Rabu, 14 Juli 2021

Metalabeling and the duality between cross-sectional and time-series factors

By Ernest Chan and Akshay Nautiyal

Features are inputs to supervised machine learning (ML)
models. In traditional finance, they are typically called “factors”, and they
are used in linear regression models to either explain or predict returns. In
the former usage, the factors are contemporaneous with the target returns,
while in the latter the factors must be from a prior period.

There are generally two types of factors: cross-sectional vs
time-series. If you are modeling stock returns, cross-sectional factors are
variables that are specific to an individual stock, such as its earnings yield,
dividend yield, etc. In our previous blog post, we described how we provide 40 such
factors to our subscribers for backtesting and live predictions. But as we
advocate using ML for risk management and capital allocation purposes (i.e. metalabeling), not for returns predictions,
you may wonder how these factors can help predict the returns of your trading
strategy or portfolio. For example, if you have a long-short portfolio of tech
stocks such as AAPL, GOOG, AMZN, etc., and want to predict whether the
portfolio as a whole will be profitable in a certain market regime, does it
really make sense to have the earnings yields of AAPL, GOOG, and AMZN as
individual features?

Meanwhile, time-series factors are typically market-wide or
macroeconomic variables such as the familiar Fama-French 3-factors:market
(simply, the market index return), SMB (the relative return of small cap vs
large cap stocks), and HML (the relative return of value vs growth stocks).
These time-series factors are eminently suitable for metalabeling, because they
can be used to predict your portfolio or strategy’s returns.

Given that there are many more obvious cross-sectional
factors than time-series factors available, it seems a pity that we cannot use
cross-sectional factors as features for metalabeling. Actually, we can –  Eugene Fama and Ken French themselves showed
us how. If we have a cross-sectional factor on a stock, all we need to do is to
use it to rank the stocks, form a long-short portfolio using the rankings, and
use the returns of this portfolio as a time-series factor. The long-short
portfolio is called a hedge portfolio.

We show the process of creation of a hedge portfolio with
the help of an example, starting with Sharadar’s fundamental cross-sectional
factors (which we generated as shown in the blog). There are 40 cross sectional factors
updated at three different frequencies - quarterly, yearly and twelve month
trailing. In this exercise, however, we use only the quarterly cross-sectional
factors. Given a factor like capex (capital expenditure), we consider the
normalized (the normalization procedure is found in the previously cited blog
post) capex of approximately 8500 stocks on particular dates from January 1st,
2010 till current date. There are 4 particular dates of interest every year
-  January 15th, April 15th, July 15th
and October 15th. We call these the ranking dates. On each of these dates we
find the percentile rank of the stock based on normalized capex. The dates are
carefully chosen to capture change in the cross sectional factors of the
maximum number of stocks post the quarterly filings.

Once the capex across stocks is ranked at each ranking date
(4 dates) each year we obtain the stocks present in the upper quartile (i.e ranked
above 75 percentile) and the stocks present in the lower quartile (i.e ranked
below 25 percentile). We take a long position on the ones which showed highest
normalized capex and take a short position on the ones with the lowest. Both
these sets together make our long-short hedge portfolio.

Once we have the portfolio on a given ranking date we
generate the daily returns of the portfolio using risk parity allocation (i.e
allocate proportional to inverse volatility). The daily returns of each chosen
stock are calculated for each day till the next ranking date. The portfolio
weights on each day are the normalized inverse of the rolling standard
deviation of returns for a two month window. These weights change on a daily
basis and are multiplied to the daily returns of individual stocks to get the
daily portfolio returns.  If a portfolio
stock is delisted in between ranking dates we simply drop the stock and not use
it to calculate the portfolio returns. The daily returns generated in this
process are the capex time series factors. This process is repeated for all
other Sharadar cross-sectional factors. 

So, voila! 40 cross-sectional factors become 40 time-series
factors, and they can be used for metalabeling any portfolio or trading
strategy, whether it trades stocks, futures, FX, or anything at all.

What about the opposite conversion? Can we turn time-series
factors into cross-sectional factors suitable for predicting the returns of
individual stocks? Actually, there is no need. You can directly add any time-series
factor to your feature set for predicting individual stock’s returns. This is
equivalent to building a linear factor model with an individual stock’s returns
as dependent variable and the time-series factor as independent variable, a
process well-known in traditional finance.

On a side note: besides these 40 time-series (and their
corresponding cross-sectional) features, we have compiled an additional 197
proprietary time-series features available to our Premium subscribers, and
available via our API.

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