Optimal rebalancing of
portfolios withtransaction costs

Note noAuthors

Date

ADMIN/01/2007
Helge Holden
Lars Holden
11th September 2007

The authors
Helge Holden is professor in mathematics at NTNU
Lars Holden is managing director at NR and has a dr. philos. in mathematics

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TitleOptimal rebalancing of portfolios withtransaction costs

Authors

Helge Holden <Helge.Holden@math.ntnu.no>Lars Holden <Lars.Holden@nr.no>Date

11th September 2007

Publication number

ADMIN/01/2007

Abstract
Rebalancing of portfolios with a concave utility function is considered. It is provedthat transaction costs imply that there is a no-trade region where it is optimal not
to trade. For proportional transaction costs it is optimal to rebalance to the boundary when outside the no-trade region. With flat transaction costs, the rebalancefrom outside the no-trade region should be to a state at an internal surface inthe no-trade region but never a full rebalance. The standard optimal portfolio
theory is extended to n-symmetric assets, general utility function, and more general stochastic processes. Examples are discussed. The paper studies rebalancingof portfolios with a general concave utility function. It is proved that transaction costs imply that there is a no-trade region in the state space where it is not
optimal to perform trading. If the transaction costs are proportional, then it is optimal to rebalance when outside the no-trade region to a state at the boundaryof this region. If the transaction costs have fixed or flat elements, then the rebalance from outside the no-trade region should be to a state at an internal surface
in the no-trade region but never a full rebalance. This extends previous resultsby considering n-symmetric assets, using a general utility function, and a larger
class of stochastic processes. We find an analytic approximation to the boundaryof the no-trade region for a quadratic utility function. The results are illustrated
with an explicit example and a numerical example with simulation. The paper
is a natural extension of the optimal portfolio theory (Markowitz, (7)) since it includes transaction costs and handles more general stochastic processes and utility
functions.
Keywords

Rebalance portfolio, portfolio optimization,transaction costs, no-trade region, utility function

Target group
Availability

Open

3

Project
Project number
Research field
Number of pages

34

Copyright © 2007

Norwegian Computing Center

Optimal rebalancing of portfolios with transaction costs4

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . .

6

2

The model . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Utility functions . . . . . . . . . . . . . . . . . . . . . . . .

18

4

An explicit example . . . . . . . . . . . . . . . . . . . . . .

20

5

A numerical example . . . . . . . .
5.0.1
Estimate U when n = 2 .
5.0.2
Estimate U when n > 2 .
5.0.3
Simulation of portfolios .

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24
25
28
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6

Closing remarks . . . . . . . . . . . . . . . . . . . . . . .

33

References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

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5

1 Introduction

Financial portfolios usually consist of several different assets and rebalancing theportfolio is necessary in order to control the weights in the different assets. Most
papers neglect transactions costs, but it is documented in, e.g., Atkinson et al.
(1), Donohue and Yip (4) and Leland (5) that this may be a significant problem.
In the present paper it is proved, assuming only a concave utility function, thattransaction costs imply that there is a no-trade region where it is not optimal toperform trading. If the transaction costs are proportional, then it is optimal to rebalance when outside the no-trade region to a state at the boundary of this region.
If the transaction costs have fixed or flat elements, then the rebalance from outside the no-trade region should be to a state at an internal surface in the no-traderegion but never a full rebalance. This extends previous results by considering nsymmetric assets, using a general utility function, and a larger class of stochastic
processes. Furthermore, we provide an approximate formula and a more precise
iterative simulation algorithm for the boundary of the no-trade region and theinternal surface it should be rebalanced to in the case of fixed elements in thetransaction costs.
The problem of optimally rebalancing a portfolio with transactions costs isstudied in several papers, see Chang (3) and Liu (6) and references herein. The
case of two assets can be analyzed analytically, see, e.g., Taksar et al. (12) and Øksendal and Sulem (10). The multi-asset problem under strong assumptions has
been studied by, e.g., Donohue and Yip (4). There are also some papers with dynamic programming algorithms for determining the no-trade region in higher
dimensions, see, e.g., Sun et al. (11) and Leland (5). Atkinson et al. (1) give anapproximate formula for the boundary. Most papers conclude that there is a notrade region where trade should not be performed, and all rebalance from a stateoutside the no-trade region is to the boundary of the no-trade region. Donohue
and Yip (4) formulate this as a general result with general n and only assuming
concave utility function, but does not prove the result. Furthermore, they consider different rebalance strategies for portfolios with one risk-free asset and up
to seven risky assets. Liu (6) gives a thorough discussion of the problem withconstant absolute risk aversion and one risk-free investment and n uncorrelated
geometric Brownian motion investments. The paper discusses both proportionaland fixed transaction costs and shows the existence of a no-trade region that is a
fixed threshold for each investment with risk. All rebalance is between an investment with risk and the risk-free investment.

Optimal rebalancing of portfolios with transaction costs

6

There are different formulations of the optimal portfolio problem. The classical paper Markowitz (7) employs a utility function defined on a portfolio. Leland (5) and Donohue and Yip (4) focus on portfolios with targets ratios and theutility function is deviance from the target ratios and the total transaction costs.
The recent paper Liu (6) builds on the work of Merton (8) that optimizes the consumption that is possible based on the portfolio. All models only seem to imvolvegeometric Brownian motion. The properties of the different formulations appear
quite similar, but they offer different mathematical challenges. The present paper
extends the framework of Markowitz (7) with n symmetric investments by also
incorporating transaction costs in addition to a more general utility function anda larger class of stochastic processes. Our formulation includes the formulationof Leland (5) and the formulation of Merton (8) if we assume the consumption
and investment are known. Our assumption with n symmetric assets generalize
the assumption in Liu (6) where there is one risk-free asset at the cost of more
complex analysis and complex boundary to the no-trade region.
The theoretical agreement regarding the existence of a no-trade region is incontrast to the current practice in most portfolios. Donohue and Yip (4) state thatthe typically reduction of transaction costs by using an optimal rebalance strategy
is 50%. It seems to be most common to rebalance to what is considered the optimalbalance at fixed time intervals, often monthly or each quarter, see, e.g., Leland (5).
Other portfolios define intervals for the weights in each asset and adjust to theboundary of these intervals either at fixed periods or continuously. Frequently,
these decision criteria are often combined with a full rebalance at certain situations. In the Norwegian Petroleum Fund (9) the rebalance is mainly performed
when deciding which new assets to buy. In addition, the portfolio is rebalanced
to the target weights if the weights are outside certain intervals over two consecutive months. The bank states that as a large investor it is an advantage that thetime of rebalances is not known in the marked and that the size of each rebalance
is not too large. The strategy presented in this paper satisfies these criteria. The
document from Norges Bank (9) includes a numerical simulation study of thetransaction costs under different rebalancing strategies.
Section 2 describes the model, and the theorem regarding the no-trade region
is proved. Different utility functions are discussed in Section 3. Sections 4 and 5
illustrate the theory with examples using different utility functions. The optimal
relative weights and the no-trade region are described in both cases. In Section
4 analytic calculations are sufficient. In this section is it also described how torebalance a portfolio to the boundary of the no-trade region. In Section 5 it is necessary with approximations and simulation. A procedure on how to determine
the no-trade region approximately is provided, and it is shown how to improve
this approximation by simulation. This example shows that the transaction costs

Optimal rebalancing of portfolios with transaction costs

7are reduced to 1/4 by the use of an optimal no-trade region. The paper is ended
by some closing remarks in Section 6.

