Inversion

Optimal Execution
of Portfolio Transactions∗
Robert Almgren† and Neil Chriss‡
December 2000

Abstract
We consider the execution of portfolio transactions with the aim of
minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact. For a simple linear cost model, we explicitly construct the efficient frontier in the
space oftime-dependent liquidation strategies, which have minimum
expected cost for a given level of uncertainty. We may then select optimal strategies either by minimizing a quadratic utility function, or
by minimizing Value at Risk. The latter choice leads to the concept of
Liquidity-adjusted VAR, or L-VaR, that explicitly considers the best
tradeoff between volatility risk and liquidation costs.

∗

Wethank Andrew Alford, Alix Baudin, Mark Carhart, Ray Iwanowski, and Giorgio
De Santis (Goldman Sachs Asset Management), Robert Ferstenberg (ITG), Michael Weber
(Merrill Lynch), Andrew Lo (Sloan School, MIT), and George Constaninides (Graduate
School of Business, University of Chicago) for helpful conversations. This paper was begun
while the first author was at the University of Chicago, and thesecond author was first
at Morgan Stanley Dean Witter and then at Goldman Sachs Asset Management.
†
University of Toronto, Departments of Mathematics and Computer Science;
almgren@math.toronto.edu
‡
ICor Brokerage and Courant Institute of Mathematical Sciences;
Neil.Chriss@ICorBroker.com

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Contents
Introduction

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1 The1.1
1.2
1.3
1.4
1.5

Trading Model
The Definition of a Trading Strategy . .
Price Dynamics . . . . . . . . . . . . . .
Temporary market impact . . . . . . . .
Capture and cost of trading trajectories
Linear impact functions . . . . . . . . .

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2 The
2.1
2.2
2.3
2.4

Efficient Frontier of Optimal Execution
The definition of the frontier . . . . . . . . .
Explicit construction of optimal strategies . .
The half-life of a trade . . . . . . . . . . . . .
Structure of the frontier . . . . . . . . . . . .

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3 The
3.1
3.2
3.3
3.4

Risk/Reward Tradeoff
Utility function . . . . . . . . .
Value at Risk . . . . . . . . . .
The role of utility in execution
Choice of parameters . . . . . .

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4 The
4.1
4.2
4.3

Value of Information
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Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Serial correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Parameter shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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5 Conclusions
A Multiple-Security Portfolios
A.1 Trading model . . . . . . . . . . . .
A.2 Optimal trajectories . . . . . . . . .
A.3 Explicit solution for diagonal modelA.4 Example . . . . . . . . . . . . . . . .
References

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