a computational scheme for optimal investment – consumption with proportional transaction costs

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Journal of Economic Dynamics & Control 31 (2007) 1132–1159

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A computational scheme for optimal investment –consumption with proportional transaction costs

Kumar Muthuraman�

School of Industrial Engineering, Purdue University, 315 N Grant St., West Lafayette, IN 47906, USA

Received 15 March 2005; accepted 5 April 2006

Available online 30 June 2006

Abstract

We consider the optimal investment – consumption strategy of an investor who can invest in

a stock and a bank. We consider the case where proportional transaction costs are present and

the objective is to maximize the discounted utility of consumption. We describe an efficient

computational scheme that transforms the arising free-boundary problem to a sequence of

fixed-boundary problems. We prove the convergence of the scheme and also show that the

converged solution is the optimal value function. Finally, we compare and contrast the results

obtained by our procedure with certain well-known results and approximations. The proposed

scheme also lends itself to optimizing portfolios with multiple risky assets.

r 2006 Elsevier B.V. All rights reserved.

JEL classification: G11; C61

Keywords: Portfolio optimization; Transaction costs; Stochastic control; Hamilton – Jacobi – Bellman

equation; Free boundary

1. Introduction

This paper considers the optimal consumption – investment strategy of an investorwho operates in a market containing one risk-free (‘bank’) and one risky asset

see front matter r 2006 Elsevier B.V. All rights reserved.

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K. Muthuraman / Journal of Economic Dynamics & Control 31 (2007) 1132–1159 1133

(‘stock’). The price process of stock is modeled as a geometric Brownian motion. Theinvestor is given an initial position in both assets. In time, he can choose to eitherconsume money from the bank, buy stock with money in the bank or add money tothe bank by selling stock.

Transacting, that is, buying or selling stock, incurs proportional transaction costs.That is, the investor pays a proportion of the value transacted to a third party thatenabled the transaction. The investor obtains utility by consuming money from thebank. We take the utility function to be either the power utility, or the log utility.The investor’s objective is to transact and consume so as to maximize the expectednet present value of utility, namely, E

R10 e�ytuðctÞdt, for some discount factor y40.

The y is also called the impatience factor since it represents the impatience of theinvestor in consuming wealth. The investor is allowed to trade in continuous timeand in infinitesimal quantities.

The model considered in this paper, without transaction costs was the focus ofMerton’s seminal paper (Merton, 1969). The optimal policy proposed by Mertoncontinuously transacts to hold fixed fractions of total wealth in various stocks andconsumes a (different) fixed fraction of wealth. Merton’s policy requires that aninfinite number of transactions be made in any finite time interval. This suggests thatin the presence of transaction costs, Merton’s policy would no longer be optimal.With transaction costs, the investor would want to make fewer transactions. Inparticular, he would make transactions only if the fraction of his stock holding is‘sufficiently’ far away from Merton’s optimal fraction to warrant the transaction.Magill and Constantinides (1976) first considered proportional transaction costs andconjectured that the optimal policy would be characterized by an interval ofinaction, such that the optimal policy would not transact when the fraction of wealthin stock lies in this interval. When the fraction lies outside the interval the optimalpolicy would be to buy or sell just enough to bring the fraction into the interval.

The problem with proportional transaction costs is now understood to be asingular stochastic control problem. Taksar et al. (1988) were the first to make thisobservation. They obtained optimal policies, for a model without consumption, thatmaximized asymptotic growth rate of portfolio. Constantinides (1986) showed thattransaction costs only have a second order effect on liquidity premium. In 1990,Davis and Norman (1990) solved the Merton problem with proportional transactioncosts for the one-stock case. They provided detailed characterization of the optimalpolicy, and conditions under which the HJB equation has a smooth solution. Theyalso provided a numerical method to calculate the optimal policy. Numericalmethods for the one-stock problem can also be found in Davis and Norman (1990);Tourin and Zariphopolou (1997). The existing numerical methods do not lendthemselves to the multiple stock case. Moreover, the computational complexity ofthese methods even for the one-stock case are extremely high. In a review of the area,Zariphopolou (1999) indicates that an implementation of these methods on aworkstation takes hours to yield satisfactory boundaries. Though finding the freeboundary translates to finding two points, this is a very hard problem as indicated bythe run-times of implementation. In comparison the method we present here takesseconds ð�30 sÞ to yield boundaries that converge within a tolerance of 10�6. Later

