OPTIMAL REBALANCING OF PORTFOLIOS WITH
TRANSACTION COSTSHELGE HOLDEN AND LARS HOLDEN

Abstract. Rebalancing of portfolios with a concave utility function is considered. It is proved that 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 flattransaction costs, the rebalance from outside the no-trade region should be to
an internal state in the no-trade region but never a full rebalance. The standard optimal portfolio theory is extended to an arbitrary number of equallytreated assets, general utility function, and more general stochastic processes.
Examples are discussed.

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 [7] and Leland [9] that this may be a significant problem.
In the present paper it is proved, assuming only concave utility function, thattransaction costs imply that there is a no-trade (or no-transaction) region 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 boundary ofthis region. If the transaction costs have fixed or flat elements, then the rebalancefrom outside the no-trade region should be to an internal state in the no-traderegion but never a full rebalance. This extends previous results by considering an
arbitrary number of equally treated assets, using a general utility function, fixed inaddition to proportional transaction costs, and a larger class of stochastic processes.
Most mathematical papers in the area use the viscosity solution of the Hamilton–
Jacobi–Bellman equation to prove the existence and properties of the solution. Our
approach is simpler. Existence is proved assuming the utility function is continuousand concave. The solution is found applying the Bellman principle at all times tomaximize the utility. Furthermore, we provide an approximate formula and a more
precise iterative simulation algorithm for the boundary of the no-trade region andthe boundary for the internal states it should be rebalanced to in the case of fixed
elements in the transaction costs.
The problem of optimally rebalancing a portfolio with transactions costs is studied in several papers, see Chang [4], Dybvig [5] and Liu [10] and references herein.
The case of two assets can be analyzed analytically, see, e.g., Davis and Norman[6], Taksar et al. [18] and Øksendal and Sulem [16]. The multi-asset problem understrong assumptions has been studied by, e.g., Donohue and Yip [7]. Akian et al. [2]solve the multi-asset problem in an investment-consumption model with hyperbolic
absolute risk aversion, Brownian motion and assuming proportional transaction
costs using the viscosity solution of a variational inequality. Benth et al. [3] prove
Date: June 17, 2011.
Key words and phrases. Rebalance portfolio, portfolio optimization, transaction costs, notrade region, utility function.
1

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H. HOLDEN AND L. HOLDEN

that the multi-asset problem with Lévy processes is well defined by using dynamic
programming as well as the theory of viscosity solutions, but do not mention the
no-trade region in the solution.
The more applied problem of finding the no-trade region in higher dimension is
also discussed in several papers. Dynamic programming is prefered in most papers
to identify the moving boundaries. Akian et al. [2], Muthuraman and Kumar [13]and Muthuraman and Zha [14] specify algorithms, but these do only work for very
low dimension. More practical solutions are described in Sun et al. [17], Leland[9], and Fabozzi et al. [8]. Atkinson et al. [1] give an approximate formula for theboundary.
Most papers conclude that there is a no-trade region where trade should not
be performed, and all rebalance from a state outside the no-trade region is to theboundary of the no-trade region. Donohue and Yip [7] formulate this as a general
result for an arbitrary number of assets 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 [10] gives
a thorough discussion of the problem with constant absolute risk aversion and one
risk-free investment and an arbitrary number of uncorrelated geometric Brownian
motion investments. The paper discusses both proportional and 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 therisk-free investment.
There are different formulations of the optimal portfolio problem. The classicalpaper Markowitz [11] employs a utility function defined on a portfolio. Leland [9]and Donohue and Yip [7] focus on portfolios with targets ratios and the utility
function is deviance from the target ratios and the total transaction costs. Davisand Norman [6] and Liu [10] build on the work of Merton [12] that optimizes theconsumption that is possible based on the portfolio. The properties of the different formulations appear quite similar, but with different mathematical challenges.
All models seem to assume geometric Brownian motion. Section 2 describes ourmodel, and the theorem regarding the no-trade region is proved. It extends theframework of Markowitz [11] with an arbitrary number of symmetric investments
by also incorporating transaction costs in addition to a more general utility function
and a larger class of stochastic processes. Different utility functions are discussed
in Section 3. Our assumption with an arbitrary number of symmetric assets generalizes the assumption in Liu [10] where there is one risk-free asset at the cost of
more complex analysis and complex boundary to the no-trade region. It includes
the formulation of Leland [9] and the formulation of Merton [12] if we assume that
consumption and investment are known.
The theoretical agreement regarding the existence of a no-trade region is incontrast to the current practice in most portfolios. Donohue and Yip [7] state thatthe typically reduction of transaction costs by using an optimal rebalance strategy is50%. 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[9]. Other portfolios define intervals for the weights in each asset and adjust to
the boundary 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 Government Pension Fund [15] the rebalance is mainly performed
when deciding which new assets to buy. In addition, the portfolio is rebalanced tothe target weights if the weights are outside certain intervals over two consecutive
months. Norges Bank (The Norwegian Central Bank) states that as a large investor
it is an advantage that the time of rebalances is not known in the marked and that

OPTIMAL REBALANCING

3

the size of each rebalance is not too large. The strategy presented in this paper
satisfies these criteria. The document from Norges Bank [15] includes a numericalsimulation study of the transaction costs under different rebalancing strategies.
Sections 4 and 5 illustrate the theory with examples using different utility functions. The optimal 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
to rebalance a portfolio to the boundary of the no-trade region. With the utility
function discussed 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 are reduced to 1/4 by the use of an optimal
no-trade region. Closing remarks are presented in Section 6.
2. The model
Consider a fixed but arbitrary number n of 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 the processes are Markovian, i.e., if Vi,t isknown, we do not get more information regarding the value at a later point in timeby knowing the value of Vi,s for all prior times, that is, for s < t. Since all assets aretreated in the same way, we consider the problem as symmetric in the n assets. Iflog(Vi,t ) are correlated Lévy processes, all the utility functions defined in Section 3
are well-defined. Lévy processes include Brownian motion.
The portfolio is given by at = (a1,t , a2,t , . . . , an,t ) where ai,t is the number ofasset i at time t in the portfolio. The value of the portfolio at any given time tequalsnX(1)
Wt (at ) =ai,t Vi,t = at · Vt .
i=1

