Mat-2.106 Independent Research Projects in Applied Mathematics

The Performance of a Relative Value Trading Strategy

Markus Ehrnrooth, 62834B21.12.2007

HELSINKI UNIVERSITY OF TECHNOLOGYDepartment of Engineering Physics and MathematicsSystems Analysis Laboratory

Contents

1 Introduction 1

2 Background 22.1 Gatev et al.’s Study of Pairs Trading . . . . . . . . . . . . . . 22.2 Considerations on Efficiency . . . . . . . . . . . . . . . . . . . 3

3 Technicalities of Pairs Trading 43.1 Pairs Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Return Calculation . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Capital Requirements and Risk measures . . . . . . . . . . . . 7

4 Empirical Results 84.1 Data and Implementation of Pairs Trading . . . . . . . . . . . 94.2 Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Liquidity Considerations . . . . . . . . . . . . . . . . . . . . . 154.5 Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Comparison With Gatev et al. (1999) 19

6 Conclusions 20

A Appendix 21

ii

1 Introduction

The usefulness of technical trading strategies is controversial. Since Fama

(1965), most academics have accepted the efficient market hypothesis, and

asset returns have been considered strictly uncorrelated. Consequently, trad-

ing rules that use patterns in past prices to generate abnormal profits have

widely been labeled as charlatanism by the academic world.

Practitioners have not been as condemning, however. In the pre-market-

efficiency era, Keynes (1936) put forth the view that investor’s decisions

“can be taken only as a result of animal spirits . . . ”, and this view continues

to be shared by many. In the 1980’s, proprietary trading desks of major

investment banks were using trading programs based on statistical analysis to

automatically execute trades, allegedly with great success. Today, a variety

of hedge-funds and investment banks claim to use quantitative methods, or

“statistical arbitrage” in their trading activities. Many of these institutions

have been very profitable.

Recently, academic interest in asset return predictability has returned. A

growing number of researchers argue that market inefficiencies give rise to

time-series patterns in asset returns, rejecting the well-established conjecture

that abnormal profits are impossible to obtain from past-information-based

trading strategies (Conrad 1998).

This study investigates the profitability of one such trading strategy com-

monly known as “pairs trading” along the line of Gatev et al. (1999). Pairs

trading is simple. Based on past returns, find two stocks that have behaved

similarly. When the spread between them widens, buy the loser and sell the

winner. If the pattern of the two stocks moving together remains, the spread

will narrow and the investor will profit. We examine the return and risk of

pairs trading with daily data for Finnish equities from 1997 to 2007.

Our study reveals significant positive returns from the strategy. Part of

the positive returns may be due to bid-ask bounce and implicit trading with

illiquid stocks, but returns still remain significant and positive after these

effects are accounted for.

The remainder of this study is as follows. Section 2 presents the study of

Gatev et al. (1999) and provides background information. Section 3 presents

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our implementation of pairs trading and risk calculations. Empirical results

are presented in Section 4, and they are compared to those of Gatev et al.

(1999) in Section 5. Section 6 concludes.

2 Background

The study by Gatev, Goezmann and Rouwenhorst (1999) contradicts the ef-

ficient market hypothesis, according to which pairs trading should not yield

positive profits, were markets effective. Because of this apparent contradic-

tion, we present some theoretical considerations about market efficiency.

2.1 Gatev et al.’s Study of Pairs Trading

Gatev et al. (1999) test pairs trading on CRSP1 daily data from 1962 to

1997. Their implementation of pairs trading is as follows;

1. Calculate cumulative returns for all stocks from a 12 month period.

2. For each stock, find a matching partner, i.e., the security that minimizes

the sum of squared deviations between the two normalized price series.

3. Trade the pairs during the following 6 months;

3.1 When the prices of a pair diverge by more than two standard

deviations, as calculated during the evaluation period, buy the

looser and short the winner.

3.2 Unwind the position at the next crossing of the prices.

3.3 Continue with 3.1 and 3.2 during the whole trading period.

3.4 If prices do not cross before the end of the trading period, gains

or losses are calculated at the last day of the trading period.