Optimal rebalancing of portfolios with transaction costs8

2 The model

Consider n assets, and let Vi,t denote the stochastic value of asset i, for i = 1, 2, . . . , n
at time t. We assume that the stochastic properties of Vi,t are known, and that theprocesses are Markovian, i.e., if Vi,t is known, we do not get more information regarding the value at a later point in time by knowing the value of Vi,s for all valuess < t. We assume these stochastic processes are sufficiently regular in order thata utility function is defined on the stochastic process. For most utility functions
this implies that the expectation and a measure for variability is defined over atime period (t, T ) with T > t. In particular, this will be satisfied if log(Vi,t ) arecorrelated Levy processes. Levy processes include Brownian motion.
The portfolio is given by at = (a1,t , a2,t , . . . , an,t ) where ai,t denotes the weightof asset i at time t. The value of the portfolio at any given time t equalsnX
Wt =ai,t Vi,t .
(2.1)
i=1

We denote by
aj,t Vj,t(2.2)
Wt
the relative weights. In order to simplify the formulation we let ai,t ∈ R, i.e.,
both positive and negative values are considered, where negative values indicate
a short position in the asset. The results will be similar if we consider only nonnegative weights. The investor may at any time rebalance the portfolio. Let thetime of rebalance be at times t1 , t2 , · · · . We will consider only one rebalance at atime and therefore omit the index in order to simplify the notation. A rebalanceat time t implies that the portfolio is changed from
n
X
Wt− =
ai,t− Vi,t
(2.3)
rj,t =

i=1

to
Wt+ =

n
X

ai,t+ Vi,t

(2.4)

i=1

where ai,t are functions of time t that are constant between each rebalancing, andai,t− and ai,t+ denote the limits when approaching t from below and above, respectively. In the case there are discontinuities in the value of the assets, Vi,t , wewill always let Vi,t denote the limit from the right, i.e., Vi,t = Vi,t+ . This also includes (2.3). The reason for this definition is that if a jump in Vi,t implies a rebalance, this is performed immediately, and obviously based on the values after the
jump, i.e., Vi,t = Vi,t+ .

Optimal rebalancing of portfolios with transaction costs

9Assume that
Wt+ = Wt− − c(Dt )(2.5)

where the function c(Dt ) ≥ 0 is the cost of selling or buying assets at time t. The
set Dt contains all relevant information or data regarding the assets up to time t,i.e.,
Dt = {(ai,s , Vi,s ) | i = 1, . . . , n, s ≤ t+}.
We will assume that the transaction costs have proportional and fixed terms, i.e.,c(Dt ) =nXci,1 |ai,t+ − ai,t− |Vi,t + ci,2 χ(ai,t+ − ai,t− )

i=1



(2.6)

where ci,j ≥ 0 for i = 1, 2, . . . , n and j = 1, 2. The function χ(ai,t+ − ai,t− ) = 1
if ai,t+ − ai,t− 6= 0, i.e., if there is a rebalance in asset i at time t, and it equals
0 otherwise. For each asset i the cost function consists of a fixed fee if ci,2 6= 0
and a cost proportional to the weight if ci,1 6= 0. This formula covers the properties of interest from a theoretical point of view, and it is trivial to extend it to
let the transaction costs, e.g., depend on whether we sell or buy an asset. If we
include tax, the cost function is more complicated depending on when the asset
was bought, the difference between value when sold and bought and depends onthe country. We have c(Dt ) = 0 if there is no rebalance at time t.
We need to define a utility function U(Wt ) as a real valued function of thestochastic process Wt . It is natural that the utility increases in E{WT } for somevalue of T > t and decreases in a measure for the variability. See Section 3 for a
discussion on possible utility functions.
At every time t the investor optimizes U(Wt ). Since the investor at each time
t focuses on the future, i.e., s > t, a different function is optimized for each valueof t. When the investor optimizes at time t, we assume the investor applies thesame strategy for each later point t1 > t when optimizing U(Wt1 ). We assume
Markov strategies, i.e., the strategy is a function that only depends on the present
situation and not the entire previous history; viz.
at+ = Sz (at− , Vt )(2.7)

where at− and at+ denotes the weights before and after a possible rebalance attime t, and Vt denotes the values of the assets at time t. The same strategy isapplied for every value of t. The set of admissible strategies are parameterized byz ∈ Z.
The simplest possible strategy is never to rebalance, that is, ai,t is constant in
time, and we denote this by z = 1. Another strategy is rebalance in order to havethe relative weights within a given interval. Denote the interval by [r̂1 , r̂2 ]. Thusr̂i1 ≤ ri,t+ ≤ r̂i2

Optimal rebalancing of portfolios with transaction costs

10

or

r̂ 2
r̂i1
Wt− ≤ ai,t+ ≤ i Wt− .
Vi,t
Vi,t

We denote this strategy as z = 2, or alternatively z = (2, r̂1 , r̂2). For this strategy
it is necessary with detailed rules to determine how this rebalance is performed,e.g., only rebalance when one weight reaches the boundary of the admissible interval. Other strategies are of course also possible.
Instead of having discrete rebalancing at times t1 , t2 , . . . , it is possible to consider the case of continuous rebalancing. For instance, one can rebalance continuously in order to have constant relative weights. This may be denoted as z = 3
or z = (3, r̂t ) if we want to include the relative weights in the notation. Thus
ri,t+ = ri,t−or
ai,t+ = ai,t−

Wt+
ri,t−
=
Wt+ .
Wt−
Vi,t

In the case of continuous rebalancing it is necessary to redefine (2.3)–(2.6) to acontinuous setting. In this case the value of the portfolio has a continuous reduction cc (Dt ) due to transaction costs where1
cc (Dt ) =nXci,1 |a′i,t |Vi,t .

(2.8)

i=1

The coefficient a′i,t denotes the derivative of ai,t . Many stochastic processes, including Brownian motion, will have unbounded transaction costs if there is continuous rebalancing in order to have constant relative weights if not all ci,1 = 0.
Since a portfolio depends on the present weights at+ and the strategy z, we
will use the notation W (at+ , z) where at+ is the weights after a possible rebalanceat time t. We will often omit the plus sign if this is clear from the context.
We want as general formulation of the utility as possible and need the following definitions:Definition 2.0.1 We collect the basic definitions below:
We are given n assets with value Vi,t , i = 1, . . . , n, at time t.
(i) A portfolio is given by weights at = (a1,t , a2,t , . . . , an,t ) of some assets where the valueof the assets Vi,t are modeled by stochastic processes. The value Wt of the portfolio is given
by equation (2.1). The investor has the opportunity to rebalance the portfolio according to
the equations (2.3)–(2.6) or (2.8) in case of continuous rebalancing.
(ii) An investment strategy is a function Sz defined by (2.7). We denote such strategies
1. The subscript c indicates continuous rebalancing.

