A Simple Algorithim for Capturing Portfolio

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A Simple Algorithm for Optimal Portfolio Selection with Fixed Transaction Costs

Author(s): Nitin R. Patel and Marti G. SubrahmanyaSource: Management Science, Vol. 28, No. 3 (Mar., 1982), pp. 303-314Published by: INFORMSStable URL: http://www.jstor.org/stable/2630883 .

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MANAGEMENT SCIENCE

Vol. 28, No. 3, March 1982

Priintedin U.S..

A SIMPLE ALGORITHM FOR OPTIMAL PORTFOLIOSELECTION WITH FIXED TRANSACTION COSTS*

NITIN R. PATELt AND MARTI G. SUBRAHMANYAMt

The general optimal portfolio selection problem witlhfixed transaction costs is a complex

mathematical programming problem. However, by placing reasonable restrictions on thevariance-covariance matrix of returns, it is possible to simplify the solutioniof the problem.

Specifically, if the structure of returns between securities is such that the pairwise correlation

coefficients are approximately the same, a fairly simple algorithm which requires little

computational effort can be employed. This method can also be extended to.the case where

changes in the information set necessitate a revision of an existing portfolio.

(FINANCE; FINANCE-INVESTMENTS; PORTFOLIO SELECTION)

1. Introduction

The pioneering work of Markowitz ([13] and [14]) in portfolio analysis has served as

the basis for the development of much of modern financial theory. Two directions of

enquiry have been pursued in the literature:one, the development of modern portfoliotheory to optimize selection of portfolios under different scenarios and the other, the

derivation of a theory of pricing of capital assets under uncertainty. One basic

implication of most models in portfolio theory is that investors are well diversified in

their holdings of securities. However, this implication of portfolio theory does not

accord with the observed holdings of securities by investors, who typically hold a small

numberof securities.'While there are alternative explanations of this phenomenon, an important factor is

the presence of transaction costs, broadly defined to include both direct costs such as

brokeragecommissions as well as the costs of analyzing securities, which are ignored in

much of the theory. Though some attention has been focussed on the question of

variable transaction costs, (see for example, Pogue [16], and Chen, Jen, and Zionts [3]),the question of undiversified portfolios remains unresolved. Variable transaction costsrender individual securities less desirable but do not inhibit the addition of more

securities to a portfolio. Fixed transaction costs such as odd-lot commissions do

provide an explanation for the phenomenon of reduced portfolio size but this market

imperfection has not been modelled in the literature, by and large. Mao [12], Jacob [9],and more recently, Levy [10] examine the fixed transaction costs problem indirectly byplacing restrictions on the number of securities in the optimal portfolios. However,

they do not explicitly relate the optimal number of securities to the transaction costs,and provide only a general rationale for the restriction.

The only detailed analysis of the fixed transaction costs problem in the portfolioselection literature is by Brennan [2].2 Assuming that the structure of the Sharpe [18],

*Acceptedby MartinJ. Gruber; eceivedMarch1980.Thispaperhas beenwiththeauthorI month or I

revision.1i ndianInstitute f Management,Ahmedabad.INew YorkUniversity.'In a study by Blume, Crockett and Friend [1] for the tax year 1971, it was foulld that investors hold

relatively undiversified portfolios. Based on a sample of individual income tax returns, it was computed that,

among dividend paying stocks, 34 per cent held only one stock, 50 per cent held no more thanitwo anid only

I I per cent listed more thani ten stocks.

2Stapleton and Subrahmanyam [19] provide an analysis of the effect of fixed transactioni costs on

equilibriulll prices. However, they are concerned more with asset prices with this market imperfection and

do not deal with the optimal portfolio selection problem directly. They therefore use an enumilerationi ethod

to solve the portfolio problem in order to determine equilibrium prices.

303

0025-1909/82/2803/0303$0 1.25Copyrigiht 0) 1982, The Itistittite of Management Scietnces



