advocated by CAPM is helpful if returns of different assets are
uncorrelated. Unfortunately, correlations between asset returns may
grow in bear markets [4]. Besides the failure to describe prolonged
bear markets, another disadvantage of CAPM is its high sensitivity to
proxy for the market portfolio. The latter drawback implies that
CAPM is accurate only conditionally, within a given time period,
where the state variables that determine economy are fixed [2]. Then
it seems natural to extend CAPM to a multifactor model.
10.3 ARBITRAGE PRICING THEORY (APT)
The CAPM equation (10.2.1) implies that return on risky assets is
determined only by a single non-diversifiable risk, namely by the risk
associated with the entire market. The Arbitrage Pricing Theory
(APT) offers a generic extension of CAPM into the multifactor
paradigm.
APT is based on two postulates. First, the return for an asset i
(i ¼ 1, ..., N) at every time period is a weighed sum of the risk factor
contributions f
j
(t) (j ¼ 1, ...,K,K< N) plus an asset-specific com-
ponent e
i
(t)
R
i
(t) ¼ a
i
þ b
i1
f
1
þ b
i2
f
2
þ ...þ b
iK
f
K
þ e
i
(t) (10:3:1)
In (10.3.1), b
ij
are the factor weights (betas). It is assumed that the
expectations of all factor values and for the asset-specific innovations
are zero
E[f
1
(t)] ¼ E[f
2
(t)] ¼ ... ¼ E[f
K
(t)] ¼ E[e
i
(t)] ¼ 0 (10 :3 :2)
Also, the time distributions of the risk factors and asset-specific
innovations are independent
Cov[f
j
(t), f
j
(t
0
)] ¼ 0, Cov[e
i
(t), e
i
(t
0
)] ¼ 0, t t
0
(10:3:3)
and uncorrelated
Cov[f
j
(t), e
i
(t)] ¼ 0 (10:3:4)
Within APT, the correlations between the risk factors and the asset-
specific innovations may exist, that is Cov[f
j
(t), f
k
(t)] and
Cov[e
i
(t), e
j
(t)] may differ from zero.
116 Portfolio Management
The second postulate of APT requires that there are no arbitrage
opportunities. This implies, in particular, that any portfolio in which
all factor contributions are canceled out must have return equal to
that of the risk-free asset (see Exercise 3). These two postulates lead to
the APT theorem (see, e.g., [5]). In its simple form, it states that there
exist such K þ 1 constants l
0
, l
1
, ...l
K
(not all of them equal zero)
that
E[R
i
(t)] ¼ l
0
þ b
i1
l
1
þ ...þ b
iK
l
K
(10:3:5)
While l
0
has the sense of the risk-free asset return, the numbers l
j
are
named the risk premiums for the j-th risk factors.
Let us define a well-diversified portfolio as a portfolio that consists
of N assets with the weights w
i
where
P
N
i¼1
w
i
¼ 1, so that w
i
< W=N
and W 1 is a constant. Hence, the specific of a well-diversified
portfolio is that it is not overweighed by any of its asset components.
APT turns out to be more accurate for well-diversified portfolios
than for individual stocks. The general APT states that if the return of
a well-diversified portfolio equals
R(t) ¼ a þ b
1
f
1
þ b
2
f
2
þ ...þ b
K
f
K
þ e(t) (10:3:6)
where
a ¼
X
N
i¼1
w
i
a
i
, b
i
¼
X
N
k¼1
w
k
b
ik
(10:3:7)
then the expected portfolio return is
E[R(t)] ¼ l
0
þ b
1
l
1
þ ...þ b
K
l
K
(10:3:8)
In addition, the returns of the assets that constitute the portfolio
satisfy the simple APT (10.3.5).
APT does not specify the risk factors. Yet, the essential sources of
risk are well described in the literature [6]. They include both macro-
economic factors including inflation risk, interest rate, and corporate
factors, for example, Return on Equity (ROE).
4
Development of
statistically reliable multifactor portfolio models poses significant
challenges [2]. Yet, multifactor models are widely used in active
portfolio management.
Portfolio Management 117

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