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Pairs Trading: performance of a relative value arbitrage strategy

Evan G. Gatev William N. GoetzmannK. Geert Rouwenhorst

Yale School of Management

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Statistical “Arbitrage”

Identify a pair of stocks that move in tandem

When they diverge: short the higher one buy the lower one

Unwind upon convergence

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P(1) - P(2)

On Off

Example

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Who does it?

Proprietary trading desks Morgan Stanley Nunzio Tartaglia - 1980’s Other investment banks

Hedge funds (Long-short) Cornerstone D.E. Shaw?

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Economic Rationale

Tartaglia: “Human beings don’t like to trade against

human nature, which wants to buy stocks after they go up, not down…”

Imperfect markets? Over-reaction Under-reaction

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Relative Pricing

Approximate APT Models Long-short “arbitrage in expectations” Self-financing Eliminate relative mispricing Silent on absolute pricing

Mechanisms risk-matched portfolios risk-matched securities

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Law of One Price

Matching state payoffs => Matching pricesNear Matching state payoffs ?=>

Chen and Knez (1995) market integrationConditions:

Errors in states that don’t matter much

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Methodology

Two stages: 1. Pairs Formation 2. Pairs Trading

Committed Capital full period when-needed no extra leverage

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Pairs Formation Period

Match on stock cumulative return index Minimize squared price error Twelve months of daily prices

Equivalent to matching on state-prices Each day is a different state Assumes stationarity Assumes a year captures all states

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Pairs Formation Period

Daily CRSP filesEliminate stocks that missed a day trading in

a yearCumulative total return index for each stockAlso restrict to same broad industry

category: Utilities, Transports, Financials, Industrials

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Related Work

“Style Analysis” via clustering algorithm Brown and Goetzmann (1997)

Bossaerts (1988) Seeking co-integration in price series

Chen and Knez (1995) market integration measures finding close pricing kernel across two markets

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Trading Period

Six-month periods: 1962-1997 starting a new “trader” each month closing all positions at end of each six month

How many pairs to use? 5, 20 and 20 after first 100, then all pairs under

distance metric

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Trading Period

Open at 2 (historical over leading year)Close upon convergence, or end of six-

month period Same-day vs. wait one day to control bid-

ask effect

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Excess Return Computation

Weakly positive payoff inside the six-month interval and:

Positive or negative payoff on last day No “marking to market”Ignore financing issuesExcess return on pair = sum of payoffs over

interval

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Excess Return

Return on committed capital Sum of payoffs over all pairs in period/# pairs Allow $1/per pair

Return on employed capital All $1/pair used

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0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96102108114120

Pair 4, Kennecott and UniroyalStarting Month 19620801

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Results for Same Day Trading

Portfolio of 5 and 20 best pairs earn an average of 6% per six month period.

Average size of stocks in pairs: 3rd to 4th decile

Utilities predominate

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Same-Day Trading Performance

T op 5 T op 2 0 20 - 1 00 A ll

6 -m onth M ean 5 .9 8 % 6 .0 1 % 4 .5 1 % 4 .1 0 %

U tilit ie s 8 1% 8 2% 33 % 9%T ransportation 1 % 1 % 2 % 3%

F in ancia l 4 % 4 % 16 % 12%In dustria l 1 3% 1 3% 48 % 76%

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Monthly Next-Day Portfolio

T op 5 T op 2 0 20 - 1 00 A llM onth ly E xc e ss

R eturn 0 .6 0 % 0 .6 4 % 0 .5 7 % 0 .4 7 %S T D 1 .34% 1 .02% 1 .04 % 0.98%

S ha rpe R a tio 0 .454 0 .636 0 .557 0 .480

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Monthly Performance

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Cumulative Excess Returns

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Systematic Risk Exposure

T o p 5 t o p 2 0 2 0 a 1 0 0 A ll E R PI n t e r c e p t 0 .0 0 5 6 0 .0 0 6 1 0 .0 0 5 6 0 .0 0 4 4 0 .0 0 7 0

U .S . E q u it y P r e m . 0 .0 1 0 2 0.0080 -0.0067 - 0 .0 1 6 7

S m a ll - B ig 0.0404 0.0511 0 .0 4 6 5 0 .0 6 8 8 0 .2 1 1 7

H ig h B /M - L o wB /M

0 .0 7 9 1 0 .0 2 1 0 0 .0 1 8 0 0 .0 6 8 9 - 0 5 8 4 7

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Ibbotson Risk Exposures

T o p 5 t o p 2 0 2 0 a 1 0 0 A ll E R PI n t e r c e p t 0 .0 0 5 9 0 .0 0 6 4 0 .0 0 5 7 0 .0 0 4 7 0 .0 0 3 5

U .S . E q u it y P r e m . - .0 2 9 6 -.0259 -0.0206 - 0 .0 4 0 8

S m a ll S t o c k P r e m . 0.0477 0.0494 0 .0 3 4 9 0 .0 4 9 6 0 .1 7 7 1

B o n d D e f a u lt P r e m . 0 .1 2 8 8 0 .1 2 6 6 0 .1 3 4 4 0 .1 6 8 0 0 .7 1 0 3

B o n d H o r iz o n r e m . 0 .0 8 1 5 0 .0 7 4 3 0 .0 4 4 0 0 .0 4 2 9 0 .6 8 0 5

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Monthly Value at Risk

Top 5 T op 2 0 N e xt 20 A ll1 % -0 .0227 -0 .0151 -0 .0198 -0 .01335 % -0 .0121 -0 .0059 -0 .0092 -0 .0066

1 0% -0 .0078 -0 .0034 -0 .0063 -0 .00431 5% -0 .0049 -0 .0015 -0 .0040 -0 .00332 0% -0 .0031 -0 .0004 -0 .0022 -0 .0021

P r re t.< 0 28 .70% 2 2.10% 2 5.60% 32 .30 %M in . -0 .1020 -0 .0618 -0 .0340 -0 .0238

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Micro-Structure

Bid-Ask Bounce conditional upon an up move, price is likely an

ask. conditional upon a down move, price is likely a

bid.J&T (1995) C&K (1998)

Contrarian profits all bounce?

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Controlling for Bid-Ask Bounce

Wait a day to open position Wait a day to close positionEffect:

Excess return drops by 240 BP

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Transactions Costs

Conservative round-trip cost estimate Same Day vs. Wait 1 Day = 200 BP 2.4 RT per pair/6 months 83 BP/RT and an effective spread of 42 BP

Net 6 month excess return: 168 to 88 BP

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Contrarian Profits?

Mean Reversion DeBondt and Thaler(1985,1987) LSV (1994) Lehman (1990), Jegadeesh (1990)

Test: If solely mean-reversion, Random pairs should be profitable. They are mostly not.

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Bootstrap for Utilities

P ortfo lio top 5 top 1 0 top 20 A llM ean E xc e ss R e tu rn 0 .0504 0 .0489 0 .0518 0 .0478

N W S ta nda rd E rro r 0 .0047 0 .0041 0 .0038 0 .0031t-statist ic 10 .70 1 1 .93 1 3 .70 1 5 .43

B oo tstrapped d ist r ibu tion o f rando m p a irs

m e a n 0 .0009 0 .0011 0 .0014 standa rd d e viation 0 .0071 0 .0051 0 .0035

p -va lue a c tua lp ro fit s

0 .0000 0 .0000 0 .0000

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Improvements

We may be opening pairs too soonWe may not be picking pairs wiselyOther sensible rules

don’t open a pair on the last day of the period

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Implications

Document relative price reversionMarginally profitable

Consistent with hedge fund businessNot simply mean reversion