The Arbitrage Pricing Theory (Chapter 10)

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The Arbitrage Pricing Theory(Chapter 10)

• Single-Factor APT Model
• Multi-Factor APT Models
• Arbitrage Opportunities
• Disequilibrium in APT
• Is APT Testable?
• Consistency of APT and CAPM

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Essence of the Arbitrage Pricing Theory

• Given the impossibility of empirically verifying
  the CAPM, an alternative model of asset pricing
  called the Arbitrage Pricing Theory (APT) has
  been introduced.
• Essence of APT
• A securitys expected return and risk are
  directly related to its sensitivities to changes
  in one or more factors (e.g., inflation, interest
  rates, productivity, etc.)

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Essence of the Arbitrage Pricing
Theory(Continued)

• In other words, security returns are generated by
  a single-index (one factor) model
• where
• or, by a multi-index (multi-factor) model

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Single-Factor APT Model(A Comparison With the
CAPM)

• CAPM (Zero Beta Version)
• Factor Market Portfolio
• Actual Returns
• Expected Returns

• APT (One Factor Version)
• Factor Your Choice
• Actual Returns
• Expected Returns

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Single-Factor APT Model(A Comparison With the
CAPM)Continued

• CAPM (Zero Beta Version)
• Continued
• Portfolio Variance

• APT (One Factor Version)
• Continued
• Portfolio Variance

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Multi-Factor APT Models

• One Factor
• Two Factors

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Multi-Factor APT Models(Continued)

• N Factors

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The Ideal APT Model

• Ideally, you wish to have a model where all of
  the covariances between the rates of return to
  the securities are attributable to the effects of
  the factors. The covariances between the
  residuals of the individual securities,
• Cov(?j, ?k), are assumed to be equal to zero.

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APT With an Unlimited Number of Securities

• Given an infinite number of securities, if
  security returns are generated by a process
  equivalent to that of a linear single-factor or
  multi-factor model, it is impossible to construct
  two different portfolios, both having zero
  variance (i.e., zero betas and zero residual
  variance) with two different expected rates of
  return. In other words, pure riskless arbitrage
  opportunities are not available.

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Pure Riskless Arbitrage Opportunities(An Example)

• Note If two zero variance portfolios could be
  constructed with two different expected rates of
  return, we could sell short the one with the
  lower return, and invest the proceeds in the one
  with the higher return, and make a pure riskless
  profit with no capital commitment.

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Pure Riskless Arbitrage Opportunities(An
Example) - Continued
Expected Return ()
D
C
B
A
E(rZ)1
E(rZ)2
Factor Beta
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Approximately Linear APT Equations

• The APT equations are expressed as being
  approximately linear. That is, the absence of
  arbitrage opportunities does not ensure exact
  linear pricing. There may be a few securities
  with expected returns greater than, or less than,
  those specified by the APT equation. However,
  because their number is fewer than that required
  to drive residual variance of the portfolio to
  zero, we no longer have a riskless arbitrage
  opportunity, and no market pressure forcing their
  expected returns to conform to the APT equation.

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Disequilibrium Situation in APT A One Factor
Model Example

• Portfolio (P) contains 1/2 of security (B) plus
  1/2 of the zero beta portfolio
• Portfolio (P) dominates security (A). (i.e., it
  has the same beta, but more expected return).

Expected Return ()
B
P
E(rP)
E(I1)
Equilibrium Line
A
E(rA)
E(rZ)
Beta
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Disequilibrium Situation in APT A One Factor
Model Example(Continued)

• Arbitrage Investors will sell security (A).
  Price of security (A) will fall causing E(rA) to
  rise. Investors will use proceeds of sale of
  security (A) to purchase security (B). Price of
  security (B) will rise causing E(rB) to fall.
  Arbitrage opportunities will no longer exist when
  all assets lie on the same straight line.

