# Optimal Turnover, Liquidity, and Autocorrelation

Bastien Baldacci111Quantitative Advisory Solutions,
<EMAIL_ADDRESS>Jerome Benveniste222Courant Institute,
NYU Tandon School, Ritter Alpha<EMAIL_ADDRESS>and Gordon Ritter333Courant
Institute, NYU Tandon School, Columbia University, Baruch College, and Ritter
Alpha<EMAIL_ADDRESS>
## Introduction

One of the central problems faced by institutional investment managers is the
proper management of trading costs. For very active strategies, the most
significant source of trading costs is usually market impact; see among others
[2, 4]. The manager may reduce overall market impact cost by reducing
turnover, but this potentially also reduces the manager’s ability to monetize
time-sensitive trading opportunities.

The proper level of turnover of a real strategy can depend on many factors.
Nevertheless, in a class of Gaussian process models with linear price
impact444This class of models has been the subject of an intensive literature,
see among others [1, 8] and the reference books [6, 10], we derive a simple
explicit relation (20) between the optimal turnover, the autocorrelation of
the trading signal, the investor’s risk-aversion, and the liquidity and
volatility of the underlying asset.

Our explicit results, while mathematically elegant, only apply to the case of
linear price impact, leading to quadratic total cost. Empirical studies
support the conclusion that the price impact of large orders is in fact
proportional to the square root of the order’s participation; see [14] and
references therein. Nevertheless, we join [7] in the belief that it is
worthwhile to develop explicit formulas which apply to the linear case,
because they can be used as heuristics, approximations to more complex models,
or to develop bounds.

Historically, one of the first explicit formulas relating approximate
investment performance to forecast accuracy was the [9] fundamental law of
active management, which asserts that $\mathrm{IR}\approx\mathrm{IC}\sqrt{N}$,
where IR is the information ratio, IC is the information coefficient defined
as the correlation of a single signal to the dependent return, and $N$ is the
effective number of independent bets. Bets may be independent either because
they pertain to statistically independent investments, or because they pertain
to different periods of time, or both.

Unfortunately the [9] formula cannot relate the turnover of the strategy to
any other meaningful quantity. The formula always prefers increasing turnover,
because if asset returns are serially independent, then trading more often
increases $N$ with no perceived cost. Liquidity of the asset is not an input
to the formula.

In order to better understand the fundamental relation between optimal
turnover, liquidity, and autocorrelation of alpha signals, we work in a
continuous-time stochastic process model. Letting $x_{t}$ denote an investor’s
holdings of a risky asset at time $t$, denominated in dollars (or any
convenient numeraire), define the _steady-state turnover_ as the following
limit

$\lim_{t\to\infty}\frac{{\mathbb{E}}|\dot{x}_{t}|}{{\mathbb{E}}|x_{t}|}$ (1)

where $\dot{x}_{t}$ denotes the time-derivative.

The steady-state turnover of the _optimal_ strategy is of clear interest to
practitioners and portfolio managers, as is the steady-state Sharpe ratio. In
what follows, we show that in a convenient Gaussian process model, the steady-
state turnover can be computed explicitly, and obeys a clear relation to the
liquidity of the asset and to the autocorrelation of the alpha forecast
signals. Thus, we contribute to the literature on optimal execution, in the
spirit of [5, 12], by providing closed-form expressions of important trading
metrics in a general linear-quadratic framework. Let $\lambda$ denote the
linear price impact coefficient in the tradition of [11], let $\kappa$ denote
the investor’s absolute risk-aversion, and let $\sigma$ denote the
instantaneous volatility. In the special case of an Ornstein-Uhlenbeck model
with mean-reversion speed $\phi$, we find that steady-state optimal turnover
is given by

$\sqrt{\frac{\sigma\left(\phi\sqrt{\kappa\lambda}+\kappa\sigma\right)}{\lambda}}.$

Before deriving these explicit formulas, we present Theorem 1, which is
arguably the most general result on optimal trading strategies for quadratic
trading-cost models in the setting of a mean-quadratic-variation objective.