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K. Muthuraman / Journal of Economic Dynamics & Control 31 (2007) 1132–11591134

Shreve and Soner (1994), in an exhaustive theoretical exposition, considered arelaxation of the Davis and Norman problem and used viscosity solution tech-niques to provide existence and uniqueness results and characterized the re-gularity of the value function. In a recent paper, Janecek and Shreve (2004)have carried out an asymptotic analysis and have provided an asymptotic ex-pansion of the value function in powers of l1=3, where l, the cost of transaction goesto zero.

Duffie and Sun (1990) studied a model with a different transaction andinformation accrual structure and concluded that in their model an investor wouldchoose to re-balance his portfolio at constant time intervals. Weiner (2000) hasstudied a model in a which the stock price process has stochastic volatility. Pliskaand Selby (1994); Morton and Pliska (1995); Atkinson et al. (1997) have studied amodel with no consumption and with transaction costs that are of the managementfee type, that is, the investor pays a fixed fraction of his entire wealth when he wantsto re-balance his portfolio. For a review of literature that treat such optimizationproblems with multiple stock refer to Muthuraman and Kumar (2006).

In this paper we propose a computational scheme to solve the portfoliooptimization problem with proportional transaction costs. The model is describedin Section 2 and a brief discussion of the value function and the Hamilton – Jacobi –Bellman (HJB) equation that characterizes the value function is provided inSection 3. Section 4 describes the computational scheme. Theoretical vali-dation of the procedure is presented in Section 5. Section 6 compares and contrastsour results with the results obtained by Davis and Norman (1990), asymptoticexpansions obtained by Janecek and Shreve (2004) and approximations suggestedby Constantinides (1986). We conclude in Section 7. The scheme described inSection 4, can easily be extended to treat the multi-dimensional version of theproblem considered here. But providing rigorous theoretical guarantees alongthe lines of Section 5 are very hard. In Muthuraman and Kumar (2006), weillustrate heuristically how the scheme can be extended to multiple dimensionsand provide exhaustive results as numerical evidence for the convergence ofthe scheme.

2. Model formulation

We consider a market consisting of one risk-free and one risky investment. Therisk-free investment, called the ‘bank’, continuously pays an interest rate r40. Theevolution of X ðtÞ, the value in the bank, can then be expressed as X ðtÞ ¼ X ð0Þ ert or,

dX ðtÞ ¼ rX ðtÞdt. (1)

The risky investment, called ‘stock’ hereafter, has a mean rate of return a. We willassume a4r. We will take a standard Brownian motion B ¼ fBðtÞ : tX0g on itsstandard filtered probability space ðO;F;PÞ, as our source of uncertainty, whereF ¼ fFðtÞ : tX0g is a right continuous filtration of s-algebras on this space thatrepresents the information revealed by the Brownian motion.

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Let Y ðtÞ denote the value in stock at time t and s2 be the variance of the stockreturns. Then,

dY ðtÞ ¼ Y ðtÞ½adtþ sdBðtÞ�. (2)

The investor is given an initial position of x dollars invested in the bank and y dollarsinvested in the stock. He must choose a consumption and trading policy to maximizehis objective. Consumption cð:Þ occurs from money in the bank and only in non-negative quantities. We assume that cð:Þ is adapted to Ft, that is, it is non-speculative. We will also require that cð:Þ be integrable for any finite t, that is,Z t

0

cðsÞdso1 8tX0. (3)

To model the transaction controls we consider two Ft-adapted processes LðtÞ andUðtÞ which are right continuous with left limits. LðtÞ represents the cumulativeamount of money spent from the bank to buy stock before incurring transactioncosts. Similarly, UðtÞ represents the cumulative amount of money obtained fromselling stock before incurring transaction costs. Thus, LðtÞ and UðtÞ are non-negativeand non-decreasing processes. Buying and selling stock incurs proportionaltransaction costs. Let lX0 and mX0 be the transaction cost for buying and sellingstock, respectively. We will assume that lþ m40 to avoid the trivial case. To bemore precise, buying a unit worth of stock will cost ð1þ lÞ of wealth from the bankand selling a unit worth of stock will result in ð1� mÞ of wealth added to the bank.

