Electronic copy available at: http://ssrn.com/abstract=2774063

Demystifying pairs trading: The role of volatility and

correlation

Stephanie Sarah Riedinger

Catholic University Eichstaett-Ingolstadt

Auf der Schanz 49, 85049 Ingolstadt, Germany

Phone: 0049 841 937 219 26, Mail to: Stephanie.Riedinger@ku.de

Abstract

This paper investigates how the two technical drivers, volatility and correlation, in-

fluence the algorithm of the pairs trading investment strategy. We model and em-

pirically prove the connection between the rule-based pair selection, the trading al-

gorithm, and the total return. Our insights explain why pairs trading profitability

varies across markets, industries, macroeconomic circumstances, and firm charac-

teristics. Furthermore, we critically evaluate the power of the traditionally applied

pair selection procedure. In the US market, we find risk-adjusted monthly returns of

up to 76bp for portfolios, which are double sorted on volatility and correlation be-

tween 1990 and 2014. Our findings are robust to liquidity issues, bid-ask spread,

and limits of arbitrage.

Keywords: Pairs trading, Relative-value arbitrage, Volatility, Limits of arbitrage

JEL Classification: G11, G12, G17

Electronic copy available at: http://ssrn.com/abstract=2774063

1. Introduction

Does the act of measuring influence the outcome of an experiment? In science, this question is

frequently asked to identify the Hawthorne effect, also called observer effect. Empirical find-

ings lose their credibility, if the applied methodology interferes with the outcome.1 We trans-

fer this issue to investing and ask whether the return of a technical, rule-based strategy is driv-

en by the algorithm itself. In this paper, we answer the initial question for the highly recog-

nized pairs trading investment strategy, which exploits arbitrage profits from temporary mar-

ket inefficiencies.2 Gatev/Goetzmann/Rouwenhorst (2006) find that certain industry portfolios

outperform others. Do/Faff (2010) explain a superior return in times of market distress, be-

tween January 2000-December 2002 and July 2007-June 2009, with a decrease of market effi-

ciency. Jacobs/Weber (2015) observe varying returns across different countries and propose

the number of eligible pairs and limited investor’s attention as possible explanations. Without

doubt, less efficient markets or industries allow more mispricing opportunities, which in turn

increase the strategy’s return. Yet, this paper’s main purpose is to demonstrate that at least

part of the additional return originates from the interaction of the algorithm with higher mar-

ket volatility and correlation levels over time, across countries, and across industries.

We extend the current knowledge about pairs trading with two important findings. Firstly, we

provide evidence that the traditional formula to select pairs privileges pairs with high correla-

tion and low stock volatility, while disregarding other pairs. However, this selection fails to

exploit the full return potential. Therefore, we also study the returns of pairs with different

pair volatility and correlation levels. Secondly, our results demonstrate that pair volatility and

correlation significantly influence the two return dimensions, the return per trade and the trad-

ing frequency, through the trading algorithm. This paper contributes furthermore to the litera-

ture on gradual information diffusion, volatility timing, and limits to arbitrage. Altogether, our

findings contribute to better understand previous academic findings by providing a technical

foundation, how the input factors, pair volatility and pair correlation, interact with pairs trad-

ing in the cross-section and over time. In this context, pair volatility and pair correlation are

1 For instance, in quantum physics it is impossible to detect an electron without interacting with a photon. How-

ever, this interaction biases the electron’s path. 2 In contrast to the classical Hawthorne effect, the observed object, the return, does not adapt its behavior as it is

observed. Rather, the observation system, that is the algorithm, influences the return.

the technical executing factors that translate illiquidity, market distress and other factors into

higher returns. Furthermore, our findings are of high relevance for practitioners, who wish to

implement profit increasing modifications. Finally, our results impressively emphasize the

importance of critically reflecting the impact of a rule-based procedure on the outcome.

The profitability of pairs trading is often explained by the information diffusion hypothesis

(Gatev/Goetzmann/Rouwenhorst (2006), Engelberg/Gao/Jagannathan (2009); Chen/Chen/Li

(2013), Jacobs/Weber (2013, 2015)). The hypothesis argues that pairs trading exploits tempo-

rary mispricing, which arises as one stock incorporates common news faster than another one.

Based thereupon, the idea of univariate pairs trading is simple. Each month, a new pairs trad-

ing period, consisting of a twelve-months identification period and a subsequent six-months

trading period starts. Gatev/Goetzmann/Rouwenhorst (2009) suggest a simple measure to se-

lect pairs for trading, the sum of squared price differences (SSD). At the beginning of each

identification period, they normalize all stock prices to one and compute the daily price dif-

ference between any possible pair combination in the subsequent twelve months. For each

pair, the SSD cumulates the daily price spreads: SSD = ∑ (Pt,a-Pt,b)2T

t=1 , where Pt,a is the nor-

malized stock price of stock a at day t. Furthermore, Gatev/Goetzmann/Rouwenhorst (2009)

compute the standard deviation of a pair’s price spread. The twenty pairs with the lowest

SSD form a trading portfolios, and they are eligible for trading in the consecutive six-months

trading period. Low SSD stocks are chosen, because a small SSD indicates similar price de-

velopment of a pair’s stocks in the past and is expected to predict further commove in the fol-

lowing trading period. At the beginning of the trading period, all prices are again normalized

to one. During the next six months, the author’s check every day whether the price spread of

any pair included in the trading portfolio exceeds two times the standard deviation , as com-

puted during the identification period (2-rule). If the 2-rule applies, they initiate a one unit

long position in the worse performing ‘loser’ stock and a one unit short position in the better

performing ‘winner’ stock. The underlying idea is that the price spread is a temporary mis-

pricing, as both stocks are expected to commove. The positions exploit the anticipated price

reversion and are neutralized as soon as both normalized prices fully converge. On average

during January 1991 and December 2014, a traditional pairs trading portfolio earns a monthly

risk adjusted return of 37bp in the US market. As the complete procedure is highly rule-

driven, we question how the strictly defined process influences the strategy’s return.

This paper challenges the traditional selection procedure and trades pairs that are sorted on

pair volatility and pair correlation. As previously mentioned, we demonstrate that the selec-

tion procedure picks highly correlated pairs with low volatility. Therefore, we are highly in-

terested how traditionally disregarded pairs, with various volatility and correlation levels, per-

form. We suspect that volatility and correlation not only influence the selection formula, but

also the trading formula, due to the technical nature of both formulas. During the twelve-

month identification period, we calculate the volatility of each stock and the correlation of all

possible stock combinations based on daily prices. We refer to this correlation as pair correla-

tion ρAB. Furthermore, we define the pair volatility σAB of a pair as the sum of stock A’s and

stock B’s individual historic volatility σAB = σA2 + σB

2 during the identification period. A port-

folio, containing highly correlated pairs with low pair volatility, earns a monthly risk adjusted

pairs trading return of 47bp, whereas a portfolio with similar pair correlation, but highly vola-

tile, stocks earns even 60% more. Even a portfolio with low pair volatility and negatively cor-

related pairs earns a return of 46bp. We wonder whether this astonishing outperformance is

incidental. Therefore, we systematically investigate the return of different levels of pair vola-

tility and pair correlation. Each month, we allocate all pair combinations based on their pair

volatility σAB and their correlation ρAB to one of five correlation quintiles and one of five pair

volatility quintiles. Afterwards, we randomly select twenty pairs out of the intersection of

each pair volatility and pair correlation quintile and include them into one of twenty-five trad-

ing portfolio. These portfolios are labelled VCS Portfolios (volatility and correlation sorted

portfolios) in the following. Pairs within these VCS Portfolios are eligible for trading in the

consecutive trading period. The most surprising result from the analysis is that twenty, out of

twenty-five, VCS Portfolios significantly outperform the return of the traditional portfolio,

with monthly risk-adjusted returns of 39bp up to 209bp. The return increases for higher pair

volatility and higher correlation, although both effects are not linear across the quintile levels.

Actually, the traditional pairs trading algorithm was designed to perfectly exploit arbitrage

opportunities. However, the demonstrated failure to outperform alternatively formed portfoli-

os emphasizes the need to better understand how the algorithm really works.

To demystify the ‘black box’ pairs trading, we model the technical interdependencies of the

rule-based selection and trading algorithm with the strategy’s return generating components in

figure 1. We argue that the algorithm influences two major components, the selection of pairs

eligible for trading and the maximum return per trade.

[Insert Figure 1]

Firstly, we claim that the strategy’s SSD selection criterion determines the stock selection.

The SSD picks stock pairs, whose historic price spread is low. Our findings indicate that pair

volatility AB and correlation ρAB explain up to 88% of the SSD’s variance. Furthermore, high

pair volatility and low pair correlation increase the SSD. Hence, we conclude that the SSD

selection procedure selects pairs with a low volatility and a high correlation level. To elimi-

nate this selection bias effect in the following analyses, we consider pairs with all pair volatili-

ty and correlation levels.

Secondly, we argue that the traditional algorithm limits the maximum return per trade via the

2σ-rule. The trading algorithm initiates positions as soon as the price difference of a pair’s

stocks exceeds 2σAB. Positions are closed afterwards, as soon as both prices fully converge.

Hence, we approximately earn the initial price spread. The return per trade is obviously high-

er, if the initial price spread 2σAB is high. We conclude that the size of 2σAB restricts the max-

imum return per trade. We elaborate that σAB equals the standard deviation of a portfolio

comprising a long and a short stock. 2σAB can be rewritten as formula with input parameters

pair volatility and pair correlation: 2σAB = 2√σA2 +σB

2 – 2ρAB

σAσB . The first derivation reveals

that high pair volatility and negative correlation enhance the return per trade. Our empirical

results validate that negatively correlated pairs earn twice as much as per trade in comparison

to highly correlated pairs (same volatility level). Furthermore, highly volatile pairs earn up to

three times more per trade than low volatile pairs (same correlation).

Together with the return per trade, the trading frequency, or more precisely the number of

trades, determine the total return of pairs trading. For instance, Do/Faff (2010) find that se-

lecting pairs with frequent stock price crossings significantly enhances the return. We propose

that the trade opening trigger of 2σAB is defined based on the computed pair volatility and

correlation during the identification period. Yet, the likelihood to open a new trade is higher,

if the pair volatility increases or the correlation decreases in the subsequent trading period.

Consistent with a mean-reversion process, our results report more volatility increases for ini-

tially low pair volatility, and more correlation decreases for initially high correlation levels.

More trades in turn increase the trading frequency, which is beneficial for the strategy’s total

return.

Notably, high pair volatility increases both, the return per trade and also the trading frequen-

cy. In contrast, high pair correlation increases the trading frequency on the one hand, but de-

crease the return per trade on the other hand. Yet, it is not obvious, whether the influence on

the return per trade or on the trading frequency dominants. This overlaying effects explain,

why there is no linear change across different pair volatility and correlation quintile levels of

the VCS portfolio’s monthly returns.