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Anticipated Versus Unanticipated Events

• Given a Single-Factor Model
• Substituting the right hand side of Equation 2
  for Aj in Equation 1

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Anticipated Versus Unanticipated
Events(Continued)

• Note If the actual factor value (I1,t) is
  exactly equal to the expected factor value,
  E(I1), and the residual (?j,t) equals zero as
  expected, then all return would have been
  anticipated
• rj,t E(rj)
• If (I1,t) is not equal to E(I1), or (?j,t) is
  not equal to zero, then some unanticipated return
  (positive or negative) will be received.

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Anticipated Versus Unanticipated Events(A
Numerical Example)

• Given
• Expected Return
• Anticipated Versus Unanticipated Return

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Anticipated Versus Unanticipated Returns(A
Graphical Display)
rj,t .115
.105
E(rj) .09
E(rZ) .06
.03
E(rZ)
I1,t
E(I1)
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Consistency of the APT and the CAPM
I1,t
I2,t
?M,I1
?M,I2
AI1
AI2
rM,t
rM,t
0
0

• Consider APT for a Two Factor Model
• In terms of the CAPM, we can treat each of the
  factors in the same manner that individual
  securities are treated (See charts above)
• CAPM Equation

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• Note that ?M,I1 and ?M,I2 are the CAPM (market)
  betas of factors 1 and 2. Therefore, in terms of
  the CAPM, the expected values of the factors are
• By substituting the right hand sides of Equations
  1 and 2 for E(I1) and E(I2) in the APT equation,
  we get

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• There are numerous securities that could have the
  same CAPM beta (?M,j), but have different APT
  betas relative to the factors (?1,j and ?2,j).
• Consistency of the APT and CAPM (an example)
• Given Factor 1 (Productivity) ?M,I1 .5
• Factor 2 (Inflation)
  ?M,I2 1.5

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• Assuming the market is efficient, all of the
  securities (1 through 6) will have equal returns
  on the average over time since they have a CAPM
  beta of 1.00. However, some would argue that it
  is not necessarily true that a particular
  investor would consider all securities with the
  same expected return and CAPM beta equally
  desirable. For example, different investors may
  have different sensitivities to inflation.
• Note It is possible for both the CAPM and the
  multiple factor APT to be valid theories. The
  problem is to prove it.

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Empirical Tests of the APT

• Currently, there is no conclusive evidence either
  supporting or contradicting APT. Furthermore, the
  number of factors to be included in APT models
  has varied considerably among studies. In one
  example, a study reported that most of the
  covariances between securities could be explained
  on the basis of unanticipated changes in four
  factors
• Difference between the yield on a long-term and a
  short-term treasury bond.
• Rate of inflation
• Difference between the yields on BB rated
  corporate bonds and treasury bonds.
• Growth rate in industrial production.

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Is APT Testable?

• Some question whether APT can ever be tested. The
  theory does not specify the relevant factor
  structure. If a study shows pricing to be
  consistent with some set of N factors, this
  does not prove that an N factor model would be
  relevant for other security samples as well. If
  returns are not explained by some N factor
  model, we cannot reject APT. Perhaps the choice
  of factors was wrong.

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Using APT to Predict Return

• Haugen presents a test of the predictive power of
  APT using the following factors
• Monthly return to U.S. T-Bills
• Difference between the monthly returns on
  long-term and short-term U.S. Treasury bonds.
• Difference between the monthly returns on
  long-term U.S. Treasury bonds and low-grade
  corporate bonds with the same maturity.
• Monthly change in consumer price index.
• Monthly change in U.S. industrial production.
• Dividend to price ratio of the SP 500.

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Haugen presents continued . . .

• Using data for 3000 stocks over the period
  1980-1997, he found that the APT did appear to
  have only limited predictive power regarding
  returns.
• He argues that the arbitrage process is
  extremely difficult in practice. Since
  covariances (betas) must be estimated, there is
  uncertainty regarding their values in future
  periods. Therefore, truly risk-free portfolios
  cannot be created using risky stocks. As a
  result, pure riskless arbitrage is not readily
  available limiting the usefulness of APT models
  in predicting future stock returns.