For the sake of readability in the rest of this paper, unless necessary, we willsuppress the dependence on time t when denoting the processes BðtÞ;X ðtÞ;Y ðtÞ;cðtÞ;LðtÞ;UðtÞ. With consumption and transaction, the controlled evolution of X andY can be described by the equations

dX ¼ ðrX � cÞdt� ð1þ lÞdLþ ð1� mÞdU , (4)

dY ¼ Y ½adtþ sdB� þ dL� dU . (5)

The initial position that the investor starts with is ðx; yÞ, that is, X ð0�Þ ¼ x andY ð0�Þ ¼ y.

The solvency region is defined as

Sl;m ¼ fðx; yÞ 2 R2 : xþminðð1þ lÞy; ð1� mÞyÞX0g.

The initial portfolio ðx; yÞ and its future evolution are restricted to lie in Sl;m, whichis the set of points from which the investor can conduct transactions to move to apoint of non-negative value in both assets. A consumption – transaction policyðc;L;UÞ is called admissible if X and Y given by Eqs. (4)–(5) lie in Sl;m for all tX0,that is,

P½ðX ;Y Þ 2Sl;m for all tX0� ¼ 1. (6)

Therefore, an admissible policy will ensure that bankruptcy does not occur in finitetime. We will use U to denote the set of all admissible policies. U is clearly non-empty, since we can construct an admissible policy ð~c; ~L; ~UÞ from any policy ðc;L;UÞ

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by terminating ðc;L;UÞ at any arbitrary time when the state is still in Sl;m, andmoving all wealth to the risk-free asset.

The utility that the investor obtains by consuming c dollars from the bank is givenby the utility function uðcÞ. We will consider two common utility functions, the logutility function and the power utility function. They are given by

Log utility : uðcÞ ¼ logðcÞ, (7)

Power utility : uðcÞ ¼cg

g; ga0; go1. (8)

Here, g is the relative risk aversion coefficient that describes the investor’s riskpreference. These utility functions are very common in modeling investor’s riskpreference and belong to a class of functions called the hyperbolic absolute risk

aversion (HARA) functions. Let y40 be the discounting factor. Then the investor’sobjective is to choose an admissible consumption – transaction policy ðc;L;UÞ so asto maximize

Jx;yðc;L;UÞ ¼ Ex;y

Z 10

e�ytuðcÞdt, (9)

subject to (4)–(6). The optimal value function V is defined as

V ðx; yÞ ¼ supðc;L;UÞ2U

Jx;yðc;L;UÞ. (10)

3. The value function and the HJB equation

The stochastic control problem described in the previous section can betransformed into a partial differential equations (PDE) problem. The value function(10) has been shown in Shreve and Soner (1994) to be a smooth solution of the HJBequation

max ~LV þmaxcðuðcÞ � cV xÞ; ð1� mÞV x � V y;�ð1þ lÞV x þ V y

� �¼ 0, (11)

where

~LV � 12s2y2V yy þ ayVy þ ðrxÞVx � yV . (12)

The first order maximizing conditions for c are

c ¼V�1x when uðcÞ ¼ logðcÞ;

V 1=ðg�1Þx when uðcÞ ¼ cg=g:

((13)

In the above equations and the rest of the paper we will use V x to denote the partialdifferential of V with respect to x, V y and V yy to denote the single and double partialdifferential with respect to y. The existence of a C2 solution to the HJB equation ingeneral is a very delicate issue. However, for the model that we are considering,

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Shreve and Soner (1994) have shown that the value function V is C2 except possiblyat x ¼ 0.

The optimal transaction policy can be described by two scalars fs and fb whichdefine a region of inaction. The region of inaction (no-transaction region) can berepresented by

O ¼ ðx; yÞ 2Sl;m :x

y2 ðfs;fbÞ

� �.

If x=yofs the optimal action is to sell stock till x=y ¼ fs and similarly if x=y4fb

the optimal action is to buy stock till x=y ¼ fb. In the region of inaction theoptimum is not to conduct any transaction. We will denote by qS and qB theboundaries x=y ¼ fs and x=y ¼ fb, respectively. Fig. 1 illustrates the variousregions.