One puzzle remains, why pairs trading is profitable for less correlated pairs, although the

strategy intends to exploit temporary mispricing of close economic substitutes. We disagree

that close economic substitutes must be highly correlated and argue, that the information dif-

fusion hypothesis demands a gradual information diffusion across two close economic substi-

tutes. Exactly this required temporary dissimilar price development causes a lower correla-

tion. Highly correlated pairs are likely to incorporate common information simultaneously and

hence offer little chance to exploit temporary mispricing.

The remaining parts of the paper are organized as follows. In section 2, we derive three re-

search propositions from theory. Section 3 introduces the data selection, the methodology of

pairs trading, the return calculation, applied modifications and the test design. Afterwards,

section 4 explores the link between the classical selection criterion, pair volatility and pair

correlation. Section 5 validates our hypotheses with empirical evidence on trade level, where-

as section 6 reports results in calendar time. Section 7 discusses the profitability of close and

non-close economic substitutes. Finally, section 8 discusses possible return sources and ad-

dresses robustness issues, before section 9 concludes.

2. Development of research questions

Although previous papers do not directly examine the influence of pair volatility and correla-

tion, their findings motivate our research. Firstly, out of fifty pairs with the lowest distance

measure (SSD), Do/Faff (2010) select the twenty pairs with the highest number of price cross-

ings during the identification period and observe a higher profitability. These authors argue

that multiple divergences and convergences of a pair within the identification period, predict

more trading opportunities. Moreover, they find an increased profitability during times of

market distress from January 2000-December 2002 and July 2007-June 2009. Both bear mar-

ket phases were accompanied by high market volatility. Secondly, the success of pairs trading

was explored in many countries, for instance Andrade/diPietro/Seashole (2005), Perlin (2009)

and Bolgün/Kurun/Güven (2010) among others. Most notable, Jacobs/Weber (2015) provide a

broad overview for 34 countries worldwide. They observe that less developed markets usually

realize a higher return. These markets are normally less liquid and hence possess higher mar-

ket volatility. Thirdly, Engelberg/Gao/Jagannathan (2009) provide a number of in-depth anal-

yses about the drivers of opening, horizon and divergence risks. Among other findings, they

demonstrate the positive influence of idiosyncratic volatility on the opening probability and a

negative impact of idiosyncratic volatility on the time till convergence and also on the diver-

gence probability. Similarly, Jacobs/Weber (2015) examine the influence of idiosyncratic vol-

atility as proxy for limits to arbitrage and find a positive influence of volatility on the return.

Finally, Huck (2015) investigates the effect of total market volatility over time. He initializes

positions, if the VIX is categorized into a certain regime, in addition to the traditional opening

signal. However, the return is only significant at times of an increasing or high 3-month mov-

ing average VIX.

We understand pair volatility and pair correlation as technical drivers of the pairs trading re-

turns. The total return consists of two return building blocks, the return per trade and the trad-

ing frequency. Firstly, the return per trade is defined as the average return per single round-

trip of a pair. Obviously, a high return per trade increases the total return. Secondly, the trad-

ing frequency is defined as the number of trades of a pair during one trading period. Earning

the return per trades more often raises the strategy’s total profitability. We proceed with intro-

ducing our three research proposals. In section 2.1, we derive why we expect pairs, which are

picked by the traditional selection criterion SSD, to possess certain pair volatility and correla-

tion levels. Afterwards, we explain how the two two drivers, pair volatility and correlation,

influence the return per trade in section 2.2 and the trading frequency in section 2.3.

2.1 The SSD selection criterion

A broad stream of literature applies the distance measure (SSD), as proposed by Gat-

ev/Goetzmann/Rouwenhorst (2006), to identify close economic substitutes. The SSD com-

putes the sum of squared differences of the normalized prices of two stocks and selects the

twenty pairs with the lowest SSD:

Distance Measure (SSD) = ∑ (PA,t-PB,t)2

=Tt=1 (1.1)

= ∑ (PA,t2 + PB,t

2 − 2ρA,B)Tt=1 (1.2)

where ρA,B is the Pearson correlation coefficient of the standardized price time series.3 From

the mathematical term in 1.2 we see that maximizing the correlation coefficient equals mini-

mizing the SSD (Krauss (2015)). Based thereupon, we conclude that the SSD typically selects

highly correlated pairs.

Moreover, we argue that low stock volatility is also crucial to minimize the SSD. Term 1.2

furthermore reveals that high stock prices for PA,t or PB,t also increase the SSD. Both prices

are normalized to one in t = 0. Thus, high values for PA,t and PB,t can only develop, if stock

volatility is high. Whether the minimizing influence of correlation on the SSD or the increas-

ing effect of stock volatility dominates, strongly depends on the magnitude of both factors.

Term 1.1 helps to develop a good intuition. For instance, consider three stocks A, B, and C,

whose prices PA, PB and Pc are all normalized to one in t = 0. PA and Pc are uncorrelated with

ρA,C = 0, but both are independent and normally distributed random variables with the same

standard deviation . Hence, the their confidence interval for the price realizations of PA and

PC is identical with [1 − 𝓏1−𝛼

2

𝜎

√𝑛, 1 + 𝓏1−

𝛼

2

𝜎

√𝑛]. PB is perfectly correlated with PA, however a

△ change in PA translates into a 4△ of PB.4 Consequently, PA and PB are perfectly correlated

with ρA,B = 1. However, the confidence interval of PB is significantly larger, so the deviation

from the expected value is usually higher. The daily price difference between PA and PC might

therefore be smaller than the price difference between PA and PC, although ρA,C = 0 and

3 The Pearson correlation coefficient equals: 𝜌𝐴,𝐵 =

1

𝑡−1∑ (

𝑃𝐴,𝑡−𝑃𝐴̅̅ ̅̅

𝑠𝑃𝐴

) (𝑃𝐵,𝑡−𝑃𝐵̅̅ ̅̅

𝑠𝑃𝐵

)𝑇𝑡=1 , where 𝑃𝑖,𝑡 =

𝑃𝑖,𝑡−𝑃�̅�

𝑠𝑃𝑖

are the

daily z-transformed normalized prices. 4 For instance, if PA raises from 1$ to 1.01$, PB raises from 1$ to 1.04$.

ρA,B = 1. For example in t = 1, Stock A rises from PA,0 = 1 to PA,1 = 1.01. Hence, Stock B

rises from PB,0 = 1 to PB,1 = 1.04. The price difference between stock A and B is: |PA,1-PB,1| =

0.04. Stock C is uncorrelated with stock A and does not move at all, PC,0 = PC,1 = 1. The price

difference between stock A and C is therefore: |PA,1-PC,1| = 0.01. If we cumulate the daily

squared price differences according to 1.1, we come to the following conclusion: The SSD for

two uncorrelated stocks (A and C) can be smaller than the SSD of two highly correlated

stocks (A and B), if the highly correlated pair includes at least one highly volatile stock. Alto-

gether, we expect a positive link between the SSD and correlation, and a reverse link for the

SSD and pair volatility.

Research proposition 1:

Pairs, with a high correlation and a low pair volatility, are associated with a low SSD.

A validation of research proposition 1 would indicate, that applying the SSD results in trading

with highly correlated pairs with little volatility. However, it is unclear, whether these pair

volatility and correlation combinations are beneficial for the return per trade and the trading

frequency.

2.2 Return per trade

We initialize positions as soon as the stock pair’s price difference exceeds two historic stand-

ard deviations 2hist, as calculated during the identification period. The return is continuously

earned while the pair is ‘open’, until stock prices fully converge. At that point in time, we

neutralize positions. We distinguish four different closing types. Firstly, trades that fully con-

verge during the trading period are denoted as ‘natural trades’. Secondly, pairs which not fully

converge on the last day of the trading period, are forcefully closed and labelled ‘incomplete

trades’. Thirdly, as we trade with highly volatile stocks, investors might be concerned about

the asymmetric return profile5 of pairs trading. Therefore, we close a pair, if the price differ-

ence exceeds 4hist on any day while a trade is open, similar to Engelberg/Gao/Jagannathan’s

(2009) 10-day-maximum strategy. Hence, the return potential is symmetric within the range

of -2ơ and +2ơ. These trades are labelled ‘overshooting trades’. Affected pairs are blocked for

5 The positive return per trade is limited to 2ơ, while pairs that further diverge might generate an infinite loss.

the rest of the trading period.6 Finally, ‘delisted trades’ are trades that automatically close if

one stock is delisted.

We conjecture that a pair always earns a positive return, if prices fully converge (natural

trade).7 The return per trade equals the stock price difference at the time of the opening. Posi-

tions are initialized as soon as prices diverge by more than 2hist. Thus, the return of a natural

trade equals 2hist plus an overshooting component. The overshooting component occurs, if

the price difference exceeds the 2ơhist trigger during the day and prices further diverge until

closing prices are set. We derive:

Return per natural trade = 2σhist+ Overshooting. Accordingly, the return of a natural trade is

higher, if the historical standard deviation is larger.

Pairs trading can be understood as a portfolio, consisting of one unit of stock A (long) and one

unit of stock B (short). Therefore, the pair’s historic standard deviation can easily be comput-

ed. A simple mathematic transformation uncovers the relationship between trade return, vola-

tility of stock A and B and the pair’s correlation coefficient pAB:

Return per natural trade = 2√σA2 +σB

2 - 2ρAB

σAσB + Overshooting, (2)

The formula validates our conjecture that the return of a successful trade is always positive.

The overshooting component is always positive, as otherwise the total price difference would

not exceed 2ơhist. As stock A is unequal to stock B, we conclude (σA-σB)2>0. Hence, the in-

fluence of volatility on the return per trade is always positive, as σA2 +σB

2 > 2ρAB

σAσB with

ρAB∈[-1,1]. Based upon this inequality, we argue that a higher volatility level of stock A or

stock B (or both) increases the historic standard deviation and thus the return per trade.

On the contrary, higher pair correlation decreases 2ơhist. Neglecting the overshooting compo-

nent, the first differentiation of the function regarding the historical correlation coefficient

6 Otherwise, a new trade would open on the next day, as the price spread is still above 2ơhist because prices are

not set back to one during the trading period. We also assume that a price spread of 4ơhist indicates a permanent

price spread and thus refrain from further trading with these pairs. 7 A possible unprofitable price development of one stock is always overcompensated by the return of the other

stock that overcomes the previous price difference. For instance, if the stock price of the short position is rising,

the return of the long position will over compensate the negative return of the short position as otherwise, stock

prices would not fully converge.

equals:

∂f(σA,σB,ρAB)

∂ρAB

= - 2σAσB

√σA2 + σB

2 - 2ρABσAσB

, (3)

The first differentiation is always negative, as the numerator and the denominator are positive.

The return should therefore increase with declining historical correlation. We conclude that

more volatile stocks and a negative or low stock correlation enhance the return per trade.

