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Lecture 6:Back-testing statistical-arbitrage

strategies

Marco Avellaneda

G63.2936.001

Spring Semester 2009

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Simulation of trading Profit/Loss

simplicityfor,0and

,0005.0bps5assumewillweTypically,

periodofstartataccountin theequity

scommission&clearingimpactmarket

stockshortoncashforreceivedinterest

stocklongoncashforpaidinterest

cashforratereferenceorrateFundsFed

periodoverstockofreturnadjusted-dividend

periodofstartat thestockininvestment

,

,

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r

nE

rr

rr

r

nR

niQ

n

ni

ni

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Examples of Leverage

Long-only:

Long-only, Reg T:(margin acct)

Long-short $-Neutral, Reg T:

:Long-short, Equal target

position in each stock

EQN

EQ

ESL

ESEL

EL

E

L

i ii max

max

22

1.63.0,3.1

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+

===

=

130-30 Investmentfunds

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Sharpe Ratio

rS

tE

EE

tN

E

EE

tN

N

n n

nn

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n n

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=

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=

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1

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periods

periods

1

2

1

1

periods

2

1 1

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periods

Expected returnover simulation period

Variance over

simulation period

Sharpe Ratio

The Sharpe ratio measures returns above the risk-free rate.

It is independent of the leverage of the strategy (dimensionless).

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( )( )

( )

( ) ( )

( ) ( )( ) ( )tdWdttXmtdX

tdXdtt

ttP

tdP

tS

tdS

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ifactors

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Modeling the Evolution of StockResiduals

Daily sampling frequency

Statistical Estimation Window=3 months (~ 60 business days)

Stock returns a sum of the market returnand a residual process

Residual= drift component (expectedexcess return above mkt.) + incrementof a stationary process

Ornstein-Uhlenbeck

AR-1 process

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Defactoring using ETFs

=

+=

0

/ jijiij

iETFjiji RR

If stock I is in sector j

otherwise

Regress returns on sectorETF returns

In some cases, we construct ``synthetic ETFs (e.g., if the ETF did not exist

in the past). These are taken to be Capitalization-Weighted

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Trading Signals

Introduce the s-score for each stock:

( )ieq

iii

mtXts,

)(

=

50.0ifpositionshortClose50.0ifpositionlongClose

25.1ifpositionshortOpen

25.1ifpositionlongOpen

+

+>

=