According to the above procedure, portfolios with different numbers of

pairs2 are traded starting at every month during the sample period, except

1Center for Research in Security Prices. CRSP maintains academic research qualitystock market databases with data for individual NYSE, AMEX and NASDAQ securitiesfrom 1926. This data is almost exclusively used in American research.

2Gatev et al. (1999) study portfolios consisting of the top 5 and top 20 pairs, pairs101-121, and all pairs from each respective evaluation period.

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the first 12 months that are initially needed for pairs formation. With six

month trading periods staggered by one month, the study has 408 observa-

tions of six month trading intervals.

Gatev et al. (1999) study two different approches. In the first approch,

pairs positions are opened on the day that prices diverge by the above men-

tioned two standard deviations amount, and closed on the day the spread

closes. In the second approch, positions are opened and closed on the follow-

ing day. Pairs trading involves buying stocks that have been doing poorly

relative to their pair, and selling those that have been doing well. The strat-

egy is thus prone to implicitly be buying at bid quotes and selling at ask

quotes. This issue is addressed by Gatev et al. by the wait-one-day return

calculations.

When it comes to risk, they study investigates the average daily dollar

change in position value across all pairs and monthly Value-at-Risk per-

centiles for the strategy. Gatev et al. also study the possibility that the

returns from pairs trading is merely due to mean reversion in stock prices by

bootstrapping the matched pairs’ returns to random pairs, and calculating p-

values for the matched-pair-returns under these bootstrapped distributions.

Transaction costs from the strategy are considered and pairs trading is tried

with pairs matched only within the four largest sector groupings by Standard

and Poors.

2.2 Considerations on Efficiency

It is important to consider the source of apparent profitability of a historical-

prices-based trading strategy, since the lack of stock price predictability is

viewed by some as synonymous with market efficiency (Fama 1970 and 1991).

Gatev et al. (1999) suggest that pairs trading corresponds to a convergence

strategy basing itself on observing deviations from “The Law of One Price”,

LOP, defined by Chen and Knez (1995) as “. . . any two portfolios generating

the same future payoff must have the same price”. The question is how

regularities such as LOP are maintained in markets.

Schwert (2002) states that market anomalies that are discovered in aca-

demic papers tend to disappear, probably because an increased number of

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investors trying to profit from these anomalies essentially erase them. Gatev

et al. (1999) give an indication of a similar pattern in pairs trading; profits

from the strategy decreased toward the end of their observation period, pre-

sumably because of the large amount of market participants trying to profit

from similar strategies. Markets in both cases thus took several months – if

not years – to achieve weak-form efficiency.

We argue, that the apparent market weak-form ineficciency arises because

of its hidden nature. The deviations from a conjectured LOP that have been

identified are not trivial. We furthermore believe that the relatively low num-

ber of quantitative-trading specialists on the Helsinki exchange contribute to

the persistence of apparent profits from pairs trading. After all, efficiency

requires an active market participation by agents, which will not be observed

unless these are convinced of the economic profit from such action. We also

point to the fact put forth by Gatev et al. (1999) that “. . . few individ-

ual investors – even day traders – have heard of pairs trading”, that is in

accordance with the hidden characteristic of the inefficiency explored.

3 Technicalities of Pairs Trading

Our implementation of pairs trading consists of two stages; stocks are matched

into pairs based on their price movements during twelve months (matching

period), and traded the following six months (trading period).

3.1 Pairs Matching

Let the matrix P ∈ Rn×m contain the closing prices of m securities through

the interval [Ti−n, Ti] with n trading days. The column n-vector P • , j =

[pj1, pj2, . . . , p

jn] thus denotes the closing prices of the j:th stock, and we call

the price vector pj a stock.