## A very general result on quadratic costs

Presently we generalize the well-known result of [8], which assumed the
return-predictor process is a Markovian jump diffusion, to any square-
integrable process. Our theorem also generalizes the main result of [1], and
thus has applications to optimal execution of algorithmic orders. Such a
general result is of intrinsic interest in its own right, but we need it
specifically later in this paper, to derive results on steady-state optimal
turnover and strategy performance. The proof is also of interest, as it shows
that the problem can be converted into a convex optimization problem in an
infinite-dimensional space, and is thus amenable to standard convex
optimization techniques as per [3].

Fix a probability measure space $(\Xi,\mathbb{P})$ where $\mathbb{P}$ is a
probability measure on a $\sigma$-algebra of events. The outcomes in this
space are various possible trajectories of the market and of our trading
within it. The filtration $\mathcal{F}=\\{\mathcal{F}_{t}:t\geq 0\\}$ denotes,
as usual, the information that is available at time $t$. An _adapted process_
is a stochastic process $x$ such that each $x_{t}$ is
$\mathcal{F}_{t}$-measurable. Since it is used often, we define a notation for
the conditional expectation:

${\mathbb{E}}_{t}[Y]:={\mathbb{E}}[Y\mid\mathcal{F}_{t}],$

where $Y$ is any random variable.

Let $\mathcal{H}$ denote the usual real Hilbert space of all adapted
$\mathbb{R}^{N}$-valued processes on $[0,\infty)$ which are integrable in the
mean-square sense:

$\mathbb{E}\int_{0}^{\infty}||\mu_{t}||^{2}dt<\infty.$

Also let $\mathcal{A}$ be the Sobolev space of ${\mathbb{R}}^{N}$-valued
adapted processes that are almost surely differentiable and whose derivative
lies in $\mathcal{H}$. We use the symbol $\cdot$ for the Euclidean inner
product in ${\mathbb{R}}^{N}$, and $\langle\,,\,\rangle$ for the standard
inner product on $\mathcal{H}$:

$\langle x,y\rangle=\mathbb{E}\int_{0}^{\infty}x_{t}\cdot y_{t}\,dt\,.$

In the following, the positive-definite matrix $\Omega$ denotes the quadratic
co-variation of the asset return process; with this interpretation, the risk
term in our objective function is integrated instantaneous variance. This
corresponds with the treatment of integrated variance in [1], and also
corresponds directly to the objective function of [8] in the zero-discounting
case.

######  Theorem 1.

Let $\mu\in\mathcal{H}$, let $\Omega$ be a positive-definite $N\times N$
covariance matrix and $\Lambda$ a positive-definite $N\times N$ market-impact
coefficient matrix. Let $\kappa>0$ be a risk-aversion coefficient. Then for
any $x_{0}\in{\mathbb{R}}^{N}$, there is a unique solution to the optimization
problem

$\max_{x\in\mathcal{A};\;x(0)=x_{0}}{\mathbb{E}}\int_{0}^{\infty}\left[\mu_{t}\cdot
x_{t}-\frac{1}{2}\dot{x}\cdot\Lambda\dot{x}-\frac{\kappa}{2}x_{t}\cdot\Omega
x_{t}\right]\,dt,$ (2)

given by the solution, with boundary condition $x(0)=x_{0}$, to the
stochastically-forced ODE system

$\dot{x}_{t}=-\Gamma x_{t}+b_{t}$ (3)

where

$\displaystyle\Gamma$ $\displaystyle=(\kappa\Lambda^{-1}\Omega)^{1/2}\text{
and }$ (4) $\displaystyle b_{t}$
$\displaystyle=\int_{t}^{\infty}e^{\Gamma(t-s)}\Lambda^{-1}\mathbb{E}_{t}\mu_{s}\,ds.$
(5)

######  Remark 1.

The objective in (2) may be unbounded if

$\mathbb{E}\int_{0}^{\infty}||\mu_{s}||^{2}\,ds=\infty$

as will be the case in many examples of interest, including stationary
processes. However, suppose that $\mu$ satisfies the weaker condition

