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The Limits of Arbitrage
ANDREI SHLEIFER and ROBERT W.VISHNY商学院 周 美 & 杜慧卿
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ABSTRACT
In reality , almost all arbitrage requires capital, and is typically risky.
Professional arbitrage is conducted by a relatively small number of highly speciali-zed investors using other’s capital.
Arbitrage becomes ineffective in extreme circumstances.
Anomalies in financial markets.
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Fundamental concepts
“The simultaneous purchase and sale of the same, or essentially similar, security in two different markets for advantage-ously different prices.”
No risk and need no capital.
“The simultaneous purchase and sale of the same, or essentially similar, security in two different markets for advantage-ously different prices.”
No risk and need no capital.
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Realistic arbitrage are more complex
Require capital good faith moneyIn short run, one may lose moneyDifferent trading hours, settlement dates,
and delivery terms.If prices are moving rapidly, the value
may differ additional risks
Require capital good faith moneyIn short run, one may lose moneyDifferent trading hours, settlement dates,
and delivery terms.If prices are moving rapidly, the value
may differ additional risks
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PBA( performance-based arbitrage)
More commonly, conducted by relatively few professional, highly specialized investors;
Outside resourcesBrains and resources are separatedAllocate funds based on past returns
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I. An agency model of limited arbitrage
Assume :① Three types: noise traders, arbitrageurs,
and investors in arbitrage funds
② Fundamental value is V;
③ Three time periods
tp
Assume :① Three types: noise traders, arbitrageurs,
and investors in arbitrage funds
② Fundamental value is V;
③ Three time periods
④ , ,
⑤
tptS
tt pSVt ][)(QN tF
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I. An agency model of limited arbitrage
Arbitrageurs‘ demand 22)2( pFQA Arbitrageurs‘ demand Aggregate demand equals the unit
supply:
22)2( pFQA
222 FSVp
Arbitrageurs‘ demand Aggregate demand equals the unit
supply:Not fully invest,
22)2( pFQA
222 FSVp
1D
Arbitrageurs‘ demand Aggregate demand equals the unit
supply:Not fully invest, ,
22)2( pFQA
222 FSVp
1D
11)1( pDQA 111 DSVp 11)1( pDQA 111 DSVp
Arbitrageurs‘ demand Aggregate demand equals the unit
supply:Not fully invest, ,Market segment, T investors with $1, so
22)2( pFQA
222 FSVp
1D
11)1( pDQA 111 DSVp
TF 2
Arbitrageurs‘ demand Aggregate demand equals the unit
supply:Not fully invest, ,Market segment, T investors with $1, so
22)2( pFQA
222 FSVp
1D
11)1( pDQA 111 DSVp
TF 2
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I. An agency model of limited arbitrage
Compete in the price the charge;Assume marginal cost constant, so
competition drives price to marginal costBayesians, allocate funds according to
past performance( PBA );An increasing function:
})()(*){(* 111121112 FDFppFDGFF
Compete in the price the charge;Assume marginal cost constant, so
competition drives price to marginal costBayesians, allocate funds according to
past performance( PBA );An increasing function:
})()(*){(* 111121112 FDFppFDGFF
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I. An agency model of limited arbitrage
Benchmark: zero returnA linear function: aaxxG 1)(
Benchmark: zero returnA linear function: ,with aaxxG 1)( 1a
Benchmark: zero returnA linear function: ,withThen
aaxxG 1)( 1a
)1()1()}()(*{ 12111111212 ppaDFFaDFppDaF
Benchmark: zero returnA linear function: ,withThenIf , gain funds; or ,lose funds The higher is , the more sensitive to
past performance
aaxxG 1)( 1a
)1()1()}()(*{ 12111111212 ppaDFFaDFppDaF
21 pp
a
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I. An agency model of limited arbitrage
An arbitrageur’s optimization problem:
q
An arbitrageur’s optimization problem:
q ;
1-q12 SSS
An arbitrageur’s optimization problem:
q ;
1-q ,12 SSS
02 S Vp 202 S Vp 2
An arbitrageur’s optimization problem:
q ;
1-q , .
Maximize:
12 SSS
02 S Vp 2
})1()*
{(*)(
})1()*
(){1(
1111
21
2
1111
1
FaDFp
pD
p
Vq
FaDFp
VDaqEW
An arbitrageur’s optimization problem:
q ;
1-q , .