Here we write Vt = (V1,t , V2,t , . . . , Vn,t ). The weights are defined byai,t Vi,t.
Wt
In order to simplify the formulation we let ai,t ∈ R, i.e., both positive and negativevalues are considered, where negative values indicate a short position in the asset.
The results will be similar if we consider only non-negative values.
The problem to be discussed in this paper consists in deciding when it is preferable to rebalance the portfolio and to determine the optimal composition of the
new portfolio.
A rebalance at time tj implies that the portfolio is changed from
(2)

(3)

ri,t =

W tj − =nXai,tj−1 Vi,tj

i=1

to
(4)

W tj =nXai,tj Vi,tj

i=1

where ai,tj denote the number of asset i in the time interval (tj , tj+1 ). In the casethere are discontinuities in the value of the assets, Vi,t , we will always let Vi,t denotethe limit from the right. This also applies to (3). The reason for this definitionis that if a jump in Vi,t implies a rebalance, this is performed immediately, andobviously based on the values after the jump. Assume that(5)

Wtj = Wtj − − c(Dtj )4

H. HOLDEN AND L. HOLDEN

where the function c(Dtj ) ≥ 0 is the cost of selling or buying assets at time tj , which
we denote by the rebalancing cost or transaction cost. In the case of a positivetransaction cost c, transactions will happen in discrete times that we denote by
tj . With no transaction costs there will be a continuous rebalance, and we will
not discuss that any further in this paper, except in Theorem 2.2 (B). The set Dt
contains all relevant information or data regarding the assets up to time t, i.e.,
Dt = {(as , Vs ) | s ≤ t}.
We will assume that the transaction costs have proportional and fixed terms, i.e.,(6)c(Dtj ) =nX

ci,1 |ai,tj − ai,tj−1 |Vi,tj + ci,2 χ(ai,tj − ai,tj−1 )

i=1where ci,j ≥ 0 for i = 1, 2, . . . , n and j = 1, 2. The function χ(ai,tj − ai,tj−1 ) = 1
if ai,tj 6= ai,tj−1 , i.e., if there is a rebalance in asset i at time tj , and it equals 0
otherwise. For each asset i the cost function consists of a fixed fee if ci,2 6= 0 and acost proportional to the change in number of assets if ci,1 6= 0. This formula covers
the properties of interest from a theoretical point of view. It is trivial to extend it
to the case when the transaction costs depend on whether we sell or buy an asset.
To include taxation in the model may complicate the cost function if taxes dependon the time of acquisition of the asset and the difference between the value whensold and bought.
The quantity as (at , z) for s > t denotes the time development (at time s) of
the portfolio which equals at at time t. The portfolio is a stochastic function anddepends on the rebalance strategies z that are applied. In a similar way we write
the value of the portfolio
Ws (as (at , z)) = as (at , z) · Vs .
There is a plethora of strategies. The simplest possible strategy is never to rebalance, that is, ai,t remains constant in time. Another natural strategy is to rebalancein order to have the weights within a given interval. We denote the said intervalby [r̂1 , r̂2 ], with r̂j = (r̂1j , · · · , r̂nj ), in the sense thatr̂i1 ≤ ri,tj ≤ r̂i2 ,
or

r̂2
r̂i1
Wtj ≤ ai,tj ≤ i Wtj .
Vi,tj
Vi,tj
For this strategy it is necessary with detailed rules to determine how the rebalanceis performed, e.g., only rebalance when at least one weight reaches the boundaryof the admissible interval. Other strategies are of course also possible. We assume
Markov strategies, i.e., the strategy z is a function only depending on the present
situation and not the entire previous history; viz.
(7)

atj = Sz (atj−1 , Vtj ).

We assume the same strategy is applied at every point in time where it is possible
to rebalance. Hence, a strategy is a complete description of when and how torebalance, and we assume the same strategy is applied now and for every momentin the future. This may imply that it is rebalanced very often as is usual in hedging
strategies. The set of admissible strategies is denoted by Z. We restrict ourselves
to Markov strategies.
The utility function U (at , z, t) is assumed to be a real-valued deterministic function of the portfolio at , the strategy z, and time t. We suppress the dependence
on the time horizon T in the notation. It is natural that the utility increases in
E{WT (aT (at , z))} for some value of T > t and decreases with the variability. See