In contrast, overshooting trades, which are closed if the price spread exceeds 4ơhist, always

yield a negative return. Recall that positions are initialized at a price spread of 2ơhist and are

closed at 4ơhist. The potential loss is hence limited to 2ơhist and therefore symmetric to the re-

turn potential of natural trades. The link between return on the one side and volatility and cor-

relation on the other side is reverse to natural trades. Hence, low volatility and high positive

correlation reduce the downside risk.

Incomplete and delisted trades neither fully converge nor diverge by more than 4ơhist. These

trades earn a positive return, if prices converge. However, they generate a loss, if prices di-

verge. In conclusion, the link depends upon the direction of the stock movement.

Research proposition 2:

Highly volatile pairs and negatively correlated pairs increase the return per trade of converg-

ing trades. In contrast, low volatility and high correlation reduce the downside risk of diverg-

ing trades.

2.3 Trading frequency

Turning now to the second return dimension, the trading frequency, we argue that a high trad-

ing frequency is beneficial for the strategy’s total return. The findings of Do/Faff (2010), who

observe a higher profitability for pairs whose prices frequently intersect each other, motivate

us to investigate the drivers of the trading frequency. Obviously, more trades with positive

return are beneficial, whereas many trades with a negative return are harmful. A pair can gen-

erate several natural trades within one trading period. Earning the price spread several times

during one trading period successively increases the total return. In contrast, incomplete

trades, overshooting trades and delisted trades always represent the last trade within a trading

period. A pair cannot open after an incomplete trade, as the incomplete trade is closed on the

last day of the trading period. Pairs are blocked for further trades after an overshooting trade,

as otherwise pairs are opened and closed daily, while the price spread still exceeds 4ơhist.

Likewise, a particular pair obviously cannot open after the delisting of one incorporated stock.

So, as unprofitable trades are limited to one, increasing the trading frequency only raises the

number of natural trades, the closing type which always yields a positive return.

The trading frequency is determined by the probability to open a pair and the time till conver-

gence. The probability to generate loss-making overshooting trades is linked to the probability

of further price divergence, while a pair is open. Engelberg/Gao/Jagannathan (2009) explore

the influence of various variables on the former probabilities, including average mean idio-

syncratic volatility. The authors observe a positive influence of idiosyncratic volatility on the

opening probability and a negative influence on the time till convergence and on the probabil-

ity of further price divergence.

We concentrate on the probability of a pair opening, and we argue that a volatility increase or

a pair correlation decrease between the identification and the trading period raise the opening

probability. Let Xt be a normally distributed random variable,8 which describes the price

spread, the difference between the normalized prices of stock A and B, on day t. The left

graph of figure 2 displays the density function of Xt during the identification period (Den-

sityID_Period) with Xt~N(0; σhist). Pairs trading expects both stocks to strictly co-move, so the

expected value μ of the price spread Xt should be zero. The distribution of Xt is symmetric, as

it is equally likely that stock A outperforms stock B and vice versa. Following the traditional

pairs trading algorithm, the trigger point to open a trade is 2ơhist. More precisely, a pair is

opened, if Xt falls below μ-2ơhist or exceeds μ+2ơhist.

[Insert Figure 2]

The dashed area in the left figure, between μ-2ơhist and μ+2ơhist, represents the density to not

exceed the trigger points. We derive from the density function that this probability is 95.45%.

If the volatility does not change between the identification and the trading period, the proba-

bility to open a new trade is hence 4.55%. However, the opening probabilities might change

8 This is a simplified assumption to demonstrate the idea. In reality, Xt is a function of Xt-1, the return of stock A,

and the return of stock B.

dramatically, if Xt’s volatility increases to ơTrade. The former trigger points of μ +/- 2ơhist,

based on the volatility during the identification period ơhist, is still active. The probability of Xt

to exceed the trigger points, illustrated by the grey area in the right figure, is significantly

higher than before, with Xt~N(0; σTrade). In an extreme case the opening probability could

increase from 4.55% to 31.73%, if the volatility doubles (ơTrade = 2ơhist) between the identifi-

cation and the trading period. On the contrary, a drop in the volatility of Xt (ơTrade < ơhist) re-

duces the probability to open a new trade.

We now turn to the drivers of Xt’s volatility. The volatility of the price spread Xt (ơhist and

ơTrade) is positively influenced by the stock volatility σA and σB. Furthermore, Xt’s volatility is

negatively affected by the correlation of stock A and B, ρA,B.9 Considering all links together,

we conjecture that an increase of stock volatility σA or σB, or a decrease of pair correlation

ρA,B, between the identification and trading period, increase the volatility of Xt. The increase

of Xt’s volatility, in turn, raises the chance to open a new trade and hence raises the trading

frequency, which is beneficial for the total return. In contrast, we expect that a decrease of

stock volatility and a pair correlation increase reduce the trading frequency. Therefore, select-

ing promising pairs is also a matter of correctly forecasting volatility and correlation changes.

In the volatility quintile, which includes the pairs with lowest pair volatility, we expect to find

some stock pairs with permanently low pair volatility. But we also reckon to find some pairs,

whose pair volatility is only temporarily low and which are hence expected to rise in the fu-

ture. Likewise, the pair volatility quintile, including the pairs with the highest pair volatility,

might also include some pairs with temporarily high pair volatility. We hypothesize that at

least some pairs with temporarily low (high) pair volatility will exhibit a pair volatility in-

crease (decrease) in the near future. Therefore, we argue that pairs within the lowest pair vola-

tility quintile are more likely to observe a favorable volatility increase, and hence generate

more trades, than pairs within the highest pair volatility quintile. We argue similarly for corre-

lation and expect to observe more correlation decreases for pairs within the highest correlation

quintile, in contrast to pairs within the lowest correlation quintile.

Research proposition 3:

9 We derive these connection from the following mathematical transformation: V(X) = V(PA − PB) = V(PA) +

V(PB) − 2Cov(PA, PB) = V(PA) + V(PA) − 2ρA,B√V(PA)√V(PB).

The number of natural trades is increased by low stock volatility and high pair correlation

during the identification period, coupled with higher volatility and lower correlation during

the trading period. Pair volatility increases are more likely for pairs with currently low pair

volatility, and correlation decreases are more likely for pairs with a currently high correla-

tion.

Altogether, we conclude that high pair volatility increases the return per trade, but decrease

the trading frequency. Low correlation increases the return per trade, but decreases the num-

ber of trades. So the puzzle remains, which volatility-correlation combination of pairs is the

best for the strategy’s total return. To address this puzzle, we compute the monthly return of

pairs trading to evaluate whether high correlation is beneficial for the total return.

3. Methodology and data selection

After introducing the three central research propositions of this paper in section 2, we pro-

gress with introducing our data selection (3.1) and a detailed description of the traditional

pairs trading algorithm (3.2). Afterwards, we precisely explain how the return is calculated on

the per trade level (3.3.1) and in calendar time (3.3.2). Finally, section 3.4 motivates applied

modifications and describes the test design. Several streams of literature coexist, which apply

different algorithms for selecting pairs or trading. Most prominent are the univari-

ate/multivariate distance approach (Gatev/Goetzmann/Rouwenhorst (2006), Engel-

berg/Gao/Jagannathan (2009), Jacobs/Weber (2015)), the cointegration approach (Vidya-

murthy (2004), Lin/McCrae/Gulati (2006)), the copula based algorithm (Stand-

er/Marais/Botha (2013)), the stochastic approach (Tourin/Yan (2013)) and mixed models. For

further reference, Krauss (2015) provides a detailed overview and evaluation.

Hauck/Afawubo (2015) and Huck (2013, 2015) empirically evaluate different approaches. We

concentrate on the most prominent univariate distance approach, first introduced by Gat-

ev/Goetzmann/Rouwenhorst (2006).

3.1 Data Selection

We obtain the historical index constituents of the S&P 1500 between January 1990 and De-

cember 2014 from the Compustat Monthly Updates - Index Constituents file. Furthermore, we

draw daily stock price data from the WRDS CCM merged database. We exclude stocks for a

particular pairs trading cycle, if any of the three: bid, ask, or closing price, is unavailable on

more than one day during the identification period. After this preselection, approximately

1900 stocks on average are eligible for trading each month, forming around 1.8 million possi-

ble pair combinations. In contrast to fellow papers, we keep stocks in our sample, which are

not traded on one day during the identification period, as bid and ask prices are available and

trading is hence possible. This modification allows us to include less liquid, more volatile

stocks, which is necessary to accurately evaluate our research propositions two and three. All

computations are executed in Stata and the implemented matrix language Mata.

Gatev/Goetzmann/Rouwenhorst (2006) and follow-up papers implement a one-day-waiting

strategy to consider a potential upward bias in returns caused by the bid-ask bounce.10

After

the initial trading signal, they wait for one day until they trade. We refrain from approximat-

ing the bid-ask-spread and trade with the exact last bid or last ask price. At trade initialization,

we buy the long position at the ask price and sell the short position at the bid price. We neu-

tralize the long position at the bid price and the short position at the ask price. Similarly, we

use these latter prices for the daily evaluation. Additional computations, like the average vola-

tility or correlation, are executed with closing prices.

3.2 The Pairs Trading Algorithm

Following Gatev/Goetzmann/Rouwenhorst (2006), one pairs trading period in our paper con-

sists of two phases – first, a 12-months identification period and a subsequent 6-month trading

period.

At the beginning of the identification period, all prices are normalized to one. During the con-

secutive twelve months, we compute three key figures for each possible pair combination AB

in the stock universe: the pair volatility σAB = σA2 + σB

2, the pair correlation ρA,B, and the tra-

ditional selection measure, the sum of squared differences of normalized prices (SSD).

As introduced in the previous section, Gatev/Goetzmann/Rouwenhorst (2006) calculate the

10

In an upward trend, the closing price most likely represents the ask price. As pairs trading sells the increasing

“winner stock”, an investor receives only the lower bid price. Likewise, the closing price of a decreasing stock is

more likely to be the bid price, which must be purchased at the higher ask price. Hence, using closing prices

might overestimate the real return.

SSD as sum of the squared deviations between the normalized price movements of two stocks

over the identification period:

Distance Measure (SSD) = ∑ (PA,t-PB,t)2T

t=1 (4)

where t is the day index during the identification period, Pa,t is the normalized price of stock A

(or B) at day t. At the end of the identification period, the authors form a portfolio, consisting

of the twenty pairs with the lowest SSD, which is traded during the next six months.

To avoid an in-sample bias, the trading period starts on the first day after the identification

period. Again, all prices are normalized to one. At the end of each day, the trading algorithm

checks for each pair, included in the trading portfolio, whether to open a new trade or close a

currently open trade. If the normalized prices of a pair diverge by more than two historic

standard deviations 2σhist, as calculated during the identification period, we initialize the self-

financing trade as follows: We sell one monetary unit of the better performing ‘winner’ stock

and simultaneously buy one monetary unit of the less successful ‘loser’ stock. As soon as both

prices completely converge, we neutralize both positions. Thereafter, a pair might open and

close several times during the trading period. Recall from section 2, that all open positions are

closed, if prices do not fully converge until the end of the trading period. Furthermore, trades

are also closed, if the price spread exceeds 4σhist (overshooting trade) or one stock is delisted

on any day, while the trade is active (delisted trade). A new pairs trading period starts every

month, so at each point in time, six portfolios are trading simultaneously11

.