Optimal rebalancing of portfolios with transaction costs

11admissible. Let the set of admissible investment strategies may be parameterized by z ∈ Z.
We write z = (z1 , z2 ) where z1 ∈ {1, 2, . . . , m}, and z2 denotes (when required) the parameter values needed in the definition of the strategy.
(iii) A utility function is a real-valued function U(Wt (at , z)) where Wt defined by theequations (2.1)–(2.6), or (2.8) in case of continuous rebalancing, is assumed to be sufficiently regular such that U is well-defined. The portfolio value Wt depends on at , theportfolio weights, the stochastic processes Vt describing the value of each asset, and theinvestment strategy Sz parameterized by z.
(iv) U(Wt ) is continuous/continuously differentiable/concave if U(Wt (at , z)) is continuous/continuously differentiable/concave in at for every value of z. The utility functionPP
is concave if for any a1 and a2 satisfying i a1,i Vi,t =i a2,i Vi,t = A where A is aconstant, we haveU(Wt (βa1 + (1 − β)a2 , z)) ≥ βU(Wt (a1 , z)) + (1 − β)U(Wt (a2 , z))(2.9)

for any constant 0 < β < 1. We say that U(Wt ) is strictly concave if (2.9) holds withstrict inequality.
(v) We say that U(Wt (at , z)) has compact level sets in the weights if for each combinationof A > 0, D ∈ R, and z ∈ Z, the set
XΩA,D,z,t = {at ∈ Rn |ai,t Vi,t = A, U(Wt (at , z)) ≥ D}i

is compact.
(vi) We say that the transaction costs are super linear in the weights if the transaction
costs due to a rebalance Dt from at− to at+ satisfyc(Dt ) ≥ B||at− − at+ ||VtP
for a constant B > 0. We use the notation ||at ||Vt = ni=1 |ai,t |Vi,t.

(2.10)(vii) We say that the utility is super linear in the weights if there exists a constant M > 0such that(2.11)
U(Wt (ât , z)) ≥ U(Wt (at , z)) + M||ât − at ||Vt
if âi ≥ ai for all i and for all values of z.
(viii) U(Wt ) is homogeneous if U(cWt ) = cU(Wt ) for all constants c.
(ix) For a strategy z and constant A > 0, we say that ãA,z,t is an optimal weight if itmaximizes the utility, that is,U(Wt (ãA,z,t , z)) =

sup

P

at
i ai,t Vi,t =A

U(Wt (at , z)).

Note that when there are transaction costs, equations (2.3)–(2.6) imply thatP
i ai,t− Vi,t .

Optimal rebalancing of portfolios with transaction costs(2.12)
Pi ai,t+ Vi,t <12

(x) An optimal investment strategy z̃ is a strategy with the following property: It requiresa rebalance at time t if and only if U(Wt (at , z)) may be increased and then it is rebalanced
to a value that optimizes U(Wt (at , z)). More precisely,U(Wt (at+ , z̃)) = sup U(W (Sẑ (at− , Vt ), ẑ))(2.13)ẑ∈Z

for all values at− . Note that at+ = Sz̃ (at− , Vt ).
(xi) A no-trade region is a region Ωt where (at− , z) ∈ Ωt if and only if it is not possible toincrease U(W (at , z)) by a rebalance including a change of strategy z and including thetransaction costs at time t. More preciselyΩt = {(at− , z) | at− = Sz (at− , Vt ) andU(Wt (at− , z)) = sup U(Wt (Sẑ (at− , Vt ), ẑ))}.

(2.14)ẑ∈Z

The no-trade region may be defined both in terms of the weights at and the relative weights
rt .
When defining optimality we use the Bellmann principle of optimality “If a strategy is optimal for each point in time at that point of time, given that an optimal
strategy will be used thereafter, then the strategy is optimal”, see Bellman (2). We
may then formulate the following theorem.
Theorem 2.0.2 Consider an admissible set of investment strategies parameterized byPz ∈ Z. Let U(Wt ) be a utility function defined on a portfolio Wt = ni=1 ai,t Vi,t , thatis continuously differentiable, concave and has compact level sets in the weights. Let
the transactions costs, c(Dt ), be of the form (2.3)–(2.6) or (2.8) in case of continuous
rebalancing.
P(A) There is an optimal weight ãA,z,t for each value of A = i ai,t Vi,t > 0 and z. The
optimal weight is unique when the utility function is strictly concave.
(B) If c(Dt ) = 0 for all values of t, then the optimal investment strategy is to rebalance
continuously, in order to get optimal weights, i.e., one chooses z = 3 and thus (ãA,3,t , 3)given by (2.12).
(C) If the transaction costs, (2.10), and the utility function, (2.11), are super linear in theweights, then there is a no-trade region, Ωt , and (ãA,z̃,t , z̃) ∈ Ωt .
(D) If there is a no-trade region and the transaction costs have ci,2 = 0 for i = 1, . . . , n,then there exists an optimal strategy where one only rebalances to the boundary of theno-trade region.
(E) If there is a no-trade region and the transactions costs have ci,2 = 0 for i = 1, . . . , n,and Vi,t for i = 1, 2, . . . , n have stationary relative increments2 and the utility function
2. The distribution of Vi,t+∆t /Vi,t is independent of t.

Optimal rebalancing of portfolios with transaction costs13

is homogeneous, then the no-trade region is time independent in the relative weights rt .
(F) If ci,j > 0 for i = 1, . . . , n and j = 1, 2 and the utility function is super linear in theweights, (2.11), there is a no-trade region. From a state outside the no-trade region it is
optimal to rebalance to a state at an internal surface in the no-trade region.
Proof 2.0.1 (A) Since U has compact level sets in the weights, there exists for each strategy z some weights ãA,z,t where the optimum is obtained. The function U is concave. The
optimal weight is unique when the utility function is strictly concave.
(B) Since U is continuous on the compact set ΩA,D,z,t, there exist parameters (ãA,z̃,t , z̃)that optimize U(Wt ). More precisely,U(Wt (ãA,z̃,t , z̃)) =sup

U(Wt (at , z))(2.15)P (at ,z)i ai,t Vi,t =A

for each value of t. When there are no transactions costs, it is optimal always to rebalance
to have these weights according to property (2.12).
(C) Consider a rebalance from at− to at+ . Define a third state ât such that âi,t = ai,t+if ai,t+ ≥ ai,t− and âi,t = ai,t+ + β(ai,t− − ai,t+ ) if ai,t+ < ai,t− where β is determinedsuch that
n
n
n
X
X
X
âi,t Vi,t =
ai,t− Vi,t =
ai,t+ Vi,t + c(Dt ).
i=1

i=1

i=1

These two equations may be rewritten such thatβX(ai,t− − ai,t+ )Vi,tiai,t− >ai,t+=X(ai,t− − ai,t+ )Vi,t +

iai,t− >ai,t+X(ai,t− − ai,t+ )Vi,t = c(Dt ). (2.16)

iai,t− <ai,t+Since c(Dt ) > 0, the second equality implies that β is positive, while the first one givesthat β ∈ (0, 1). Thus âi,t ≥ ai,t+ and âi,t is between ai,t+ and ai,t− including the endpoints for all i. Let Dt be a rebalance from at− to at+ . Since β < 1 and the transaction
costs are super linear we have the following inequalities||ât − at+ ||Vt ≥ c(Dt ) ≥ B||at− − at+ ||Vt
for a constant B > 0.
According to the item (A) above, there exist optimal weights ãA,z,t for each value of t.
However, when there are transactions costs, it is not necessarily optimal to rebalance tothese parameter values. At the optimal weights ãA,z,t , the derivative of U with respect to atPvanishes on the set i ai,t Vi,t = A, since we assume that U is continuously differentiable.
PP
We have that i âi,t Vi,t = i ai,t− Vi,t = A. The derivative will also vanish along each

