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
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 security’s expected return and risk are
directly related to its sensitivities to changes
in one or more factors (e.g., inflation, interest
rates, productivity, etc.)
Essence of the Arbitrage Pricing Theory
(Continued)
– In other words, security returns are generated by a
single-index (one factor) model:
rj,t  A j  β1, jI1,t  ε j,t
where:
I1,t  Val u eof Factor (1) i n pe riod(t)
β1, j  be taof se cu rity(j) with re spe ctto Factor (1)
– or, by a multi-index (multi-factor) model:
rj,t  A j  β1, jI1,t  β 2, jI 2,t  . . . + βn, jIn, t  ε j,t
Single-Factor APT Model
(A Comparison With the CAPM)
CAPM (Zero Beta Version)
Factor = Market Portfolio
APT (One Factor Version)
Factor = “Your Choice”
Actual Returns:
Actual Returns:
rj,t  A j  β jrM, t  ε j,t
rj,t  A j  β1, jI1,t  ε j,t
Expected Returns:
Expected Returns:
E(r j )  E(rz )  [E(rM )  E(rz )]β j
E(r j )  E(rz )  [E(I 1 )  E(rz )]β1, j
FactorPrice*
Market Risk Premium
* Note: In thetext,lambda ( ) denotes
factorprice.
Single-Factor APT Model
(A Comparison With the CAPM)
Continued
CAPM (Zero Beta Version)
Continued
Portfolio Variance:
σ 2 (rp )  βp2 σ 2 (rM )  σ 2 (ε p )
APT (One Factor Version)
Continued
Portfolio Variance:
2
2
2
σ 2 (rp )  β1,
σ
(I
)

σ
(ε p )
p
1
m
wh e re βp 
m
x β
j j
j1
wh e re β1,p 
x β
j 1, j
j1
m
σ 2 (ε p ) 

m
x 2jσ 2 (ε j )
j1
Assu m i n gC O V(ε j , ε k )  0
σ 2 (ε p ) 

x 2jσ 2 (ε j )
j1
Assu m i n gC O V(ε j , ε k )  0
Multi-Factor APT Models
 One Factor
rj, t  A j  β1, jI1,t  ε j,t
E(r j )  E(rz )  [E(I1  E(rz )]β1, j
2
2
2
σ 2 (rp )  β1,
σ
(I
)

σ
(ε p )
p
1
 Two Factors
rj,t  A j  β1, jI1,t  β 2, jI 2,t  ε j,t
E(r j )  E(rz )  [E(I1 )  E(rz )]β1, j  [E(I 2 )  E(rz )]β 2, j
2 2
2 2
2
σ 2 (rp )  β1,
σ
(I
)

β
σ
(I
)

σ
(ε p )
p
1
2,p
2
Multi-Factor APT Models
(Continued)
 N Factors
rj,t  A j  β1, jI1,t  β 2, jI 2,t  . . . + βn, jI n, t + ε j,t
E(r j )  E(rz )  [E(I1 )  E(rz )]β1, j
 [E(I 2 )  E(rz )]β 2, j
+....
+ [E(I n )  E(rz )]βn, j
2 2
2
2
2
2
2
σ 2 (rp )  β1,
σ
(I
)

β
σ
(I
)

.
.
.
+
β
σ
(I
)
+
σ
(ε p )
p
1
2,p
2
n, p
n
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.
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.
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.
Pure Riskless Arbitrage
Opportunities
(An Example) - Continued
Expected Return (%)
0.25
C
D
B
A
E(rZ)1
E(rZ)2
0
-0.5
0
0.5
1
Factor Beta
1.5
“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.
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:
β1,P  .5 β1,B  .5 β1,Z  .5(2.0) .5(0) 1.0
 Portfolio (P) dominates security (A). (i.e., it has the same
beta, but more expected return).
Expected Return (%)
B
20
P
E(rP)
E(I1)
Equilibrium Line
E(rA)
A
E(rZ)
Beta
0
0
0.5
1
1.5
2
2.5
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.
Anticipated Versus Unanticipated Events
 Given a Single-Factor Model:
rj,t  A j  β1, jI1,t  ε j,t
{Equ ation#1}
S inceE( ε)  0, we can stateth at:
E(r j )  A j  β1, jE(I 1 )
or
A j  E(r j )  β1, jE(I 1 )
{Equ ation#2}
 Substituting the right hand side of Equation #2 for Aj in
Equation #1:
rj,t  E(r j )  β1, jE(I 1 )  β1, jI1,t  ε j,t
rj,t

E(r j )
+
β1, j[I1,t  E(I 1 )]  ε j,t
Actual
Anticipated
Unanticipated
Return
Return
Return
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.
Anticipated Versus Unanticipated Events
(A Numerical Example)
 Given:
E(rZ )  .06 E(I 1 )  .12 I1,t  .15 ε j,t  .01 β1, j  .5
 Expected Return:
E(r j )  E(rZ )  [E(I1 )  E(rZ )]β1, j
= .06 + [.12- .06].5
= .09
 Anticipated Versus Unanticipated Return:
rj,t  E(r j )  β1, j[I1,t  E(I 1 )]  ε j,t
= .09 + .5[.15- .12]+ .01
=
.09
Anticipated
.015+ .01
Unanticipated
Anticipated Versus Unanticipated Returns
(A Graphical Display)
rj,t = .115
.105
E(rj) = .09
E(rZ) = .06
.03
0
0.03
0.06 0.09
E(rZ)
0.12
E(I1)
0.15
I1,t
0.18
0.21
Consistency of the APT and the CAPM
I1,t
I2,t
M,I1
M,I2
AI1
AI2
rM,t
0
0
rM,t
 Consider APT for a Two Factor Model:
E(r j )  E(rZ )  [E(I1 )  E(rZ )]β1, j  [E(I 2 )  E(rZ )]β2, j
 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:
E(r j )  E(rZ )  [E(rM )  E(rZ )]βM, j
 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:
E(I 1 )  E(rZ )  [E(rM )  E(rZ )]β M, I 1 Equation1
E(I 2 )  E(rZ )  [E(rM )  E(rZ )]β M, I 2 Equation2
 By substituting the right hand sides of Equations 1 and 2
for E(I1) and E(I2) in the APT equation, we get:
E(r j )  E(rZ )  ([E(rM )  E(rZ )]β M, I 1 ) β1, j
+ ([E(rM )  E(rZ )]β M, I 2 ) β 2, j
E(r j )  E(rZ )  [E(rM )  E(rZ )](β M, I 1β1, j  β M, I 2β 2, j )
Note th at: β M, j  β M, I 1β1, j  β M, I 2β 2, j
S i n ce: E(r j )  E(rZ )  [E(rM )  E(rZ )]β M, j
 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
Security  1,j
______ ____
1
0
2
.4
3
.8
4
1.2
5
1.6
6
2.0
 2,j
____
.667
.534
.400
.267
.134
0
 M,I1 1,j +  M,I2 2,j =  M,j
___________________
.5(0) + 1.5(.667) = 1.00
.5(.4) + 1.5(.534) = 1.00
.5(.8) + 1.5(.400) = 1.00
.5(1.2) + 1.5(.267) = 1.00
.5(1.6) + 1.5(.134) = 1.00
.5(2.0) + 1.5(0) = 1.00
 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.
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.
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.
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 S&P 500.
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.