OPTIMAL REBALANCING

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Section 3 for a discussion on possible utility functions. At each rebalance time t
the investor optimizes U (at , z, t). Many utility functions depend only on the valueof the portfolio Ws (as (at , z)) for s > t. But if we include deviance from a reference
portfolio in the utility, then also the number of assets, at , is important in itself.
We want as general formulation of the utility as possible and need the following
definitions:Definition 2.1. We collect the basic definitions below:
We are given n assets each with value Vi,t , i = 1, . . . , n, at time t. All assets aretreated in the same manner. The values of the assets are given by stochastic Markov
processes with or without jumps.
(i) A portfolio is given by the collection of assets at = (a1,t , a2,t , . . . , an,t ). The
value Wt of the portfolio is given by equation (1). The investor has the opportunity to rebalance the portfolio according to the equations (3)–(6) at arbitrary times
t1 , t2 , . . . by choosing the optimal points in time for rebalance and the optimal rebalance each time.
(ii) Let z ∈ Z denote the investment strategy. We only consider Markov strategies
atj = Sz (atj−1 , Vtj ), i.e., the state after the rebalance depends only on the stateprior to the rebalance, namely atj−1 , and the value of each asset, i.e., Vtj at thetime tj of rebalance. We assume the same strategy is applied at each possible decision point.
(iii) A utility function is a real-valued deterministic function U (at , z, t) only depending on at , the strategy z, and time t. We assume the stochastic processes Vt
and the strategy z are sufficiently regular such that U is well-defined.
(iv) The utility U (at , z, t) is continuous/continuously differentiable/concave if U (at , z, t)
is continuous/continuously differentiable/concave in at for every value of z ∈ Z.
The utility function is concave in at if for any a1 and a2 satisfying a1 · Vt =a2 · Vt = A where A is a constant, we have(8)

U (βa1 + (1 − β)a2 , z, t) ≥ βU (a1 , z, t) + (1 − β)U (a2 , z, t)

for any constant 0 < β < 1. We say that U (at , z, t) is strictly concave if (8) holdswith strict inequality.
(v) The utility U (at , z, t) has compact level sets in the number of assets, at , if foreach combination of A > 0, D ∈ R, and z ∈ Z, the setΩA,D,z,t = {a ∈ Rn | a · Vt = A,U (a, z, t) ≥ D}

is compact.
(vi) The transaction costs are super linear in at if the transaction costs due to arebalance Dtj from atj−1 to atj satisfyc(Dtj ) ≥ B||atj−1 − atj ||VtPn
for a constant B > 0. We use the notation ||a||Vt = i=1 |ai |Vi,t .

(9)(vii) The utility is super linear in a if there exists a constant M > 0 such that(10)

U (â, z, t) ≥ U (a, z, t) + M ||â − a||Vt

if âi ≥ ai for all i and for all values of z ∈ Z.
(viii) The utility U (a, z, t) is homogeneous if U (ca, z, t) = cU (a, z, t) for all constants c.

6

H. HOLDEN AND L. HOLDEN(ix) The utility U (a, z, t) is translation invariant in time1 for a strategy z if U (a, z, t) =cU (b, z, t + ∆t) when a, b ∈ Rn are such that ai Vi,t /Wt = bi Vi,t+∆t /Wt+∆t fori = 1, . . . , n for a constant c that only depends on t, t + ∆t, Wt , and Wt+∆t .
(x) For a strategy z and constant A = Wt (ãA,z,t ) > 0, we say that ãA,z,t is an
optimal portfolio if it maximizes the utility, that is,(11)

U (ãA,z,t , z, t) = sup U (a, z, t).
aa·Vt =A(xi) An optimal investment strategy z̃ is a strategy with the following property: It
requires a rebalance at time tj if and only if U (atj−1 , z, tj ) may be increased andthen it is rebalanced to a value that optimizes U (atj , z, tj ). More precisely,(12)

U (atj , z̃, tj ) = sup U (Sẑ (atj−1 , Vtj ), ẑ, tj )ẑ∈Zwhere atj = Sz̃ (atj−1 , Vtj ).
Note that when there are transaction costs, equations (3)–(6) imply that Wtj <W tj − .
(xi) A no-trade region is a region Ωt where a ∈ Ωt if and only if it is impossible toincrease U (a, z, t) by a rebalance including a change of strategy z and including thetransaction costs at time t. More preciselyΩt = {a | a = Sz̃ (a, Vt ) and(13)

U (a, z̃, t) = sup U (Sẑ (a, Vt ), ẑ, t)}.
ẑ∈Z

The no-trade region may be defined both in terms of the number of assets, a, andthe weights, r.
(xii) We say that the distribution has stationary relative increments if Vi,t+∆t /Vi,t
is independent of t for i = 1, . . . , n.
If the utility function has stationary relative increments, it may be more natural
to use the weights r rather then the number of assets, a, to describe the portfolio.
If the portfolio has infinite time horizon, it is natural to choose a strategy z and autility function U that is translation invariant in time. Section 3 illustrates sometranslation invariant utility functions.
We may then formulate the following theorem.
Theorem 2.2. Let U (at , z, t) be a utility function defined on a portfolio at anda Markov strategy z, that is continuously differentiable, concave and has compact
level sets in the weights. Let the transactions costs, c(Dt ), be of the form (3)–(6).
Then the following statements hold:
P(A) There is an optimal portfolio ãA,z,t for each fixed value of A = i ai,t Vi,t > 0of the portfolio and each given z. The optimal portfolio is unique when the utility
function is strictly concave.
(B) If there are no rebalancing costs, i.e., c(Dt ) = 0 for all values of t in (5), thenthe optimal investment strategy z̃ is at any time to rebalance to the optimal portfolioãA,z̃,t given by (11) at every point t in time. Here A = at · Vt .
(C) If both the transaction costs, (9), and the utility function, (10), are super linear
in the number of assets at , then there is a non-trivial no-trade region, Ωt , andãA,z̃,t ∈ Ωt . The no-trade region contains an open set containing ãA,z̃,t for a given
1This essentially states that the utility function is only a function r , and a change in time ort
the value of the portfolio Wt only results in a proportional change in the utility.