3.3 Return Calculation

3.3.1 Return per Trade Calculation

To empirically assess research proposition 2, we calculate the return per trade, which is the

payoff of one particular trade. A pairs trader earns a positive return, if the underlying price of

the short position decreases or the long position rises. Consider the following example: The

historical standard deviation of a pair’s price spread is σhist = 0.05. Thus, we open the pair as

soon as the difference (|PA - PB|) between the normalized price of stock A (PA) and the nor-

malized price of stock B (PB) exceeds 0.1 (=2 σhist). For instance, PA,0 = 1 and PB,0 = 1 in t = 0,

11

As a result of the overlapping trading periods, at each point in time one of the portfolios starts to trade in the

particular month, the second one is active for already two months, the third one for three months and so on.

and PA,1 = 1.05 and PB,1 = 0.95 in t = 1. Thus, we open the trade in t = 1. We initiate a short

position in the ‘winner’ stock A with a capital commitment of CA,1 = (-1)$ and a long position

in the ‘loser’ stock B with a capital commitment of CB,1 = +1$. In t = 2, the normalized prices

of stock A and B fully converge to one: PA,2 = 1 (decrease of -4.76%) and PB,2 = 1 (increase of

5.26%). The capital commitments are CA,2 = −1$ ∗ (1 + (−0.0476)) = −0.9524, and

CB,2 = +1$ ∗ (1 + 0.0526) = 1.0526. As both normalized prices of stock A and B fully

converge, we close the trade and receive a payoff of: (CA,2 − CA,1) + (CB,2 − CB,1) =

0.05$ + 0.05$ = 0.10$. The return of 0.10$ represent the return per trade.

3.3.2 Calendar Time Return calculation

Furthermore, we also calculate the monthly return in calendar time. We closely follow Gat-

ev/Goetzmann/Rouwenhorst (2006) return calculation: All positions are marked-to-market

daily. We calculate the return 𝑟𝑖,𝑡, which is the return that stock i realizes between day t-1 and

t, for all stocks in our portfolio. The stock weight wi,t can be interpreted as capital investment

in stock i at day t-1 or as buy-and-hold strategy, which reinvested daily returns until day t-1.

At the opening day t = 1, the stock weight wi,t is defined as follows:

wi,1 = I = {

1 for a long position−1 for a short position0 if the pair is closed

(5)

At the following days t >1, the stock weigths wi,t are calculated as the product of the previous

days’ capital investment:

wt,i = I ∗ wt−1,i ∗ (1 + rt−1,i) = I ∗ (1 + rt−1,i) … (1 + r1,i) = I ∗ ∏ (1 + rt−1,i)t−1t=1 (6)

The pairs trader’s payoff of stock i on day t equals the weight (or capital investment in t-1) wi,t

multiplied with the stock return ri,t between t-1 and t:

Payofft,i = wt,i ∗ rt,i (7)

For instance, the stock commitment in stock A in t-1 is +1.25$ (long position) and stock A

increased from 20$ to 21$ (5% increase). Hence, the stock commitment in A in t is +1.25$ ∗

(1 + 0.05) = 1.3125$. The payoff is: 1.3125$ − 1.25$ = 1.25$ ∗ 0.05 = 0.0625$.12

The daily return of a pair P, which includes stocks 𝑖 ∈ 𝑃, equals:

rP,t =∑ wt,i∗rt,ii∈P

∑ |wt,i|i∈P=

∑ payofft,ii∈P

∑ |wt,i|i∈P (8)

The daily pair return can be understood as sum of the daily payoffs divided by the total pair

capital commitment.13

The daily returns are cumulated to monthly returns afterwards. Follow-

ing Gatev/Goetzmann/Rouwenhorst (2006) and Do/Faff (2010), we report the return on com-

mitted capital. The measure scales the portfolios payoff by the number of actively traded

pairs.14

Six portfolios, each starting in a subsequent month, are traded simultaneously at any time. As

stock prices are likely to diverge further from their normalized prices over time, pair are more

likely to open late in the trading period. Therefore, we average the monthly return of the six

simultaneously traded portfolios. Figure 3 illustrates this process.

[Insert Figure 3]

Gatev/Goetzmann/Rouwenhorst (2006) interpret the return of pairs trading as payoffs to a

proprietary trading desk, where different traders manage six pairs trading portfolios, whose

identification and trading periods are each staggered by one month. Alternatively, the return

can be interpreted as average return of one active pair across all open pairs within the same

portfolio and across six portfolios, which trade simultaneously, but started in consecutive

month. We refer to Gatev/Goetzmann/Rouwenhorst (2006) for further details on the return

calculation.

12

Another example for a short position: The stock commitment in stock B in t-1 is -2$ (short position) and stock

B decreases from 40$ to 37.6$ (6% decrease). Hence, the stock commitment in B in t is: −2$ ∗ (1 + (−0.06)) =

−1.88$. The payoff is: −1.88$ − (−2$) = −2$ ∗ −0.06 = 0.12$. 13

The pair return of our example pair P1, including stock A and B, equals: r1,t =1.25$∗0.05+(−2$∗−0.06)

|1.25$|+|−2$|=

0.0625$+0.12$

|1.25$|+|−2$|=

0.1812$

3.25$= 0.0562.

14 Inactive pairs are neglected, as they do not require a capital commitment and can hence be invested in a risk-

free asset.

The calendar time return might considerably differ from the return per trade for three reasons.

Firstly, the monthly return reflects the combined effect of trading frequency and return per

trade. Secondly, we scale the monthly return to the number of included active pairs in the

portfolio. Thirdly, a trade might be open for several months. Therefore, the pair earns only a

fraction of the return per trade within one month.

3.4 Modifications and test design

To systematically investigate the impact of volatility and correlation, we slightly modify the

classical pair selection algorithm: Firstly, we compute the pair correlation ρA,B

and a proxy for

the combined volatility, the pair volatility (σA2 +σB

2 ) during the twelve month identification

period. Afterwards, we define five pair correlation and five pair volatility quintiles and classi-

fy pairs accordingly. Quintile Corr_Q1 (Corr_Q5) includes the pairs with the lowest (highest)

pair correlation, quintile Vola_Q1 (Vola_Q5) includes the pairs with the lowest (highest) level

of pair volatility. Thirdly, we construct 25 pair groups from the intersection of the five pair

volatility and five pair correlation quintiles groups. For example, Corr_Q1/Vola_Q1 includes

the pairs, whose pair correlation and also pair volatility are among the lowest 20% compared

to all other pair combinations. The quintile and group affiliation of a pair is updated every

identification period. Fourthly, we randomly pick twenty pairs out of each group and include

them into our pair volatility and pair correlation double-sorted (VCS) portfolios. These pairs

are included in the portfolio during the subsequent six month trading period. For each trading

period, we define one trading portfolio for each of the 25 volatility-correlation-quintile com-

binations. Finally, we repeat the random pair selection in the previous step ten times to elimi-

nate a selection bias.

Altogether, we analyze 1200 pairs of each group, at any point in time (20 pairs in each portfo-

lio, 6 simultaneously traded portfolios resulting from the overlapping trading periods, and 10

sets with randomly chosen pairs per group). If not otherwise stated, we report the average

monthly return of the ten simultaneously traded sets (time series) or alternatively the average

return per trade for all trades realized by the 1200 selected pairs (cross-section) in the follow-

ing sections. Following Engelberg/Gao/Jagannathan (2009), the former definition is labelled

calendar time, whereas the latter one is defined as event time.

Table 1 reports the average pair correlation and pair volatility per quintile during the identifi-

cation period. For instance, the average pair correlation of all Corr_Q5 pairs (highest pair cor-

relation quintile) is 0.794.

[Insert Table 1]

The average pair correlation ranges from -0.468 to 0.794. So, the sample also includes uncor-

related (Corr_Q2) and negatively correlated pairs (Corr_Q1). The average pair volatility rang-

es from 0.016 to 0.545. We use normal volatility for the classification, instead of idiosyncratic

volatility, as it is more convenient for practitioners to apply a simple, observable measure. To

calculate idiosyncratic volatility, we follow Xu/Malkiel’s (2003) direct decomposition method

and extract the residuals, obtained from a time series regression of daily stock returns on

Fama/French’s (1993) 3-factor model. We compare the results for normal and idiosyncratic

volatility and find that the allocation to quintiles is almost identical.

4 Link between the classical selection distance measure, volatility

and correlation

What are the implications of applying the traditional pair selection criterion (SSD)? This sec-

tion investigates whether low pair volatility and high correlation influence the SSD, and if to

what extent. The results are important to understand. A significant influence of both factors

indicates that selecting the twenty pairs with the smallest SSD equals selecting pairs with a

specific pair volatility-correlation combination. This specific combination could in turn influ-

ence the trading behavior and hence the return potential. Furthermore, we benchmark our

monthly returns of a traditional pairs trading portfolio in the US between 1990 and 2014 with

the returns of fellow studies.

The classical distance measure was introduced by Gatev/Goetzmann/Rouwenhorst (2006) and

applied by several other authors. It calculates the sum of the squared deviations between the

normalized price movements of two stocks. Research proposition 1 derives, that low pair vol-

atility and high correlation minimize the SSD. We regress the SSD on pair volatility and cor-

relation to investigate the explanation power of both factors. Our sample includes all possible

pair combinations across all identification periods. We utilize period and pair clustered stand-

ard errors. The results are displayed in Table 2, Panel A.

[Insert Table 2]

Pair volatility and correlation explain 88% of the SSD’s total variation, verifying a strong

influence of both factors. As expected, the SSD and correlation are positively linked, whereas

SSD and volatility are inversely linked. It is unclear from the SSD’s mathematical formula,

whether the effect of pair correlation or the effect of single volatility dominates. The standard-

ized regression coefficients reveal that low pair volatility dominates the effect of high correla-

tion. To check the robustness of our results, we conduct the same regression with our ten re-

duced data sets and confirm the previous results separately for each set. We conclude that the

traditional selection algorithm picks pairs with low pair volatility and high correlation. More-

over, we examine into which volatility and correlation quintiles the twenty pairs with the low-

est SSD are classified. Unsurprisingly, almost 67% of all SSD selected pairs are classified as

Corr_Q5/Vola_Q1, the portfolio with the lowest pair volatility and highest correlation. The

remaining pairs are all allocated to one of the other Corr_Q5 or Vola_Q1 portfolios.

The SSD portfolio allows a direct comparison with selected fellow studies. We exclude papers

that apply major modifications, conceal raw returns or one-day-waiting returns15

. Do/Faff

(2010) detect a declining trend in pairs trading return over time. Therefore, we must consider

the covered time period, when comparing the returns. Furthermore, minor adaptions of the

original algorithm might cause small differences, like the pairs preselection in Papa-

dakis/Wysocki (2007). Table 2, Panel B summarizes the monthly U.S. returns for similar

stock universes. Our return of 37bp is consistent with the return of Do/Faff (2010) of 37bp for

1989-2002 and 24bp for 2003-2009, as well as 36bp of Papadakis/Wysocki (2007) for 1994-

2006, suggesting the reliability of our computation and our bid-ask price approach.

