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3. Data and Empirical Properties of Alphas
In this section we describe empirical properties of our formulaic alphas based on data
proprietary to WorldQuant LLC, which is used here with its express permission. We provide as
many details as possible within the constraints of the proprietary nature of this dataset.
For our alphas we take the annualized daily Sharpe ratio , daily turnover , and cents-per-
share . Let us label our alphas by the index , where is the number of
alphas. For each alpha,
,
and
are defined via
Here:
is the average daily P&L (in dollars);
is the daily portfolio volatility;
is the average
daily shares traded (buys plus sells) by the -th alpha;
is the average daily dollar volume
traded; and
is the total dollar investment in said alpha (the actual long plus short positions,
without leverage). More precisely, the principal of
is constant; however,
fluctuates due to
the daily P&L. So, both
and
are adjusted accordingly (such that
is constant) in Equation
(4). The period of time over which this data is collected is Jan 4, 2010-Dec 31, 2013. For the
same period we also take the sample covariance matrix
of the realized daily returns for our
alphas. The number of observations in the time series is 1,006, and
is nonsingular. From
we read off the daily return volatility
and the correlation matrix
(where
Note that
, and the average
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daily return is given by
.
Table 1 and Figure 1 summarize the data for the annualized Sharpe ratio
, daily turnover,
, average holding period
, cents-per-share
, daily return volatility
, annualized
average daily return
, and pair-wise correlations
with .
3.1. Return v. Volatility & Turnover
We run two cross-sectional regressions, both with the intercept, of
over i)
as
the sole explanatory variable, and ii) over
and
. The results are summarized in
Tables 2 and 3. Consistently with [Kakushadze and Tulchinsky, 2015], we have no statistically
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Here the average is over the time series of the realized daily returns.
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