It is straightforward to show that the optimal value function is concave andsatisfies the ‘homothetic property’, that is, for ant r40

V ðrx; ryÞ ¼1

ylogðrÞ þ V ðx; yÞ when uðcÞ ¼ logðcÞ,

V ðrx; ryÞ ¼ rgV ðx; yÞ when uðcÞ ¼ cg=g.

Moreover, due to the concavity of the value function and the homothetic property, itcan be argued (as in Davis and Norman, 1990; Shreve and Soner, 1994) that O is acone in R2. Therefore, by defining a function W ðxÞ

W ðxÞ ¼ V ðx; 1Þ

we can reduce the dimensionality of the problem. If we can find W ðxÞ 8x 2T �ðm� 1;1Þ we can use the homothetic property to obtain V ðx; yÞ, 8ðx; yÞ 2Sl;m. For

Fig. 1. The solvency region.

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example, in the log utility case

V ðx; yÞ ¼ Vx

y; 1

� �þ

1

ylogðyÞ ¼W

x

y

� �þ

1

ylogðyÞ. (14)

In T, the sell region is ðm� 1;fsÞ, the region of inaction O � ðfs;fbÞ and the buyregion is ðfb;1Þ.

In terms of W ðxÞ, the HJB equation (11) becomes

max LW þmaxcðuðcÞ � cW 0Þ; SW ; BW

� �¼ 0, (15)

where

LW � b3x2W 00 þ b2xW 0 þ b1W þ b4, (16)

SW �W 0 � js, (17)

BW � �W 0 þ jb. (18)

For the log utility (uðcÞ ¼ logðcÞ) case

b3 ¼12s2, (19)

b2 ¼ r� aþ s2, (20)

b1 ¼ �y, (21)

b4 ¼1

ya�

s2

2

� �, (22)

js ¼1

yð1þ x� mÞ, (23)

jb ¼1

yð1þ xþ lÞ(24)

and for the power utility ðuðcÞ ¼ cg=g) case,

b3 ¼12s2, (25)

b2 ¼ s2ð1� gÞ þ r� a, (26)

b1 ¼ �12s2gð1� gÞ þ ag� y, (27)

b4 ¼ 0, (28)

js ¼gW

1þ x� m, (29)

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jb ¼gW

1þ xþ l. (30)

When in the one-dimensional context, as above, W 0 and W 00 represent the first andsecond derivatives of W .

4. The computational scheme

We will use a two-step procedure to solve (15). We begin by choosing an arbitraryregion of inaction, O0 � ðf0

s ;f0bÞ. For the transaction policy corresponding to O0, we

calculate the optimal consumption c0 and the associated value function W 0. In thenext step we use a boundary update procedure that uses O0 and W 0 to obtain a newregion of inaction O1 � ðf1

s ;f1bÞ and thus a new transaction policy. We will repeat

the procedure to get a sequence of regions of inaction O0;O1;O2; . . ., correspondingconsumptions c0; c1; c2; . . . and corresponding value functions W 0;W 1;W 2; . . .. Inessence, the procedure transforms the free-boundary problem (15) into a sequence offixed-boundary problems which are easier to solve numerically. Step 1 solves thefixed-boundary problem while step 2 creates the sequence of regions of inaction.

We assume that the arbitrarily chosen O0 is large enough so that the optimalregion of inaction, O�, is a subset of O0. However, if O0 is not large enough weprovide a condition (Eq. (44)) that will fail, in which case one can restart the iterationwith a larger O0.

4.1. Step 1

In step1, given On we seek cn and W n. cn and W n solve

LW þmaxcðuðcÞ � cW 0Þ ¼ 0 in On (31)

with boundary conditions

SW ¼ 0 at fs (32)

and

BW ¼ 0 at fb. (33)

Again, for readability we will suppress iteration number n in the subscript when theindex is obvious. The maximum is achieved in (31) by c ¼ ðW 0Þ

�1 (log utility) orc ¼ ðW 0Þ

1=ðg�1Þ (power utility). This makes (31) a non-linear elliptic ODE problem.We will use an iterative scheme to solve (31)–(33) as follows. Start with a guess valuefor consumption. Given a consumption cn;m

LW þ ðuðcn;mÞ � cn;mW 0Þ ¼ 0 in On (34)

is a linear elliptic equation and can be solved along with (32) and (33) to obtainW n;m, where W n;m would be the value function given a transaction policy On and aconsumption cn;m.

