portfolio management with transaction costs

Portfolio Management with Transaction CostsAuthor(s): Colin Atkinson, Stanley R. Pliska and Paul WilmottSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 453, No. 1958 (Mar.8, 1997), pp. 551-562Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/53094 .

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Portfolio management with transaction costs

BY COLIN ATKINSON1, STANLEY R. PLISKA2 AND PAUL WILMOTT1'3

1Department of Mathematics, Imperial College of Science, Technology and Medicine, London SW7 2BZ, UK

2 College of Business Administration, University of Illinois at Chicago,

Chicago, IL 60607-7124, USA 3 The Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB, UK

It is known that the optimal trading strategy for a certain portfolio problem featuring fixed transaction costs is obtained from the solution of a free boundary problem. The latter can only be solved with numerical methods, and computations become formidable when the number of available securities is larger than three or four. This paper shows how a transformation of the free boundary problem together with an asymptotic analysis (performed about the solution when the transaction cost is zero) leads to solutions which are shown to be good approximations for cases which can be solved by numerical methods. These approximately optimal trading strategies are easy to compute, even when there are many risky securities, as is illustrated for the case of the 30 Dow Jones Industrials.

1. Introduction

There is a large literature on the mathematical modelling of portfolio management problems. The idea behind such modelling is to start with a stochastic process model of the underlying securities, build up a model for the value of the portfolio as a function of the trading strategy, and then, depending on the stated objective, solve the stochastic optimization problem for the optimal trading strategy. The traditional methodology involves stochastic control theory and dynamic programming (see, for example, Ingersoll 1987; Merton 1990), whereas a modern approach involves martin- gale methods and convex optimization theory (see Karatzas 1989; Pliska 1986). The result is an optimal trading strategy that tells the portfolio manager how much of each security to hold at each point in time.

Unfortunately, virtually all of this literature assumes that the transaction costs are negligible. We say 'unfortunately' because, except for very large investors who own very liquid securities, the transaction costs can have a profound effect on portfolio performance. In practice, many portfolio managers seem to turn over their portfolios fairly often, say every year. For average (that is, not wealthy) individual investors who maintain diversified portfolios and for large investors who own stock in small or even medium-sized companies, the resulting transaction costs can reduce the portfolio's rate of return by several percent a year.

Only a modest number of research studies have dealt explicitly with transaction costs because the mathematical difficulties are large. For example, Davis & Norman (1990) and Dumas & Luciano (1991) studied investors maximizing total discounted

Proc. R. Soc. Lond. A (1997) 453, 551-562 ? 1997 The Royal Society Printed in Great Britain 551 T)FX Paper



C. Atkinson, S. R. Pliska and P. Wilmott

utility of consumption, Taksar et al. (1988) studied investors maximizing asymptotic rate of growth of portfolio value, and Duffie & Sun (1990) analysed a model where the transaction cost is a fixed fraction of portfolio value and there is also a proportional cost for withdrawal of funds for consumption, but the security prices are not observed between transactions. Fleming et al. (1990) carried out a similar study. While all these models differ in their specifications, they tend to share a common feature: the computation of optimal trading strategies is very difficult, especially when the number of securities is allowed to rise to a range (20 or 30, say) compatible with practical use. The researchers had to work very hard just to get solutions for two securities, and efforts to proceed further were apparently abandoned.

With the objective of obtaining better computational results, Morton & Pliska

(1995) and Pliska & Selby (1995) developed a new model (Schroder (1993) inde- pendently developed the special case where there is a deterministic bank account and a single risky security). Since this present paper is based directly on the Mor-

ton/Pliska/Selby model, we shall now briefly describe it, referring the interested reader to Morton & Pliska (1995) for additional details. We will assume that the reader already has some familiarity with optimal portfolio models; readers without this familiarity should first consult Ingersoll (1987), Merton (1990), or a similar basic reference.

Morton, Pliska and Selby assume that available for trading are a deterministic bond or bank account with time t price Z4 = Z? exp(rt) and constant interest rate r as well as N risky assets with prices Zk, k = 1,..., N. These N assets are modelled as correlated geometric Brownian motions according to

/ N

dZt = Z dt + kl dW),

where p is an N-vector, Aij is the (i,j) entry in the N x N matrix A, and W1,..., WN are independent Brownian motion processes. They assume that the variance-covariance matrix AAT is of full rank.