304 N. R. PATEL AND M. G. SUBRAHMANYAM

Lintner [11] and Mossin [15] capital asset pricing model holds even under fixed

transactions costs, Brennan presents a model for determining the optimal number ofsecurities in the portfolio. Initially, he assumes that all securities have the samesystematic risk and the same residual variance, and hence, in equilibrium, the same

expected return, and derives the optimal number of securities. Subsequently, he relaxes

the assumptions of identical residual variance, and through the capital asset pricing

model, identical systematic risk.There are several difficulties with the Brennan formulation. Firstly, he needs to

assume that the capital asset pricing model holds. While this may be a reasonable

approximation, this pricing relationship is itself affected when several investors operate

under fixed transaction costs as shown by Levy [10]. More importantly, this begs the

question of superior access to information and consequently different estimates of the

parameters of security returns, which imply that the standard capital asset pricing

model may not be relevant to that investor. Indeed, if the capital asset pricing modelwere always valid, portfolio selection models are of dubious merit, except for the

residual variance component which may not be significant, even with a relatively small

number of securities. Secondly, the Brennan model focusses on the number of

securities rather than on individual candidate securities themselves. Hence, the attrac-tiveness or otherwise of an individual security cannot be examined except in the trivial

sense that securities with lower residual variance are preferred, ceterisparibus. Finally,

the computation of the optimal number of securities is quite complicated, an unattrac-

tive feature of many portfolio selection models from the viewpoint of practicing

portfolio managers.The general optimal portfolio selection problem with fixed transaction costs is a

complex mathematical programming problem. But, by placing reasonable restrictionson the variance covariance matrix of returns, it is possible to simplify the problem.Interestingly, the solution of the problem can be reduced to a fairly simple algorithm

involving very little computational effort. In ?2, the model is discussed and a

simplified form of the objective function is presented. The next section derives the

algorithmand proves that the simple rules derived are optimal. A numerical example is

worked out and explained in ?4. ?5 deals with the case where changes in the

information set and/or the wealth constraint necessitate a revision in the portfolio. In

this case, the relationship of additional securities to the existing portfolio has to be

taken into account.

2. The Model

The portfolio selection problem of the investor in a mean-variance context with

fixed transaction costs can be written as:

11 s1 17 n

Max xiRi + xoR- ai x x1sis1p1- t Yii= Z1 i1 j=1 i =1 (1)

subject o xi+?xO=Wi=l

where the xi are unrestricted in sign, yi = 0 or 1, and xi # 0 only if yi = 1. The

definition of the variables are as follows:= 1 if security i is part of the portfolio; i = 1,2, . .. , n.

= 0 otherwise.

xi = amount invested in (risky) security i; i = 1,2, ... , n.

xo= amount invested in the riskless security.

Ri= 1?

expected rate of return on security i; i =1, 2, . . ., n.



OPTIMAL PORTFOLIO SELECTION WITH FIXED TRANSACTION COSTS 305

R = 1 + rate of return on the riskless security.

a = risk aversion factor of the investor (a > 0).

Si= standard deviation of the rate of return on security i; i = 1,2, . , n.

pij = correlation coefficient between the returns on securities i and j; i, j = 1,

2, . . .,n.

t = fixed transaction cost incurred for each risky security included in the portfolio.

This cost is assumed to be incurred at the end of the period. Equivalently, the fixed

transaction cost per security at the beginning of the period is equal to t/R.

W = initial wealth of the investor.The first term refers to the mean end of period cash flow to the investor from

holdings of risky assets. The next term relates to the return from the riskless asset. The

third term is the variance of end of period cash flow, while the last term refers to the

fixed transaction cost. The above formulation appears to be similar in form to the

standard Markowitz model with a riskless asset, except for the term involving the fixedtransaction cost.3 However, there is a basic difference; in this model, the computation

of the mean return and the variance of the return of the portfolio takes into account

only those securities that are in the portfolio rather than all the securities available in

the market. This aspect of the problem complicates it considerably, since it becomes a

mixed-integer quadratic programming problem for which no simple solution method

exists. At this point, there are two possibilities. The first would be to simplify the

structure of the model by placing some restrictions on the parameters. The other

method would be to obtain an approximate solution to the problem as formulated. We

have chosen the former alternative.One possible restriction on the structure of returns is that all pairwise correlation

coefficients between security returns are equal. While this assumption is difficult to

justify on any conceptual grounds, it has been shown to be a reasonable assumption,

empirically. In a study of alternative methods of forecasting the correlation structure

of returns, Elton and Gruber [4] demonstrate that this assumption produces better

estimates of the future correlation structure of security returns than do correlation

coefficients based on historical returns which may differ across pairs of securities, the

most general specification possible. Surprisingly,another intuitively appealing simplifi-

cation, the single index approximation of Sharpe [17] does not perform as well as the

constant correlation assumption. In the Elton and Gruber study, the assumption of aconstant correlation coefficient produced forecasts that were more accurate than nine

other specifications that were tested. In a more recent study by Elton, Gruber, and

Urich, [8], the forecasting performance of the constant correlation coefficient assump-

tion is tested using a variety of statistical procedures. The performance of this

assumption is consistently superior to that of all the alternatives in each of the

statistical tests performed. The constant correlation coefficient assumption will be usedin the rest of this paper.4

Suppose we assume that

pUj= p for all i, j; i #/ j (p > 0).

