APT AND MULTIFACTOR MODELS OF RISK AND RETURN

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CHAPTER 9

• APT AND MULTIFACTOR MODELS OF RISK AND RETURN

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• Arbitrage
• Exploitation of security mispricing, risk-free
  profits can be earned
• No arbitrage condition, equilibrium market prices
  are rational in that they rule out arbitrage
  opportunities

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9.1 MULTIFACTOR MODELS
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Single Factor Model

• Returns on a security come from two sources
• Common macro-economic factor
• Firm specific events
• Focus directly on the ultimate sources of risk,
  such as risk assessment when measuring ones
  exposures to particular sources of uncertainty
• Factors models are tools that allow us to
  describe and quantify the different factors that
  affect the rate of return on a security

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Single Factor Model

• ri Return for security I
• Factor sensitivity or factor loading or
  factor beta
• F Surprise in macro-economic factor
• (F could be positive, negative or zero)
• ei Firm specific events
• F and ei have zero expected value, uncorrelated

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Single Factor Model

• Example
• Suppose F is taken to be news about the state of
  the business cycle, measured by the unexpected
  percentage change in GDP, the consensus is that
  GDP will increase by 4 this year.
• Suppose that a stocks beta value is 1.2, if GDP
  increases by only 3, then the value of F?
• F-1, representing a 1 disappointment in actual
  growth versus expected growth, resulting in the
  stocks return 1.2 lower than previously expected

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Multifactor Models

• Macro factor summarized by the market return
  arises from a number of sources, a more explicit
  representation of systematic risk allowing for
  the possibility that different stocks exhibit
  different sensitivities to its various components
• Use more than one factor in addition to market
  return
• Examples include gross domestic product, expected
  inflation, interest rates etc.
• Estimate a beta or factor loading for each factor
  using multiple regression.
• Multifactor models, useful in risk management
  applications, to measure exposure to various
  macroeconomic risks, and to construct portfolios
  to hedge those risks

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Multifactor Models

• Two factor models
• GDP, Unanticipated growth in GDP, zero
  expectation
• IR, Unanticipated decline in interest rate, zero
  expectation
• Multifactor model Description of the factors
  that affect the security returns

Factor betas
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Multifactor Models

• Example
• One regulated electric-power utility (U), one
  airline (A), compare their betas on GDP and IR
• Beta on GDP U low, A high, positive
• Beta on IR U high, A low, negative
• When a good news suggesting the economy will
  expand, GDP and IR will both increase, is the
  news good or bad ?
• For U, dominant sensitivity is to rates, bad
• For A, dominant sensitivity is to GDP, good
• One-factor model cannot capture differential
  responses to varying sources of macroeconomic
  uncertainty

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Multifactor Models

• Expected rate of return13.3
• 1 increase in GDP beyond current expectations,
  the stocks return will increase by 11.2

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Multifactor Security Market Line

• Multifactor model, a description of the factors
  that affect security returns, what determines
  E(r) in multifactor model
• Expected return on a security (CAPM)

Compensation for bearing the macroeconomic risk
Compensation for time value of money
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Multifactor Security Market Line

• Multifactor Security Market Line for multifactor
  index model, risk premium is determined by
  exposure to each systematic risk factor and its
  risk premium

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9.2 ARBITRAGE PRICING THEORY
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Arbitrage Pricing Theory

• Stephen Ross, 1976, APT, link expected returns to
  risk
• Three key propositions
• Security returns can be described by a factor
  model
• Sufficient securities to diversify away
  idiosyncratic risk
• Well-functioning security markets do not allow
  for the persistence of arbitrage opportunities

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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

• Law of One Price
• If two assets are equivalent in all economically
  relevant respects, then they should have the same
  market price
• Arbitrage activity
• If two portfolios are mispriced, the investor
  could buy the low-priced portfolio and sell the
  high-priced portfolio
• Market price will move up to rule out arbitrage
  opportunities
• Security prices should satisfy a no-arbitrage
  condition

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Well-diversified portfolios

• Well-diversified portfolio, the firm-specific
  risk negligible, only systematic risk remain
• n-stock portfolio

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Well-diversified portfolios

• The portfolio variance
• If equally-weighted portfolio , the
  nonsystematic variance
• N lager, the nonsystematic variance approaches
  zero, the effect of diversification

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Well-diversified portfolios

• This is true for other than equally weighted one
• Well-diversified portfolio is one that is
  diversified over a large enough number of
  securities with each weight small enough that the
  nonsystematic variance is negligible, eP
  approaches zero
• For a well-diversified portfolio

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Betas and Expected Returns

• Only systematic risk should command a risk
  premium in market equilibrium
• Well-diversified portfolios with equal betas must
  have equal expected returns in market
  equilibrium, or arbitrage opportunities exist
• Expected return on all well-diversified portfolio
  must lie on the straight line from the risk-free
  asset

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Betas and Expected Returns

• Only systematic risk should command a risk
  premium in market equilibrium
• Solid line plot the return of A with beta1 for
  various realization of the systematic factor (Rm)

Expected rate10,completely determined by Rm
Subject to nonsystematic risk
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• B E(r)8. beta1 AE(r)10. beta1
• Arbitrage opportunity exist, so A and B cant
  coexist
• Long in A, Short in B
• Factor risk cancels out across the long and short
  positions, zero net investment get risk-free
  profit
• infinitely large scale until return discrepancy
  disappears
• well-diversified portfolios with equal betas must
  have equal expected return in market equilibrium,
  or arbitrage opportunities exist

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• What about different betas
• A beta1,E(r)10
• C beta0.5,E(r)6
• D 50 A and 50 risk-free (4) asset,
• beta0.510.500.5, E(r)7
• C and D have same beta (0.5)
• different expected return
• arbitrage opportunity