We define the log-return matrix corresponding to the price matrix P as

R ∈ Rn−1×m, where

Ri , j = ln

(P i , j

P i−1 , j

), i > 1. (1)

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Correspondingly, the cumulative log-return matrix is S ∈ Rn−1×m

Si , j =∑k≤i

Rk,j. (2)

We use two different distance metrics in matching stocks into pairs; a square-

sum metric and an absolute-value-sum metric. Let the corresponding “dis-

tances” between two stocks pl and pk be

dssi (pl , pk) = (sli − ski )2 (3)

dabsi (pl , pk) = |sli − ski | (4)

‖pl − pk‖ss =n−1∑i=1

dssi (pl , pk) =n−1∑i=1

(sli − ski )2 (5)

‖pl − pk‖abs =n−1∑i=1

dabsi (pl , pk) =n−1∑i=1

|sli − ski | (6)

where sji = Si , j. For each stock pl, we select one matching partner pk, k 6= l,

that minimises the distance (5), and one that minimises the distance (6).

Pairs are formed separately for each matching period [Ti−n, Ti].

3.2 Trading

We trade the top five and ten pairs of matched stocks with the smallest

historical distance. Starting on the day after the last day of the matching

period, the pairs are traded for six month.

Let P t ∈ Rh×m be the closing-price-matrix for the trading period, and

say stock pl was found to be a distance minimizing partner to stock pk during

the preceding matching period. We denote the closing prices of these two

stocks during the trading period by tl and tk.

A position is opened whenever the one-day distance between tl and tk as

defined by (3) or (??), depending on according to which metric the pair was

formed, exceeds two historical standard deviations as measured during the

matching period. More precisely, assume the pair was formed according to

the absolute value metric (6). If dabsi (tl , tk) exceeds 2σabs, defined as

σabs =1

n− 1

n−1∑i=1

((dabsi − d̄absi )2

) 12

, (7)

5

where d̄absi denotes the sample mean, a position is opened by shorting the

winner stock of the pair, tj, j ∈ {l, k} for which sji > s−ji , and buying the

loser stock t−j. The position is closed when the cumulative returns cross

again, i.e. when s−jm ≥ sjm.

Positions are re-opened in the same way if a return-spread re-emerges

during the trading period. Any position open on the last day of the trading

period is closed, regardless of the returns of the pair in question.

Because opening or closing a position on the same day that a wide spread

emerges or a spread closes may not be possible in real life, we also investigate

trading so that positions are opened and closed on the day following the

observing of a spread or crossing in returns.

3.3 Return Calculation

We assume that an equal monetary amount is invested into the short and long

position on opening a long-short position. We let the size of each position

be 1 monetary unit and denote the stock which is sold short by t− and the

stock which is bought by t+. Assuming that a long-short position is opened

at time-index i and closed on time-index k, the investment yields cash flows

+1 , −r−i,k and

−1 , +r+i,k

for the short and long position, respectively. Here r+/−i,k denotes the return for

the stock t+/− over the period [i, k], calculated as exp(∑k

j=i R+/−j ), where R

is the log return matrix corresponding to the closing prices from the trading

period P t as defined by (1). We calculate the return from the single “pair-

trade” asr+i,k

r−i,k. (8)

Note that the strategy is self financing by construction, requiring no capital

as initial cash-flows cancel. Thus, any returns may directly be considered

excess returns.

Any pair that opens and closes before the end of the trading interval

will yield a positive cashflow. Each pair thus yields (none, one or several)

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randomly distributed positive cash flows during the trading interval, and

(possibly) a positive or negative cash flow at the end of the trading interval.

For each stock-pair, we calculate the return over the whole trading period

as the sum of the separate returns. Specifically, by the return for one pair

over an entire trading period we mean

1 +∑

(r+

r−− 1), (9)

where the sum is taken over all sub-periods when the pair-position has been

open. We thus assume that the same one monetary unit is invested each time

the pair opens, even though the pair may already have yielded a positive cash

flow by already having opened and closed before the end of the trading period.

For a portfolio of several pairs, we assume that a fixed monetary amount

is invested in the portfolio and assigned evenly to each pair. Each pair thus

has an equal amount assigned to it, which is invested if the pair opens. The

return from the portfolio is calculated as the sum of the returns from each

pair in the portfolio. We thus have no flexibility in the allocation of assets,

and some of the capital allocated to the portfolio may never actually be

invested. We do not account for inflation nor do we discount cash flows, so

the return on a pair that does not open is 1.