3In this formulation, we have chosen to model the fixed transaction cost as leading to a reductioll in the

mean end of period caslhflow. An alternative formulation would be to treat it as a reduction in the resources

available for investment, with the cost appearing in the wealth constraint. As will be clear in the analysis

later on, the two approaches are formally equivalent.

4The fixed transaction costs problem using the simplified structure suggested by the single index model of

Sharpe [17] will be analyzed in a subsequent paper. However, it turns out that the solutioni technique for this

case is more complex than the case studied here. The structureof the single-index model does not permit the

decomposition analyzed in this paper, since the interactions due to the integer constrainitsbecome more

difficult to handle.




The variance of the end of period cash flow can be written as

n n n1

p xXjjsjsj + E X12Sii=1 j=1 i=1

i#&jn n1 fl

=p>E E j + (_1p) E)2SI.i=lj=l i=1

The portfolio selection problem of the investor becomes

17 n 17 n *Max >x,RJ? xOR-t E yi-a p x1x1ssj + (1-p) E xsi2

Xi i=1 i=1 i=1 y =1 i=1 2

subject to E xi +x0 = W

i= 1

and yA= 0 or 1; xo, xi are unrestricted in sign with the proviso that xi #/ 0 only if

Yi= 1.

If the securities chosen to form the portfolio belong to some set S c (1, 2, .. , n,

then yj = 1 for j C S and yj = Xj= 0 for j 4 S. Eliminating xo using the wealth

constraint, the objective function becomes a concave function of xj, j C S. Taking

derivatives gives us the following necessary and sufficient5 conditions for optimal

levels of xj, given S.

n

(R1- R) - 2aps sixiS-2a(1 _p)s,.x1=0, ] C S, (3)

or

x. = 2 (lp)s j4S

0O, jI 4 S.

Sincey1 = 1 if and only if xj #0 ,

Rjs( p) PEi pj y,iji 1,25 ... ., n. (5)

Multiplying (5) by sj and adding over all j E S, we get

ES jS R.- R MEmp wherem=jSjsx, E

2as1I -p) i- iS

so that

iES 2a[mp + (I -p)] S s (6)

Substituting (6) into (5) yields

=(Rj - R)yj pyjEiEs(Ri -R)si

2a(l _ p)S>2 2a( - p)[(m- I)p + I ]sj

djy] pyjziE~sciyi j 1 5..n 72a(1-p)s? 2a(1-p)[(m-1) +p1]j=I1s n (7)

sThe first order condition is sufficient since the objective function after substituting the constraint isconcave.




where cj = (R - R)/sj represents the excess return of securityj over the riskless rate

of return as a ratio of the standard deviation of the return of the security.

Multiplying (3) by Xj,adding over j E S, and rearranging,we get

SiS ~~~(RjR)p 2 sjs~xix~?j+I -p) s]xj2=i~~Sj~~S j~s, jx S 2a 7.(8)e Sje s y es ye s

Substituting for x0 in the objective function, using the wealth constraint, and using (8),

the objective function becomes

n

2, (Rj - R)xjl2- tm + WR. (9)j=1

Substituting for xj from (7) the objective function expressed in terms of y. becomes

4a(1l_-p) 1 -' Yj- [(m- ? I - tm - WR. (10)

For fixed m, since a, p, t, w, R are constants, our problem is reduced to

Max cj2y1

j=ll [(in-I)p + 1](J

subject to yj =0,l and yj = m.j=l

We assume that the securities have been ranked so that cl > C2 . .. > c,7- Note if

two securities have the same ci, they may be treated as a single security in our analysis,

so that the strict inequalities above do not imply any loss of generality.

3. The Algorithm

We are now in a position to derive a simple algorithm for portfolio selection and

show that it is optimal.

THEOREM 1. Let S be the set of optimal securities for a given number of securitieschosen, m, m >.2 (i.e., ISj = m and S = (k IYk = 1}). Then there is noj E S such that

both] - 1 andj + 1 X S.

We argue from contradiction. Suppose there is such a j. Let S' = S - (j}. Let

OFV ( Q) denote the objective function value for any set of indices Q.

