1. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 1
Testing Mean Reversion:
Evaluation of a Statistical Arbitrage Strategy Over Time
Taweh Beysolow II
Advisor: Yusif Simaan
Fordham University

2. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 2
I. Introduction
History of Statistical Arbitrage
From the 1980s forward, statistical arbitrage strategies yielded astounding returns
that were attractive to both fund managers and investors. This was generally the case
until the end of the 20th
century and beginning of the 21st
century, in which several
incidents of considerable magnitude caused strategies to perform less well at best and
terribly at worst. Statistical arbitrage funds failed catastrophically, investors withdrew
funds, and many places that did practice statistical arbitrage in part stopped completely.
While there has been resurgence of the discipline to some degree in the late 2000s and
early 2010s, it is still neither as largely practiced nor as profitable as it was in the 1980s.
While it is uncertain whether the profitability of this discipline in the future will ever
reach its former levels, the study of the discipline’s rise, fall, and how it can be improved
provides useful information for aspiring quantitative analysts, researchers, and traders.
What is arbitrage and how is it different from statistical arbitrage?
Arbitrage is defined as the act of taking advantage of inefficiencies in the price
discovery process to obtain a “risk-free” profit. Typically, this involves the trading of
stocks, options, or fixed income products. In traditional arbitrage, this involves trading
between two or more markets, with the objective of buying assets in one market and
selling them to make a profit, often requiring an instantaneous buy and sell. Traditional
arbitrage usually involves investing nothing, takes no risk, but is able to make money on
this discrepancy. In theory, asset prices adjust quickly and eliminate arbitrage
opportunities.

3. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 3
Statistical arbitrage seeks to take advantage of temporary mispricing consistently
and yielding positive expected profit. As such, it is not the same as the traditional form of
arbitrage.
What is the market?
In statistical arbitrage, the market usually is described as an index, or benchmark
of some sort, and the equity would be considered a subset of the market. An example of
this would be the SPY and YHOO – the former is a proxy for the market, here being the
S&P 500, and the latter is a subset of the S&P 500.
II. Mechanics and Assumptions of Statistical Arbitrage
Trending vs. Reverting: What Causes Both
Stocks generally are described to move, broadly speaking, in two distinct patterns:
trending and reverting. Trending stock prices usually are caused by fundamental changes
in the stock or the market, such as positive/negative news, earnings calls, and or changes
in systemic factors within the market. These changes are typically considered to be long-
term, and as such won’t be the primary focus of the study. We would like to focus on the
short-term changes, which typically are classified as reversion.
In this experiment, we consider that stocks are priced relative to the market, and
therefore that they can be either in alignment with the market or be misaligned, either
positively or negatively. When systematic changes occur in the market, we should in
theory observe that all prices of stocks within the market move in a constant proportion
with the market. This is true regardless of whether the market rises or falls. However, if

4. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 4
we observe a movement of a stock that is considerably large in either direction, and the
market did not move accordingly, we would consider this a temporary disequilibrium. As
such, we flag these instances and predict there will be a reversionary movement towards
the price that is closer towards its equilibrium price relative to the market.
To illustrate the concept, consider that we are looking at a mining stock. It is just
announced that there was major accounting fraud committed by the company, in which
they overstated earnings. Long-term downward price movements would be expected.
Contrastingly, imagine we are looking at the same mining stock but for some reason
today, it gained 20% when there was no outwardly visible reason that should have
happened. We do not expect this move to be permanent, and as such this would be
considered a temporary mispricing. In short, fundamental changes cause permanent
moves, where as temporary changes do not.
III. Experiment
In this experiment, we will download the open and close data for all stocks in the
S&P 500 from yahoo Finance using the quantmod package in Yahoo Finance. From here,
we will calculate returns as such:
𝐶𝑙𝑜𝑠𝑒𝑅'(
=
*+(,-
*+(
− 1 (1)
𝑂𝑝𝑒𝑛𝑅'(
=
*+(4-
*+(
− 1 (2)
𝐴𝑣𝑔. 𝑅'(
=
9
:
𝛴:
𝑅'(
− 1 (3)
𝑊ℎ𝑒𝑟𝑒 𝑥 = 𝑠𝑡𝑜𝑐𝑘 𝑥, 𝑦 = 𝑠𝑡𝑜𝑐𝑘 𝑦, 𝑡 = 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 1,2, … 𝑛 ,
𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠, 𝑎𝑛𝑑 𝑃 𝑥 𝑡
= 𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑆𝑡𝑜𝑐𝑘 𝑥 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡

5. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 5
Each ticker is a data frame (a generalized matrix of data), and each data frame is an
element within the list containing all of the stock data. After we extract the open and
close column data of each stock and calculate the returns, we store this data in a list of its
own, from which we will calculate all further statistics. The subsequent statistics are all
rolling window calculations, with the window being 60 days. Other practitioners may feel
free to change the rolling window size and observe differences in the strategy’s
performance.
We use equations 1 for the first 60-day period moving window, as it represents the
observed before trading begins. However, once we are assumed to be trading with real
time data, we must use equation 2. The reasoning is that if we want to trade in period t,
we only at that time can observe data prior to that point, not ahead. As such, returns are
calculated open to open. After these vectors are reorganized, we take a 60-day moving
average of the returns, and proceed to perform a 60-day moving window calculations of
the following statistics:
𝜎' =
Q(S+(
T UV+(
)X
:T9
(4)
𝜌',Z[
=
]^_ ',Z[
`+`ab(
(5)
𝛽'(
=
𝜌',Z[
`+
`ab(
(6)
𝑊ℎ𝑒𝑟𝑒 𝜎' = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛, 𝜌',Z[ = 𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠, 𝑅'(
= 𝑟𝑒𝑡𝑢𝑟𝑛 , 𝑎𝑛𝑑 𝛽'(
= 𝐵𝑒𝑡𝑎

6. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 6
The standard deviation (eq.4) will represent the risk associated with each asset,
the correlation (eq.5) will be a measure of the market and the stock’s respective
correlation, and beta (eq.5) is a measure of how risky an asset is relative to the market.
These equations play several critical roles in our experiment. One of those functions is
determining the optimal amount by which we should hedge our portfolio:
𝐻 = −𝛽 𝑃𝑜𝑟𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝐻 = ℎ𝑒𝑑𝑔𝑒 𝑟𝑎𝑡𝑖𝑜, 𝑎𝑛𝑑 𝑃𝑜𝑟𝑡 = 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑉𝑎𝑙𝑢𝑒
As mentioned before, hedging is an integral component to statistical arbitrage
strategies. In the context of this experiment, the hedge ratio is the percentage of our
portfolio investment that we devote to short selling the market. We hedge in statistical
arbitrage because we wish to lock the gains we observe from a temporary positive
disequilibrium with the stock relative to the market.
For example, say we observe a price rise substantially while the market overall
did not move. As per the concept of reversion, we expect that over the next period that
this stock will fall and go closer towards its equilibrium price relative to the market.
Because we wan to preserve the positive return yielded from this disequilibrium, we short
the market to offset the losses, from the stock being held through this period.
One-Factor Return Model
For the experiment, we will use the following model:
𝐿𝑒𝑡 𝑦
∗
= 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑟𝑒𝑡𝑢𝑟𝑛 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡, 𝛼 = 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 ,
𝜖 = 𝑛𝑜𝑖𝑠𝑒 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡
𝑦
∗
= 𝛼 + 𝛽 𝑅Z[ + 𝜖 (7)

7. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 7
The only systemic factor in returns is the market. As such, when we subtract the
portion of the returns attributed to the market, we are left with the following:
𝛼 = 𝑅'(
− 𝛽' 𝑅Z[(
(7)
𝑊ℎ𝑒𝑟𝑒 𝛼 = 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙
The residual (eq.7) is defined as the returns specific to the stock, where it is
here an average. To trigger a trade due to very high or very low residuals, we
calculate the following statistics as well:
𝜎m+(
= 𝜎'(
n − 𝛽'(
n 𝜎'(
n (8)
𝑠𝑖𝑔𝑛𝑎𝑙 =
m
`o+(
(9)
𝑊ℎ𝑒𝑟𝑒 𝜎m+(
= 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 𝑟𝑖𝑠𝑘 (𝑠𝑡𝑑. 𝑑𝑒𝑣. 𝑜𝑓 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠)
Residual Risk (eq.8) essentially is the in period standard deviation of the residuals
themselves. Plainly spoken, when we take the ratio of the residual to the residual risk, we
get a number that tells us how many standard deviations from the mean the alpha is,
giving us our trading signal (eq.9).
Why do we focus on residuals?
The two things that affect residuals are 1) long term fundamental changes and 2)
trading imbalances that cause the residual to be artificially high/low. Fundamental
changes to a stock, such as an increase in earnings or a structural change in the sector of
the stock, cause longer term changes to the price levels that a stock will gravitate around.

8. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 8
As such, this doesn’t necessarily help us for finding intraday trading opportunities. Being
that we want to look at the short term, we want to focus on non-systemic factors only,
and arguably factors that are relatively short lived in their effects.
The signal has a lower limit of negative infinity and an upper limit of infinity. We
consider our statistical significance level in this experiment to be 95%, or that any
absolute value of the signal greater than or equal to two represents an outlier. However,
further research can be performed on individual stocks and/or portfolios to determine an
optimal significance level.
Trading Algorithm
Due to the significance level being 95% in this experiment, the bounds between
which we will hold a cash position in our account is -2 < signal < 2. When signal < -2, we
say that statistically the residual is artificially low and that the following period is likely
to feature a signal value closer to 0. As such, we will place an un-hedged investment in
the stock for this period. When the signal is > 2, we say that statistically the residual is
artificially high, and that the following period is likely to feature a signal value closer to
zero. As such, we will hedge our long investment in the stock by shorting the market.
In the case of our experiment, the market is the SPY. Our portfolio will be a
subset of the market, dependent on which stocks are statistically most likely to exhibit
reversionary moves. Plainly stated, if a stock is likely to make a reversionary move, it
will be traded. If not, we will opt to hold a cash position instead.

9. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 9
IV. Results and Discussion
Summary Statistics:
Total Number of Days 504
Total Trades 317,649
Average Daily Trades 138
Minimum Daily Trades 9
Maximum Daily Trades 362
Average Return
0.37%
Std. Dev. Of Returns 0.36%
Cumulative Return 403.80%
Sharpe Ratio .99

10. Testing Mean Reversion: Evaluation of a Statistical Arbitrage Strategy Over Time 10
Returns in Theory vs. Reality
It is important to note that while these returns are certainly quite appealing, that it
is highly improbable that they would ever be realized should this strategy be employed in
live trading activities. The reasoning behind this is that we are observing prices, and
ultimately the returns derived from them, close to close. However, to properly execute
this strategy, we cannot hold onto these equities overnight, putting us in a position where
we must decide when to sell and buy. Secondly, it is not guaranteed that we can buy at
the prices we observe, nor is it guaranteed that we can short at the prices we observe. For
example, the demand to short in instances where the signal is significantly higher than 2,
will be very high. This will in turn create higher costs for the shares that are available and
therefore impact profitability. We also assumes our broker can even find the shares to
short, when it is possible that our broker in some instances has none or at least not the
optimal amount we want to short. Moreover, there are transaction costs for each trade that
would accumulate over this period, which would also impact the profitability.