Both of the above measures are conservative, as positive cash flows gener-

ated during the trading period could either be re-invested, or their time-value

could be accounted for. We, however, interpret the apparent conservativism

as a sort of capital-cushion for the strategy employed, in a way that will be

made clear below.

3.4 Capital Requirements and Risk measures

Since pairs trading involves short selling, some practicalities arise. We as-

sume that any brokerage firm requires a minimum 120% collateral on stocks

borrowed3. In our strategy, the long positions and the positive cash flows

3The 120% collateral figure reflects the requirements of the online brokerage Nordnetin November 2007. The figure is calculated daily on closing prises of borrowed shares, andcan thus be seen as equivalent to a mark to market practise. If collateral levels are notmet, the short positions are closed.

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earned during the trading period serve as collateral. It is to note that the

amount of collateral depends on how the short position is doing.

Let γi denote the aggregate market value of the short positions in a pair-

portfolio on the day with time index i, µi denote the aggregate market value

of long positions in a pair-portfolio and πi denote the sum of cash-flows earned

so far during the trading period. We define the daily portfolio situation to

be

κi = µi + πi − γi. (10)

A strategy needs additional capital whenever κi < 1.2γi. In the case of a

hedge-fund, for instance, positive capital requirement does not mean that

additional funds have to be committed to the strategy. It means that the

fund has to have valuable enough long positions in some assets, devoted to

some other strategy, deposited with the broker to cover the 120% capital

requirement of the short positions in the pairs trading strategy.

We will use historical percentile measures of pairs portfolio performance

as our main risk measure, which is analogous to a historical Value-at-Risk.

Let X = [x1, x1, . . . , xn], x1 ≤ x2 ≤ . . . ≤ xn be the collection of observed

daily returns or portfolio situations, ordered in ascending order. We calculate

historical VaR as

V aRα = min{xi |xi ≥ xj ∀ j < dαne} , α ∈]0, 1]. (11)

4 Empirical Results

We test pairs trading as defined in Section 3 on historical closing prices

of stocks listed on the Helsinki Exchange. Our study is in most regards

similar to that of Gatev et al. (1999) in order to be able to make significant

comparisons between results. In terms of timespan, our study continues from

Gatev et al.’s end; our data spans from 1997 to 2007.

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4.1 Data and Implementation of Pairs Trading

Closing prices of 129 stocks4 listed on the Helsinki Exchange in November

2007 are used in testing pairs trading. Closing prises for these stocks through

the period 1 January 1997 to 30 November 2007 are studied. Some of the

stocks studied have been listed since 1997 and consequently these stocks do

not have observations during the whole period studied. In finding matching

pairs for stocks, only stocks with observations for every day of the matching

period were matched.

Since only data on stocks that were listed at the end of our observation

period was studied, our data suffers from “bankruptsy-biase”; companies that

have gone bankrupt or been de-listed during our observation period have been

omitted. In studying a directional strategy this would be a clear potential

source for biased results, but since we study a relative-value strategy the

impact is smaller. One may still reason that the stock of a company near

bankruptsy is more likely to be the loser-stock in a pair, which our strategy

buys. Consequently the “bankruptsy-biase” in our data makes potentially

bad bets impossible, possibly introducing an upwards biase in our results.

Portfolios of top five and top ten pairs as measured by historical distance

are traded starting every month during our observation period, except for

the twelve first months which are needed for the first matching of pairs. A

total of 115 trading periods are studied.

Examples of two pairs and the evolution of their returns during the match-

ing and trading period is showed in Figures 1 and 2. Here, the return evo-

lution of the two stocks is remarkably similar during the evaluation period,

and that the similarity does not remain quite as strong during the trading

period. This is understandable, because we explicitly have snooped the data

during the matching period for best matches. It is however an informative

observation, since it implies that the spread of two historical standard devia-

tions is in fact a small spread during the trading period, and suggests a much

larger spread could be used to trigger the opening of positions. This would

yield less trading (with thus smaller transaction costs) but potentially higher

returns from each trade. We have not explored this possibility further.

4See tables 6 and 5 in Appendix A for a list of the included stocks and those notincluded.