OFV (S)= 1.c7 - Ec1)S iES [(m-I)p + 1(is /

- Ec - ( ( E ci) (11)

-S[(i l)p 1][ [ilM )p? 1] cj* (12)

OFV(S'U{]-1})= cI

2pcki v [(m-2)p + 13




Since OFV(S) - OFV(S'U { - 1}) > 0

[ (m-1)p + 1 (jP -lcj) iE c - [ (m -2)p + I ] (CJ_lc /-) > O

or

(ci- I - c.) t2p ci - [(m - 2)p + I](cj-, + cj)) > O (14)

and since cj- l - cj > 0, the expression within the curly brackets above (call it (a)) must

be greater than or equal to zero.

A similar argument using OFV(S) - OFV(S'U{j ? 1}) > 0 leads to

(cj+I-cj){2p E ci-[(m-2)p + (cI + c) > 0

i E-sI

and since cj+I - cj < 0, the expression within the curly brackets above, (call it (b)),

must be less than or equal to zero.

Subtracting (b) from (a) we have

[ (m -2)p + I I(Ci+ I- ci- ) >1 .

For m > 2, the first factor above > 0 which implies cj+I > cl which is a contradic-

tion. Q.E.D.We conclude from Theorem 1 that for a given m, the optimal S is of the form

{1, 2, ... , u, 1, + 1, . . ., n} where u + n-I + = m. In fact, an optimal portfoliocan be represented by the single number u, since 1 = n - m + u + 1. We shall

therefore say that an optimal portfolio is u* to mean that the securities in the portfolio

are {1,2, . . ., u*,l*,l* + 1, . . ., n} where l*= n-m + u* + 1; if u* = m, l* = n + 1

and the portfolio represented in this case will be {1,2, . .. , m}.

THEOREM . Let u,*, U*1+ be optimalportfoliosof sizes m and m + 1 respectively.Let

1 = n-rn +?u* + 1 and i+H = n (m + 1) + u*1 + 1 =n-rm + u* + . Then u* +

1 u,+ > U,* if p > 0 and c1 > 0.

(i) We first show u,*7 1 > u,s*+. We argue from contradiction; suppose u,*+?

? u* ?2.

Since u,* is optimal, increasing 1/*7and u,* both by 1 cannot increase the objective

function value. From this we conclude

(Cl - cl*){2p (Eci ? ) [ (i - 2)p + 1] (c,,,*+ ? cl*) ? 0.

C \i=,*)*,+)I )

Since the first factor above > 0, the expression in curly brackets (call it (c)) > 0.Under the hypothesis, U*[ + > u* + 2, so that

1*l= n-m +,+ l n- nm + U,, + 2 > n - m + U,, ? 1 =,+ .

Also, since u,*7? is optimal, if we exchange security u,* + 1 with security 1*, the

objective function value cannot irncrease.From this, we conclude

0 t{221 [/tI1+I fl

(c,* - c,*?1){2(12p c + ci + ci [( - 2)p + 1](cl C"* > ?.t i=l i= ?2 i=l,*,+,




The first factor above being < 0, implies that the expression in the curly brackets (call

it (d)) < 0.

Subtracting (d) from (c), we have

2p ci - 2 ci > 0.i = l,,+ I i=U,*,+2)

Since p > 0,

+ I'-, + X n-rn + tu,*+ Xtil +I

E ci- E ci= E ci- E ci> 0.

i l,+ ==,* +2 i = n7-77n + ti* +2 i = t* +2

However, since ci > 0, the above expression is less than

n-rnm+, * U,*

2 ci1- ci E (C,11+i-

ci) <0

i=n-m+u,*, +2 i=u*+2 i=ti*+2

since c,, 1+ i < ci-

This leads to a contradiction. It follows that u,*+ 2 > u*1 and hence, we conclude

that u* + 1 > u*+

(ii) To show that u, + > u*, we follow an exactly analogous argument but decreas-

ing u + and ',*t+Il by 1. Q.E.D.

This theorem tells us that if we have u,*,then to find

u*we need

onlyto check

for which value of the pair u,*,u,*+ 1 the objective function is larger.

Theorems 1 and 2 directly suggest an algorithm for finding the optimal portfolios for

different values of m, for m = 1,2, . .. , m. This algorithm is flow-charted in Figure 1.

It is easy to show that the computational complexity of the algorithm is 0(n) and is,

therefore, very efficient. (If we include the effort involved in sorting the securities in

increasing order of ci, the algorithm is O(n log n).)

