another set of pairs was created by randomly pairing the securities with one
another. The trades that would have been executed based on the empirical
pairing approach were then pitted against the trades where the securities
were randomly paired. The difference in the returns between the two groups
was found to be statistically significant, and the return generated by the me-
thodically paired set was better than the randomly paired sample set.
Unlike the purely empirical approach, the methodology that we sub-
scribe to comprises theoretical valuation concepts that are then validated
with empirical models and data. We will later show that the theoretical val-
uation approach helps us to easily identify pairs based on the fundamentals
of the firm. It also leads naturally to the formula used to measure the spread,
the degree of mispricing between the two securities. Our theoretical expla-
nation for the comovement of security prices stems from arbitrage pricing
theory (APT). According to APT, if two securities have exactly the same risk
factor exposures, then the expected return of the two securities for a given
time frame is the same. The actual return may, however, differ slightly be-
cause of different specific returns for the two securities. It is important to
note at this point that APT for the two securities has to be valid in all time
frames. Let the price of securities AandBat time tbe , and at time
t+ibe , respectively. The return in the time period ifor the two
securities is given as.^1
Now let us say that we have the prices of both securities at the current
time. The return on both securities is expected to be the same in all time
frames. In other words, the increment to the logarithm of the prices at the
current time must be about the same for both the securities at all time in-
stances in the future. This, of course, means that the time series of the loga-
rithm of the two prices must move together, and the spread calculation
formula is therefore based on the difference in the logarithm of the prices.
Having explained our approach, we now need to define in precise terms
what we mean when we say that the price series or the log price series of
the two securities must move together. Fortunately for us, the idea of co-
movement of two time series has been well developed in the field of econo-
metrics. We discuss it in the following section on cointegration.
Cointegration
In the introduction to time series we briefly discussed the preprocessing step
for nonstationary series. The series is typically transformed into a stationary
log(pptA)−−log(tiA++) and log( )pptB log(tiB)pptiA++ and tiBpptA and tBOverview 75
(^1) The value as calculated here is approximately equal to. Thus, return can
be thought of as the increment in the logarithm of the prices.
pp
p
ti t
t
- −