$\mathbb{E}\int_{t}^{\infty}e^{-\epsilon
s}||\mathbb{E}_{t}\mu_{s}||^{2}\,ds<\infty$

for any $t$ and any $\epsilon>\Gamma$. This condition ensures that $b_{t}$ is
well-defined for all $t\geq 0$, and is satisfied by most realistic forecasts;
for instance, those with at most linear growth. Then the truncated process
$\mu_{s}^{K}=\mathbbm{1}_{\\{s<K\\}}\mu_{s}$ is in $\mathcal{H}$, hence
Theorem 1 gives an optimal strategy $x^{K}_{s}$ corresponding to
$\mu_{s}^{K}$. Moreover, one may prove that on every bounded interval $[0,T]$,
$x^{K}\to x$ as $K\to\infty$ where $x$ is the solution to the ODE (3) for the
forecast process $\mu$, and convergence is in the mean-square sense on $[0,T]$
for the process and its first derivative.555A proof of this fact is available
from the authors on request. Thus an investor with an infinite horizon and no
discounting will arrive at a well-defined investment strategy by using (3), or
equivalently by considering finite-horizon problems and letting the horizon
tend to infinity.

######  Remark 2.

The system (3) is said to be stochastically-forced because, when viewed from
time $t=0$, $b_{t}$ is a stochastic process due to the presence of
${\mathbb{E}}_{t}\mu_{s}$, which is a conditional expectation in the
filtration.

######  Remark 3.

Note that the solution (3) can be rewritten so that we only take the square
root of symmetric and positive semi-definite matrices using the fact that

$\Gamma=\kappa\Lambda^{-1/2}\big{(}\Lambda^{-1/2}\Omega\Lambda^{-1/2}\big{)}^{1/2}\Lambda^{1/2}.$

_Proof of Theorem 1._ Let $K$ be the integral operator defined by

$(Ku)_{t}=\int_{0}^{t}u_{s}\,ds,$

which allows us to write the original problem as a convex optimization
problem:

$\max_{u}\big{[}\langle Ku,\mu\rangle-\frac{1}{2}\langle u,\Lambda
u\rangle-\frac{\kappa}{2}\langle Ku,\Omega Ku\rangle\big{]}\,.$ (6)

Here $u$ is an auxiliary process defined so that $x_{t}=(Ku)_{t}$. By convex
analysis, any solution to (6) must also solve

$K^{*}\mu-\Lambda u-\kappa K^{*}\Omega Ku=0.$ (7)

where the operator $K^{*}$ is given by

$K^{*}u=\int_{t}^{\infty}\mathbb{E}_{t}u_{s}ds$

and so (7) takes the form

$\int_{t}^{\infty}{\mathbb{E}}_{t}\mu_{s}\,ds-\Lambda
u_{t}-\kappa\int_{t}^{\infty}\int_{0}^{s}\Omega{\mathbb{E}}_{t}u_{z}\,dzds=0\,.$
(8)

Fix a time $t_{0}$, let $u$ be the solution of (7), let $x=Ku$, and let

$x^{t_{0}}_{t}=\mathbb{E}_{t_{0}}x_{t},\;\text{ for }t>t_{0}.$

Applying the operator $\mathbb{E}_{t_{0}}$ to (8) and differentiating with
respect to $t$ gives the following linear second-order ODE for
$x^{t_{0}}_{t},\;t\geq t_{0}$:

$\frac{d^{2}x^{t_{0}}_{t}}{dt^{2}}=\kappa\Lambda^{-1}\Omega
x^{t_{0}}_{t}-\Lambda^{-1}\mathbb{E}_{t_{0}}\mu_{t}.$ (9)

Solving a linear second-order ODE is standard, and in this case yields:

$x^{t_{0}}_{t}=e^{-\Gamma(t-t_{0})}x^{t_{0}}_{t_{0}}+\int_{t_{0}}^{t}e^{-\Gamma(t-s)}\int_{s}^{\infty}e^{-\Gamma(z-s)}\Lambda^{-1}\mathbb{E}_{t_{0}}\mu_{z}\,dzds$
(10)

and therefore

$\left.\frac{dx^{t_{0}}_{t}}{dt}\right|_{t=t_{0}}=-\Gamma
x_{t_{0}}^{t_{0}}+b_{t_{0}}.$