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Figure 1: The top graph depicts the return of the Atria and Finnair stock during thematching period 4 January 1999 to 3 January 2000 and the bottom graph during thetrading period 5 January 2000 to 3 July 2000. The historical standard deviation of theabsolute value of the difference between the cumulative log-returns (see equation (7)) is0.0308. A position is opened on 19 January 2000, and the position is closed on the 3rd ofFebruary 2000. A new position is opened on the 7th of February 2000, and this positionis closed on June 20th. The pair yields a profit of 23.27% during the trading period.

4.2 Returns

The results from pairs trading can be found in Table 1. All variations of

pairs trading studied yields significantly positive returns. The best perform-

ing portfolio is the top 5 absolute-value-distance-pair portfolio, which yields

average annualized returns of 22.94% when positions are opened and closed

on observing a spread in returns or a re crossing of returns, “no Wait” and

13.89% when positions are opened on the day after the observing of a spread

or crosing in returns, “Wait”.

The drop in returns is significant when waiting is introduced in the cal-

culation of returns for all portfolios of pairs, but still remains significantly

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Figure 2: The top graph depicts the return of the UPM and Store Enso stock during thematching period 3 April 2006 to 2 April 2007 and the bottom graph during the tradingperiod 4 April 2007 to 3 October 2007. The historical standard deviation of the absolutevalue of the difference between the cumulative log-returns (see equation (7)) is 0.0118.A position is opened on 25 April 2007, and the position is only closed at the end of thetrading interval. The trade incurs a loss of 11.58%.

positive. This observation gives some indication of that we may be implicitly

be buying at bid quotes and selling at ask quotes when no waiting is done,

since we are effectively buying a loosing stock and selling a winning stock.

The drop in 6-month returns is between 400 and 300 percentage points for

the different portfolios.

The portfolios with pairs matched according to the squared deviation

metric (see equation (5)) on average perform worse than the portfolios with

pairs matched according to the absolute deviation metric (see equation (6)).

One reason for this is that they on average trade less. Regardless of this, they

are not less risky in the sense of having less loss incuring observations (see

Table 2) or having higher minimum returns. Since the two distance measures

largely match up the same stocks, this suggests that frequent trading with

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Table 1: Return statistics for 6-month trading periods during which pairs trading hasbeen implemented with top 5 and top 10 pairs. Two different distance metrics have beenused in forming pairs (see Section 3.1). 114 6-month trading periods were observed. The“mean pos openings” row displays the average number of positions opened during thetrading intervals for the portfolio, and the “mean open trigger” -row displays the averagevalue of the two standard deviation daily distance between pairs that when is reachedimplies the opening of a position.

Absolute-Value-Distance PairsWait no Wait

Top 5 Top 10 Top 5 Top 10Mean portfolio return 1.0672 1.0624 1.1088 1.1037t-value 113.7810 139.2857 102.6179 118.7807Min portfolio return 0.8699 0.8437 0.8791 0.8595Max portfolio return 1.3387 1.3193 1.4009 1.3565Mean pos openings 13.4696 25.9739 13.5478 26.1217Mean open trigger 0.0544 0.0590 0.0544 0.0590

Squared-Deviation-Distance PairsWait no Wait

Top 5 Top 10 Top 5 Top 10Mean portfolio return 1.0550 1.0522 1.0927 1.0871t-value 123.5483 161.8731 101.4652 136.2105Min portfolio return 0.8439 0.8657 0.8687 0.8746Max portfolio return 1.3744 1.2805 1.3962 1.3236Mean pos openings 10.2348 19.9391 10.3043 20.0435Mean open trigger 0.0054 0.0064 0.0054 0.0064

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smaller bets corresponding to smaller spreads in returns when positions are

opened is a better strategy than taking positions at bigger spreads5.

It is interesting to note that the worst 6-performance of a pair portfolio

with waiting that is observed is a 15.61% loss, while the best performance is

a whipping 37.4% gain.

Histograms over the returns from the top 5 and top 10 absolute value

pair portfolio, calculated with one day waiting can be found in figure 3.