Optimal rebalancing of portfolios with transaction costs

14

line in this plane. Then a Taylor expansion in one variable gives that for all ε > 0, there
exists δ > 0 such that
U(W (ât , z)) ≤ U(W (at− , z)) + ε||at− − ât ||Vt
for all ||at− − ãA,z,t ||Vt < δ, ||ât − ãA,z,t ||Vt < δ and all values of z.
Using that the utility function increases at least linear in the weights and that δ may
be arbitrary small, we haveU(W (at+ , z)) ≤ U(W (ât , z)) − M||ât − at+ ||Vt
≤ U(W (at− , z)) + ε||at− − ât ||Vt − MB||at+ − at− ||Vt
≤ U(W (at− , z)).
Hence, it is not possible to increase the utility by rebalancing from at− to at+ . The above
calculation is valid for any value of z and hence also for the optimal strategy z̃. Since it
is not possible to increase the utility by a rebalance from at− , there must be a no-traderegion containing (ãA,z̃,t , z̃).
(D) We will prove that if the transaction costs only have proportional elements andnot flat elements (i.e., ci,1 > 0 and ci,2 = 0), then all rebalance is to the boundary of theno-trade region. In the following argument, we will assume that we apply the optimal
strategy z̃. Let a be outside the no-trade region and assume the optimal rebalance is toanother state d. By definition, the state d cannot also be outside the no-trade region, sincethen it will be possible to increase the utility by rebalancing from d. If d is in the interiorof the no-trade region, we may define a third point b which is on the line between a and d
and is on the boundary of the no-trade region. Let ca,b, be the transaction costs between aand b. When the transaction costs are proportional, then ca,d = ca,b + cb,d . Then we may
consider a rebalance from a to d to first be a rebalance from a to b and then from b to d.
From the assumptions we have U(W (a, z̃)) < U(W (d, z̃)). Since b is on the boundary of
the no-trade region and d is inside the region, we have U(W (b, z̃)) ≥ U(W (d, z̃)). Thisimplies that we will not be able to improve the rebalance by rebalancing to the interior of
the no-trade region compared to a rebalance to the boundary of the no-trade region.
(E) If all the elements that determine the no-trade region are time independent, thenalso the no-trade region is time independent. The transaction costs (2.6) are time independent if it is proportional with the transaction and there are no flat elements (i.e, ci,2 = 0).
It is necessary to assume that the utility function is homogeneous in order to change
corresponding to proportional transaction costs. Hence, if Vi,t has stationary relative increments, then the no-trade region is time independent in the relative weights rt .
(F) When all ci,j > 0, the transaction costs are super linear and the existence ofthe no-trade region follows from item (D). When the utility function is continuous andci,2 > 0, it cannot be optimal to perform infinitesimal changes in the weights. Hence froma state outside the no-trade region it is optimal to rebalance to a state inside the no-traderegion. When the fixed transaction costs are paid, there still are proportional transaction

Optimal rebalancing of portfolios with transaction costs

15costs and item (D) above implies that there is another no-trade region. Hence the optimal
rebalance is to a point at the boundary of this interior no-trade region. There may be aninterior no-trade region for each combination of assets to sell and buy.
In a practical problem the investor will often increase or decrease the investment. This is an opportunity to rebalance to a lower additional cost than when
there are no investment or consumption. Hence, these changes are important forthe strategy to optimize the utility including reducing the transaction costs. If
these changes are known in advance, the size of the no-trade region will increase
when approaching the time when there is a change. The result may be that weonly rebalance at these time-points.
In general, rebalancing implies that the investor sells assets that have increasedin value over the last period. However, if the knowledge about the change in thevalue of an asset over a period implies a change in the expected further performance of the value of the asset, e.g., due to time dependent variance, it is criticalthat this is modeled in a satisfactory way and included in the rebalancing strategy. It is well known that the optimal weights ãA,z,t are sensitive to small changes
in the parameters in the stochastic process of the assets. If our expectation regarding the stochastic processes may change, e.g., due to new information, thenthis uncertainty should be included in the model. This will imply that the utility
is more stable in at , the no-trade region will be larger and the probability that
we rebalance to weights that we soon after find out are far from optimal due tochanges in the expected performance of the stochastic processes, is smaller.
Usually, there are some assets that have high expectations and high uncertainty. If these increase as expected, we may expect to sell these assets regularly
in order to maintain the relative weights. However, if these assets have decreasedin value, it may be optimal to wait and hope that these assets will increase invalue such that a rebalancing will not be necessary. This asymmetry is shown in
Tables 5.1 and 5.4. See also discussion at the end of Section 5.0.1. The significanceof this effect depends on the difference in expectation and the variability. With theutility function discussed in Section 4 however, the no-trade region is symmetric.
This utility function only considers properties at the moment it is evaluated andnot expected development later. Therefore, the difference in expectation between
the different assets does not influence the no-trade region.
The size of the no-trade region depends on the importance of the transaction
costs relative to the importance that the weights in the portfolio are close to the
optimal weights, ãA,z,t. By making the no-trade region larger, transactions will beless frequent and the associated costs go down. However, this also implies thatthe weights in the portfolio may be further from the optimal weights. The optimal size of the no-trade region may be difficult to assess, but it is not critical that
we know the exact position since the derivative of the improvement vanishes at

Optimal rebalancing of portfolios with transaction costs

16this point as we will see later in Figure 4.1. Assuming there is only proportionaltransaction costs, most of the reduction in transaction costs may be ensured if one
rebalances only to the estimated boundary of the no-trade region, rather than afull rebalance to the optimal weights. In the case with fixed cost, there is obtained
a similar improvement by rebalancing to an internal surface in the no-trade region rather than a full rebalance to the optimal weights.

Optimal rebalancing of portfolios with transaction costs

17

3 Utility functions

The definitions in this section are motivated by Markowitz (7). An investor wants
high expectation and low variability in the portfolio. Hence, the utility should
increase in E{WT } for some value of T > t and decrease with the variability. Someauthors argue that the utility function also should be decreasing in the expected
deviance from a reference portfolio. If we know that we will end the portfolio attime T , it may be natural with the utility function
1

U(Wt ) = E{WT } − d(var(WT )) 2

(3.1)

or

var(WT ).
(3.2)
Wt
Then the end date is coming closer each day, implying that the no-trade region
is gradually increasing, since the effect of more optimal weights decreases. If we
do not know when the portfolio is ended, it is natural to weight the time withexp(−βt). Then the relative weights of the different time-points in the future isalways the same and the no-trade region is stationary if the assets have stationary
relative increments. This leads to the following utility function
Z ∞1
U(Wt ) =(E{Ws } − d(var(Ws )) 2 ) exp(−βs)ds(3.3)
U(Wt ) = E{WT } − dtfor constants d > 0 and β > 0. Other alternatives include value at risk VaRα W =P (W < α), e.g.,
Z ∞U(Wt ) =(E{Ws } − dVaRα (Ws )) exp(−βs)ds,(3.4)
t

or expected shortfall E{Sα (W )} = E{W |W < α}Z ∞U(Wt ) =(E{Ws } − dE{Sα (Ws )}) exp(−βs)ds.

(3.5)

t

The threshold α may depend on the time s and the value of the portfolio Ws′ fors′ < s, e.g.,
αs = 0.8 ′ max{Ws′ }.
t≤s ≤max{t,s−1}

It is possible to increase the investment K(t) > 0 or consume part of the investment K(t) < 0 by the following equation
Wt+ = Wt− + K(t) − c(Dt ).