OPTIMAL REBALANCING

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A = at · Vt .
(D) Consider the case when the transaction costs have ci,2 = 0 for i = 1, . . . , n.
Then there exists an optimal strategy z̃ with the following property: It rebalances
whenever the portfolio is outside the no-trade region, and in that case it rebalances
to the boundary of the no-trade region.
(E) Consider the case when the transaction costs have ci,2 = 0 for i = 1, . . . , n, andVi,t for i = 1, 2, . . . , n have stationary relative increments and the utility function
is homogeneous and translation invariant in time. Then the no-trade region is time
independent in the weights rt .
(F) Consider the case when the transaction costs have ci,j > 0 for i = 1, . . . , n
and j = 1, 2 and the utility function is super linear in the number of assets at ,(10). From a state outside the no-trade region it is then optimal to rebalance to aninternal state in the no-trade region.
Proof. (A) Since U has compact level sets in the number of assets, at , there exists
for each strategy z a portfolio ãA,z,t where the optimum is obtained. The optimumis 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 (a, z, t). More precisely,(14)

U (ãA,z̃,t , z̃, t) =supU (a, z, t)(a,z)∈Rn ×Z
a·Vt =Afor each value of t. When there are no transactions costs, it is optimal for eachpoint in time t > 0 to rebalance to the optimal number for each asset.
(C) Consider a rebalance from atj−1 to atj . Observe first thatΩt ⊇ {ãA,z̃,t | A ∈ R}
where ãA,z̃,t is defined by (14). We want to show that if atj−1 is sufficiently closeto ãA,z̃,tj , then atj−1 is inside a no-trade region. Define a third state âtj such thatâi,tj = ai,tj if ai,tj ≥ ai,tj−1 and âi,tj = ai,tj + β(ai,tj−1 − ai,tj ) if ai,tj < ai,tj−1
where β is determined such thatâtj−1 · Vtj = atj−1 · Vtj = atj · Vtj + c(Dtj ),where Dtj denotes the rebalance from atj−1 to atj . These two equations may be
rewritten such thatX(15) β(ai,tj−1 − ai,tj )Vi,tj
iai,tj−1 >ai,tj=

Xiai,tj−1 >ai,tj(ai,tj−1 − ai,tj )Vi,tj +X(ai,tj−1 − ai,tj )Vi,tj = c(Dtj ).

iai,tj−1 <ai,tj

Since c(Dtj ) > 0, the second equality implies that β is positive, while the first
one gives that β ∈ (0, 1). Thus âi,tj ≥ ai,tj and âi,tj is between ai,tj and ai,tj−1
including the endpoints for all i. Since β < 1 and the transaction costs are superlinear, we have the following relations||âtj − atj ||Vtj = c(Dtj ) ≥ B||atj−1 − atj ||Vtj
for a constant B > 0.
We have that âtj · Vtj = atj−1 · Vtj = A. Consider now(16)

U (ãA,z̃,tj , z̃, tj ) = sup U (a, z̃, tj ).
aa·Vt =A

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H. HOLDEN AND L. HOLDENAt the maximum point we know from the theory of Lagrange multipliers that thegradient of the utility is parallel to the gradient of the constraint, i.e., the gradientof the value of the portfolio. In mathematical terms∇a U (ãA,z̃,tj , z̃, tj ) = µVtj ,for some constant µ. From the Taylor formula we see that (for some unspecified
point ξ on the straight line between âtj and atj−1 )U (âtj , z̃, tj ) = U (atj−1 , z̃, tj ) + ∇a U (ξ, z̃, tj ) · (âtj − atj−1 )= U (atj−1 , z̃, tj ) + µVtj · (âtj − atj−1 ) + (∇a U (ξ, z̃, tj ) − µVtj ) · (âtj − atj−1 )= U (atj−1 , z̃, tj ) + ∇a U (ξ, z̃, tj ) − U (ãA,z̃,tj , z̃, tj ) · (âtj − atj−1 ).
Given ε > 0, there exists δ > 0 such that˜ z̃, tj ) − U (ãA,z̃,t , z̃, tj ) ||V||∇a U (ξ,
j

tj

<ε

for all ξ˜ such that ||ξ˜ − ãA,z̃,tj ||Vtj < δ. If both âtj and atj−1 satisfy||âtj − ãA,z̃,tj ||Vtj < δ,||âtj−1 − ãA,z̃,tj ||Vtj < δthen also ||ξ − ãA,z̃,tj ||Vtj < δ, and we infer that(17)

U (âtj , z̃, tj ) ≤ U (atj−1 , z̃, tj ) + ε||âtj − atj−1 ||Vtj .

Using that the utility function increases at least linearly in the weights and thatε may be arbitrary small, we haveU (atj , z̃, tj ) ≤ U (âtj , z̃, tj ) − M ||âtj − atj ||Vtj≤ U (atj−1 , z̃, tj ) + ε||atj−1 − ât ||Vtj − M B||atj − atj−1 ||Vtj≤ U (atj−1 , z̃, tj ).
Hence, it is not possible to increase the utility by rebalancing from atj−1 to atj .
The above calculation is valid for any strategy. Since it is not possible to increase
the utility by a rebalance from atj−1 , there must be a no-trade region containingãA,z̃,tj . The same argument implies that all portfolios on the line between atj−1 andatj with the same value of the portfolio also are inside the no-trade region. Hence,
the no-trade region contains an open set containing ãA,z̃,tj for a given A = a · Vt .
(D) We will prove that if the transaction costs only have proportional elements
and no flat elements (i.e., ci,1 > 0 and ci,2 = 0), then all rebalances are to theboundary of the no-trade region. We will first show that there is an optimal rebalance strategy. The definition of no-trade region implies that the optimal strategy
is to rebalance whenever outside the no-trade region. The rebalance will always
be to a state inside or on the boundary of the no-trade region, since otherwise it
will immediately rebalance to another state. The utility has compact level sets fora given value A of the portfolio. The no-trade region is a subset of such a levelset. Furthermore, the no-trade region is a closed set. Thus the no-trade region iscompact, and the utility function obtains its supremum inside the no-trade region.
The transaction costs also vary continuously. Hence, the utility function, after amove into the no-trade region, including the transaction cost for the move, is acontinuous function that obtains its maximum in the no-trade region. Hence, wehave proved the existence of the optimal strategy. It is to rebalance whenever outside the no-trade region. In that case it rebalances to the supremum of the utility
after a move, including the transaction costs of the move in the compact area of
the no-trade region. The optimal strategy is denoted 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-traderegion, since then it will be possible to increase the utility by rebalancing from d.