With Vt denoting the value of the portfolio at time t and with e = 1- a a specified small non-negative number, Morton, Pliska and Selby assumed that a transaction at time t leads to a reduction in the value of the portfolio from Vt to aVt. In other words, the transaction cost equals a fixed fraction e of the portfolio's value. Then under the objective of maximizing the asymptotic growth rate

lim E[ln VT]/T, T->oo

they showed that the optimal trading strategy can be fully specified by an N-vector b coupled with a stopping rule r. Here bk represents the fraction of the wealth that is held in security k, in which case the fraction 1 - >k bk of the portfolio's value is held in the riskless bond. In general, these various fractions can be negative or greater than one. The times between successive transactions are governed by the stopping rule T, which is the first passage time of the 'risky fraction' process to a certain boundary. When a transaction occurs, the transaction cost is paid and the remaining funds are rebalanced among the assets and bank account so as to restore the actual fractions to the desired values in the vector b. The investor then steps back and observes the fluctuations of these fractions (the 'risky fraction process'), waiting to do the next transaction according to the stopping rule T, and then the transaction cost cycle repeats itself.

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The optimal values of b and r are found as the solution of an optimal stopping problem or, equivalently, a free boundary problem. In particular, Morton & Pliska

(1995) showed that the free boundary problem is given by the following linear com- plementarity problem for the function H(b):

?(H) < R - r, H + log( - 1Tb) ? 0,

(?(H) - R + r) . (H + log(1 - 1Tb)) = 0.

Here 1 is the N-vector consisting of a column of ls and

(H) = 1 Hi jbb (eT - bT)AAT(ej - b) i,j

+ Hibi(eT - bT) ( t- rl - AATb). (1.1) i

Subscripts on H denote a derivative, R is the growth rate of the portfolio, to be determined and ei is the N-vector with a single 1 in the ith row. The function H and all of its first derivatives must be continuous everywhere. This differential operator corresponds to the 'risky fraction' process, which represents the fractional allocation of money among the various securities in the absence of transactions. The system is closed by specifying

sup{log(1 - e) + log(1 - 1Tb) + H(b)} = 0. (1.2) b

The function H(b) is interpreted as the value function for optimally stopping the risky fraction process when it starts at the point b, where the reward for stopping is - ln(1 - 1Tb), but up until the time of stopping a 'continuation fee' of R- r per unit time is paid. The optimal b is the vector which maximizes (1.2). The optimal stopping rule r is equal to the first exit time by the risky fraction process from the 'continuation region' Q, which is the region where H(b) + ln(1 - lTb) > 0.

In one dimension with a single risky security, the continuation region Q is simply specified by two points and Morton and Pliska (1995) easily solved this case with numerical methods. But complications rapidly escalated when they moved to the case of two risky securities, as infinitely many points were required to specify Q. Moreover, conventional numerical procedures broke down due to the nonconstant coefficients in the differential operator. Pliska & Selby (1995) rescued the situation for the case of two risky securities by transforming the original stopping problem (1.1) to one featuring constant coefficients (i.e. corresponding to ordinary Brownian motion). This enabled Morton & Pliska (1995) to compute solutions for the two risky security cases by using a discrete time Markov chain approximation. However, it was not clear whether comparable transformations could be obtained for cases involving three or more risky securities, and it was apparent that computational effort would grow exponentially with respect to the number of securities, so their approach offered little hope for cases involving more than, say, four or five securities.

At this point Atkinson & Wilmott (1995) looked at the Morton-Pliska-Selby work and recognized that an asymptotic analysis would provide good approximations to the optimal solutions, provided the transaction cost fraction e is small (but still of realistic size). They performed a local analysis about the solution that arises when the transaction cost is zero, a solution that is called the 'Merton' solution, because it was first reported by Merton (1971). Very simply, the Merton solution is fully


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specified by the N vector

b (AAT)-l(,- rl).