Optimal rebalancing of portfolios with transaction costs

(3.6)

18

New investments and consumption are opportunities to rebalance with smaller
additional transaction costs than to transfer between different assets. The properties of K(T ) for T > t may be known at time t, may be stochastic or may be partof the optimization.
Merton (8) formulates the following utility function
U(Wt ) = E{

Z T

U1 (Kc , s)ds + U2 (WT , T )}

(3.7)

t

where the two utility functions U1 is strictly concave in −Kc (i.e., consumption)and U2 is concave in WT . The problem is both to rebalance the portfolio and to findthe consumption Kc ≤ 0 that optimizes U. It is assumed to be one risk-free asset
with no transaction costs. Liu (6) finds the constant thresholds of the no-traderegion for each asset in this model, assuming uncorrelated geometric Brownian
motion. All rebalance is a transfer between the risk-free asset and one of the otherassets.
Leland (5) and Donohue and Yip (4) define ideal weights r̃i and use the utility
function
Z T
n
X
X
bi
U(Wt ) =(ri,s − r̃i )2 ds −c(Dtj )
(3.8)
i=1

t

j

for constants bi > 0 and where the last sum is over all times tj where there is arebalance. Instead of (2.5), it is assumed that transaction costs are paid by additional contributions. Leland (5) finds an approximation to the corner points in the
no-trade region with this utility function assuming geometric Brownian motion,
proportional transaction costs and one risk-free asset.
Note that the utility functions (3.1), (3.2), (3.3), (3.4), and (3.5) satisfy the assumptions in the theorem. The theorem is also valid for (3.7) if we fix the consumption, and for (3.8) if we set c(Dt ) = 0 and include the transaction costs inthe utility function U. Properties (B)–(F) in the theorem depend on the transaction costs, and, in particular, whether the transaction costs (i) are zero, or (ii) is
not identically zero; or (iii) does not contain fixed elements. If there are fixed elements in the transactions costs, it is optimal to rebalance to a point inside the notrade region in order to avoid a new rebalance too soon. But the rebalance should
never be to the optimal value ãA,z,t when the transaction costs have proportional
elements in addition to the fixed elements, since the marginal improvement vanishes when the weights approach ãA,z,t .

Optimal rebalancing of portfolios with transaction costs

19

4 An explicit example

In this section we illustrate the theorem by an example that is made so simple
that it is possible to estimate the no-trade region mostly by analytic formulas. Let
the utility function beU(Wt , rt ) = (1 −nXdk (rk,t − r̃k )2 )Wt .

(4.1)

k=1

Note that we here have included the relative weights in the list of arguments of
U. The optimal values of the relative weights are obviously r̃ = (r̃1 , r̃2 , · · · , r̃n ),but in a rebalance we need to consider how much the value of the portfolio Wt is
reduced compared to the improvement due to better relative weights.
We do not know the exact form of the no-trade region. It is natural to rebalance when the relative weights of one asset ri,t is high and the relative weights ofanother asset rj,t are low. If a rebalance involves more than two assets, it may beconsidered as several independent rebalances only involving two assets, when
we assume there is only proportional transaction costs, i.e., ck,2 = 0 for all values of k. This makes it natural to assume that we may approximate the no-traderegion by a region given by
H = {r | −Dj,i < ri − rj < Di,j for all i, j where i 6= j, andnXri = 1}

(4.2)

i=1

when there are only proportional transaction costs. We want to determine theconstants Di,j .
Consider a rebalance from asset i to asset j with i 6= j. Assume furthermorethat before the rebalance we have ri,t− −rj,t− > Di,j , and that we want to rebalancetori,t+ − rj,t+ = Di,j .

(4.3)

We will assume the new weights are at+ = at− + b where bi < 0, bj > 0, andbk = 0, for k 6= i, j. With these restrictions on b equations (2.5) and (2.6) giveWt+ = Wt− − ci,1 |bi |Vi,t − cj,1 |bj |Vj,t − ci,2 − cj,2 .

(4.4)

This equation has the solution, using equations (2.3) and (2.4),bi =−η − ci,2(1 − ci,1 )Vi,t

Optimal rebalancing of portfolios with transaction costs

(4.5)

20

and
bj =

η − cj,2(1 + cj,1 )Vj,t(4.6)

for a constant η > cj,2 . We find η by combining the above expression with (4.3).
This gives the solution
c

c

η=

c

c

ai,t− Vi,t − aj,t− Vj,t + 1−ci,2i,1 − 1+cj,2j,1 − Di,j (Wt− + 1−ci,2i,1 + 1+cj,2j,1 )Di,j ( 1−c1 i,1 + 1+c1 j,1 ) − 1−c1 i,1 + 1+c1 j,1

.

(4.7)

The above formula is useful when we want to determine the new weights in arebalance.
Equation (4.4) combined with the expression for the relative weights (2.2)
gives the following expression for the utility function after a rebalance
U+ = (1 −nX
k=1dk (rk,t+ − r̃k )2 )Wt− (1 − ri,t− ci,1 + rj,t− cj,1) − ci,2 − cj,2.
1 − ri,t+ ci,1 + rj,t+ cj,1

(4.8)

According to Theorem 2.0.2, it is optimal to rebalance whenever the relative
weights are outside the no-trade region, and then it is rebalanced to the boundaryof the no-trade region. We find Di,j for the values of ri,t+ and rj,t+ where
∂U
∂U
−
= 0.
∂ri,t+ ∂rj,t+(4.9)

If we neglect higher-order terms in ci,1 and cj,1 , we get, using (4.8), that
Di,j =

ci,1 + cj,1
+ r̃i − r̃j .
di + dj(4.10)

Since we are able to find an expression for Di,j that is independent of rk,t for
k 6= i, j, by neglecting higher order terms in ci,j , this indicates that (4.2) is a good
approximation to the no-trade region.
We will then consider the case with also a fixed transaction fee, i.e., ck,2 > 0.
For simplicity we neglect the case where it is optimal to rebalance more than two
assets at the same time. Assume the no-trade region may be approximated by aregion on the form
G = {r | −Ej,i < ri − rj < Ei,j for all i, j where i 6= j andPn

i=1 ri = 1 }.

(4.11)In this case it is optimal to rebalance whenever the relative weights are outside G,
and then it should be rebalanced to the border of H since this is the optimal value
when the flat fee is paid. We find Ei,j from the values of ri,t− and rj,t− where theequation U− = U+ is satisfied. Here U− is defined by (4.1) and U+ is defined by(4.8) with ri,t+ − rj,t+ defined by (4.3) and (4.9). This gives
1 c + c
 c +c(ci,1 + cj,1 )2
i,2
j,2
i,1
j,1
2 2
B+
(1 − B)B + r̃i − r̃j (4.12)
Ei,j = 2+2(di + dj )Wt−
4(di + dj )di + dj

Optimal rebalancing of portfolios with transaction costs

21

1.00
0.99
0.98

Utility

0.97
0.96

0.10

0.15

0.20

0.25

0.30

r

Figure 4.1. The top curve is the utility as a function of r and Wt = 1. This has optimum forr̃ = 0.2. The two other curves show the utility that may be obtained from a rebalance fromrt− = 0.3. The top one of these has only proportional transaction fee while the lowest
also has flat fee. Notice that both of these curves have optimum for the same value r thatis the upper bound of the no-trade region with only proportional costs. The maximum of
the lowest curves is equal to U (0.3), illustrating that the boundary of the no-trade region
including both proportional and flat transaction costs is 0.3. Since the two curves at thebottom are quite flat close to the optima, we see that the exact position of the no-traderegion is not critical.