Recalling that $x^{t_{0}}_{t_{0}}=x_{t_{0}}$, that

$\left.\frac{dx^{t_{0}}_{t}}{dt}\right|_{t=t_{0}}=\dot{x}_{t_{0}},$

and that $t_{0}$ was arbitrary gives the desired result. $\Box$

This result is helpful due to its level of generality. Theorem 1 represents a
strict generalization of [8] Proposition 1. We generalize this result to
predictor processes which are not necessarily Markov, and to the case of zero
discounting. For Markov predictors with zero discounting, our solution agrees
with [8] exactly. The rate matrix presented in [8] simplifies to

${\bar{M}}^{\text{rate}}=(\kappa\Lambda^{-1}\Omega)^{1/2}=\Gamma.$

Under these conditions, the matrix denoted $b$ in [8] is also equal to the
rate matrix. We then have from (5),

$\displaystyle{\bar{M}}^{\text{aim}}$
$\displaystyle=\Gamma^{-1}\Lambda^{-1}\int_{t}^{\infty}e^{\Gamma(t-s)}\mathbb{E}_{t}\mu_{s}\,ds$
$\displaystyle=\Gamma\int_{t}^{\infty}e^{\Gamma(t-s)}\mathbb{E}_{t}[(\kappa\Omega)^{-1}\mu_{s}]\,ds$

The latter expression agrees with equation (12) of [8].

Theorem 1 enables researchers to derive explicit optimal trading strategies
for a wide range of models by relating them to equivalent ODE systems.
Moreover, the maximization in (2) is not simply over static, predetermined
trading plans, but over the substantially larger class of all admissible
stochastic processes. This contributes to the literature on optimal execution
in a linear-quadratic framework in the spirit of [12]. In our case, Theorem 1
facilitates the derivation of simple explicit formulas for the steady-state
turnover and information ratio of the optimal strategy, as we shall present in
the coming sections.

## Steady-state calculations for Gaussian processes

In the sequel we are interested in detailed calculations for single-asset
trading paths and in particular, on account of (3), we are interested in
studying a Gaussian process $x$ defined by the stochastically-forced linear
ODE:

$\dot{x}_{t}=-\gamma x_{t}+a_{t},$

for some constant $\gamma>0$, where $a_{t}$ is a process satisfying

${\mathbb{E}}_{t}a_{s}=e^{-\theta(t-s)}a_{t}\text{ for all }s>t,$ (11)

for some $\theta>0$. We shall see later that certain of the processes (5) from
the theorem must necessarily satisfy the relation (11), and hence are valid
choices for $a_{t}$. It follows from (11) that
$\lim_{s\to\infty}{\mathbb{E}}_{t}[a_{s}]=0$, so the steady-state mean of the
process $a$ vanishes, and hence the same must hold for $x$ and $\dot{x}$. For
simplicity let us assume that the unconditional mean of $a_{t}$ is zero for
all $t$.

Under these assumptions, the Gaussian process $x_{t}$ has certain steady-state
properties which can be derived in closed form. Since $x_{t}$ is prototypical
solution of an optimal trading problem that we discuss later on, the steady-
state properties of $x_{t}$ are clearly of crucial importance to understanding
the optimal steady-state turnover.

For notational convenience, let us define the following three functions:

$h(t)={\mathbb{E}}a_{t}x_{t},\ g(t)={\mathbb{E}}x^{2}_{t},\text{ and
}v(t)={\mathbb{E}}\dot{x}^{2}_{t}.$ (12)

We are interested in the steady-state behaviour of these functions.

Note that $h$ has time-derivative

$\displaystyle\dot{h}$
$\displaystyle={\mathbb{E}}[\dot{a}_{t}x_{t}+a_{t}\dot{x}_{t}]$
$\displaystyle=-\theta{\mathbb{E}}a_{t}x_{t}+{\mathbb{E}}a_{t}^{2}-\gamma{\mathbb{E}}a_{t}x_{t}$
$\displaystyle=c_{0}-(\theta+\gamma)h.$

where $c_{0}:={\mathbb{E}}a_{t}^{2}$ denotes the steady-state variance of
$a_{t}$. Then the steady-state value of $h(t)$ is