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Once we solve the linear problem we update our consumption with the first ordermaximization condition in Eq. (31), that is,

cn;mþ1 ¼½W 0

n;m��1 for log utility and

½W 0n;m�

1=ðg�1Þ for power utility.

8<: (35)

As is evident, in the representation cn;m, n represents the iteration index of theboundary update sequence, while m represents the index of the consumptioniteration. Fig. 2 shows the iterative procedure. A good guess for the initialconsumption cn;0 would be the converged consumption in iteration n� 1 and forn ¼ 1 a good guess is the Merton consumption fraction, that is, the optimalconsumption fraction when no transaction costs are present

cn;0 ¼cmp if n ¼ 1;

cn�1 if n41:

((36)

The Merton consumption cmp is given by,

cmp ¼

yð1þ xÞ for log utility and

1þ x

1� gy� gr�

gða� rÞ2

2s2ð1� gÞ

� �for power utility:

8><>: (37)

Fig. 2. Summary of iteration procedure.

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We terminate the iterative procedure when

supy2Onjcn;mþ1ðyÞ � cn;mðyÞjo�

for some tolerance parameter �. In order to implement this procedure we need tosolve (34) along with (32) and (33) which constitutes a linear elliptic ODE. Solvingsuch linear elliptic ODEs are quiet straightforward and can be done using variousnumerical methods including the finite difference method. We solve this using thefinite element method (FEM) particularly due to its ability to tract with ease regionsthat are not of any regular shape. Moreover, FEMs have very good stability andaccuracy. For a general introduction to FEM, we direct the readers to Oden andReddy (1978) and Hughes (1987).

4.2. Step 2: the boundary update procedure

We need to establish a boundary update procedure that can transform the free-boundary problem into a sequence of fixed-boundary problems. Specifically, we seekthe new region of inaction Onþ1 given On and W n calculated using step 1. First, recallfrom the earlier arguments that,

LW n � ðlogðW0nÞ þ 1Þ ¼ 0 in On (38)

with boundary conditions

SW n �W 0n � js ¼ 0 (39)

and

BW n � �W 0n þ jb ¼ 0. (40)

We also define an operator Q for notational convenience. The operator Q maps theset of real-valued functions onto the same set. For z : R! R

QzðxÞ ¼1

b3x2ðuðzðxÞ1=ðg�1ÞÞ � b2xzðxÞÞ. (41)

With g ¼ 0 for log utility.If we can create a boundary update sequence that could give us an Onþ1 fromfOn;W ng such that W nþ14W n and also the assurance that the sequence of Osconverge, then we have effectively converted the free-boundary problem into aconverging sequence of fixed-boundary problems. Such an update procedure isdescribed by the following equations for fnþ1

s and fnþ1b ,

fnþ1s ¼ inffx4fn

s j ðSW nÞ0¼ Qjs � QW 0

ng (42)

and

fnþ1b ¼ supfxofn

b j ðBW nÞ0¼ Qjb � QW 0

ng. (43)

This is equivalent to moving the boundary fns ðf

nbÞ towards the interior to the first

point where the gradient of SW n (BW n) is equal to Qjs � QW 0n (Qjb � QW 0

n).

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Notice that the boundary update procedure shown above moves the boundaries ina monotonic fashion. Hence the generated sequence of O’s are nested, that is,Onþ1 � On. This makes it obvious that for the boundary update procedure to workwe require that our initial guess, O0, contains the optimal no-transactions region O�.For any given On and W n, the following condition assures that O� � On:

SW 0jf0soSW 0jf0

sþt

and

BW 0jf0boBW 0jf0

b�t, (44)

for some d40 and all t 2 ð0; dÞ. The above conditions simply say that it is necessarythat the derivative of SW n (BW n) is positive (negative). Therefore, if O0 and W 0

satisfy the above condition, it guarantees that the arbitrarily chosen O0 was largeenough. If either of the above conditions fail, then it indicates that the arbitrarilychosen O0 was not large enough. A restart of the procedure with a larger O0 isrequired. A good way to choose a larger O0 in such cases is to move each boundaryhalf way between the old position and the boundary of the solvency region and check(44) again. In the next section we will show that the movement of boundariesdictated by (42)–(43) is well defined, provided (44) is satisfied.

