# The Limits of Arbitrage

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"The Limits of Arbitrage"

```					      The Limits of Arbitrage

Andrei Shleifer & Robert Vishny (JF 1997)

Presented by Steven Laufer
December 4, 2007
Motivation

Absence of arbitrage has been an underlying assumption in
most/all of papers so far: Existence of SDF, Compact budget
sets.
In reality, many funds practice professional arbitrage.
Why don’t the actions of these funds eliminate arbitrage
Overview

Agency Model of Limited Arbitrage
Model Results
Authors’ Discussion
My Discussion
Model: General Assumptions

Arbitrage requires capital.
Arbitrage practiced by managers who manage investors’
money. Investors may withdraw capital. Investors do not
understand the arbitrage mechanism and can only evaluate
managers based on past performance, not future expected
returns.
Arbitrageurs highly specialized. Only have expertise to
practice in one market. Take large enough positions to aﬀect
prices.
Risk-free arbitrage: arbitrage strategy makes money with
probability one.
Mis-pricing may deepen before it is resolved.
Model: Setup
Market for one security
Three periods:
Period 3: Price goes to fundamental value V (p3 = V ). V
known to arbitrageur from start.
Period 2: With probability (1-q), nose traders realize the
mis-pricing and price goes to fundamental value V (p2 = V ).
Otherwise . . .
Periods 1 & 2:
Prices pt determined in equilibrium.
Noise traders have “pessimism shock” St which causes them
to underprice asset. If p2 = V , S2 > S1 so mis-pricing worsens.

QN(t) = [V − St ]/pt
Model: Setup

Arbitrageur in periods 1 & 2
Arbitrageur has available funds Ft . F1 given exogenously. F2
depends on t=1 returns. Investors may withdraw funds
between periods 1 and 2.
Arbitrageurs invest Dt ≤ Ft . Security in unit supply so
equilibrium prices given by

pt = V − St + Dt

Assume F2 < S2 so p2 < V . (Mis-pricing survives because of
lack of capital available for arbitrage.)
If p2 = V , D2 = Ft because prices surely rise to p3 = V .
If p2 = V , arbitrageur invests in cash.
The interesting period is t=1.
Performance-based Arbitrage
Investors use gross returns from t=1 to t=2 to judge whether
to withdraw or invest more capital (even though lower returns
predict increased future returns).

F2 = F1 · G ([D1 · (p2/p1) + (F1 − D1 )]/F1 )

G (x) = ax + 1 − a, a ≥ 1
a controls sensitivity of investment to past performance.
G (1) = 1, G = a ≥ 1, G ≤ 0
If mis-pricing worsens at t = 2, investors see loss and
withdraw funds. F2 = F1 − aD1 (1 − p2 /p1 ).
Arbitrageurs maximize expected third-period wealth
D1 · V
EW = (1 − q) a             + F 1 − D1     + (1 − a)F1
p1
V         D1 · p2
+q          a           + F 1 − D1    + (1 − a)F1
p2          p1
Model Results
First order condition for arbitrageur
(1 − q) (V /P1 ) + q (p2/p1 − 1) V /p2 ≥ 0
Equality if D1 < F1 . Inequality if D1 = F1 .
Proposition 1: For a given V ,S1 , S2 ,F1 , and a, there is a q ∗
s.t. for q > q ∗ , D1 < F1 and for q < q ∗ , D1 = F1 .
Proposition 2: At the corner solution (D1 = F1 ),
dp1 /dS1 < 0, dp2 /dS2 < 0, and dp1 /dS2 = 0. At the interior
solution dp1 /dS1 < 0, dp2 /dS2 < 0, and dp1 /dS2 < 0.
No general results about eﬀect of a on pricing. Depends on
whether prices can approach true value in t = 2, which
determines whether arbitrageurs might be given more or less
funds in period 2.
Proposition 3: If D1 = F1 and S2 > S1 , then for a > 1,
F2 < D1 and F2 /p2 < D1 /p1 . (D1 = F1 is a suﬃcient but not
necessary condition).
Proposition 4: If D1 = F1 , dp2 /dS2 < −1 and
d 2 p2 /dadS2 < 0.
Author’s Discussion

Why is PBA plausible? Contractual restrictions on withdrawls
are not common. Creditors demand immediate repayment
when value of collateral falls.
Arbitrageurs may liquidate positions following losses if fear
that future deepening mis-pricings would cause dramatic
future investment withdrawls.
Arbitrageurs may be uncertain about proﬁtability of strategy.
Author’s Discussion

Less arbitrage in equity and foreign exchange markets than in
bond markets because true value in more uncertain.
PBA model explains why arbitrage is harder in more volatile
markets. If traders are specialized, they cannot diversify away
from idiosyncratic risk, no risk premium for extra volatility.
Alternatively, in segmented market, idiosyncratic risk may be
priced. Arbitrageurs would earn higher expected returns in
more volatile assets, even if volatility not correlated with
systematic risk. Not in data.
Arbitrage strategies more attractive in markets where
resolution of mis-pricing is faster and more certain.
Example: Stocks of commercial banks were underpriced in
1990-91 but funds had to liquidate before could realize
returns. Liquidation put further downward pressure on prices.
Discussion

Important to understand why arbitrage opportunities persist.
Limited capital available for arbitrage and uncertainty about
timing of resolution of mis-pricings are interesting
explanations.
BUT
These are not the main mechanisms of this paper.
Discussion: PBA Model

Investors evaluate arbitrageurs on time scale shorter than the
time expected to proﬁt from arbitrage.
Investors judge based on past performance even though past
and future returns are perfectly negatively correlated.
Investors are not behaving rationally!
Do we really expect mis-pricings to become worse?
Mechanism of PBA reducing ability of arbitrageurs to aﬀect
prices relies on this assumption!
Discussion: Importance of PBA

Limited capital reduces ability of arbitrageurs to correct
mis-pricings. PBA further reduces that ability. Evidence for
Why is PBA more appealing than risky arbitrage? What
about just uncertainty in timing of resolution of mis-pricing?
Authors were going to consider a=1 but don’t.
Discussion of volatility doesn’t rely on PBA. If large probability
of future deepening of mis-pricing, capital-limited ﬁrms will
hold more cash to take advantage of future opportunities.
Conclusion

Limited capital available for arbitrage and uncertainty about
timing of resolution of mis-pricings are interesting
explanations.
Not convinced about importance of PBA. Particularly with
modeling of investors and assumption of deepening
mis-pricing.
Need better description/model of evolution of mis-pricings.
If the price gets further from true value, is that evidence that
the mis-pricing will persist? Compare that to PBA model.

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