In conclusion, this section’s empirical results confirms research proposition 1. This finding

implicates that an investor commits to trade with highly correlated pairs with low pair volatili-

ty, when applying the traditional SSD measure. The results of the following analyses will re-

veal whether the SSD’s pair selection is optimal regarding the return per trade, the trading

frequency, and the monthly return.

15

Previous studies apply a one-day-waiting strategy to model the bid-ask-spread (see Gatev/Goetzmann/

Rouwenhorst (2006) for further details). In contrast, we directly use closing bid prices for selling and closing ask

prices for buying stocks.

5 The effect of volatility and correlation on the return per trade

and the trading frequency

The return of pairs trading consists of two major building blocks, the return per trade and the

trading frequency. The return per trade equals the payoff of one particular pair’s trade. The

return can be earned over a time period between one day and six months. The trading fre-

quency equals the number of trades of one specific pair within one six months trading period,

and it states the frequency of desired trading opportunities. To answer the paper’s main re-

search question, whether part of the return is explained by the pairs trading algorithm itself,

this section separately investigates the influence of the two factors, pair volatility and correla-

tion, on both return building blocks. A significant influence of both factors reinforces our the-

sis, that specific levels of pair volatility and correlation explain at least part of higher observed

returns across industries, countries, and over time.

The previous section uncovered that the traditional selection procedure picks pairs with low

pair volatility and high correlation. In this section’s analysis, we are however interested in the

isolated effect of pair volatility and correlation on the trading algorithm. Therefore, we study

the trading behavior of all twenty-five pair volatility and correlation sorted portfolios (VCS

portfolios), which were previously introduced in section 3. This procedure allows us to ana-

lyze the pure influence of both factors on the trading procedure itself, without the distracting

interference of the SSD’s preselected pair volatility and correlation combination. Section 5.1

starts with investigating the effect of pair volatility and correlation on the first return building

block, the return per trade. Afterwards, section 5.2 analyzes the effect on the second return

building block, the trading frequency.

5.1 Return per Trade

Research proposition 2 conjectures a positive influence of high volatility and a negative influ-

ence of high correlation for profitable natural trades and vice versa for loss-making overshoot-

ing trades. The influence on the return per trade of incomplete and delisted trades depends on

whether prices converge or diverge. Like natural trades, high volatility and low correlation are

beneficial for converging prices, whereas low volatility and high correlation are preferable for

diverging prices.

Table 3 compares the return per trade for each VCS Portfolio separately for each closing type

in panel A to D. Panel A reports that the average return per natural trade of a Vo-

la_Q1/Corr_Q5 pair (lowest volatility and highest correlation quintiles) is 0.14 units for a one-

unit commitment in the long and the short position. The average return per natural trade rang-

es between 0.1441 units for Vola_Q1/Corr_Q5 pairs and 0.7256 units for Vola_Q5/Corr_Q1

pairs. As expected, the return per trade for natural trades rises for higher volatility levels

(within each correlation quintile) and decreases for higher correlation (within each volatility

quintile). Although incomplete trades are forcefully closed on the last day, Panel B reports an

average positive return per trade. Similar to natural trades, higher volatility and lower correla-

tion significantly enhance the profitability of incomplete successful trades. The return ranges

between 0.0118 units for Vola_Q1/Corr_Q5 and 0.0338 units for Vola_Q5/Corr_Q1. Panel C

displays that higher volatility and lower correlation increase the loss per trade for overshoot-

ing trades. The return per trade ranges between -0.4668 units, for Vola_Q5/Corr_Q1, and -

0.1125 units, for Vola_Q1/Corr_Q5. As expected, the link between volatility, correlation and

return of overshooting trades is reverse to natural trades. Panel D reports that delisted trades

are on average negative for highly volatile and low correlated pairs. Yet, the return is increas-

ing and positive for less volatile and stronger correlated pairs. The average return per trade

ranges between -0.1322 units, for Vola_Q5/Corr_Q1, and 0.0190, for Vola_Q2/Corr_4. How-

ever, the return of 8 out of 25 portfolios is insignificant, indicating that there is no clear vola-

tility and correlation impact pattern for delisted trades.

[Insert Table 3]

Altogether, our findings support research assumption 2. As expected, high volatility and low

correlation levels increase the return of converging pairs, but also increase the loss of diverg-

ing pairs. We learn that the volatility and correlation levels during the identification period

dictate the average return level per trade. Hence, pairs in more volatile markets or in times of

general higher market volatility should earn more per trade. Furthermore, our findings imply

that the SSD’s preselection of highly correlated pairs with low volatility earn less per trade

than less correlated pairs with higher volatility.

5.2 Trading Frequency

After examining the return per trade in the previous section, we now turn to the second return

building block, the trading frequency. This section explores the link between pair volatility,

correlation, and trading frequency to evaluate research proposition 3. We argue that raising

the trading frequency translates into earning the return per trade multiple times during one

trading period. Recall that a high trading frequency increases only the number of always prof-

itable, closing type 1 trades (natural trades). Closing type 2, 3 and 4 are always the last trade

within a specific trading period and are therefore unaffected by a higher trading frequency.

The first analysis investigates the influence of pair volatility and correlation on the probability

of a level shift. A level shift describes one pair’s change of pair volatility or correlation, be-

tween the identification period and the trading period. We expect to register more volatility

decreases for pairs with high pair volatility (Vola_Q5 quintile) and less volatility decreases

for pairs with little volatility (Vola_Q1 quintile). Likewise, we argue that correlation increases

are more likely for low correlated pairs (Corr_Q1) and less likely for originally high strongly

correlated pairs (Corr_Q5). The price spread Xt, which is defined as the difference between

the two normalized stock prices of the pair on day t, determines, whether a new trade is ini-

tialized. The opening probability rises, if the volatility of Xt is low during the identification

period and high during the trading period. As derived for research proposal 3, we conjecture

that volatility increases and a correlation decreases raise the volatility of Xt, which in turn

raise the desired opening probability.

We first compute the average pair volatility and the correlation coefficient of each pair during

the identification period and repeat the computation separately for the trading period. Based

thereupon, we generate two dummies for each pair: the Volatility Dummy (Correlation Dum-

my) indicated whether the pair experienced a pair volatility increase (correlation decrease) or

not. Afterwards, we merge the information with our trade dataset, where each trade represents

one observation. So we link each trade observation with the information, whether the trade

generating pair experienced a volatility increase or correlation decrease. Finally, we calculate

for each VCS Portfolios the average percentage of trades, which were generated by pairs with

a desired volatility increases and correlation decreases. Table 4, Panel A reports the results for

volatility increases. For instance, in the Corr_Q1/ Vola_Q1 Portfolio 61.08% of all trades are

executed by pairs, whose volatility level increased. The far right column represents the differ-

ence between the quintile with the lowest and the highest correlation, for a given pair volatili-

ty level. Statistical significance of a two sample mean comparison test at the 10%, 5%, and

1% level is indicated by *, **, and ***. We observe across all volatility levels, that pair vola-

tility decreases are more likely for reversely correlated pairs (Corr_Q1 quintile), as Corr_Q5 -

Q1 is negative for all volatility levels. The bottom row calculates the difference between the

quintile with the lowest and the highest pair volatility, for a given pair correlation level. Con-

sistent with the mean-reversion theory, pair volatility increases are more likely for low pair

volatility levels (Vola_Q1) across all correlation levels as Vola_Q5 - Q1 < 0. Overall, the ta-

ble suggests that most trades are generated by pairs, with a low pair volatility and low correla-

tion.

[Insert Table 4]

Furthermore, we conjecture that correlation decreases raise the trading frequency. Similar to

Panel A, Panel B displays the percentage of trades, originated by pairs whose correlation de-

creased between the identification and trading period. For instance, in the Corr_Q1/ Vola_Q1

Portfolio 23.68% of all trades are executed by pairs, whose correlation level increased. Again,

the far right column and the bottom row show the inter-quintile differences. Beneficial corre-

lation decreases are more likely for pairs with high pair correlation (Corr_Q5 - Q1 > 0) and

more likely for pairs with low volatility (Vola_Q5 - Q1 > 0). Notably, the inter-quintile dif-

ference between the highest and lowest correlation quintile is especially high, indicating an

especially strong effect of correlation on the probability of correlation decreases. Altogether,

we conclude that high volatility and high correlation raise the trading frequency.

So far, we learnt that high pair volatility and high correlation are positive for the correlation-

decrease-effect, but harmful for the volatility-increase-effect. These findings raise the obvious

question, whether the aggregated influence of both factors on the trading frequency is positive

or negative. To answer this question, we assess whether the influence of the correlation-

decrease-effect on the trading frequency dominates the volatility-increase-effect or the other

way round. Therefore, we count the number of trades over all periods for each VCS Portfolio.

The data in Panel C reveals that high pair volatility portfolios generate fewer trades than low

pair volatility portfolios. High correlation portfolios produce significantly more trades than

low correlation portfolios. We conclude that the effect of low pair volatility in the volatility-

increase-effect dominates the effect of high pair volatility in the correlation-decrease-effect.

Furthermore, the effect of high correlation in the correlation-decrease-effect is stronger than

the effect of low correlation in the volatility-increase-effect. Altogether, low pair volatility

and high correlation raise the total number of trades. The SSD, which selects pairs with low

pair volatility and high correlation, is a convenient selection measure to increase the trading

frequency. So far this paper has analyzed the influence of pair volatility and correlation on the

two return building blocks, return per trade and trading frequency. The next section continues

to analyze the aggregated effect of both factors on the total return.

6 Empirical results in calendar time

Our previous results are antithetic so far. High pair volatility and negative correlation increase

the return per trade on the one hand (section 5.1), but reduce the trading frequency on the oth-

er hand (section 5.2). The trading algorithm conceals which effect dominates. However, only

the aggregated effect is of importance for an investor. This section therefore takes a closer

look at the monthly pairs trading returns. The monthly return represents the payoff to an in-

vestor, and it considers the aggregated effect of pair volatility and correlation on the return per

trade and the trading frequency. It provides not only the possibility to compute a risk-adjusted

return, but also allows us to compare the monthly return of a traditionally formed portfolio

and portfolios with alternating volatility-correlation combinations. Additionally, the results

help us to ultimately judge whether the SSD is superior in selecting profitable pairs.

The analysis of the monthly returns between January 199116

and December 2014 uncovers,

whether the positive effect of high pair volatility and low correlation on the return per trade,

or the negative on the trading frequency dominates. Table 5, Panel A reports the average

monthly return for each VCS Portfolio between January 1991 and December 201417

. All re-

turns are positive, significant and range between 19bp to 234bp. For instance, portfolio Vo-

la_Q1/Corr_Q5 (low volatility/high correlation) yields an average monthly return of 46bp.