We first provide a heuristic justification for the update procedure. For a given On,consider the W n that is the solution to Eqs. (38)–(40). If this W n where to satisfy,

max LW n þmaxcðuðcÞ � cW 0

nÞ;SW n;BW n

� �¼ 0, (45)

that is, Eq. (15), then the corresponding On � O�. But in general an arbitrarilypicked On is not the optimal region of inaction, hence Eq. (45) will be violated. Ifconditions (44) hold, then the violation of equation will be in the interior of On asshown by the gray region in Fig. 3. Moving each boundary to any point in theviolation region that is adjacent to it, is bound to yield an improvement in the valuefunction. This suggests that the update procedure of (42)–(43) is a policyimprovement procedure.

We still need to decide how far we need to move into the violation regions. Aconsideration for where we wish to move is the desire for a procedure that yields anested sequence of regions of inaction, that is, Onþ1 � On. If we can choose such anupdate procedure, this would give us tremendous computational advantage becausewe no longer need to calculate W n in the entire state space and can restrict our

Fig. 3. The boundary update.

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attention to calculating it only within the region of inaction On. Such a point isprovided by the update (42)–(43). This choice of boundary update is motivated bythe so-called principle of smooth pasting and results in nested sequence of regions ofinaction in closely related problems as will be shown in Theorem 1.

The stopping criterion for terminating the boundary iterations can be set in twoways—either in terms of the convergence of the region of inaction, that is,

maxðfns � fn�1

s ;fn�1b � fn

bÞo�f (46)

or the convergence of the value functionZ fnþ1b

fnþ1s

½W nþ1ðxÞ �W nðxÞ�dxo�W . (47)

Implementation wise, testing the convergence of the boundaries is easier, so ourimplementation determines convergence in terms of convergence of the boundaries.

5. Convergence of the update procedure

In this section we prove the convergence of the computational scheme. We will onlyprovide the treatment for the log utility case. Most of the analogous results for thepower utility can be arrived with similar reasoning. Lemma 1 shows that the region ofinaction will entirely lie in the positive quadrant, assuming as in Davis and Norman(1990) that the Merton line lies in the positive quadrant (that is, ða� rÞ=s2o1). This isproved by showing that the solution, f , to Eq. (50) on any O ¼ ðfs;fbÞ with fs ¼ 0,would have ðSf Þ040. Which, in the light of condition (44), that is justified in Theorem1, implies that policy improvement can be achieved by increasing fs.

Theorem 1 proves that by using the update procedure (42)–(43), each step of thecomputational scheme (a) is well defined (b) results in policy improvement (c) yieldsa nested sequence of regions of inaction and (d) converges to the optimal valuefunction. This is true, provided the initial guess for the region of inaction containedthe optimal region of inaction.

The innovation that the proposed scheme brings, lies in the boundary updatesequence in step 2. The particular method suggested in the previous section to solvethe non-linear fixed-boundary problem (step 1) is simply one possible methodamongst many that are available. Hence we do not provide any theoretical claim tothe convergence of the iteration in step 1. We simply require that step 1 solves Eq.(31) with (32)–(33). Our focus in this section will be the theoretical treatment of theboundary update sequence described in step 2.

Step 2 of the scheme, for log utility, strives to solve

max b3x2W 00 þ b2xW 0 þ b1W þ b4 � 1� logðW 0Þ;W 0 �

1

yð1þ x� mÞ;

�

�W 0 þ1

yð1þ xþ lÞ

�¼ 0. ð48Þ

Eq. (48) is simply obtained by substituting for optimal c in (15).

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Lemma 1. Let W 2 C2ðfs;fbÞ be the solution to,

SW �W 0 �1

yð1þ x� mÞ¼ 0 in ðm� 1;fsÞ, (49)

b3x2W 00 þ b2xW 0 þ b1W þ b4 � 1� logðW 0Þ ¼ 0 in ðfs;fbÞ � O, (50)

BW � �W 0 þ1

yð1þ xþ lÞ¼ 0 in ðfb;1Þ, (51)

with boundary conditions given by

W ðxþÞ ¼W ðx�Þ for x ¼ fs;fb (52)

and

W 0ðxþÞ ¼W 0ðx�Þ for x ¼ fs;fb. (53)

If fs ¼ 0 and ða� rÞ=s2o1, then

ðSW Þ0jfsþ40.

