Final_1.pdf

must be perfectly correlated. We also established that the common factor re-
turn of the APT model might be interpreted as the innovations derived from
the common trends. The correlation between the innovation sequences is
therefore the correlation between the common factor returns. The closer the
absolute value of this measure is to unity, the greater will be the degree of co-
movement. The distance measure we propose is therefore exactly that: the
absolute value of the correlation of the common factor returns.
The formula for the distance measure is therefore given as

(6.12)

In APT terms, if xAandxBare the factor exposure vectors of the two stocks
AandB, and Fis the covariance matrix, the distance measure may be cal-
culated as

(6.13)

Note that in Equation 6.13 we have used only common factor terms. The
specific variance contribution is not used.

Interpreting the Distance Measure

In previous discussions we hinted that the perfect alignment of the factor
exposure vectors, that is, a zero angle between them, is indicative of co-
integration. In this section we will show that the correlation measure as cal-
culated in Equation 6.13 could actually be interpreted as the cosine of the
angle between transformed versions of the factor exposure vectors corre-
sponding to the two stocks.
But why do we need to transform the factor exposure vectors? Can we
not directly measure the angle between them? We could do that but for the
fact that all factors in the multifactor model are not created equal. Returns
are more sensitive to changes in some factors versus the others. We therefore
need to transform from the space of factor exposure to the space of returns
and then measure the angle between the transformed vectors. We will dis-
cuss the exact nature of this transformation and show that correlation can
indeed be interpreted as the cosine of the angle.

ρ =

()()

xFx

xFx xFx

AB

ABBB

dA B

rr

rr

AB

AB

(,)

cov ,

var var

==

()

() ()

ρ

94 STATISTICAL ARBITRAGE PAIRS