$\bar{h}=\frac{c_{0}}{\theta+\gamma}.$

Similarly,

$\displaystyle\dot{g}$
$\displaystyle=2{\mathbb{E}}\dot{x}_{t}x_{t}=2{\mathbb{E}}a_{t}x_{t}-2\gamma{\mathbb{E}}x_{t}^{2}$
$\displaystyle=2h-2\gamma g,$

in steady state, that is

$\dot{g}=2\bar{h}-2\gamma g,$

so the steady-state value for $g$ is

$\bar{g}=\frac{\bar{h}}{\gamma}.$

Finally, we have

$\displaystyle v(t)$ $\displaystyle={\mathbb{E}}[(a_{t}-\gamma x_{t})^{2}]$
$\displaystyle={\mathbb{E}}[a^{2}_{t}-2\gamma a_{t}x_{t}+\gamma^{2}x_{t}^{2}]$
$\displaystyle=c_{0}-2\gamma h(t)+\gamma^{2}g(t),$

which, in steady state is:

$\displaystyle\bar{v}$
$\displaystyle=c_{0}-2\gamma\bar{h}+\gamma^{2}\bar{g}=c_{0}-\gamma\bar{h}$
$\displaystyle=c_{0}\left(1-\frac{\gamma}{\theta+\gamma}\right)$
$\displaystyle=\frac{c_{0}\theta}{\theta+\gamma}=\bar{h}\theta.$

Since differentiation is a linear operator, the derivative of a Gaussian
process is another Gaussian process (see [15] section 9.4 and references
therein). Since $\\{x_{t}\\}$ is a Gaussian process, each $x_{t}$ is a
Gaussian random variable, as is each $\dot{x}_{t}$. In our example both
$x_{t}$ and $\dot{x}_{t}$ have mean zero. From classical properties of the
mean-zero normal distribution, one then has
${\mathbb{E}}[|x_{t}|]^{2}=(\pi/2){\mathbb{E}}[x_{t}^{2}]$, and the same holds
for $\dot{x}_{t}$ with the same constant, $\pi/2$. Hence the steady-state
turnover, as defined by (1), exists and is given by

$\lim_{t\to\infty}\frac{{\mathbb{E}}|\dot{x}_{t}|}{{\mathbb{E}}|x_{t}|}=\frac{\sqrt{\bar{v}}}{\sqrt{\bar{g}}}=\sqrt{\theta\gamma}.$
(13)

## Single-asset trading strategies

Suppose an investor with risk-aversion $\kappa>0$ is trading a single
financial asset and solves

$\sup_{x}{\mathbb{E}}\left[\int_{0}^{\infty}\mu_{t}x_{t}-\frac{1}{2}\kappa\sigma^{2}x_{t}^{2}-\frac{1}{2}\lambda{\dot{x}_{t}}^{2}\,dt\right]$
(14)

where $x,\mu$ are assumed to satisfy the technical hypotheses in Theorem 1,
and the supremum is over all such $x$. The use of $\lambda$ to denote a linear
price impact coefficient is inspired by, and consistent with, the notation of
[11]; indeed such a coefficient is often referred to as _Kyle’s lambda_. We
are interested in the steady-state properties of solutions to (14), for which
the key result is the following corollary of Theorem 1.

######  Corollary 2.

The optimal trading strategy solving (14) is the solution of the ODE with
stochastic coefficients:

$\dot{x}_{t}=-\gamma x_{t}+m_{t},$ (15)

where:

$\displaystyle\gamma$ $\displaystyle=\sqrt{\frac{\kappa\sigma^{2}}{\lambda}};$
(16) $\displaystyle m_{t}$
$\displaystyle=\int_{t}^{\infty}\lambda^{-1}e^{-\gamma(s-t)}{\mathbb{E}}_{t}\mu_{s}\,ds.$
(17)

Corollary 2 is a straightforward consequence of applying Theorem 1 to the
single-asset case, where $N=1$ and the $N\times N$ matrices $\Lambda,\Omega$
are just scalars $\lambda,\sigma^{2}$. In particular (16) may be seen as the
reduction of (4).