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An arbitrage opportunity
A/C/D, well-diversified portfolio, D 50 A and
50 risk-free asset, C and D have same beta
(0.5), different expected return, arbitrage
opportunity
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• M, market index portfolio, on the line and beta1
  
• no-arbitrage condition to obtain an expected
  return-beta relationship identical to that of
  CAPM

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• EXAMPLE
• Market index, expected return10Risk-free
  rate4
• Suppose any deviation from market index return
  can serve as the systematic factor
• E, beta2/3, expected return42/3(10-4)8
• If Es expected return9, arbitrage opportunity
• Construct a portfolio F with same beta as E,
• 2/3 in M, 1/3 in T-bill
• Long E, short F

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One-Factor SML

• M, market index portfolio, as a well-diversified
  portfolio, no-arbitrage condition to obtain an
  expected return-beta relationship identical to
  that of CAPM
• three assumptions a factor model, sufficient
  number of securities to form a well-diversified
  portfolios, absence of arbitrage opportunities
• APT does not require that the benchmark portfolio
  in SML be the true market portfolio

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9.3 A MULTIFACTOR APT
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Multifactor APT

• Use of more than a single factor
• Several factors driven by the business cycle that
  might affect stock returns
• Exposure to any of these factors will affect a
  stocks risk and its expected return
• Two-factor model
• Each factor has zero expected value, surprise
• Factor 1, departure of GDP growth from
  expectations
• Factor 2, unanticipated change in IR
• e, zero expected ,firm-specific component of
  unexpected return

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A MULTIFACTOR APT

• Requires formation of factor portfolios
• Factor portfolio
• Well-diversified
• Beta of 1 for one factor
• Beta of 0 for any other
• Or Tracking portfolio the return on such
  portfolio track the evolution of particular
  sources of macroeconomic risk, but are
  uncorrelated with other sources of risk
• Factor portfolios will serve as the benchmark
  portfolios for a multifactor SML

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A MULTIFACTOR APT

• Example Suppose two factor Portfolio 1, 2,
• Risk-free rate4
• Consider a well-diversified portfolio A ,with
  beta on the two factors
• Multifactor APT states that the overall risk
  premium on portfolio A must equal the sum of the
  risk premiums required as compensation for each
  source of systematic risk
• Total risk premium on the portfolio A
• Total return on the portfolio A 9413

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A MULTIFACTOR APT

• Factor Portfolio 1 and 2, factor exposures of any
  portfolio P are given by its and
• Consider a portfolio Q formed by investing in
  factor portfolios with weights
• in portfolio 1
• in portfolio 2
• in T-bills
• Return of portfolio Q

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A MULTIFACTOR APT

• Suppose return on A is 12 (not 13), then
  arbitrage opportunity
• Form a portfolio Q from the factor portfolios
  with same betas as A, with weights
• 0.5 in factor 1 portfolio
• 0.75 in factor 2 portfolio
• -0.25 in T-bill
• Invest 1 in Q, and sell in A, net investment
  is 0, but with positive riskless profit
• Q has same exposure as A to the two sources of
  risk, their expected return also ought to be
  equal

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9.4 WHERE TO LOOK FOR FACTORS
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Multifactor APT

• Two principles when specify a reasonable list of
  factors
• Limit ourselves to systematic factors with
  considerable ability to explain security returns
• Choose factors that seem likely to be important
  risk factors, demand meaningful risk premiums to
  bear exposure to those sources of risk

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Multifactor APT

• Chen, Roll, Ross 1986
• Chose a set of factors based on the ability of
  the factors to paint a broad picture of the
  macro-economy
• IP change in industrial production
• EI change in expected inflation
• UI change in unexpected inflation
• CG excess return of long-term corporate bonds
  over long-term government bonds
• GB excess return of long-term government bonds
  over T-bill
• Multidimensional SCL, multiple regression,
  residual variance of the regression estimates the
  firm-specific risk

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Multifactor APT

• Fama, French, three-factor model
• Use firm characteristics that seem on empirical
  grounds to proxy for exposure to systematic risk
• SMB return of a portfolio of small stocks in
  excess of the return on a portfolio of large
  stocks
• HML return of a portfolio of stocks with high
  book-to-market ratio in excess of the return on a
  portfolio of stocks with low ratio
• Market index is expected to capture systematic
  risk

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• Fama, French, three-factor model
• Long-standing observations that firm size and
  book-to-market ratio predict deviations of
  average stock returns from levels with the CAPM
• High ratios of book-to-market value are more
  likely to be in financial distress, small stocks
  may be more sensitive to changes in business
  conditions
• The variables may capture sensitivity to
  risk-factors in macroeconomy

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9.5 THE MULTIFACOTOR CAPM AND THE APT
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APT and CAPM Compared

• Many of the same functions give a benchmark for
  rate of return.
• APT
• highlight the crucial distinction between factor
  risk and diversifiable risk
• APT assumption rational equilibrium in capital
  markets precludes arbitrage opportunities (not
  necessarily to individual stocks)
• APT yields expected return-beta relationship
  using a well-diversified portfolio (not a market
  portfolio)

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APT and CAPM Compared

• APT applies to well diversified portfolios and
  not necessarily to individual stocks
• 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

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The Multifactor CAPM and the APM

• A multi-index CAPM
• Derived from a multi-period consideration of a
  stream of consumption
• will inherit its risk factors from sources of
  risk that a broad group of investors deem
  important enough to hedge, from a particular
  hedging motive
• The APT is largely silent on where to look for
  priced sources of risk