The Single Index Model

• Advantages
• Reduces the number of inputs for diversification
• Easier for security analysts to specialize
• Drawback
• the simple dichotomy rules out important risk
  sources (such as industry events)

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Single Factor Model

• Returns on a security come from two sources
• Common macro-economic factor
• Firm specific events
• Possible common macro-economic factors
• Gross Domestic Product Growth
• Interest Rates

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Single Factor Model

• ßi index of a securitys particular return to
  the factor
• F some macro factor in this case F is
  unanticipated movement F is commonly related to
  security returns

• Assumption a broad market index like the SP500
  is the common factor

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Single Index Model
ai stocks expected return if markets excess
return is zero bi(rM-ri) the component of
return due to market movements ei the component
of return due to unexpected firm-specific events
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Risk Premium Format
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Components of Risk

• Market or systematic risk risk related to the
  macro economic factor or market index
• Unsystematic or firm specific risk risk not
  related to the macro factor or market index
• Total risk Systematic Unsystematic

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Measuring Components of Risk

• ?i2 total variance
• ?i2 ?m2 systematic variance
• ?2(ei) unsystematic variance
• Cov(Ri,Rj)?i?j ?m2

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Examining Percentage of Variance

• Total Risk Systematic Unsystematic

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Index Model and Diversification
Portfolio
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Risk Reduction with Diversification
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Security Characteristic Line
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Regression Results
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Industry Prediction of Beta

• BMO Nesbitt Burns and Merrill Lynch examples
• BMO NB uses returns not risk premiums
• a has a different interpretation a rf (1-b)
• M. Lynchs adjusted b2/3(sample b)1/3(1)
• Forecasting beta as a function of past beta
• Forecasting beta as a function of firm size,
  growth, leverage etc.

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Multifactor Models

• Use factors in addition to market return
• Examples include industrial production, expected
  inflation etc.
• Estimate a beta for each factor using multiple
  regression
• Chen, Roll and Ross
• Returns a function of several macroeconomic and
  bond market variables instead of market returns
• Fama and French
• Returns a function of size and book-to-market
  value as well as market returns

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Multifactor Model Equation

• Ri E(ri) BetaGDP (GDP) BetaIR (IR) ei
• Ri Return for security i
• BetaGDP Factor sensitivity for GDP
• BetaIR Factor sensitivity for Interest Rate
• ei Firm specific events

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Multifactor SML Models

• E(r) rf BGDPRPGDP BIRRPIR
• BGDP Factor sensitivity for GDP
• RPGDP Risk premium for GDP
• BIR Factor sensitivity for Interest Rate
• RPIR Risk premium for GDP

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Summary Reminder

• Objective To discuss the nature and illustrate
  the use of arbitrage. To introduce the index
  model and the APT.
• The Single Index Model
• The Arbitrage Pricing Theory

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Arbitrage Pricing Theory

• Arbitrage - arises if an investor can construct
  a zero investment portfolio with a sure profit
• Since no investment is required, an investor can
  create large positions to secure large levels of
  profit
• In efficient markets, profitable arbitrage
  opportunities will quickly disappear

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Arbitrage Example

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Arbitrage Portfolio

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Arbitrage Action and Returns
Action Short 3 shares of D and buy 1 of A, B
C to form portfolio P Returns You earn a higher
rate on the investment than you pay on the short
sale
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APT Well-Diversified Portfolios

• F is some macroeconomic factor
• For a well-diversified portfolio eP approaches
  zero
• The result is similar to CAPM

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Portfolio Individual Security Comparison
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Disequilibrium Example
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Disequilibrium Example

• Short Portfolio C
• Use funds to construct an equivalent risk higher
  return Portfolio D
• D is comprised of A Risk-Free Asset
• Arbitrage profit of 1

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APT and CAPM Compared

• APT applies to well diversified portfolios and
  not necessarily to individual stocks
• With APT it is possible for some individual
  stocks to be mispriced - not lie on the SML
• APT is more general in that it gets to an
  expected return and beta relationship without the
  assumption of the market portfolio
• APT can be extended to multifactor models