Figure 3: Histograms over top 5 and top 10 absolute value pairs portfolio returns over6-month trading intervals. Positions have been opened and closed with a lag of one dayafter observing the opening spread or closing re-crossing of prices.

4.3 Risk

We consider the risk characteristics of pairs trading in light of historical VaR

quantiles. The quantiles for portfolios with one day waiting before opening

positions are shown in Table 2, and the results for portfolios without waiting

can be found in Table 3. We show statistics for both 6-month trading period

portfolio returns, as well as for daily portfolio situations. Note that the Daily

Portfolio Situation -numbers include the positive cashflows earned from pairs

that have opened and closed before the end of the trading period.

The 6-month trading period results are rather robust. For returns calcu-

lated with one day waiting before opening and closing positions, in only 1%

5We have not tested the validity of this claim. Thus we may only conclude that withopen triggers close to two historical standard deviations, increasing the open trigger slightlydoes not seem to be profitable. It may however be the case that a significant increase inthe open trigger would be profitable.

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Table 2: Historical VaR quantiles for 6-month trading period portfolio returns and dailyportfolio situations. All returns have been calculated with a one day waiting before openingand closing positions. We have 114 observations for the 6-month trading period portfolioreturns and 14375 observations for daily portfolio situation.

6-month trading period portfolio returnsabs-Value pairs Squared-Deviations pairs Hex Indexa

Top 5 Top 10 Top 5 Top 100.5% 0.8703 0.8452 0.8669 0.8667 0.70451% 0.8731 0.8565 0.9035 0.8742 0.70625% 0.9038 0.9274 0.9709 0.9459 0.743310% 0.9324 0.9728 0.9912 0.9668 0.8058Share of returns ≤ 1 0.2174 0.2000 0.2696 0.2000 0.4435

Daily Portfolio Situationb

abs-Value pairs Squared-Deviations pairs Hex Indexc

Top 5 Top 10 Top 5 Top 100.5% 0.8710 0.8460 0.8234 0.8669 0.71571% 0.9003 0.8976 0.8621 0.9035 0.73575% 0.9732 0.9812 0.9652 0.9709 0.814010% 0.9941 0.9974 0.9899 0.9912 0.8626Share of days ≤ 1 0.1322 0.1202 0.1505 0.1558 0.4618Min 0.7651 0.6900 0.6972 0.7624 0.6543Max 1.5677 1.5831 1.5478 1.5514 1.5390

aCalculated as the six-month return on an equally weighted portfolio of all stocks inour data sample.

bIncludes cash-flows from pairs that have closed during the trading period.cCalculated from daily situations of an equally weighted portfolio of all stocks in our

data sample. The portfolio is held six months at the time so that results are comparableto those from our pairs trading strategy.

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of the periods studied did pairs trading parform worse than a 14.35% loss,

and in only 10% of trading periods was the loss greater than between 7% and

1% for the different portfolios. In all, roughly one fifth of periods produced

a loss. The pairs-portfolios are less volatile according to a historical VaR

measure than the Hex Index.

For the daily portfolio situations, we see that only 0.5% of days showed

losses of more than between 17.6% and 12.9%. The worst observed portfolio

situation was a loss of 31%. With 120% collateral required on short positions

and assuming the loss is equally distributed between decrease in value of long

positions and increase in value of short positions, additional capital of 54.1%

of the value invested in the pairs trading portfolio6 would be needed to keep

the portfolio afloat.

4.4 Liquidity Considerations

Our implementation of pairs trading has so far assumed that all of the stocks

studied may be sold and bought at any time. This may not be the case

for some of the less liquid stocks included in our study, and we might thus

be booking profits from trades that are not practically feasible, at least not

with big stakes. To test this, we re-try the pairs trading with a sample of

only liquid stocks. We define a stock as liquid if the average monthly volume

traded in the period September 2006 through August 20077 exceeds EUR

500 000. An overview of the companies considered liquid is given in table

6 in Appendix A. The performance statistics from pairs trading with liquid

stocks only are displayed in Table 4.

With the liquid stocks, we only study characteristics of absolute-deviation-

distance pairs, see (6), and returns calculated when positions are opened and

closed on the day after the opening spread or re-crossing of pairs is observed.