The most striking result to emerge from Panel A is the successive increase in monthly returns

for higher volatility levels from Vola_Q1 to Vola_Q4 in correlation quintiles Corr_Q2 to

Corr_Q5. The return increase is particularly stronger in higher correlation levels.

16

The monthly returns start in January 1991, as the first 12-months identifications period starts in January 1990. 17

The first identification period starts in January 1990.

[Insert Table 5]

Notably, even correlation quintile Corr_Q1 portfolios earn significant returns. In other words,

pairs trading with negatively correlated pairs is also profitable. The return differences between

Corr_Q5 and Corr_Q1 portfolios are insignificant in three out of five cases, indicating equally

high portfolio returns. This finding contradicts the classical pairs trading idea of exploiting a

temporary mispricing of two close economic substitutes. We address this puzzle in the next

section.

Furthermore, twenty-four out of twenty-five VCS Portfolios outperform the SSD portfolio

with a monthly return of 37bp. Even the direct peer portfolio Vola_Q1/Corr_Q5 (low volatili-

ty/high correlation) earns 10bp more. We speculate that the superior performance of the Vo-

la_Q1/Corr_Q5 portfolio results from randomly selecting twenty out of approximately 72000

pairs within the particular quintile intersection group. Therefore, the selected pairs exhibit an

average level of correlation and volatility within the VCS group. In contrast, the pairs of the

SSD portfolio observe the absolutely lowest volatility and the highest correlation. We con-

clude that even slightly higher volatility and lower correlation levels increase the pairs trading

return.

To account for well-established return patterns (Fama/French (1993), Conrad/Kaul (1989),

Jegadeesh/Titman (1993), Carhart (1997)), we regress monthly returns on a six factor model,

including Fama/French’s three factor model, a momentum factor and a short-term-reversal

factor. All data is obtained from Kenneth French’s website. As our sample includes stocks

with different liquidity levels, we further extend our factor model with Pastor/Stambaughs’

(2003) liquidity factor. Panel B reports the alphas of the regression for each VCS Portfolio.

Like fellow papers, we use Newey-West Standard errors with lag 6. The results confirm our

previous findings. All alphas are positive and almost always significant. Again, high volatility

and strong correlation increase alpha.

Altogether, the results suggest that pairs with high pair volatility and high correlation exploit

the maximal return potential. We conclude that the positive volatility effect on the return per

trade and the positive effect of strong correlation on the trading frequency dominate. This im-

plicates that the SSD, which selects pairs with low volatility, is suboptimal to identify the

most promising pairs. The monthly return of the traditionally selected portfolio with 37bp is

considerably lower than the risk adjusted return of most alternatively formed VCS Portfolios.

7 The information diffusion hypothesis and non-close economic

substitutes

The previous section revealed that also portfolios, consisting of low or negatively correlated

pairs, earn a significantly positive return. At first sight, this finding contradicts the natural idea

of exploiting temporary mispricing of close economic substitutes. We argue that close eco-

nomic substitutes must not necessarily be perfectly correlated. Seemingly uncorrelated pairs

may also share a common economic link, like industry affiliation or the dependence on similar

input factors or identical customers. In the light of the widely accepted information diffusion

hypothesis, pairs trading with negatively correlated pairs can also be successfully, if link spe-

cific information is gradually incorporated. We rather argue that successful pairs trading re-

quires pairs, which are not perfectly correlated. Highly correlated pairs incorporate common

news exactly at the same time, and hence they fail to offer any temporary mispricing. In sum,

we derive that successful pairs trading must share a common sensitivity to certain risk factors.

This section will therefore examine, whether seemingly non-close economic substitutes share

these common links. To quantify the economic closeness of two stocks, we develop the close

economic substitute score (CESS), which indicates the extent to which two stocks react similar

to five risk factors. Furthermore, we are interested whether pairs with many common links

trade more often within a given portfolio than pairs with less common links. This insight is

important to understand why the SSD might be suboptimal. We expect to observe more trades

by less correlated pairs within a portfolio, if perfectly correlated pairs indeed create fewer

trading opportunities.

To measure the economic closeness of two stocks, we develop the close economic substitute

score (CESS), which considers multiple pricing factors. Previous studies like Chen/Chen/Li

(2013) concentrate on similar firm characteristics. However, the demanded stock comovement

is generated by similar price sensitivities to exogenous shocks. We apply the highly recog-

nized five factor model including Fama and French’s three factors model (1993), the WML

factor and the Short-Term Reversal factor to quantify price sensitivities. All data is obtained

from Kenneth French’s website. Firstly, we regress daily closing prices of all stocks on the

five factor model and extract the regression coefficients, which indicate the sensitivity to the

respective factor. Each factor represents one pricing category. Secondly, we calculate the ab-

solute difference between the factors of stock A and B for each category, for all possible pair

combinations, and all identification period. The factor difference quantifies the similarity of

both stock’s price reaction to the respective pricing factor. Thirdly, we sort the absolute factor

difference and allocate the pair accordingly to one of five quintiles each month. This step is

repeated for each of the five categories. For example, a pair is allocated to the lowest market

news quintile. So, this pair consists of two stocks, which react more similar to market shocks

than 80% of all other pair combinations. We denote to the quintile as Ci,n,t, where i is the cate-

gory (i= Market, SMB, HML, Mom and STR), n the particular pair, and t the current identifi-

cation period. As we form five quintiles for each category, Ci,n,t takes on values between 1 and

5. Finally, we calculate the CESS score for each pair combination n in each identification pe-

riod t. The score cumulates the quintile numbers Ci,n,t und subtracts 5.

CESSn,t= ∑ Ci,n,t-55i=1 (9)

The score ranges between zero (the difference between the price sensitivities is among the

lowest 20% in all five categories) and twenty (the difference between the price sensitivities is

among the highest 20% in all five categories). The pair’s CESS score is low, if both stocks

react very similar to common shocks in all five categories, and it is high if both stocks react

differently. For example, the score is equal to ten, if the quintiles of all five categories equal

three. The CESS score is incremented by one point, if the factor difference in one category

increases so that the pair is allocated to the next higher quintile. The CESS measures the simi-

larity to common pricing factors, whereas the correlation coefficient quantifies the pure math-

ematical relationship. As both measures are closely related, we expect to observe a higher

CESS for negatively correlated pairs.

We argued previously that even negatively pairs might share some common links and might

hence exploit temporary mispricing. The previous section revealed that all VCS Portfolio,

including the portfolios with negatively correlated pairs, yield a positive return. Therefore, we

are now interested whether these pair’s stocks react similar to at least common risk factors,

measured by the CESS. Table 6, Panel A reports the average equally weighted CESS for all

VCS Portfolio across all pairs and over time. The average CESS ranges from 7.16

(Corr_Q5/Vola_Q1) to 13.48 (Corr_Q5/Vola_Q5). The CESS of the SSD portfolio is 7.61 and

smaller than 92% of the other portfolios. So, the SSD indeed selects pairs, with similar sensi-

tivity to common risk factors, which can also be called close economic substitutes. The CESS

score is higher for portfolios with negatively correlated pairs (Corr_Q1). The average CESS

score of Corr_Q1 portfolios is between 9.85 and 13.48. A pair with no similarities would be

classified into the highest quintile in each category and have a CESS score of 20. Hence, a

CESS between 9.85 and 13.48 indicates that the pair shares at least some common links. The

mathematical negative correlation of Corr_Q1 pairs is an aggregated measure, which might

conceal that stocks temporary co-move. We argue that as long as a common link exists, link

specific news might create trading opportunities. Therefore, pairs trading with negatively

pairs might also be profitable.

[Insert table 6]

To assess whether close economic substitutes trade more often than non-close economic sub-

stitutes within a portfolio, the next analysis compares the portfolio CESS with the average

CESS per trade.

Therefore, we assign a pair’s CESS to each executed trade by the pair. Afterwards, we com-

pute the average CESS across all trade within a portfolio. The CESS per portfolio can be un-

derstood as equally weighted average of all pairs included in the portfolio. The CESS per

trade in contrast the trade weighted average CESS. The CESS per trade and the portfolio

CESS are equal, if all pairs are equally likely to trade, irrespective of whether they are close

economic substitutes or not. In contrast, if the CESS per trade and the portfolio CESS are un-

equal, than a certain group of pairs is more likely to generate new trades. We derive that close

economic substitutes (lower CESS) trade more frequently, if the per trade CESS is smaller

than the portfolio CESS and vice versa. For the next analysis, we calculate the difference be-

tween the portfolio CESS and per trade CESS, and divide the difference by the portfolio

CESS to scale it. Panel B reports this percentage difference between portfolio CESS and per

trade CESS. The data reveals that close economic substitutes trade more frequently than non-

close economic substitutes, as the percentage is negative for all VCS Portfolios. For instance,

in the Corr_Q1/Vola_Q1 portfolio the trade weighted CESS is 4.27% smaller than the average

CESS of all pairs included in the portfolio. Hence, relatively closer economic substitutes must

trade more frequently than relatively less close economic substitutes to decrease the trade-

weighted CESS. However, we observe a reverse pattern for the SSD portfolio with a CESS

increase of +19.70%. So, within a portfolio of highly correlated pairs, pairs which are relative-

ly less similar trade more often. We conclude that perfectly correlated pairs indeed generate

less trades than less correlated pairs. Furthermore, we found that even negatively correlated

pairs share common links and hence may earn a positive pairs trading return.

8 Robustness checks

The previous results naturally raise a number of questions. Section 8.1 discusses the question,

whether arbitrage opportunities are actually exploitable or only theoretical. Furthermore, we

investigate whether the returns of highly volatile or low correlated portfolios simply represent

a liquidity premium. Afterwards, section 8.2 evaluates, whether specific attributes of non-

close substitutes generate the volatility and correlation effect. Finally, section 8.3 answers the

question, whether the return is mainly driven by the short position, because of short selling

constraints.

8.1 Limits of arbitrage

The literature on limits of arbitrage suggests that market frictions deprive arbitrageurs from

practically exploiting trading opportunities (Shleifer/Vishny (1997)) due to price inefficien-

cies. In the absence of an economical foundation, one possible explanation for abnormal re-

turns is the persistence of such unexploitable trading opportunities. In the context of pairs

trading, Engelberg/Gao/Jagannathan (2009) emphasize the importance of trading costs, short

selling constraints and idiosyncratic volatility. Do/Faff (2012) investigate whether pairs trad-

ing is profitable, after controlling for commission, market impact and short selling constraints.

They find decreasing but still significant positive returns.

However, some of our pairs might be more volatile and possibly more and hence demand

higher trading costs. We address the most crucial issue of the bid-ask bounce, by directly trad-

ing with bid and ask prices. This procedure is more accurate than the traditional one-day-

waiting procedure.