OPTIMAL REBALANCING

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If d is in the interior of 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 a and b. When the transaction costs areproportional, 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 (a, z̃, t) < U (d, z̃, t). Since b is on the boundary of the no-trade region
and d is inside the region, we have U (b, z̃, t) ≥ U (d, z̃, t). This implies that we will
not be able to improve the rebalance by rebalancing to the interior of the no-traderegion 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,
then also the no-trade region is time independent. Let ri,tj − = atj−1 − Vi,tj /Wtj . If
the transaction costs are proportional with the transaction and there are no flat
elements (i.e., ci,2 = 0), thenc(Dtj ) =nX
i=1

|ai,tj − ai,tj−1 |Vi,tj = WtjnX|ri,tj − ri,tj − |,i=1

that is, the transaction costs are proportional with the value Wtj .
If the utility is homogeneous, then the utility is proportional to Wt . If Vi,t has
stationary relative increments, then the development of the portfolio is independentof the value of Vi,t ; it only depends on the weights ri,t . Then it is natural to choosea strategy that only depends on the weights rt . If the utility is translation invariant
in time, then the utility essentially only depends on rt . Under these assumptions,the no-trade region will only depend on the weights rt . If two portfolios have equal
weights initially at t = t1 , that is, r̂i,t1 = ri,t1 for i = 1, . . . , n, then they will havethe same weights r̂i,t = ri,t for all t > t1 if the same strategy z is applied.
(F) When all ci,j > 0, the transaction costs are super linear. When we also
assume that the utility is super linear, then the existence of the no-trade region
follows from item (D). If ci,2 = 0 and the portfolio is outside the no-trade region,
then it is optimal to rebalance to the boundary of the no-trade region. But thenthere may be a new rebalance very soon, if the portfolio moves outside the notrade region once more. When ci,2 > 0, it cannot be optimal to perform manysmall rebalances. When outside the no-trade region, it is by definition optimal toperform a rebalance. In order to avoid a new rebalance soon, it is necessary torebalance to an internal point in the no-trade region.

Notice that even a small transaction cost may change the strategy significantly.
Without transaction costs, it may be optimal with continuous rebalancing, whilewith transaction costs, there is a no-trade region.
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 for the 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 or the end point of the portfolio. The result may
be that we only 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 performanceof the value of the asset, e.g., due to time dependent variance, it is critical that
this is included in the model and the rebalancing strategy. It is well known thatthe optimal number of assets ãA,z,t is sensitive to small changes in the parametersin the stochastic process of the assets. If our expectation regarding the stochastic

10

H. HOLDEN AND L. HOLDEN

processes may change, e.g., due to new information, then this 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 weightsthat we soon after find out are far from optimal due to changes 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 orderto maintain the weights. However, if these assets have decreased in value, it may
be optimal to wait and hope that these assets will increase in value in order that arebalancing will not be necessary. This asymmetry is shown in Tables 1 and 4. See
also discussion at the end of Section 5.1. The significance of this effect depends onthe difference in expectation and the variability. With the utility function discussed
in Section 4 however, the no-trade region is symmetric. This utility function only
considers properties at the moment it is evaluated and not 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 number of assets, ãA,z,t . By making the no-trade region larger, transactions
will be less frequent and the associated costs go down. However, this also implies
that the 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 criticalthat we know the exact position since the derivative of the improvement vanishes
at this point as we will see later in Figure 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 state in the no-trade region
rather than a full rebalance to the optimal weights.
If any of the parameters changes, the optimal number of assets ãA,z,t changes
and also the no-trade region changes position. If we neglect this change in positionand consider an increase in volatility, then we may in general expect the no-traderegion to increase in order not to rebalance too often and be more symmetric withrespect to the optimal weights. See discussion at end of Section 5.1.
3. Utility functions
The definitions in this section are motivated by Markowitz [11]. An investor
wants high expectation and low variability in the portfolio. Hence, the utility
should increase in E{Ws } for s ≥ t and decrease with the variability. Some authors
argue that the utility function also should be decreasing in the expected deviancefrom a reference portfolio. If the portfolio ends at time T , it may be natural withthe utility function(18)

U (at , z, t) = E{WT (at , z)} − d(var(WT (atj , z)))1/2

or
(19)

U (at , z, t) = E{WT (at , z)} − dvar(WT (at , z)).
Wt

The end date T is coming closer each day, implying that the no-trade region is
gradually increasing, since the effect of more optimal number of assets decreases.
If we do not know when the portfolio is ended, it is natural to weigh the timewith exp(−βt). Then the weights of the different time-points in the future are

OPTIMAL REBALANCING

11

always the same and the no-trade region is stationary if the assets have stationary
relative increments. This leads to the following utility function
Z ∞(E{Ws (at , z)} − d(var(Ws (at , z)))1/2 ) exp(−βs)ds(20)
U (at , z, t) =t

for constants d > 0 and β > 0. This utility function is proportional with a change
in time. Other alternatives include value at risk VaRα W = P (W < α), e.g.,
Z ∞(E{Ws (at , z)} − dVaRα (Ws (at , z))) exp(−βs)ds,(21)
U (at , z, t) =tor expected shortfall E{Sα (W )} = E{W | W < α}Z ∞(22)
U (at , z, t) =(E{Ws (at , z)} − dE{Sα (Ws (at , z))}) exp(−βs)ds.
t

The utilities (20)–(22) are translation invariant for strategies z that only dependon the weights r, rather than the assets a, assuming the threshold α is suitably
scaled. The threshold α may depend on the time s and the value of the portfolio
Ws0 for s0 < s, e.g.,
αs = 0.8 0 max{Ws0 }.
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(23)

Wt = Wt− + K(t) − c(Dt ).