Maximize:
12 SSS
02 S Vp 2
})1()*
{(*)(
})1()*
(){1(
1111
21
2
1111
1
FaDFp
pD
p
Vq
FaDFp
VDaqEW
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II. Performance-based Arbitrage and Market Efficiency
The case of 1aThe case of
first order condition:
1a
0)1()1(-121
2
1
p
V
p
pq
p
Vq)(
The case of
first order condition:
Inequality :
1a
0)1()1(-121
2
1
p
V
p
pq
p
Vq)(
11 FD
The case of
first order condition:
Inequality : ;Equality :
1a
0)1()1(-121
2
1
p
V
p
pq
p
Vq)(
11 FD
11 FD
The case of
first order condition:
Inequality : ;Equality : .
The initial displacement is large and will recover with a high probability; if they fall, it can’t be large.
1a
0)1()1(-121
2
1
p
V
p
pq
p
Vq)(
11 FD
11 FD
The case of
first order condition:
Inequality : ;Equality : .
The initial displacement is large and will recover with a high probability; if they fall, it can’t be large. fully invested at 1
1a
0)1()1(-121
2
1
p
V
p
pq
p
Vq)(
11 FD
11 FD
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II. Performance-based Arbitrage and Market Efficiency
Proposition 1:For a given , , , ; and , there is a such that, for ,
, and for , .
Proposition 2: At the corner solution( ),
, , and . At theinterior solution, , , and .
It shows that arbitrageurs ability to bear mispricing is limited, larger shocks, less efficient.
V 1S S 1F
a *q *qq
11 FD *qq 11 FD
11 FD
011 dSdp 02 dSdp 01 dSdp
011 dSdp 02 dSdp 01 dSdp
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II. Performance-based Arbitrage and Market Efficiency
Uncertainty of the effect: a higher a could make market less
efficient, by withdrawing funds; A higher a will make prices adjust quickly
by giving more funds after a partial reversal of the noise shock.
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II. Performance-based Arbitrage and Market Efficiency
Consider about the extreme circumstances:
two ways: 21 FD
Consider about the extreme circumstances:
two ways: ? ?Proposition 3 : If arbitrageurs are fully inv
ested at , and noise trader mispe-rceptions deepen at , then, for ,
, and .
Fully invested arbitrageurs may face equity withdrawals and liquidate.
21 FD 2211 pFpD
1t
2211 pFpD
1t
2t
2211 pFpD
1t
2t 1a
2211 pFpD
1t
2t 1a21 FD
2211 pFpD
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II. Performance-based Arbitrage and Market Efficiency
Proposition 4: At the fully invested equili-brium, and .12 dSdp 02
2 dadSpd12 dSdp 022 dadSpd
Proposition 4: At the fully invested equili-brium, and .
This shows: prices fall more than one for one with the noise trader shock at time 2,when fully invested at time 1.
A market driven by PBA can be quite in-effective in extreme circumstances.
12 dSdp 022 dadSpd
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III. Discussion of Performance-based Arbitrage
We are uncertain about the significance of PBA
Funds decline with a lag Contractual restrictions expose more risk Agency problem inside Arbitrageurs are risk-averse
So the efficiency of arbitrage will be limited.
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III. Discussion of Performance-based Arbitrage
PBA supposes that all arbitrageurs have the same sensitivity, and will invest all funds when asset mispriced.
In reality, they differ. resources indepen-dent and invest more when price diverge further; not need to liquidate.
Big shock need more to eliminate, if not, mispricing gets deeper.
Little fresh capital available to stabilize.So PBA is likely to be important .
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IV. Empirical Implications
A. Which markets attract arbitrage reso-urces?
large funds concentrated in a few markets, bond markets & foreign exch-ange market;
the ability to ascertain value;
specialized arbitrageurs avoid volatile;
short horizons may be more relevant
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IV. Empirical Implications
B. Anomalies• higher historical returns.• EMH: compensation for higher risk impl-
ausible: large number of diversified arbitrageurs.
• few specialized arbitrageurs care about total risk fundamental or idiosyncratic.
• failing to recognize price-revisal .
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IV. Empirical Implications
• In extreme circumstances, lose enough money and liquidate;
• Investors become knowledgeable about the strategies, diminish withdrawals; but it will be slowly for investors take action.
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V. Conclusion
PBA may not be fully effective in bringing prices to fundamental values;
Specialized professional arbitrageurs may avoid extremely volatility;
The avoidance suggests a different approach to understanding persistent excess returns.
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Thank you !