Quantitative Finance for Physicists

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 6¼ 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