Optimal rebalancing of portfolios with transaction costs

22whereB=4(di + dj ) − (ci,1 + cj,1)2≈ 1.
4(di + dj )(1 − r̃i ci,1 + r̃j cj,1) − 2(ci,1 + cj,1)2This expression has the property that Ei,j > Di,j andlim

ci,2 +cj,2 →0

Ei,j = Di,j .

Note that ci,2 + cj,2 scales with Wt− . Since the expression for Ei,j is independent ofrk for k 6= i, j, this indicates that (4.11) is a good approximation for the no-traderegion.
If we wanted to include rebalance of three assets at a time, we should define
the set33G3 = {r | − Ei,k,j< ri − rj − rk < Ei,j,k

for all different i and j > k and(4.13)
Pn

i=1 ri = 1}.

We could define similar sets for rebalances involving 4, 5, · · · assets at the sametime. The no-trade region would then be defined as the intersection of G and theGi sets. Then each set Gi contributes to a reduction of the no-trade region if there
is a corresponding rebalance contribution to an increase in the utility. However,the computational work increases significantly by including rebalances involving
more than two assets.
This utility function is illustrated in Figure 4.1 with n = 2. The figure shows
three utility curves, the utility as a function of r1,t and two utility curves that may
be obtained with a rebalance from a r1,t− = 0.3 assuming either only proportionaltransaction costs or both proportional and flat transaction costs. We have chosen
constants r̃1 = 0.2, ck,1 = 0.04 and ck,2 = 0.0054 for k = 1, 2. This implies theno-trade regions
H = {r1 | 0.18 < r1 < 0.22}
and
G = {r1 | 0.1 < r1 < 0.3}
when only proportional transaction fee and both proportional and flat transaction
fee.

Optimal rebalancing of portfolios with transaction costs

23

5 A numerical example

We will use the utility function
Z ∞var(Ws |Wt ) exp(−β(s − t))ds.
U(Ws |Wt ) =E{Ws |Wt } − d
Wt
t

(5.1)

The expression Ws |Wt denotes the conditional distribution for Ws for s > t given
Wt . This utility function is a function of Ws for s > t. When we evaluate it at timet, then Wt is known and used for scaling the variance. The utility is homogeneousand hence gives a stationary no-trade region Ωt in the relative weights rt if theassets have stationary relative increments. Many other utility functions will have
similar properties. In this example, we have chosen to be more precise for thisutility function instead of making the text more genera,l but also more technical.
We assume the portfolio is evaluated once each day, and only then it is decided if
one wants to rebalance.
We will first show how to find the optimal weights for this utility function.
Then we will find an approximation for the utility function that may be used inorder to find an approximate no-trade region. We will show that this approximation gives good estimates for the no-trade region. However, simulations may give
better estimates.
We will assume all assets are lognormal distributed and find optimal weights
and an approximation for the no-trade region. Let Xt = (Xi,t , X2,t , · · · , Xn,t ) whereXt ∼ N(µX t, ΣX t).
2
The correlation matrix is ΣX = {σX,i,j}i,j . Let

Vi,t = exp(Xi,t ),and2
νi = µX,i + σX,i,i/2.

Then we have the following
E{Vi,t } = exp(νi t),
and2
Var(Vi,t ) = exp(2νi t)(exp(σi,it) − 1).

Recall that
Wt =X

ai,t Vi,t .

iOptimal rebalancing of portfolios with transaction costs

24Assume we continuously rebalance such thatri,t =

ai,t Vi,tWt

is constant in time. Defineλ=Xri,t νX,i

(5.2)

X

(5.3)

i

and
γ=2

X
i

ri,t νX,i +2ri,t rj,t σX,i,j.

i,jThen we have
E{Wt } = exp(λt),(5.4)

E{Wt2 } = exp(γt).

(5.5)

and
The utility function (5.1) gives
U=

 1
1 
1
.
−d
−
β−λ
β − γ β − 2λ(5.6)

The optimal relative weights r̃i,t are the n variables that optimize (5.6), i.e., whereP∂U/∂ri,t = 0 under the constraint that i ri,t = 1. When there are no transaction
costs it is optimal to rebalance continuously to the optimal relative weights r̃i,t .
When there are transaction costs, there is a no-trade region according to Theorem 2.0.2. We assume there is only proportional transaction costs and that theno-trade region has the form (4.2). The optimal values of Di,j may only be foundby CPU intensive simulation. However, we will find an approximation that does
not need simulation. This approximation is described for n = 2 and later it isshown how to apply it for n > 2. In Section 4 it is described how to rebalance tothe boundary of the no-trade region.
5.0.1 Estimate U when n = 2
For n = 2 it is possible to prove that the no-trade region is
H2 = {r1 | D− < r1 < D+ },(5.7)

only assuming that the utility function is concave and that the transaction costs
has the form (2.6). In a simulation the ratio r1 will vary between the two limits D−
and D+ . If the ratio moves outside the interval, it is rebalanced to the boundary.
We will first describe how r varies in this interval. Fix the time t and let V1,∆t andW∆t represent the change from t to t + ∆t in V1 and W , respectively. We have
V1,t+∆t = V1,t V1,∆t = V1,t exp(X1,∆t )Optimal rebalancing of portfolios with transaction costs

25

and
Wt+∆t = Wt W∆t = Wt (r1 exp(X1,∆t ) + (1 − r1 ) exp(X2,∆t ))≈ Wt (exp(r1 X1,∆t + (1 − r1 )X2,∆t )).
This gives
r1,t+∆t =V1,t+∆t
V1,∆t= r1,t≈ r1,t (exp((1 − r1,t )(X1,∆t − X2,∆t )).
Wt+∆t
W∆t

This implies with ∆r = r1,t − r1,t−∆t thatE{∆r} ≈ r1,t−∆t (E{exp((1 − r1,t−∆t )(X1,∆t − X2,∆t ))} − 1),

(5.8)

2
E{(∆r)2 } ≈ r1,t−∆t
E{(exp((1 − r1,t−∆t )(X1,∆t − X2,∆t )) − 1)2 }.

(5.9)

and
We will approximate the distribution of ∆r with a normal distribution φ(∆r) ∼N(µ, σ 2 ). Let p(r1 ) be an approximation to the distribution of r1 in the interval[D− , D+ ]. We want a p(r1 ) that satisfiesp(r1 )φ(µ) = p(r1 + µ)φ(−µ),

r1 , r1 + µ ∈ (D− , D+ ).