In the absence of transaction costs it is optimal to trade continuously so as to keep the portfolio fractions aligned with b, thereby producing asymptotic growth rate

r+ 1ibT_ AATb

Atkinson & Wilmott (1995) showed that the optimal b coming from (1.1) and (1.2) should be approximated by b and that the optimal continuation region Q should be approximated by an N-dimensional ellipsoid, the specifications of which are easily computed, even when the number of risky assets is moderately large. Of particular interest was their observation that the lengths of the axes of this ellipsoid are

0(e1/4), with the first passage time thus being 0(e1/2). In two dimensions (i.e. two

risky securities), they showed that their solutions can be good approximations of some of the optimal solutions computed by Morton & Pliska (1995). However, other approximations were not so good, for a reason which will now be explained.

The N-dimensional Euclidean space is divided up into regions according to the hyperplanes defined by the second-order coefficients in the differential operator equal- ing zero. Once the risky fraction process starts in a particular region, it will remain there indefinitely. (For example, if you start with positive amounts of money in every security and you have money left over which you put in the bank account, then your risky fraction process will always have positive fractions which sum to less than one.) When the Merton solution is away from the boundary of its risky fraction region, then the two-dimensional continuation regions computed by Morton & Pliska

(1995) resemble ellipses, and the Atkinson-Wilmott approximations are good ones. But when the Merton solution is near a boundary of its risky fraction region, the second order coefficients become important and distort the continuation region; it can no longer be well approximated by an ellipse. The Atkinson-Wilmott asymptotic analysis breaks down.

We are now in a position to explain the main contribution of this paper. We trans- form the free boundary problem from one involving nonconstant coefficients to one involving constant coefficients, and simultaneously we carry out the asymptotic analysis. The transformations are carried out for a general number of securities, not just two risky securities as in Pliska & Selby (1995). The results will resemble Atkinson & Wilmott (1995), in that the asymptotic analysis will produce a continuation region that is an ellipsoid. However, this continuation region will be for a transformed problem, and when this continuation region is transformed back to the original problem one will get a continuation region Q2 that may not resemble an ellipsoid, especially when the Merton solution is close to a boundary of its risky fraction region. This will be illustrated with some example comparisons involving two risky securities, and computational experience will be reported for an example involving the 30 Dow Jones Industrials.

Before beginning the analysis, it should be remarked that we will assume, without loss of generality, that the components of the Merton solution b are all strictly positive with 1Tb < 1. Hence the risky fraction process starting at b will always remain in the region where its components are strictly positive and sum to less than one. This assumption was also made by Morton & Pliska (1995). The analysis for cases where the Merton solution falls outside of this region is similar and left to the reader. For example, if the Merton solution is such that the components of b are all positive and


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1Tb > 1 (so that you want to borrow money at the risk-free rate), then you will focus on the region where the components of b are all positive and 1Tb > 1, and the sup in (1.2) will be over the same region.

The plan of this paper is as follows. After developing the transformation of variables, we carry out the local analysis on the transformed problem. This results in a leading order problem, the explicit solution of which is then derived. We also de- velop approximations to the mean first passage time from the continuation region, thereby providing an estimate of the mean time between transactions. We conclude by presenting three kinds of computational results: (1) comparisons with Morton & Pliska's continuation regions for some two-security cases; (2) comparisons with Mor- ton & Pliska's mean time between transactions for some two-security cases; and (3) computation of the optimal solution for the case of the 30 Dow Jones Industrials.

2. Transformed variables

We shall see shortly that when the transaction cost coefficient e is small, the dominant parts of the differential operator (1.1) are the second derivative terms. In b-space the coefficients of these terms are nonconstant. Unfortunately, it is this property that resulted in the one poor result of Atkinson & Wilmott (1995). As we discussed in the introduction, we shall change to a coordinate system in which these coefficients are constant. We shall perform this transformation in stages.