New investments and consumption are opportunities to rebalance with smaller additional transaction costs than to transfer between different assets. The propertiesof K(T ) for T > t may be known at time t, may be stochastic or may be part ofthe optimization.
Merton [12] formulates the following utility function
Z T(24)
U (at , z, t) = E{
U1 (Kc , s)ds + U2 (WT , T )}twhere the utility function 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 find theconsumption Kc ≤ 0 that optimizes U . It is assumed to be one risk-free asset withno transaction costs. Liu [10] finds the constant thresholds of the no-trade region
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 other assets.
Leland [9] and Donohue and Yip [7] define ideal weights r̃i and use the utility
function
Z T
n
X
X
(25)
U (at , z, t) = E{bi(ri,s (at , z) − r̃i )2 ds −c(Dtj )}
i=1tj

for constants bi > 0 and where the last sum is over all times tj where there is arebalance. Instead of (5), it is assumed that transaction costs are paid by additional
contributions. Leland [9] finds an approximation to the corner points in the no-traderegion with this utility function assuming geometric Brownian motion, proportionaltransaction costs and one risk-free asset.
Note that the utility functions (18), (19), (20), (21), and (22) satisfy the assumptions in the theorem. The theorem is also valid for (25) and (24) if we fix theconsumption. Properties (B)–(F) in the theorem depend on the transaction costs,
and, in particular, whether the transaction costs (i) are zero, or (ii) are not identically zero; or (iii) do not contain fixed elements. If there are fixed elements in thetransactions costs, it is optimal to rebalance to a point inside the no-trade region

12

H. HOLDEN AND L. HOLDEN

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 inaddition to the fixed elements, since the marginal improvement vanishes when thenumber of assets approaches ãA,z,t .
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 be
U (at , z, t) = E{(1 −

(26)

n
X

dk (rk,t (at ) − r̃k )2 )Wt (at )}.

k=1

Here r̃k for k = 1, . . . , n are given weights. With this utility function we always lookfor rebalances from at− to at such that the right-hand side of the expression above
increases. The optimal values of the 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 weights.
We do not know the exact form of the no-trade region. It is natural to rebalance
when the weight of one asset ri,t is high and the weight of another asset rj,t is low.
Consider a rebalance from asset i to asset j with i 6= j at time t where thesubscript t− denotes the values just before the rebalance. Assume furthermore that
before the rebalance we have ri,t− − rj,t− > Di,j , and that we want to rebalance tori,t − rj,t = Di,j .

(27)
PHere ri,t− = ai,t− Vi,t / j aj,t− Vj,t while ri,t is defined in (2) after the rebalance.
We will assume the new number of assets is at = at− + b where bi < 0, bj > 0, andbk = 0, for k 6= i, j. With these restrictions on b equations (5) and (6) give
Wt = Wt− − ci,1 |bi |Vi,t − cj,1 |bj |Vj,t − ci,2 − cj,2 .

(28)This equation has the solution, using equations (3) and (4),
(29)

bi =

−η − ci,2(1 − ci,1 )Vi,t

bj =η − cj,2(1 + cj,1 )Vj,t

and
(30)

for a constant η > cj,2 . We find η by combining the above expression with (27).
This gives the solution
c

(31)

η=

c

c

c

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

Di,j ( 1−c1 i,1 + 1+c1 j,1 ) − 1−c1 i,1 + 1+c1 j,1.

The above formula is useful when we want to determine the new weights in arebalance.
If a rebalance involves more than two assets, it may be considered as several
independent rebalances only involving two assets, when we assume there is only
proportional transaction costs, i.e., ci,2 = 0 for all values of i. This makes it
natural to assume that we may approximate the no-trade region by a region givenby(32)

H = {r | −Dj,i < ri − rj < Di,j for all i, j where i 6= j, and

n
X
k=1

rk = 1}OPTIMAL REBALANCING

13

when there are only proportional transaction costs. We want to determine theconstants Di,j . Liu [10] proves in a similar model that the optimal no-trade region
has corners with a similar geometry as H.
Equation (28) combined with the expression for the weights (2) gives the following expression for the utility function after a rebalancenX
Wt− (1 − ri,t− ci,1 + rj,t− cj,1 ) − ci,2 − cj,2(33)
U = (1 −.
dk (rk,t − r̃k )2 )1 − ri,t ci,1 + rj,t cj,1
k=1

According to Theorem 2.2, it is optimal to rebalance whenever the weights areoutside the no-trade region, and then it is rebalanced to the boundary of the notrade region. We find Di,j for the values of ri,t and rj,t where
∂U
∂U
−
= 0.
∂ri,t
∂rj,t

(34)

If we neglect higher-order terms in ci,1 and cj,1 , we get, using (33), thatci,1 + cj,1
(35)
Di,j =+ r̃i − r̃j .
di + dj
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, this indicates that (32) 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
Pn(36) G = {r | −Ej,i < ri − rj < Ei,j for all i, j where i 6= j and i=1 ri = 1 }.
In this case it is optimal to rebalance whenever the weights are outside G, and thenit should be rebalanced to the border of H since this is the optimal value when theflat fee is paid. This is according to Theorem 2.2, (F). We find Ei,j from the valuesof ri,t− and rj,t− where the equation U− = U+ is satisfied. Here U− is defined by(26) and U+ is defined by (33) with ri,t − rj,t defined by (27) and (34). This gives
 c +c1/2 c + c(ci,1 + cj,1 )2
i,1
j,1
i,2
j,2
2
(37) Ei,j = 2
B+
(1
−
B)
+
B + r̃i − r̃j(di + dj )Wt−
4(di + dj )2
di + dj
where

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 andB=limci,2 +cj,2 →0

Ei,j = Di,j .