Proof. Evaluating (50) at t40 and at 0þ, taking the difference and dividing by t, wehave

b3tW 00ðtÞ þ b2W 0ðtÞ þb1tðW ðtÞ �W ð0þÞÞ �

1

tðlogW 0ðtÞ � logW 0ð0þÞÞ ¼ 0.

Now taking the limit as t! 0þ ¼ fsþ,

ðb2 þ b1ÞW0ðfsþÞ �

W 00ðfsþÞ

W 0ðfsþÞ¼ 0.

Now since b240 and W ðfs þ Þ0a0,

W 00ðfsþÞ ¼ ðb2 þ b1Þ W 0ðfs þ Þ2

¼ ðb2 þ b1Þ1

y2ð1� mÞ2

4b11

y2ð1� mÞ2

¼�1

yð1� mÞ2.

Which implies that

W 00 þ1

yð1þ x� mÞ2

� ��fsþ

¼ ðSW Þ0jfsþ40: &

Theorem 1 considers a function f n that solves the HJB equation (50) in On.Provided conditions (54) and (55) hold, it shows that Onþ1 � ðfnþ1

s ;fnþ1b Þ is well

defined and a subset of On. If f nþ1 is the solution to the HJB equation (50) on Onþ1,Theorem 1 also shows that f nþ14f n. Further it shows that conditions (54) and (55)hold for f nþ1 as well. This makes it clear that further policy improvement is possible

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by stepping in, that is, O� � Onþ1. Eqs. (56) and (57) are the simplified forms of Eqs.(42) and (43), for the log utility case. Conditions (54) and (55) are the same ascondition (44).

Theorem 1. Assume that the Merton point is less than 1, that is, ða� rÞ=s2o1. Say,f n 2 C2ðOnÞ solves (49)–(53) in On � ðfn

s ;fnbÞ and also that

ðSf nÞ0jfn

sþ40 (54)

and

ðBf nÞ0jfn

b�o0. (55)

If fnþ1s and fnþ1

b are defined by,

fnþ1s ¼ inf x4fn

s j ðSf nÞ0¼

1

b3x2log

f 0njs� b2xðjs � f 0nÞ

� �� (56)

and

fnþ1b ¼ sup xofn

b j ðBf nÞ0¼

1

b3x2log

f 0njb� b2xðjb � f 0nÞ

� �� , (57)

then

1.
fnþ1s ;fnþ1
b exist,
2. fnþ1
s ofnþ1b and Onþ1 � On,

3.
f nþ14f n in the solvency region ðm� 1;1Þ, 4. ðSf nþ1Þ
0jfnþ1

s þ40 and ðBf nþ1Þ

0jfnþ1

b�o0,

n � n �
5. fs ! fs and fb! fb.
Say f solves (49)–(53) in O� � ðf�s ;f�bÞ, then

6.
f 2 C2ðm� 1;1Þ 7. maxðLf þmaxc uðcÞ � cf 0 ;Sf ;Bf Þ ¼ 0 in ðm� 1;1Þ, 8. f ¼ V , the optimal value function defined by Eq. (10), (Davis and Norman, 1990).
Proof. 1. Consider Sf n at fnb. Since lþ m40 and both lX0 and mX0, we have

Sf njfnb¼ f 0n �

1

yð1þ x� mÞ

��fnb

¼1

yð1þ fnb þ lÞ

�1

yð1þ fnb � mÞ

o0.

Now, Sf njfns¼ 0, Sf njfn

bo0 and ðSf nÞ

0jfn

sþ40. Therefore, there exists at least a x0

such that ðSf nÞ0jx0¼ 0 with Sf njx0

40. Let xs represent the infimum of such x0’s.We have

ðSf nÞ0jfn

sþ40 and ðSf nÞ

0jxs¼ 0. (58)

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Consider

1

b3x2log

f 0njs

� �� b2xðjs � f 0nÞ

� �.

From the boundary condition at fns and because Sf njxs

40 we have

1

b3x2log

f 0njs


� ��fns

¼ 0 and1

b3x2log

f 0njs


� ��xs

40.

