Final_1.pdf

rather tall order and is seldom satisfied in practice. Unless the stocks are
class A and class B shares of the same firm, it is unlikely that the stocks’ fac-
tor exposures will be perfectly aligned.
If two stocks do not have their factor exposures perfectly aligned, then
any long–short portfolio composed of the two stocks will have a nonzero
component for the common factor returns. Our model for cointegration re-
lies on a zero value for the common factor returns, and the violation of this
represents deviation from ideal conditions for cointegration. Stating the
above in formula form, we have

(6.23)

(6.24)

The value of in Equation 6.24 is nonzero. Consequently, following the
logic from the observation in the section Common Trends and APT, the
common factor spread is also a nonzero quantity. It is helpful to write this
in equation form.

(6.25)

The common factor spread may very well be nonstationary, violating the
cointegration condition of spread stationarity. But can we still make do with
less than perfect conditions of cointegration? How do we quantify the devi-
ation? Let us say that the spread series is composed of a stationary compo-
nent (typically the specific spread) and a nonstationary component (typically
the common factor spread). Let the variances of the two components be
and. Note that the variance of the nonstationary com-
ponent is specified for a time horizon T. Also, let tbe the trading horizon. If
the change in the nonstationary component of the spread is small, we could
treat it more or less as a constant and say that we have a cointegrated pair.
A measure of the deviation from cointegration is captured in the signal-to-
noise ratio as given in Equation 6.26.

(6.26)

The ideal is to have the nonstationary component as close to zero as possi-
ble. If it is exactly zero, then the signal-to-noise ratio would be infinity. In
practice, a very large number for the ratio would make our assumption of
cointegration reasonable. Given that the variance of the nonstationary com-

SNR

t

=

σ

σ

stationary

nonstationary,

σstationary^2 σnonstationary,^2 T

spreadtt=+spreadcf spreadtspec

rporcft

rrrportf =+portcf portcf

rrr r r rABA−=−γγ γ()cf Bcf +−()ABspec spec

98 STATISTICAL ARBITRAGE PAIRS