Assume now that the alpha-forecast process $\mu_{t}$ is an Ornstein-Uhlenbeck
(O-U) process:

$d\mu_{t}=-\phi\mu_{t}dt+\nu\,dW.$ (18)

In this context $\phi$ is called the _speed of mean-reversion_ , and
$\ln(2)/\phi$ is the half-life. Then $m_{t}$ in (17) satisfies the
exponential-decay condition (11), with $\theta=\gamma+\phi$:

${\mathbb{E}}_{t}m_{s}=e^{-(\gamma+\phi)(s-t)}m_{t}.$ (19)

Hence the calculations leading to (13) apply. In particular, the optimal
steady-state turnover exists and is given by (13) which becomes:

$\text{optimal turnover}=\gamma\sqrt{\phi/\gamma+1}.$ (20)

Plugging in (16) it is easily verified that (20) is equal to
$\sqrt{{\sigma(\phi\sqrt{\kappa\lambda}+\kappa\sigma)}/{\lambda}}$, and that
$\gamma$ is the scalar version of the Garleanu-Pedersen rate matrix.

Eq. (20), while only strictly valid in the case of quadratic costs, can be
used by practitioners as an easily-remembered rule of thumb. One can see, e.g.
from (19), that $\gamma$ and $\phi$ each have dimensions of inverse time, and
so optimal turnover is also in the same units as $\gamma$, inverse time. As
such it represents the fraction of steady-state book size which is traded in
the given time unit. For example, if $\kappa=10^{-6}$, then for an asset with
$\sigma=0.01/\mbox{day}$, $\lambda=\mbox{10 bps per 1\% of ADV}$, and average
daily volume (ADV) is $\$\,10\,\mbox{million}$, then one has $\gamma=0.1$/day.
If $\phi=0.2$/day, corresponding to a half-life about 3.5 days, then
$\phi/\gamma=2$ and optimal turnover (20) is about $\gamma\sqrt{3}\approx$
17.3%/day.

From (17) and the fact that

${\mathbb{E}}_{t}\mu_{s}=e^{-\phi(s-t)}\mu_{t},$

we get the relation

$m_{t}=\lambda^{-1}(\gamma+\phi)^{-1}\mu_{t}.$ (21)

Thus, using (12) with $m_{t}$ in place of $a_{t}$, and also using (21) to
solve for $\mu_{t}$, we find that the expected _ex ante_ rate of profitability
is

${\mathbb{E}}[x_{t}\mu_{t}]=\lambda(\gamma+\phi){\mathbb{E}}[x_{t}m_{t}]=\lambda(\gamma+\phi)h(t)$
(22)

We define the _steady-state information ratio (IR)_ as

$\text{IR}=\lim_{t\to\infty}\frac{{\mathbb{E}}[x_{t}\mu_{t}-(\lambda/2)\dot{x}_{t}^{2}]}{\sqrt{{\mathbb{E}}[\sigma^{2}x_{t}^{2}]}}$

The same calculations that were used in deriving (13) now allow us to find, in
the special case of the O-U process given in (18) for $d\mu_{t}$, that

$\text{IR}=\frac{\nu}{2\sigma}\sqrt{\frac{\gamma}{2\phi(\phi+2\gamma)}}$ (23)

In the numerical example from above, with $\phi/\gamma=2$ one has has
$\text{IR}=\nu/(8\gamma\sigma)\approx 0.5\nu/\sigma$.

The extension to a portfolio of $N$ _statistically independent_ assets, all
governed by mean-reverting dynamics with similar half-lives and liquidity
parameters, may be approximated by multiplying (23) by $\sqrt{N}$ as per [9].
In general the assets in a portfolio will not be statistically independent,
but there are special cases which may be approximated using the independence
assumption. For example, in a market where the covariance matrix is well-
described by a multi-factor model [13], then a strictly factor-neutral
strategy may be modeled as trading synthetic “residual assets,” which are
defined as baskets that realize the factor model’s residual returns. For
example, if one starts with a position in stock $i$, and then adds an
appropriate combination of pure factor portfolios to completely hedge stock
$i$’s exposures to all common factors, one has a representation of the $i$-th
residual asset as this hedged basket.

Residual assets will be statistically independent if the factor model’s
covariance matrix represents the true covariance of the data-generating
process. For strategies modeled as dynamic portfolios of residual assets, the
approximation of multiplying (23) by $\sqrt{N}$ is reasonable. Thus one
obtains the closed-form generalization of the classic “fundamental law of
active management” to a modern context, in which the trader is aware of market
impact and acts in a dynamically optimal way with respect to a mean-quadratic-
variation objective.

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