Once the optimal portfolios S*, S*, . . . f, S, for each value of m are obtained, we

can determine that best portfolio for a given a and t by evaluating expression (10) for

S*, S*. .., S,1.If expression (10) has a maximum value for Sk*ithen Sk is the optimal

portfolio. Next we prove an interesting characteristic of optimal portfolios.

THEOREM . Let S,*,= {l,2, . .. , u, 1,1+ 1, . . ., n} be the optimal set of securities

for a fixed m (l > u + 1). Then securities 1, 2, ..., u are bought long and 1,1 + 1, . . ., n

are sold short.

Since S* is optimal, its objective function value is not less than for the portfolio

Sm1={l, 2, ,u -l, I l, 1, ...,n}.

OFV(S*,) - OFV(S,?I)= (c,1 I c,) {2p E ci - [((m - 2)p + I] (c,l + c,,)} 0

where S' = 1,2,. . ,u- l, l,.. .,n.

Now (c l- cl,) < 0, so that

2p E ci- [(m -2)p + I ](cI- ,+ cl,)< 0i (=sI

or

[(m-2)p + IlI(cl _I + cl,) > 2p ci.




Start

OFVt= OFV( I)

FIUE.Te Algorthm

Sincec11 > ,<,~~~~~~~~~~~~~~~~~~~~~~1

- , + +1') > OFV( S,, U1[( - + 1]=-

es

~~~~~NoS,*a+,( S-*p)U[( -*l + I -p)+[( I Ul)*p 1]

|OFV,*, +, OFV( S,*,, ,

Nol

Yes

FIGURE 1. The Algorithm.

Since c,, > cl_ I,

[ (m-l2)p+l1]ci - l)>p + ci]i Is

From (7)

c,,[(M-l)p + I ]-p PZE -s,*c; c,,[(m -2)p + I]pZE s c

IIl 2as"(I1-p) [ (m-1 I)p + I 2as"(I1-p) [ (m-1 I)p + I

Since both numerator and denominator aregreater

thanzero,

it follows thatxl*

0.

Now for all j < u,

Cj[ (M -1).p+ I>c"[ (M - )p + I>p E Cj

i (E s,*,

so that Xj*> O for all j = 1, 2, ... ., u.

An analogous argument shows that x* < O and hence Xj*< O for all j = 1,1I+ 1,. n. Note also that s ,xl > s X* > s X* > ***> S-XI* and s,x* > s,+ x*

> Sl 2X*+2>*** SX,. Q.E.D.(The proo>olw

..ill fro (7)).



OPTI-MALPORTFOLIO SELECTION WITH FIXED TRANSACTION COSTS 311

4. A NuI11erical xaInple

We give below illustrative results for a hypothetical problem in which n = 20,

p = 0.4, t = 1, a = 0.01 and sj = 1 forj = 1,2, . . ., 20. The Cjwere chosen randomly in

the interval (0, 1).

= 0.875, c2 = 0.839, C3= 0.719,C4 = 0.585, C5=

0.464,6 = 0.429, C7 = 0.413, c8 = 0.331, c9 = 0.326, c10 = 0.312,

cX = 0.245, C12 =0.231, c13 = 0.209, C14 =0. 168, C15 = 0. 152,

C16 = 0. 142, C17 = 0. 120, C18= 0.1 I cI C19 0.080- C20 = 0.062.

in] Sfl* OFV,* Expression(10)

I [1] 0.4594 18.14082 [1,2] 0.8823 24.2574

3 [1,2,20] 1.3904 29.1859

4 [I to 3,20] 1.8888 31.7720

5 [I to 3? 19? 20] 2.5393 35.6939

6 [I to 3, 18 to 20] 3.1414 37.6303

7 [I to 3, 17 to 20] 3.7304 38.7151

8 [I to 4, 17 to 20] 4.3901 40.1371

9 [1 to4, 16 to 20] 5.0278 40.8788

10 [I to 4, 15 to 20] 5.6496 41.1735

11 [I to 4, 14 to 20] 6.2477 41.0642

12 [I to 5, 14 to 20] 6.8688 41.0002

13 [I to 5, 13 to 20] 7.4439 40.4762

14 [Ito 6, 13 to 20] 8.0365 40.0009

15 [I to 7, 13 to 20] 8.6090 39.3495

16 [I to 7, 12 to 20] 9.1894 38.6991

17 [I to7, 11 to 20] 9.7539 37.9204

18 [I to 8, 11 to 20] 10.2828 36.9295

19 [I to 9, 1Ito 20] 10.8108 35.9329

20 [I to 20] 11.3384 34.9341

The optimum value of m is 10 with the best portfolio S*, given by

S= [1,2,3,4,15,16,17,18,19,20].

From (7) the optimal investments in the securities are:x = 46.2137,x 2* 43.2137,

X = 33.2137,X=*= 22.0471,x = - 14.0362,x = - 14.8696,

X*= - 16.7029,x*= - 17.4529,x*= -20.0362,x*0= -21.5362.

The optimal investment in the riskless security is

20

W- jxi = W- 40.0542.i. =I




5. The Portfolio Revision Problem

The analysis presented in the previous section deals with the case where an investoroptimizes his portfolio holdings of risky securities with fixed transaction costs under

the assumption that the investor's initial wealth is in the form of cash. While thisaccurately describes the initial portfolio problem of the investor, it does not properlyreflect the costs of changing the optimal holdings over time. Specificially, there may be

