Arbitrage Pricing Theory and Multifactor Models of Risk and Return

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Arbitrage Pricing Theory and Multifactor Models
of Risk and Return

• CHAPTER 10

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Outline of the Chapter

• Arbitrage Pricing Theory
• Arbitrage
• Single factor APT
• The Security Market Lines
• Compare APT and CAPM
• Multifactor Models
• Multifactor APT

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Arbitrage Pricing Theory

• Arbitrage Pricing Theory (APT) was developed by
  Ross (1976).
• APT predicts a security market line as CAPM and
  shows a linear relation with expected return and
  risk of a security.
• According to APT
• Security returns are described by a factor model
• There are sufficient securities to diversify away
  idiosyncratic risk
• Well-functioning security markets do not allow
  for the persistance of arbitrage opportunities

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Arbitrage Pricing Theory

• An arbitrage opportunity arises when an investor
  can earn riskless profits without making a net
  investment.
• The law of one price states that if two assets
  are equivalent in all economically relevant
  respects, then they should have the same market
  price.
• Otherwise there is a chance for arbitrage
  activity-simultaneously buying the asset where it
  is cheap and selling where it is expensive.
• During the arbitrage activity, investors will bid
  up the price where it is low and force it down
  where it is expensive. As a result they eliminate
  the arbitrage opportunities
• Security prices should satisfy a no-arbitrage
  condition.

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Arbitrage Pricing Theory (Continued)

• In a well-diversified portfolio nonsystematic
  risk across firms cancels out. Thus only factor
  risk (systematic risk of the portfolio) affects
  the risk premium on the security in market
  equilibrium.

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Arbitrage Pricing Theory (Continued)

• The solid line indicates a well diversified
  portfolio, A with an expected return of 10 and
  ßA1.
• The dahsed line also indicates a well diversified
  portfolio , B, with an expected return of 8 and
  ßB1.
• Could they coexisted?
• Arbitrage opportunity
• Well-diversified portfolios with equal betas must
  have equal expected returns in market equilibrium.

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Arbitrage Pricing Theory (Continued)

• The risk-premiums of well-diversified portfolios
  with different betas should be proportional to
  their betas.
• The risk premium (difference between the expected
  return on the portfolio and the risk-free rate)
  increases in direct proportion to ß.
• The expected return on all well-diversified
  portfolios must lie on the straight line from the
  risk-free asset.
• The equation of the line will also show the
  expected return on all well-diversified
  portfolios.

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Arbitrage Pricing Theory (Continued)

• Take M, market index portfolio as a
  well-diversified portfolio.
• The systematic factor, F, is the unexpected
  return on that portfolio.
• Since M is well-diversified, should be on the
  line and its beta is 1.
• Thus, the equation of the line is

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Arbitrage Pricing Theory (Continued)

• The no-arbitrage condition leads us to the
  equation that shows an expected return-beta
  relationship, which is identical to that of the
  CAPM.
• There are only three assumptions employed this
  time to obtain the same relationship as CAPM
• A factor model describing security returns
• A sufficient number of securities to form
  well-diversified portfolios
• Absence of arbitrage opportunities
• This approach under new assumptions is called
  Arbitrage Pricing Theory.

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Arbitrage Pricing Theory (Continued)

• In addition,
• APT does not require the benchmark portfolio on
  SML to be the true market portfolio.
• Thus, the problems related to have an
  unobservable market portfolio in CAPM are not
  problems in APT.
• Also, the index portfolio can easily be employed
  as a benchmark portfolio since it is
  well-diversified in APT even though it is not
  true market portfolio.

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Individual Assets and the APT

• Remember
• Imposing no-arbitrage condition on a
  single-factor security market implies maintenance
  of the expected return-beta relationship for all
  well-diversified portfolios and for all but
  possibly a small number of individual securities.

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Individual Assets and the APT (Continued)

• APT vs CAPM
• APT applies to well diversified portfolios and
  not necessarily to individual stocks.
• APT gives a benchmark rate of return to be
  employed in capital budgeting, security
  valuation, or investment performance evaluation
  such as CAPM.
• APT is more general in that it gets to an
  expected return and beta relationship without the
  assumption of the market portfolio.
• Although CAPM holds even for securities, with APT
  it is possible for some individual stocks to be
  mispriced - not lie on the SML.

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Multifactor Models An Overview

• Factor models are employed to describe and
  quantify the different factors that affect the
  rate of return on a security.
• In multifactor models stocks exhibit different
  sensitivities to the different components of
  systematic risk.
• Two-factor Model
• Suppose there are two most important
  macroeconomic sources of risk are
• Uncertainties surrounding the state of the
  business cycle (unanticipated growth in GDP)
• Unexpected changes in interest rates

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Multifactor Models An Overview (Continued)

• Factor sensitivities (loadings, betas) measure
  the sensitivity of the security returns to the
  systematic factors.
• By using these mutifactor models different
  responses of the securities to varying sources of
  macro economy are captured.
• The question is where E(r) comes from?
• Security Market Line of CAPM shows the
  relationship between expected return and the
  asset risk
• This time we have more than one risk factors.

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Multifactor Models An Overview (Continued)

• Based on the same idea with CAPMs SML we can say
  that in the two factor model the expected rate of
  return on a security will be the sum of
• The risk-free rate of return
• The sensitivity to GDP risk (GDP beta) the risk
  premium for bearing GDP risk
• The sensitivity to interest rate risk (interest
  rate beta) the risk premium for bearing interest
  rate risk

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A Multifactor APT

• Factor Portfolios A well-diversified portfolio
  constructed to have a beta of 1 on one of the
  factors and a beta of zero on any other factor.
• Returns on factor portfolios are correlated to
  one source of risk but totally uncorrelated with
  the other sources of risk.

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Where Should We Look for Factors?

• The mutlifactor APT does not say anything about
  the determination of relevant risk factors and
  their risk premiums.
• Still we want to narrow the set
• Limit ourselves to the systematic factors with
  considerable ability to explain security returns
• Choose factors that seem likely to be important
  risk factors

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Where Should We Look for Factors? (Continued)

• Chen, Roll and Ross (1986)
• change in industrial production, change in
  expected inflation, change in unanticipated
  inflation, excess return of long-term corporate
  bonds over long-term government bonds, and excess
  return of long-term government bonds over T-bills.

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Where Should We Look for Factors? (Continued)

• Fama and French three-factor model (1996)
• They use firm characteristices to capture the
  effects of systematic risk.
• They expect that the firm-specific variables
  proxy for yet-unknown more fundamental variables.

where SMB Small Minus Big, i.e., the
return of a portfolio of small stocks in excess
of the return on a portfolio of large stocks HML
High Minus Low, i.e., the return of a portfolio
of stocks with a high book to-market ratio in
excess of the return on a portfolio of stocks
with a low book-to-market ratio