Engelberg/Gao/Jagannathan (2009) examine divergence risk as possible trading barrier. Inves-

tors might be confronted with margin calls, if the short stock further declines. Subsequently,

some arbitrageurs must liquidate their positions to meet the demanded additional capital

commitment. In our analysis, divergence risk affects overshooting trades (closing type 3) in

particular. Similar to Table 3, Panel C, we examine the volatility and correlation effect on the

trading frequency of overshooting trades. Our unreported results indicate, that high volatility

and low correlation reduce the number of overshooting trades and hence divergence risk. This

finding is consistent with Engelberg/Gao/Jagannathan (2009), who find that high idiosyncratic

volatility decreases the divergence probability. Furthermore, the maximum stock divergence

is limited to 4hist, thus investors are unaffected by margin calls beyond this barrier. We con-

clude that divergence risk is not stronger for our pair combinations compared to other papers.

D’Avolio (2002) finds that only 16% of all stocks included in the monthly CRSP file can

eventually not be shorted. 91% of all stocks, including almost all S&P 500 constituents, cost

less than 1% to borrow and have a value-weighted mean fee of 17%. The remaining 9%

stocks, also called “special stocks”, have a mean fee of 4.5% per annum. Less than 1% of the

special stocks demand negative rebate rates and charge a fee of up to 50%. Not surprisingly,

smaller stocks demand higher fees. However, these special stocks account for less than 1% of

the market by value. We assume that the probability of “specials” in our dataset is relatively

low for two reasons. First, our stock universe is restricted to current or former members of the

S&P 1500. These stocks cover around 90% of the total market capitalization and should be

under regular investor’s attention. Second, our stock universe is restricted to stocks with a

listed bid and ask price. These restrictions secure the liquidity and tradability of our stocks.

A further threat, while shorting a stock, is the probability of a stock recall before prices fully

converge. Recalls are more likely, when prices are falling. So in pairs trading when prices

start to converge, the shorted stock of a pair is expected to fall and is hence more likely to be

recalled. However, D’Avolio (2002) observes a low recall rate of around 2% per month. Fur-

thermore, pairs traders can still earn a profit, if the price already converged until the time

where the stock was recalled.

To decrease the probability of hard-to-short stocks and recalls, we test our strategy with large

stocks included at the time in the S&P 100 or the NASDAQ 100. Stocks are excluded, if they

lose their index membership to consider possible liquidity declines. This robustness check

tests, whether our findings persist in an almost limits of arbitrage free setting, as all stocks are

liquid, easy-to-short, and highly efficient due to high analyst cover. Table 7, Panel A (B)

shows the results for the S&P 100 (NASDAQ 10018

).

[Insert Table 7]

All returns are smaller compared to previous returns. So, part of the previous returns can be

explained with short selling constraints, a liquidity premium, and efficiency issues. However,

all returns are still positive and significant. High correlation levels decrease the returns of low

volatility portfolios. The volatility effect persists, especially for strong correlation portfolios.

Moreover, the NASDAQ 100 portfolios outperform the corresponding S&P 100 portfolios.

This result is consistent with our previous results, as NASDAQ 100 stocks are more volatile

than the S&P 100 stocks. Taking all results into consideration, we conclude that the volatility

and correlation effect are robust to limits of arbitrage and liquidity issues.

8.2 The influence of volatility and correlation on Close Economic Substitutes

As fellow papers predominantly trade with close economic substitutes, this section investi-

gates, whether specific attributes of non-close economic substitutes cause the observed vola-

tility and correlation effect. Therefore, we construct a stock pair universe which exclusively

comprises close economic substitutes. The difference between all price sensitivities of both

stocks must be among the lowest 20% in each pricing factor category (CESS = 0), under the

constraint that both stock companies operate within the same 49 Fama/French industries. Sim-

ilar to the original procedure, we divide pairs into four volatility and four correlation quartiles

each month. Afterwards, sixteen double sorted groups are formed. 20 pairs are arbitrarily

drawn from each group and are eligible for trading in the next trading period. The trading pro-

cedure and return calculation remain identical.

Table 8, Panel A displays the monthly mean returns for all portfolios. The returns are all sig-

nificant, positive, and consistent with previous results. Most volatility interquartile differences

are highly positive and significant. At the most, the return incline exceeds 100% between the

lowest Q1 and the highest Q4 volatility quartile. Two conclusions can be drawn. Firstly, the

results establish the positive link between volatility and profitability. Secondly, volatility sig-

18

Bid and ask prices are reported starting in Jan 1995.

nificantly increases the return potential of close economic substitutes, while not contradicting

the original idea of trading with close economic substitutes.

[Insert Table 8]

In contrast, the differences between correlation quartile Q4 (high correlation) and Q1 (low

correlation) and also Q2 (low-medium correlation) are not significant. This observation is

little surprising, as our rigorous pair selection only picks highly correlated pairs. Hence, the

variation of correlation might be too small to cause a significant effect. Although the inter-

quartile differences are not significant, lower correlation slightly increases the return especial-

ly for low and medium volatility. This finding weakly supports the inverse link between cor-

relation and return per trade.

As a consequence of the insignificant correlation effect, we refrain from controlling for corre-

lation in the next analysis. This allows us to increase the number of volatility levels to explore

the volatility effect more precisely. For Panel B, we divide our restricted stock universe into

ten volatility deciles and repeat the previous selection and trading procedure. Again, all

monthly returns are significant and positive, ranging between 49bp in decile Q1 (lowest vola-

tility) and 125bp in decile Q9 (second highest volatility). In comparison to table 5 in section 6,

volatility decile Q1 combined with decile Q2 can be regarded as former Q1 quintile, and dec-

ile Q9 combined with decile Q10 as the former Q5 quintile.

Altogether, the inter-decile difference is sometimes insignificant. Larger inter-decile differ-

ences are however usually significant. Even the decile Q6 return (medium volatility) signifi-

cantly outperforms the decile Q1 and Q2 returns. These results confirm the volatility effect

even for close economic substitutes.

8.3 Short vs. long leg

Stambaugh/Yu/Yuan (2015) argue that return asymmetry of long-short strategies might origi-

nate from short selling constraints of overpriced stocks compared to easily exploitable under-

priced stocks. In this case, the isolated short leg return exceeds the long leg return. Alterna-

tively, Gatev/Goetzmann/Rouwenhorst (2006) ponder that long leg profits might represent a

compensation for an unrealized bankrupt. In this alternative case, the long leg contributes

more to the total return than the short leg. To shed further light on this topic, we compare the

isolated return of the short and the long leg, similar to Gatev/Goetzmann/Rouwenhorst (2006)

and Jacobs (2015).

Table 9 reports the median percentage contribution of the long leg to the total return. The per-

centage is below 50% in twenty-four out of twenty-five portfolios, indicating a higher contri-

bution of the short leg. The long legs of Vola_Q5/Corr_Q1&2 portfolios contribute relatively

little with a share of 31.64%-32.60%. As the return asymmetry is especially pronounced for

highly volatile stocks, the short leg contribution might originate from practically not exploita-

ble mispricing. However, the median long leg of all other VCS Portfolios contributes more

than 40% to the total return. It is therefore unlikely, that the return solely arise from unex-

ploitable mispricing.

[Insert Table 9]

9 Conclusion

We demonstrate that part of the pairs trading return is determined by the applied rule-based

algorithm itself. The strategy’s total return consists of the return per trade and the trading fre-

quency. We link volatility and correlation levels with both return building blocks. Our empiri-

cal results reveal that reverse correlated pairs earn twice as much as highly correlated pairs

per converging trade. Higher volatility can even triple the average return per converging trade.

Furthermore, we observe eight times more trades for pairs with extremely high correlation and

low volatility compared to negatively correlated pairs. Overall, the effect of high volatility

and low correlation is positive for the return per trade, but negative for the trading frequency.

To investigate the aggregated effect of pair volatility and correlation across both return build-

ing blocks, we calculate the monthly return. We find that high correlation (beneficial for the

return per trade) and high correlation (beneficial for the trading frequency) dominate. These

portfolios with high volatility and high correlation are the most superior ones, with a monthly

risk-adjusted return of up to 76bp in the US market between 1990 and 2014. The traditionally

formed portfolio yields only a monthly risk-adjusted return of 37bp during the same time pe-

riod. The results are robust to short selling constraints, liquidity issues, and exposure to com-

mon risk factors.

We find that perfectly correlated pairs are not necessarily the best pairs, and that close eco-

nomic substitutes must not be perfectly correlated. All portfolios include pairs which share at

least some similarities, which can generate temporary mispricing, if similarity specific infor-

mation is incorporated gradually. Merely, the distribution of high and low close economic

substitute pairs varies across portfolios. The comparison between average CESS and CESS

per trade reveals that pairs need economic similarities, but also some dissimilarity, to create

trading opportunities. Further research could concentrate on the specific attributes of news

that generate trades of less close economic substitutes.

Focusing on the stocks of one particular industry or country, or applying a certain selection

procedure like the SSD results in a preselection of pairs eligible for trading. This preselection

of pairs involves the determination of a given pair volatility and correlation combination. The

determination, in turn, affects the total return via the return per trade and the trading frequen-

cy.

Altogether, we learn that some part of the variation of pairs trading returns across industries,

countries, and over time, can be explained by the variation in pair volatility and correlation. In

the future, we should always consider the role of pair volatility and correlation, before deriv-

ing any economical deductions from pairs trading research.

References

[1.] Andrade, Sandro C.; di Pietro, Vadim; Seasholes, Mark S. (2005): Understanding the

Profitability of Pairs Trading. Working paper, UC Berkeley Haas School and Northwest-

ern University.

[2.] Antón, Miguel; Polk, Christopher (2014): Connected Stocks. The Journal of Finance,

Vol. 69, pp.1099-1127.

[3.] Bolgün, Kaan E.; Kurun, Engin; Güven, Serhat (2010): Dynamic Pairs Trading Strategy

for the Companies Listed in the Istanbul Stock Exchange. International Review of Ap-

plied Financial Issues and Economics, Vol. 2, No. 1, pp. 37-57.

[4.] Brennan, Michael; Jegadeesh, Narasimhan; Swaminathan, Bhaskaran (1993): Investment

Analysis and the Adjustment of Stock Prices to Common Information. The Review of Fi-

nancial Studies, Vol. 6, No. 4, pp. 799-824

[5.] Carhart, M. (1997): On Persistence in Mutual Fund Performance. The Journal of Finance,

Vol. 52, pp. 57–82.

[6.] Chen, H.; Chen, S.; Li, F. (2013): Empirical investigation of an equity pairs trading strat-

egy. Working Paper, University of British Columbia, University of Michigan.

[7.] Cohen, Lauren; Lou, Dong (2012): Complicated firms. Journal of Financial Economics,

Vol. 104, pp. 383–400.

[8.] Conrad, Jennifer; Kaul, Gautam (1989): Mean Reversion in Short-horizon Expected Re-

turns. Review of Financial Studies, Vol. 2, pp. 225–240.