(5.10)The motivation for this is that density at r1 multiplied by the probability to increase r1 to r1 + µ should be equal to the density at r1 + µ multiplied by the probability to decrease from r1 + µ to r1 since p(r1 ) is independent of time. Equation(5.10) has as solution
p1 (r1 ) = d1 exp(4µr1/σ 2 )on the open interval (D− , D+ ). In addition, there is a positive probability that r1
is equal the endpoints D− and D+ . Therefore, we assume p(r1 ) has the formp(r1 ) = d1 exp(4µr1 /σ 2 ) + d2 δ(D− ) + d3 δ(D+ ),(5.11)

where
d3 = d3 P (∆r > 0) + P (r1,t > D+ | r1,t−∆t < D+ )Z D+ Z ∞= d3 P (∆r > 0) +p1 (r1 )φ(∆r)d(∆r)dr1.
D−

D+ −r1

We approximate p1 (r1 ) in the double integral with p1 (D+ ) and getR∞
p1 (D+ ) 0 φ(∆r)∆rd(∆r).
d3 =1 − P (∆r > 0)
Similarly, we findR0
−p1 (D− ) ∞ φ(∆r)∆rd(∆r)
d2 =.
1 − P (∆r < 0)

Optimal rebalancing of portfolios with transaction costs

26

R
The three constants d1 , d2 and d3 are scaled such that p(r1 )dr1 = 1.
Then the transaction costs due to the upper limit are
Z D+ Z ∞p(r1 )φ(r∆t )(r1 + r∆t − D+ )(c1,1 + c2,1 )d(∆r)dr1
D−

D+ −r1= (c1,1 + c2,1 )

Z ∞

Z D+

φ(∆r)
p(r1 )(r1 + ∆r − D+ )dr1 d(∆r)
D+ −∆rZ ∞1φ(∆r)(∆r)2 d(∆r).
≈ (c1,1 + c2,1 )p(D+ )200Using a similar calculation for the transaction costs due to the lower limit, gives
the following expression for the transaction costs in a time step
1
2
C∆t = (c1,1 + c2,1 )(p(D− ) + p(D+ ))E{r∆t}.
2
The expectation and variance of Wt are calculated as follows, assuming it follows the properties of lognormal distributions,qqq
E{Wtq } = (E{W∆t})t/∆t = (E{WC,∆t(1 − C∆t )q })Nd t = (1 − C∆t )qNd t (E{WC,∆t})Nd t(5.12)
where q = 1, 2. In this calculation, we have assumed the same relative increasein Wt in each time step. The variable WC,∆t is the increase in W in one time step∆t when we neglect the reduction due to transaction costs in this time step. Let
Nd = 1/∆t be number of time steps in a year. Let furthermoreX
λ̂2 = Nd log(
pi (Ri E{V1,∆t } + (1 − Ri )E{V2,∆t })) + Nd log(1 − C∆t )iandXγ̂2 = Nd log(
pi E{(Ri V1,∆t + (1 − Ri )V2,∆t )2 )} + 2Nd log(1 − C∆t )iwhere Ri and pi for i = 1, 2, · · · , m is a discretization of p(r) for r ∈ (D− , D+ ). Ri
denotes values in the interval and pi the corresponding probability.
The expression exp(λ̂2 t) may be used as an estimate for E{Wt } and the expression exp(γ̂2 t) as an estimate for E{Wt2 }. Then we may estimate the utility function
U by
1
1
1
−
Û2 =
− d(
).
(5.13)
β − γ̂2 β − 2λ̂2β − λ̂2
Instead of finding the two parameters D− , D+ , it is more stable to find theparameters for the length of the no-trade region D+ − D− and for the position−.
relative to the optimal weight r̃, e.g., P = Dr̃−D+ −D−
Assume asset 1 has higher expected increase than asset 2. Then µ > 0 and thefunction p defined in (5.11) is increasing. Since the object function is quite symmetric around the optimal weights, it is more important that the right end point of

Optimal rebalancing of portfolios with transaction costs

27the no-trade region is closer to the optimal weights than the left end point. Hence,
the relative position will satisfy P > 0.5. This implies that when asset 1 has lower
relative weight than the optimal, we are more reluctant to impose a rebalance
than for asset 2, since asset 1 is more likely to increase without a rebalance.
Optimization of the above approximation does not give the optimal no-traderegion for the utility function U. However, experiments show that the above approximation gives almost as large values of the utility function as when applying
the optimal no-trade region. The approximation may also be used as a first guesson the no-trade region. Then it is possible to adjust these values based on a simulation if wanted.
5.0.2 Estimate U when n > 2
We approximate the no-trade region by a region on the form (4.2). For n > 2
we use the same technique for each pair ri and rj . The quantity ri − rj varies
between two boundaries −Dj,i < ri − rj < Di,j . If we find an approximation to
the pair of boundaries −Dj,i , Di,j separately from the other Dk,m , we may find
−Dj,i, Di,j similarly as we found D− , D+ . The optimal values may only be foundby a simulation of all the boundaries simultaneously.
Let us fix i and j and set ∆r = ri,t −rj,t −ri,t−∆t +rj,t−∆t . Similar to the argumentfor n = 2, we have for n > 2∆r = ri,t−∆t



Vi,∆t
Vj,∆t
− 1 − rj,t−∆t
−1 .
W∆t
W∆t

This givesE{(∆r)q } = E{(ri,t−∆t (

Vi,∆t
Vj,∆t
− 1) − rj,t−∆t(− 1))q }.
W∆tW∆t(5.14)

We may use the approximation
n
X
Vi,∆t
≈ exp(Xi,∆t −rk,tXk,∆t )W∆t
k=1(5.15)in order to find expressions for E{(∆r)q }.
For n = 2, we defined discrete values Ri in the interval (D− , D+ ). Similarly,
we may define
111Ri,k = (−Dj,i − r̃i + r̃j ) +(k − )(Di,j + Dj,i )22m2
and Rj,k = −Ri,k + r̃i − r̃j for k = 1, 2, · · · , m. Then Ri,k − Rj,k varies in the interval(−Dj,i, Di,j ). In order to simplify some expressions below we define Rq,k = r̃q for
q 6= i, j and for k = 1, 2, · · · , m. This is a first order approximation to the averagevalue of rq . Let pk be the probability for the discrete value Ri,k defined using (5.14)

Optimal rebalancing of portfolios with transaction costs

28and an expression similar (5.11) for n > 2. Note that we find pk for each pair i, j.
We then have the following approximations to λ and γ using (5.12)mnXXλ̂ = Nd log(pk
Rv,k E{Vm,∆t }) + Nd log(1 − C∆t )
v=1

k=1

and
n X
n
m
X
X
pk
Rv,k Ru,k E{Vv,∆t Vu,∆t }) + 2Nd log(1 − C∆t ).
γ̂ = Nd log(
k=1

v=1 u=1

We have
2
E{Vv,∆t } = exp(µ∆X,v + σ∆X,v
/2)

and
2
2
2
E{Vv,∆ Vu,∆t } = exp(µ∆X,v + µ∆X,u + (σ∆X,v+ σ∆X,u)/2 + σ∆X,v,u),2
2
where µ∆X,v , σ∆X,vand σ∆X,v,ucorrespond to the expectation and variance forvariable v and the correlation between v and u per time step.
The approximation Û to U is calculated from the formula

1 
1
−
Û =
−d
.
β − γ̂ β − 2λ̂
β − λ̂
1

(5.16)

This approximation may by used in order to find an approximation to the
optimal boundaries (−Dj,i , Di,j ). In this approach we find the boundaries for eachpair i, j separately. Also in this case, it is stable to parameterize the length of theinterval Dj,i + Di,j and the position relative to the difference between the optimalr̃ −r̃ +D
weights r̃i − r̃j , e.g., Di i,jj+Dj,ij,i . Application of the above approximation will notgive the optimal no-trade region for the object function U. Experiments show that
optimizing the approximation gives almost the optimal value. The approximation
may also be used in order to find a first guess on the no-trade region. Then it ispossible to adjust these values based on simulation if wanted.
5.0.3 Simulation of portfolios
In this section we find the no-trade region by simulation using the utility function(5.1). The value of the assets Vi,t is modeled by logarithmic Brownian motion. In
the simulation it is rebalanced to the boundary of the no-trade region when theportfolio is outside the region using equation (4.7).
Table 5.1 shows the no-trade region for n = 2 for different values of the proportional transaction costs and with and without correlation between the different assets. Note that when adding correlation, it is necessary to change d in orderto have optimum for r̃ = 0.2. Table 5.2 compares the utility of four different strategies. It is shown that optimal rebalance gives highest utility.