Note that ci,2 + cj,2 scales with Wt− . Since the expression for Ei,j is independentof rk for k 6= i, j, this indicates that (36) is a good approximation for the no-traderegion.
If we wanted to include rebalance of three assets at a time, we should define the
set
(38)

3
3
G3 = {r | − Ei,k,j< ri − rj − rk < Ei,j,k

for all different i and j > k andPn

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14

H. HOLDEN AND L. HOLDEN

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

computational work increases significantly by including rebalances involving morethan two assets.
This utility function is illustrated in Figure 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

15

5. A numerical example
We will use the utility function
Z ∞var(Ws (as (at , z)) (39) U (at , z, t) =E{Ws (as (at , z))} − dexp(−β(s − t))ds.
Wt (at )t
The utility is homogeneous and proportional with a change in time and hence
gives a stationary no-trade region Ωt in the weights rt if the assets have stationary
relative increments. Many other utility functions will have similar properties. In
this example, we have chosen to be more precise for this utility function instead ofmaking the text more general. 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. Thenwe find an approximation for the utility function that may be used in order tofind an approximate no-trade region. We show that this approximation gives good
estimates for the no-trade region. However, simulations may give better estimates.
Assume all assets are lognormally 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 .

iAssume we rebalance as often as possible such thatai,t Vi,t.
ri,t =
Wt
DefineX(40)
λ=ri,t νX,i
i

and
(41)

γ=2

X
i

ri,t νX,i +X

2ri,t rj,t σX,i,j.

i,jThen we have
(42)

E{Wt } = exp(λt),

and
(43)

E{Wt2 } = exp(γt).

Hence, the utility function (39) with no transaction costs and continuous rebalance
to the optimal weights has the utility value
 1
1 
1
−d
−
.
(44)
U=
β−λ
β−γ
β − 2λ16

H. HOLDEN AND L. HOLDEN

The optimal weights r̃i,t are theP n variables that optimize (44), i.e., where ∂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 weights r̃i,t .
When there are transaction costs, there is a no-trade region according to Theorem 2.2. We assume there is only proportional transaction costs and that theno-trade region has the form (32). 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.
5.1. Estimate U when n = 2. Since the no-trade region is connected, the no-traderegion for n = 2 isH2 = {r1 | D− < r1 < D+ },(45)

only assuming that the utility function is concave and that the transaction costs
has the form (6). In a simulation the ratio r1 varies 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 )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
V1,t+∆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 that(46)

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

and
(47)

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

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 that p(r1 ) satisfies(48)

p(r1 )φ(µ) = p(r1 + µ)φ(−µ),

r1 , r1 + µ ∈ (D− , D+ ).

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 (48) has
the solution(49)

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 form(50)

p(r1 ) = d1 exp(4µr1 /σ 2 ) + d2 δ(D− ) + d3 δ(D+ ),where
d3 = d3 P (∆r > 0) + P (r1,t > D+ | r1,t−∆t < D+ )

OPTIMAL REBALANCING

17

Z D+ Z ∞p1 (r1 )φ(∆r)d(∆r)dr1 .

= d3 P (∆r > 0) +D+ −r1D−

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)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

Z ∞Z D+
p(r1 )(r1 + ∆r − D+ )dr1 d(∆r)

φ(∆r)= (c1,1 + c2,1 )D+ −∆r0

1≈ (c1,1 + c2,1 )p(D+ )2

Z ∞φ(∆r)(∆r)2 d(∆r).

0

Using 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,
(51)
q
q
q
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 twhere q = 1, 2. In this calculation, we have assumed the same relative increase in
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 }. Hence, an estimate for the value of theutility function U when we rebalance outside the no-trade region defined in (45) is
1
1
1
−
(52)
Û2 =
− d(
).
β − γ̂2β − λ̂2β − 2λ̂2Instead 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 (50) is increasing. Since the object function is quite symmetricaround the optimal weights, it is more important that the right end point of the

18

H. HOLDEN AND L. HOLDEN

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
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. Notice also that if
σ in (49) increases, then the slope of p decreases. Then P approaches 0.5, making
the no-trade interval more symmetric.
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 guess on
the no-trade region. Then it is possible to adjust these values based on a simulation
if wanted.
5.2. Estimate U when n > 2. We approximate the no-trade region by a region
on the form (32). 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 otherDk,m , we may find −Dj,i , Di,j similarly as we found D− , D+ . The optimal values
may only be found by 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 theargument for n = 2, we have for n > 2


Vj,∆t
Vi,∆t
− 1 − rj,t−∆t−1 .
∆r = ri,t−∆t
W∆tW∆tThis gives
Vi,∆t
Vj,∆t
(53)
E{(∆r)q } = E{(ri,t−∆t (− 1) − rj,t−∆t (− 1))q }.
W∆tW∆t
We may use the approximation
n
X
Vi,∆t
(54)
≈ exp(Xi,∆t −rk,t Xk,∆t )
W∆t
k=1

q

in order to find expressions for E{(∆r) }.
For n = 2, we defined discrete values Ri in the interval (D− , D+ ). Similarly, wedefine111(k − )(Di,j + Dj,i )Ri,k = (−Dj,i − r̃i + r̃j ) +2
2m
2
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 (53)and an expression similar (50) for n > 2. Note that we find pk for each pair i, j.
We then have the following approximations to λ and γ using (51)mnXXλ̂ = Nd log(pk
Rv,k E{Vm,∆t }) + Nd log(1 − C∆t )
k=1

v=1

and
γ̂ = Nd log(

m
X

k=1

pk

n X
n
X

Rv,k Ru,k E{Vv,∆t Vu,∆t }) + 2Nd log(1 − C∆t ).