First, let us introduce the new y-space given by

i bi

1EN bk 1 - C,--k=l From the chain rule we have

tab = E N bi bij

N 9 -bj ( b i= (1-Nk=l bk) 1- k=1 bk Yi

and

02 NN i ( bl +

ObiObj ~ ~ b -- ib EN bk

6i, 1=1 m=l \ =lbk

x( i k bk +

sjm) 0y20ym + first derivative terms,

where 6ij is the Kronecker delta. After some elementary manipulation we find that the differential operator (1.1)

becomes N N 2H E IY ym(AA T)lm d y+ first order terms. (2.1)

1=1 m=l Oy Oym

Now we simply let

xi = log y. Thus our differential equation in the no-transaction region is

N N H

Z Z (AAT)lm 1? + first order terms = R- r. (2.2) 1=1 m=l


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The solution of this must join smoothly with the reward function, that is, N

H = log (1 + Ee), on xE x QQ k=l

and N

VH=Vlog (1 + e on , x E? 9. k=l

Our differential operator now has second derivative terms with constant coefficients. As we shall see, this improves the accuracy of our later asymptotic analysis in certain cases.

In the next section we shall perform the asymptotic analysis of equation (2.2) in the neighbourhood of the Merton point.

3. Local analysis

As shown in Atkinson & Wilmott (1995) the continuation region is located within an O(C1/4) zone around the Merton point. For this reason we write

= - + e/4X,

with x being the Merton point in our new coordinate system. In the no-transaction

region we have

N N 22 -1/2 E E (AAT)lm XlHxm + first order terms = R - r. (3.1)

1=1 m=l

It is convenient to introduce G by letting N

H log +Ee xk + G(X). k=1

That is, we subtract off the reward function from H. This expression is substituted into (3.1)

Now the second derivative terms in G are 0(e1/2) and the first derivatives are

0(e3/4). Thus the second derivative terms dominate. We shall neglect the first derivative terms.

We further introduce the N x N matrix B with constant entries that is the local transformation from the X-space to the b-space. That is, to leading order,

b-b b-b = BX. C1/4

The entries Bij in the matrix B are given by

Bj b ~Sij - VV.

After some tedious algebra we find that the leading order equation for the adjusted value function G is

1VT(AAT)VG = R+ XTBTAATBX, (3.2)


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where

R = R + e1/2R.

Observe how the right-hand side of equation (3.2) is quadratic in X. Both the left- and right-hand sides of this equation are O(el/2); this prompts our choice of scaling for R - R.

The boundary conditions for the function G are simply

G=0 and VG = , on aQ, (3.3)

and

sup {G(X)} = 1. (3.4)

4. The explicit solution of the leading order problem

Problem (3.2)-(3.4) can be solved more quickly and efficiently than the original, full problem. This is especially important because of the practically large dimension- ality of the problem. We are particularly fortunate that there is a very simple explicit solution. This solution is of the form

G(X) = (XTMX - 1)2 (4.1)

for some positive definite symmetric matrix M to be determined. The no-trade region is bounded by the ellipsoid

XTMX = 1.

The solution (4.1) clearly satisfies (3.3) and (3.4), with the maximum value being at X = 0, the optimal rebalance point to leading order.

Using the relationships VXTMX = 2MX

and

VTAAt MX = Tr(AATM), it is easy to show that G = (XTMX - 1)2 is a solution of (3.2) provided that

8MAAT M + 4 Tr(AAT M)M = (B-1)TAATB-1 (4.2)

and

R = -2Tr(AATM). Equation (4.2) is to be solved for the symmetric matrix M, a task that is easy to accomplish with standard numerical methods.

We note, without including any of the details, that equation (4.2) has a unique positive definite solution for M provided that the matrix E = A(B-1)TAATB-lA is positive definite. (This is easily shown by observing that the matrices ATMA and E have the same eigenvectors. It is possible to then derive the eigenvalues for the former in terms of the latter.)

In practice, and as we see in the final section, we solve for the matrix M iteratively so there is little point in diagonalizing the problem.