Optimal rebalancing of portfolios with transaction costs

29

d
2.72
2.72
2.15

ci,1
0.01
0.001
0.01

ρ
0
0
0.3

r̃
0.2
0.2
0.2

D−
0.165
0.186
0.167

D+
0.212
0.203
0.212

Table 5.1. The critical values for n = 2 with proportional transaction. The two assets
are geometric Brownian motion with annual expectation and standard deviation equal toµV = (1.08, 1.02) and σV = diag(0.2, 0.04), and discounted by β = − log(0.8).

strategy
no rebalancemonthly rebalance
optimal rebalance (approximate)
optimal rebalance (simulated)

U4.80894.92304.94574.9460EW1
1.03251.03071.03181.0316U1
1.02541.02401.02511.0254

Cn
0
12
12
13

Cc
0
0.00190.000390.00047

Table 5.2. Comparison of four different rebalancing strategies with n = 2. The parameters
are as in the top row of Table 5.1 including the threshold for the optimal rebalance. The
quantity U is an estimate for the utility function, E{W1 } is the expected value and U1 =E{W1 } − dvar(W1 ) is another evaluation of the portfolio after 1 year, Cn and Cc denotethe annual number of transactions and annual transaction costs, respectively.

Referring to Table 5.2, we see that the simulated optimal rebalance has as expected highest utility function, but using the approximation to the no-trade region gives only slightly lower utility. The optimal no-trade region, assuming theform (4.2), is small adjustments of the no-trade region given by approximate formulas and computed by simulation. The case of no rebalance increases the ratioof the high volatile asset giving higher expected value of portfolio E{W1 } but ata cost of higher variance. The difference in variance increases faster than the difference in expected value. This is seen by comparing no rebalance and optimal
rebalance after 1 year when we use the same d value as in the utility function.
Optimal rebalance reduces the transaction costs to 1/4 compared to monthly rebalance.
The second example with n = 5 correlated assets and proportional transaction
costs is shown in Tables 5.3–5.5. Table 5.3 shows the parameters for the five assets
with values that are assumed realistic for the Norwegian stock marked, international stock marked, Norwegian real estate, Norwegian bonds, and internationalbonds. The table shows the optimal relative weights for the five assets with theutility function (5.1) and three different values of d. Table 5.4 shows the no-traderegion for d = 2. Note how the length L varies between the different combinations of assets and the position in some cases is far from the symmetric 0.5 value.
Table 5.5 is similar to Table 5.2 but for the example with five assets. The table

Optimal rebalancing of portfolios with transaction costs

30

i
1
2
3
4
5

µi
1.1
1.09
1.05
1.035
1.035

σi
0.22
0.20
0.12
0.04
0.04

ρi,1
1
0.7
0.1
0.3
0.1

ρi,2
0.7
1
0.05
0.1
0.2

ρi,3
0.1
0.05
1
0
0

ρi,4
0.3
0.1
0
1
0.3

ρi,5
0.1
0.2
0
0.3
1

r̃i,d=0.5
0.223
0.175
0.255
0.071
0.0277r̃i,d=1
0.152
0.129
0.201
0.193
0.325

r̃i,d=2
0.083
0.092
0.157
0.306
0.362

Table 5.3. The critical values for n = 5 with proportional transaction costs ci,1 = 0.01 andthree different d values. The parameters are expectation, standard deviation, correlation,and the optimal values for each asset for the different values of d.

i/j
1
2
3
4
5

1
—
0.69
0.52
0.75
0.50

2
0.094
—
0.59
0.73
0.73

3
0.094
0.069
—
0.67
0.51

4
0.044
0.053
0.12
—
0.67

5
0.041
0.058
0.10
0.14
—

Table 5.4. The optimal no-trade region for the example shown in Table 5.3 with d = 2.
Above the diagonal is the length L of the interval for ri − rj and below the diagonal is
the position P of the interval. The no-trade region is then r̃i − r̃j − LP < ri − rj <r̃i − r̃j + L(1 − P ).

strategy
no rebalancemonthly rebalance
optimal rebalance (approximate)
optimal rebalance (simulated)

U5.27455.36925.39885.4000E{W1 }
1.04761
1.04549
1.04693
1.04696U1
1.04228
1.04092
1.04239
1.04240

Cn
0
12
36
36

Cc
0
0.00220.000490.00050

Table 5.5. The table is exactly as Table 5.2 except that n = 5 and we have used data
as in Table 5.3. It compares the strategies, no rebalance, monthly rebalance, and optimal
rebalance with no-trade region found by approximation and by simulation. The optimal
no-trade region is as in Table 5.4.

Optimal rebalancing of portfolios with transaction costs

31compares four different strategies and shows that the optimal rebalance strategy
gives highest utility. The approximation to the no-trade region gives almost as
good results as the method based on simulation. The no-trade region is slightly
different in the two cases and the result is not sensitive to the exact position. Notethat we also here get a reduction of transaction costs to 1/4 compared to monthly
rebalance. But in this case the expected annual number of rebalances is 36 which
is much larger than in monthly rebalance.
The calculation in both examples is based on 10000 simulations until 10 years
and then estimated tail for t > 10. For n = 5 one such simulation takes about 3
hours using the statistical package R on a standard desk top computer. Finding
the optimal no-trade region by estimating the 20 parameters from a good starting
point, requires at least 100 simulations which gives about 2 weeks of simulation
time. The approach based on approximation took about 20 seconds which is an
improvement compared with simulation of the order 105.

Optimal rebalancing of portfolios with transaction costs32

6 Closing remarks

This paper discusses optimal rebalance of portfolios with transaction costs. We
have shown that for n symmetric assets and a general utility function, there isa no-trade region. If the transaction costs are proportional, with no flat or fixed
elements, it is optimal to rebalance to the boundary of the no-trade region whenever the portfolio is outside the no-trade region. If the transactions costs have flat
elements, it is optimal to rebalance to an internal surface in the no-trade region
whenever the portfolio is outside the no-trade region. It is never optimal to havea full rebalance or a calendar-based rebalance.
The theory is illustrated on two examples; one using analytic calculations andapproximations and one using simulations. The last example is simulated for n =2 and n = 5. Three different rebalance strategies, namely, no rebalance, monthly
rebalance, and optimal rebalance are tested using simulations. Both for n = 2
and n = 5 the transaction costs are reduced by a factor 4 compared to monthly
rebalance. These figures are slightly better than other papers on optimal rebalance
for a particular utility function. The reduction in transaction costs is probably
mainly due to the fact that we rebalance to the boundary of the no-trade region
instead of a full rebalance. The reduction in transaction costs is probably not very
critical to the exact position of the boundary of the no-trade region. But in order to
optimize the utility, it is critical to have an optimal no-trade region. The example
shows that the size of the no-trade region depends heavily on the properties ofthe stochastic processes, not only the size of the transaction costs.
Acknowledgement. The authors thank Steinar Holden for valuable comments.

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