6The value of short positions increase with 15.5% and the value of long positions de-crease by 15.5%. 120% of the value of the short position is thus 138.6% of the initialvalue invested in the short position, and 84.5% of this is covered by the value of the longposition.

7As stated in the OMX monthly report“Equity Trading by Company and InstrumentAugust 2007”, available at the time of writing fromhttp://www.omxgroup.com/nordicexchange/uutisetjatilastot/tilastotanalyysit/details/?releaseId=297610&lang=EN

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Table 3: Historical VaR quantiles for 6-month trading period portfolio returns and dailyportfolio situations. All returns have been calculated so that positions are opened and closedon the same day as spreads or recrossing of prices is observed. We have 114 observationsfor the 6-month trading period portfolio returns and 14375 observations for daily portfoliosituation.

6-month trading period portfolio returnsabs-Value pairs Squared-Deviations pairs Hex Indexa

Top 5 Top 10 Top 5 Top 100.5% 0.8804 0.8602 0.8691 0.8749 0.70461% 0.8899 0.8657 0.8720 0.8776 0.70625% 0.9171 0.9699 0.9038 0.9666 0.743310% 0.9548 0.9946 0.9499 0.9884 0.8058Share of returns ≤ 1 0.1304 0.1391 0.1913 0.1304 0.4435

Daily Portfolio Situationabs-Value pairs Squared-Deviations pairs Hex Indexb

Top 5 Top 10 Top 5 Top 100.5% 0.8869 0.8598 0.8424 0.8867 0.71571% 0.9101 0.9142 0.8760 0.9184 0.73575% 0.9818 0.9885 0.9704 0.9819 0.814010% 0.9992 1.0011 0.9950 0.9969 0.8626Share of days ≤ 1 0.1067 0.0935 0.1292 0.1260 0.4618Min 0.8272 0.7031 0.7272 0.7574 0.6543Max 1.6304 1.6255 1.6015 1.6085 1.5390

aCalculated as the six-month return on an equally weighted portfolio of all stocks inour data sample.

bCalculated from daily situations of an equally weighted portfolio of all stocks in ourdata sample. The portfolio is held six months at the time so that results are comparableto those from our pairs trading strategy.

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Table 4: Return and risk statistics for pairs trading with liquid stocks only. All results arefor pairs matched according to the minimum absolute value distance metric, see equation6. Positions have been opened one day after the opening spread or crossing of returns isobserved, corresponding to the “Wait” results in previous tables. we have 114 observationsfor 6-month trading portfolios, and 14185 for daily portfolio situation.

top 5 top 10mean portfolio return 1.0587 1.0571t-value 131.1267 165.2306min portfolio return 0.8314 0.9236max portfolio return 1.3481 1.3106mean pos openings 11.6957 22.2609mean open trigger 0.0313 0.0342

6-month Portfolio Returns Daily Portfolio Situationtop 5 top 10 top 5 top 10

0.5% 0.8357 0.9236 0.8805 0.84261% 0.8685 0.9238 0.9051 0.89225% 0.9309 0.9455 0.9720 0.970210% 0.9563 0.9768 0.9928 0.9934share ≤ 1 0.2348 0.1913 0.1387 0.1407min 0.8314 0.9236 0.7115 0.6553max 1.3481 1.3106 1.6404 1.4962

17

We consider this the most realistic and best measure of how pairs trading

may actually be implemented. We see that average returns drop slightly from

the levels of the not-necessarily liquid pairs, although remaining significantly

positive. The top 5 portfolio still yields an annualized return of 12.08%, and

the top 10 portfolio of 11.75%. Histograms over the portfolio returns can be

seen in Figure 4.

Figure 4: Histograms over top 5 and top 10 portfolio returns over 6-month trading intervals. Onlyliquid stocks have been paired together, and positions have been opened and closed with a lag of one dayafter observing the opening spread or closing re-crossing of prices.