5. First exit times

The expected time between trades is of particular interest. The expected first exit time from the continuation region satisfies, in the original variables, the differential


557




equation

L(F(b)) = -1, with

F(b) = 0, on Q. In the new scaled variables this becomes, to leading order,

1VTAATVE = -1,

with

E(X) = 0, on a9, where

E(X) =-1/2F(b). The solution is easily verified to be

E(X) 1_ XTMX) Tr(AATM) (-)

From this we can derive an expression for the mean time between transactions. Since the portfolio is rebalanced to the point X = 0, the mean time between transactions is

el/2 /E(O) =

I(AATM)

6. Computational results

In order to evaluate the accuracy of our approach, we compared our solutions with those of Morton & Pliska (1995). They computed solutions in two dimensions using Markov chain approximations. With their numerical solutions, the optimal rebalance vector b was always observed to coincide closely with Merton's optimal rebalance vector, so our interest focuses on the comparison of the continuation regions 2.

Figures 1-4 show a comparison of the continuation regions for cases 1-3, and 5 of Morton & Pliska, respectively. Parameters are given in table 1, mi, m2 and m3 are the two diagonal and one off-diagonal components of the matrix AAT, respectively. In each case the interest rate is 7%, and in all but case 3, e = 0.001. In case 3, e = 0.005. The agreement is very good for the first three cases, which are seen to have optimal rebalance points well in the interior of the state space. Moreover, the fits are better than the ellipses computed by Atkinson & Wilmott (1995). In case 5 (see figure 4) the optimal rebalance point is near the boundary of the state space, and the agreement is seen to be only fair. Our continuation region Q seems to have about the right shape (which is rather different from an ellipse), but it is somewhat bigger in size. Overall, the agreement is quite good, and it is a clear improvement over the ellipses computed by Atkinson & Wilmott (1995).

We also compared the mean times between transactions for the two-security ex- amples in Morton & Pliska (1995). In table 1 we show results for cases 1-3 and 5 of Morton & Pliska. The last two columns show the numerical and asymptotic estimates for the mean first exit times. Again the agreement is excellent.

Finally, we evaluated our computational procedure in the context of more mean- ingful problems by applying it to the case of the 30 Dow Jones Industrials. Using daily price data we first computed the variance-covariance matrix AAT. We then


558




0 0.4 0.8

Figure 1. A comparison between our asymptotic results (hollow triangles) and the Morton & Pliska numerical results (solid triangles) for case 1 of Morton & Pliska. The solid square is the Merton point.

0.8 -

0.4 - A

A. \

~I0 0.4 0.8 0 0.4 0.8



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0 0.4 0.8


0 0.4 0.8



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Table 1. A comparison of numerical and asymptotic estimates for the mean first exit time r

case AIl IL2 ml3 T asymp. r

1 0.13 0.15 0.10 0.17 0.07 3.02 2.89 2 0.14 0.15 0.10 0.17 0.07 3.25 3.26 3 0.13 0.15 0.10 0.17 0.07 7.02 6.48 5 0.13 0.15 0.13 0.20 0.14 2.47 2.51

used the Capital Asset Pricing Model (Ingersoll 1987) formula to provide values for the appreciation rate vector ,u. However, this led to a Merton solution b which, for all reasonable values of the interest rate, failed to have all components strictly positive with a sum less than one. We therefore perturbed the components of the initial vector 1u so that the Merton solution was satisfactory.

Taking 7% for the interest rate and with e = 0.001, we then solved this 30-security problem. The symmetric matrix M was computed iteratively from the equation

M =M- T) ((B-1)T AAB-1 -8MAAT M).

This is a simple rearrangement of equation (4.2). The initial guess was M = 0 and convergence to 1% accuracy was obtained in 10 iterations using Q = 20. The cal- culation was instantaneous. The expected time between rebalances was found to be 4.4046 years. Here and in table 1 the computed expected times between rebalances seem surprisingly large. However, this is a result of the 'mean reverting' behaviour of the risky fraction process. As discussed by Morton & Pliska (1995), the Merton solution is a stable point for the risky fraction process, with this diffusion process always tending to drift towards this point. Consequently, it is optimal to intervene and pay a transaction cost only when this process has deviated relatively far from the Merton solution, thereby leading to a large mean time between rebalances. To some extent, this finding is supported by the tendency of some portfolio managers to adopt buy-and-hold strategies. P.W. thanks the Royal Society for their support. S.P. thanks the Isaac Newton Institute for Mathematical Sciences for its support.

References

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Received 22 May 1995; revised 2 January 1996; accepted 29 May 1996


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