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),OPTIMAL REBALANCINGdci,12.72 0.01
2.72 0.001
2.15 0.01

ρ
r̃
0 0.2
0 0.2
0.3 0.2

D−
0.165
0.186
0.167

19

D+
0.212
0.203
0.212

Table 1. The critical values for n = 2 with proportional transaction. The two assets are geometric Brownian motion with annual
expectation and standard deviation equal to µV = (1.08, 1.02) andσV = diag(0.2, 0.04), and discounted by β = − log(0.8).

22where 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.
Hence, an approximation Û to the value of U when we rebalance outside theno-trade region ( 32) is
 1
1
1 
−d
.
(55)
Û =
−
β − γ̂
β − λ̂
β − 2λ̂This 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 each pair i, j
separately. Also in this case, it is stable to parameterize the length of the interval
Dj,i + Di,j and the position relative to the difference between the optimal weightsr̃i −r̃j +Dj,ir̃i − r̃j , e.g., D. Application of the above approximation will not give thei,j +Dj,ioptimal no-trade region for the object function U . But experiments show that theapproximation gives almost the optimal value. The approximation may also be
used in order to find a first guess on the no-trade region that is improved using
simulation.
5.3. Simulation of portfolios. In this section we find the no-trade region bysimulation using the utility function (39). The value of the assets Vi,t is modeledby logarithmic Brownian motion. In the simulation it is rebalanced to the boundaryof the no-trade region when the portfolio is outside the region using equation (31).
Table 1 shows the no-trade region for n = 2 for different values of the proportionaltransaction costs and with and without correlation between the different assets.
Note that when adding correlation, it is necessary to change d in order to have
optimum for r̃ = 0.2. Table 2 compares the utility of four different strategies. It isshown that optimal rebalance gives highest utility.
Referring to Table 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 the form (32),is only small adjustments of the no-trade region given by approximate formulas
and is computed by simulation. The case of no rebalance increases the ratio of thehigh volatile asset giving higher expected value of portfolio E{W1 } but at a 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 3–5. Table 3 shows the parameters for the five assets withvalues that are assumed realistic for the Norwegian stock marked, international
stock marked, Norwegian real estate, Norwegian bonds, and international bonds.
The table shows the optimal weights for the five assets with the utility function(39) and three different values of d. Table 4 shows the no-trade region for d = 2.

20

H. HOLDEN AND L. HOLDEN

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 2. Comparison of four different rebalancing strategies with n =2. The parameters are as in the top row of Table 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 denote theannual number of transactions and annual transaction costs, respectively.

i
1
2
3
4
5

µi
1.1
1.09
1.05
1.035
1.035

σi ρi,10.22 10.20 0.7
0.12 0.1
0.04 0.3
0.04 0.1

ρi,2
0.7
1
0.05
0.1
0.2

ρi,3 ρi,4
0.1 0.3
0.05 0.1
1
0
0
1
0
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 3. The critical values for n = 5 with proportional transaction
costs ci,1 = 0.01 and three 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
3
4
0.094 0.094 0.044—
0.069 0.053
0.59
—
0.12
0.73
0.67
—
0.73
0.51 0.67

5
0.041
0.058
0.10
0.14
—

Table 4. The optimal no-trade region for the example shown in Table
3 with d = 2. Above the diagonal is the length L of the interval forri − 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
1.04696

U1
Cn
1.04228 0
1.04092 12
1.04239 36
1.04240 36

Cc
00.00220.000490.00050

Table 5. The table is exactly as Table 2 except that n = 5 and wehave used data as in Table 3. It compares the strategies, no rebalance,monthly rebalance, and optimal rebalance with no-trade region foundby approximation and by simulation. The optimal no-trade region is asin Table 4.

OPTIMAL REBALANCING

21

Note how the length L varies between the different combinations of assets and theposition in some cases is far from the symmetric 0.5 value. Table 5 is similar to
Table 2 but for the example with five assets. The table 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 optimal
rebalance. The no-trade region is slightly different in the two cases and the result
is not sensitive to the exact position. Note that also here there is a reductionof transaction costs to 1/4 compared to monthly rebalance. But in this case theexpected 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 .
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 is ano-trade region. n symmetric assets means that all assets are treated similarly withno particular assumption on any of the assets. If the transaction costs are proportional, with no flat or fixed elements, it is optimal to rebalance to the boundaryof the no-trade region whenever the portfolio is outside the no-trade region. If thetransactions 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 have a full rebalance or a calendar-based rebalance.
The theory is illustrated by 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. It is probably not very critical to the exact positionof the boundary of the no-trade region. But in order to optimize the utility, it iscritical to have an optimal no-trade region. The example shows that the size of theno-trade region depends heavily on the properties of the stochastic processes, notonly the size of the transaction costs.
Acknowledgment. The authors thank our brother Steinar Holden for valuablecomments.
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(H. Holden)
Department of Mathematical Sciences, Norwegian University of Science and Technology, NO–7491 Trondheim, Norway, andCentre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern,
NO–0316 Oslo, Norway
E-mail address: holden@math.ntnu.no
URL: http://www.math.ntnu.no/~holden/(L. Holden)
Norwegian Computing Center, P. O. Box 114 Blindern, NO–0314 Oslo, Norway
E-mail address: lars.holden@nr.no