4.5 Transaction Costs

We may use the difference in returns calculated with and without waiting

one day before opening and closing positions as a measure of bid-ask spread

and hence transaction costs. Consider the extreme case where not waiting

before taking a position always implies first buying losers at bid prices and

selling winners at ask prices, and then selling losers at ask prices and winners

at ask prices. If the following day prices (used when waiting one day) are

equally likely to be bid or ask prices, waiting one day will reduce returns

on average by half the bid-ask spread of both stocks on opening a position

and half the spread on closing it. The average drop in returns can thus be

interpreted as a possible measure8 of transaction costs.

8According to the above reasoning, this would actually be a lower bound of transactioncosts since it is unlikely to be the case that the full bid-ask spread is observed when notwaiting.

18

We observe a drop of 3.77% and 4.16% in returns for squared-deviation-

top-five pairs and absolute-value-top-five pairs per six month trading of a

pair portfolio. Since on average 10.23 and 13.46 pairs-positions are opened

for top-five portfolios (square and abs-pairs), this translates into trading costs

of roughly 33 basis points per pair per roundtrip. If the decrease in returns

is interpreted as trading costs, returns of 1.75% and 2.08% thus remain from

the pairs trading strategy. Although significantly smaller, these returns are

still statistically significantly positive.

5 Comparison With Gatev et al. (1999)

Our study is technically very similar to that of Gatev et al. (1999). Our data

is different however. What comes to timespan, our study continues where

Gatev et al. ends. Geographically, we study stocks on a smaller and less

active and developed market.

In matching pairs we use two different distance measures. The pairs

matched according to the absolute value distance metric not used in Gatev

et al. (1999) outperform the ones matched according to their squared devia-

tions metric. Whether this effect is simply due to the more frequent trading

of absolute value pairs as a result of relatively small historical standard devi-

ations for this measure or because these pairs actually are “better matches”

has not been explored at length. We suspect the former since the pairs

formed are largely the same.

We find better results from pairs trading than Gatev et al. (1999) do.

They record average six-month returns of 5.98% from top-five-pair portfolios

calculated without waiting one day (our top-five portfolios earns on average

10.88% and 9.27% for abs pairs and squared deviation pairs, respectively),

and 3.68% when one-day waiting is introduced (6.72% and 5.50% in our

study). The drop in returns when waiting is introduced is remarkably similar

in relative magnitude (38.46% for Gatev et al., 38.10% and 40.67% for us).

If the drop is due to bid-ask spread as Gatev et al. (1999) hypothesize,

this would indicate that bid-ask-spread characteristics between winning and

loosing stocks are similar in US and Finnish markets.

We find clear differences in VaR quantiles calculated in the two studies.

19

Gatev et al. (1999) find the 1% VaR measure to be a loss of just 2.27%

and a 5% VaR measure to be a loss of 1.21%, which are both extremely low

figures. Our comparable findings are significantly higher; losses of 12.68%

and 9.62%. This is in line with the findings of Gatev et al. that the pairs

trading strategy was much more volatile toward the end of their data sample.

6 Conclusions

The performance of a simple trading strategy when applied in retrospect

on daily closing price data from the Helsinki stock exchange was evaluated

in this study. The strategy appears to be profitable, even after considering

trading costs and liquidity issues, and less risky than the market portfolio.

The results have to be interpreted with caution, however. First, the

omitting of bankrupt and de-listed companies in our data is a potential source

of upwards biase in observed results. Second, if the strategy was to be realized

in practice with large sums, the liquidity our strategy requires would probably

translate into higher transaction (i.e. liquidity) costs, since all trades could

probably not be booked at the same quotes.

Comparing our results from pairs trading to results from Gatev et al.

(1999) indicate that the strategy was more profitable on the Helsinki ex-

change than on the NYSE, and remained profitable in Helsinki after declin-

ing in profitability on the NYSE. We believe this is due to relatively little

quantitave trading activity on the Helsinki Exchange, whereas the increase

of such acivity on the NYSE erases profit potential there.

20

A Appendix

Table 5: Companies listed in Helsinki on 03.12.2007 not included in ourstudy.

Interavanti Nokia New Shares Nordea Bank (publ) FDROMX SRV Group Suomen TerveystaloTeliaSonera

21

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22

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23