a reluctance on the part of the investor to decrease the holding of a particular securityin light of adverse information on, say, its expected return since there are fixed

transaction costs incurred in selling this security as well as investing the proceeds in

one or more other securities. It may be reasonably hypothesized that the optimalholdings of securities in the portfolio after revision will be biased in favour of securities

that are initially included in the portfolio.

It turns out that the portfolio revision problem can be modelled with minor changesin the earlier analysis. Suppose xi is the amount already invested in the security i (thepresent portfolio) and di is the change in this amount (the new portfolio). The

maximization problem in equation (2) can be modified as

n fl

Max (xi +di)Ri+ (xo + dO)R t yidi i=Ii I

n n1s(afl 12 x 1(2+dyss l-)E(x js | (5

n

subject to Ed + do= W'

where W' is the additional amount available for investment and with the proviso that

Yi= 1 if di #- 0, i = 1, 2, . .. , n. It should be noted that W' could be zero, positive or

negative and does not impose any restrictions on the problem. If the problem is one of

portfolio revision merely due to changes in the information set, W' would be zero; if

there are additions to or redemptions from the portfolio, from other income or due toconsumption needs, W' would be positive or negative respectively.

The portfolio problem can be rewritten by grouping separately the terms involvingthe di's, the decision variables, in (15).

n n

Max d1Ri+ doR-t ydi= I

-a tpL (dxjsisj+ d)x1s.s1+ didsisj) + (I p) 2dixis2 + d2SJ2j + K

n

subject to d + do= W',i = I

Y= I if di::--0, i= 1,2, .. n, (16)

where

12 n 1 1 A

K xiRi+ xoR7-a p) Xi2si2

;~~~~~i_ss i _ 1) ;




which is unaffected by the portfolio revision. Equation (16) can be rewritten as

Max,1d,[R - 2aps(1 xs- 2a(1 -p)s,xij

1 r 11 11 11 A

+ doR-t fi-af p~ 2 d,dj,s1sj+ 1 -p) 2 aj2s7 9(17)

subject o E= 4j+d '1=1

y= 1 if d,i#0, i =1,2, ... , n.

This is of the same form as the original problem in (2), except that

= [R - 2apsi( x - 2a(I - A),2x1

The rest of the analysis would be similar and the same procedure as before can be used

to solve for the optimal d1's.Intuitively, the only difference between the initial problem

and the portfolio revision problem is that the interactions of each security with the

existing portfolio as reflected in the definition of R,' have to be taken into account in

the latter case. It pays to revise the existing holdings of a particular asset only if the

round trip transaction costs can be recouped by revising the portfolio.

6. CoIiclusions

This paper derives simple decision rules for optimal portfolio selection under fixed

transaction costs. Assuming that the pairwise correlation coefficient is the same across

all securities, it is shown that the optimal portfolio can be chosen without directly

solving the complex mixed-integer quadratic programming problem. The interesting

feature of the model is that securities can be ranked on the basis of a simple index, theratio of the excess returnto the standard deviation of return of the security. Thus, even

fairly large problems can be solved by hand, as convergence to the optimal solution is

rapid. Even for cases where the assumption of equal correlation coefficients is notvalid, perhaps an intuitive understanding to the solution of a fairly complex problem

can be gained. The portfolio revision problem is shown to be similar in structureto theproblem of the portfolio optimization starting with cash holdings, the difference being

that in the former case, round trip transaction costs lead to a reluctance to revise the

portfolio unless the gains are correspondingly greater.

References

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A Simple Algorithim for Capturing Portfolio

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