[9.] D'Avolio, G. (2002): The Market for Borrowing Stock. Journal of Financial Economics,

Vol. 66, pp. 271-306.

[10.]Do, Binh; Faff, Robert (2010): Does Simple Pairs Trading Still Work? Financial Ana-

lysts Journal, Vol. 66, No. 4, pp. 83–95.

[11.]Do, Binh; Faff, Robert (2012): Are Pairs Trading Profits Robust to Trading Costs? The

Journal of Financial Research. Vol. 35, No. 2, pp.261–287.

[12.]Engelberg, Joseph; Gao, Pengjie; Jagannathan, Ravi (2009): An Anatomy of Pairs Trad-

ing: The role of idiosyncratic news, common in-formation and liquidity. Working Paper,

University of North Carolina, University of Notre Dame, Northwestern University.

[13.]Fama, Eugene; French, Kenneth (1993): Common risk factors in the returns on stocks

and bonds. Journal of Financial Economics, Vol. 33, pp. 3–56.

[14.]Krauss, Christopher (2015): Statistical arbitrage pairs trading strategies: Review and out-

look. IWQW Discussion Paper Series, No. 09/2015.

[15.]Gatev, Evan; Goetzmann, William N.; Rouwenhorst, K. Geert (2006): Pairs Trading:

Performance of a Relative-Value Arbitrage Rule. The Review of Financial Studies, Vol.

19, No. 3, pp. 797–827.

[16.]Huck, N. (2013): The high sensitivity of pairs trading returns. Applied Economics Let-

ters, Vol. 20, No. 14, pp. 1301–1304.

[17.]Huck, Nicolas (2015): Pairs trading: does volatility timing matter? Applied Economics,

forthcoming.

[18.]Jacobs, Heiko; Weber, Martin (2015): On the determinants of pairs trading profitability.

Journal of Financial Markets, Vol. 23, pp. 75-97.

[19.]Jacobs, H. (2015): What explains the dynamics of 100 anomalies? Journal of Banking &

Finance, Vol. 57, pp. 65–85.

[20.]Jegadeesh, N.; S. Titman (1993): Returns to Buying Winners and Selling Losers: Impli-

cations for Stock Market Efficiency. Journal of Finance, Vol. 48, pp. 65–91.

[21.]Lin, Y.-X.; McCrae, M.; Gulati, C. (2006): Loss protection in pairs trading through min-

imum profit bounds: A cointegration approach. Journal of Applied Mathematics and De-

cision Sciences, Vol. 1, pp. 1–14.

[22.]Lo, Andrew; MacKinlay, Craig (1990): When are Contrarian Profits Due to Stock Mar-

ket Overreaction. The Review of Financial Studies, Vol. 3, No. 2, pp. 175-205.

[23.]Papadakis, George; Wysocki, Peter (2007): Pairs Trading and Accounting Information.

Working Paper, Boston University School of Management/ MIT Sloan School of Man-

agement

[24.]Pastor, L.; Stambaugh, R. (2003): Liquidity risk and expected stock returns. Journal of

Political Economy, Vol. 111, pp. 642-685.

[25.]Perlin, Marcelo S. (2009): Evaluation of pairs-trading strategy at the Brazilian financial

market. Journal of Derivatives & Hedge Funds, Vol. 15, No. 2, pp. 122-136.

[26.]Stander, Y.; Marais, D.; Botha, I. (2013): Trading strategies with copulas. Journal of

Economic and Financial Sciences, Vol. 6, No. 1, pp. 83–107.

[27.]Shleifer, Andrei; Vishny, Robert (1997): The Limits of Arbitrage. The Journal of Fi-

nance, Vol. 52, pp. 35-55.

[28.]Stambaugh, Robert; Yu, Jiangeng; Yuan, Yu (2015): Arbitrage Asymmetry and the Idio-

syncratic Volatility Puzzle. The Journal of Finance, Vol. 70, pp. 1903-1948.

[29.]Tourin, A.; Yan, R. (2013): Dynamic pairs trading using the stochastic control approach.

Journal of Economic Dynamics and Control, Vol. 37, No. 10, pp. 1972–1981.

[30.]Vidyamurthy, G. (2004): Pairs trading: Quantitative methods and analysis. John Wiley &

Sons, Hoboken, N.J.

[31.]Xu, Yexiao; Malkiel, Burton (2003): Investigating the Behavior of Idiosyncratic Volatili-

ty. The Journal of Business, Vol. 76, No. 4, pp. 613-645.

Appendix

Figure 1: The interdependencies of the pairs trading algorithm and the strategy’s return

Figure 1 illustrates the interdependencies of pairs trading. The algorithm influences the pairs

selection and the maximum return per trade. The pairs selection, in turn, determines the corre-

lation and common volatility level. These levels influence the return per trade and the trading

frequency. The strategy’s total return is a function of the return per trade and the trading fre-

quency.

Figure 2: Distribution of the prices spread Xt during the identification and trading period

Figure 2 shows the density function of Xt during the identification period (left & right hand

side) and during the trading period (right hand side). Xt is the spread of the two normalized

prices of a pair on day t. 2hist is calculated as two times the standard deviation of Xt during

the identification period, and it sets the trigger barrier to open a new trade.

Figure 3: The return calculation of the monthly portfolio return

Figure 3 displays the calculation of the monthly portfolio returns. The daily return of a portfo-

lio is averaged across all active pairs within the portfolio. Afterwards, the daily returns are

cumulated to the monthly return. This return is again averaged across the six parallel traded

portfolios, where the start of each trading portfolios is staggered by one month.

Table 1: Correlation and volatility levels over quintiles

Table 1 reports the average correlation and the average common volatility per quintile for all

selected pairs as computed during the identification periods between January 1990 and June

2014. The sample includes all pairs of all twenty-five VCS portfolios of all 10 repetitions. The

volatility of a pair is defined as sum of stock A’s and B’s volatility.

Table 2: Classical selection criterion (SSD)

Table 2, Panel A displays findings from a panel regressions of the SSD on the standardized

pair volatility (Vola_AB_std) and the standardized correlation coefficient (CorrelCoef_std).

We control for pair fixed and time fixed effects. Standard errors are clustered at the pair level

and p-values (pvar) are reported in parentheses and adjusted for heteroscedasticity and clus-

tered by pair combination. ***, **, * denote significance at the 1%, 5% and 10% level.

Panel B benchmarks the monthly excess returns of the SSD portfolio with the returns of fellow

papers.

Statistical significance at the 10%, 5%, and 1% level is indicated by *, **, and ***, respec-

tively.

Table 3: Return per trade

Table 3 shows the average return per trade for double sorted portfolios on volatility and correlation

(VCS portfolios) between January 1990 and December 2014.

Panel A displays the average return per trade for natural trades, which close after the full conver-

gence of both stocks within the trading period.

Panel B displays the average return per trade for incomplete trades, which are forcefully closed on

the last day of the trading period.

Panel C displays the average return per trade for overshooting trades, which are closed if price

diverge by more than 4hist.

Panel D displays the average return per trade for delisted trades, which are close if one stock is

delisted while the pair is open. The trade universe includes all trades of pairs from a particular

VCS Portfolio of all pair sets over time.

Statistical significance at the 10%, 5%, and 1% level is indicated by *, **, and ***, respectively.

Table 4: Level shifts and frequency

Table 4, Panel A (B) reports the percentage of pairs that experience a volatility increase (correla-

tion decrease) between the identification and the trading period for each double sorted portfolios

on volatility and correlation (VCS portfolio). The analysis includes all trades of 57600 pairs for

each VCS portfolio (20 pairs per portfolio*10 sets*288 identification periods). Corr Q5-Q1 (Vola

Q5-Q1) shows the return of a long position in the highest correlation (volatility) quintile portfolio

and a short position in the lowest correlation (volatility) quintile portfolio. Statistical significance

at the 10%, 5%, and 1% level is indicated by *, **, and ***, respectively.

Panel C reports the total number of trades for each VCS Portfolios over all trading periods.

Table 5: Average monthly trading return

Table 5, Panel A reports the average monthly raw returns of trading portfolios for each double

sorted portfolios on volatility and correlation (VCS portfolio) between January 1991 and Decem-

ber 2014. For each quintile level of volatility and correlation, Q5-Q1(2) reports the monthly return

of a strategy that invests in the Q5 portfolio and sells the according Q1(2) portfolio.

Panel B displays the alphas from a time-series regression of the one-month pairs trading return on

a six factor model, including Fama/French’s three factor model, a momentum factor, a short term

reversal factor, and Stambaugh/Pastor’s liquidity factor. We use Newey-West Standard errors with

lag 6. Statistical significance at the 10%, 5%, and 1% level is indicated by *, **, and ***, respec-

tively.

Table 6: Close Economic Substitute Score

Table 6, Panel A displays the average Close Economic Substitute Score (CESS) within each dou-

ble sorted portfolio on volatility and correlation (VCS portfolio) and the SSD portfolio for all in-

cluded pairs between January 1991 and December 2014. The CESS is defined on a scale between 0

(highly similar price sensitivity to pricing factors) and 20 (highly dissimilar price reaction to com-

mon pricing factors). Common pricing factors include Fama/French’s three factor model, the mo-

mentum factor and the short-term reversal factor.

Panel B displays the percentage difference between the average CESS of a VCS or SSD portfolio

and the average CESS of all pairs that actually traded within the respective VCS or SSD portfolio.

A negative percentage indicates that pairs with a lower CESS traded more often than the average

pair within a portfolio.

Table 7: Pairs trading in highly liquid markets

Table 7, Panel A reports the average monthly raw returns of trading portfolios for each double

sorted portfolio on volatility and correlation (VCS portfolio) between January 1991 and December

2014. The sample includes only at that time current members of the S&P 100 between January

1990 and December 2014. For each quintile level of volatility and correlation, Q5-Q1(2) reports

the monthly return of a strategy that invests in the Q5 portfolio and sells the according Q1(2) port-

folio. Likewise, Panel B displays the average monthly return of NASDAQ 100 members.

Statistical significance at the 10%, 5%, and 1% level is indicated by *, **, and ***, respectively.

Table 8: Pairs trading with close economic substitutes

Table 8, Panel A reports the pairs trading returns of sixteen portfolios double sorted on volatility

and correlation between January 1990 and December 2014. All included stock pairs must be close

economic substitutes with the lowest possible CESS of zero. Furthermore, both stocks of a pair

must operate within the same industry (Fama/French 49 industry classification).

Panel B shows the return of ten portfolios solely sorted on volatility. The corresponding p-values

are reported in brackets. Q-Q1 refers to the return difference between the observed portfolio and

the portfolio with the lowest volatility Q1. Likewise, Q-Q2 refers to the return difference between

the observed portfolio and the portfolio with the second lowest volatility Q2. Statistical signifi-

cance at the 10%, 5%, and 1% level is indicated by *, **, and ***, respectively.

Table 9: Return per short and long leg

Table 9 investigates the contribution of the short and long leg separately and displays the median

percentage contribution of the long leg to the total return between January 1991 and December

2014.