(59)

Due to the assumed smoothness of f n in On, (58) and (59) imply that ðSf nÞ0 and

1

b3x2log

f 0njs


� �,

intersect in ðfns ;xsÞ at least once. fnþ1

s being defined as the infimum of suchintersections does exist. Similarly, fnþ1

b 2 ðxb;fnbÞ can be argued to exist. Here, xb

denotes the supremum of all such points such that ðBf nÞ0jxb¼ 0 with Bf njxb

o0.2. From the definitions of Sf n and Bf n in On

ðSf nÞ0þ ðBf nÞ

0¼ f 00n �

�1

yð1þ x� mÞ2� f 00n þ

�1

yð1þ xþ lÞ2

¼1

yð1þ x� mÞ2�

1

yð1þ xþ lÞ2

40.

Since ðBf nÞ0o0 in ðxb;f

nbÞ, ðSf nÞ

040 in ½xb;fnbÞ. Which implies that xsoxb. We

therefore have, fnsofnþ1

s oxsoxbofnþ1b ofn

b.3. Define gðxÞ in Onþ1 as f nþ1ðxÞ � f nðxÞ. At fnþ1

s , Sf n40 and Sf nþ1 ¼ 0. Whichimplies that

g0ðfnþ1s Þ ¼Sf nþ1jfnþ1

s�Sf n

��fnþ1so0.

Similarly, by evaluating Bf nþ1 �Bf n at fnþ1b , we get g040 at fnþ1

b .Therefore, gðxÞ solves

b3x2g00 þ b2xg0 þ b1g� log

f 0nþ1f 0n

� �¼ 0 in Onþ1 � ðfnþ1

s ;fnþ1b Þ (60)

with boundary conditions g0 ¼ �K s and g0 ¼ Kb at fnþ1s and fnþ1

b , respectively, forsome constants K s40 and Kb40.

Since, g0ðfnþ1s Þo0 and g0ðfnþ1

b Þ40, the minima of g are achieved in the interior ofOnþ1. There exists at least a global minima, say at x, such that g0ðxÞ ¼ 0; g00ðxÞ40.Also logðf 0nþ1ðxÞ=f 0nðxÞÞ ¼ 0 since g0ðxÞ ¼ 0. Therefore, from Eq. (60)

b3x2g00ðxÞ ¼ �b1gðxÞ.

Since, b340 and �b140, g00ðxÞ40 implies gðxÞ40. Therefore,

gðxÞXgðxÞ40 8x 2 Onþ1. (61)

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Next consider the interval ðfns ;f

nþ1s Þ. Sf nþ1 ¼ 0 and Sf n40 in ðfn

s ;fnþ1s Þ. Which

imply that

g0 ¼Sf nþ1 �Sf no0 in ðfns ;f

nþ1s Þ. (62)

Also, since Sf n ¼Sf nþ1 ¼ 0 in ðm� 1;fns Þ we have

g0 ¼ 0 in ðm� 1;fns Þ. (63)

Now from (62) and (63) since g0p0 in ðm� 1;fnþ1s Þ and g40 at fnþ1

s we have thatg40 in ðm� 1;fnþ1

s Þ. Similarly, one can show that g40 in ðfnþ1b ;1Þ.

4. Rearranging Eq. (60) yields

b3x2g00 þ b2xg0 � log

f 0nþ1f 0n

� �¼ �b1g in Onþ1. (64)

Since b1o0 and gðfnþ1s Þ40, the RHS of (64) is positive at fnþ1

s , which implies that

b3x2g00 þ b2xg0 � log

f 0nþ1f 0n

� ��fnþ1s þ

40. (65)

From the definition of fnþ1s , at ðfnþ1

s þÞ, b3x2ðSf nÞ0¼ logðf 0n=jsÞ � b2xðjs � f 0nÞ. By

rearranging we have at ðfnþ1s þÞ,

logðf 0nÞ ¼ logðjsÞ þ b2xðjs � f 0nÞ þ b3x2ðSf nÞ

0

¼ logðjsÞ þ b2xðjs � f 0nÞ þ b3x2 f 00n þ

1

yð1þ x� mÞ2

� �.

Substituting the above in (65),

b3x2g00 � logðf 0nþ1Þ þ logðjsÞ þ b2xðf0nþ1 þ js � 2f 0nÞ

þb3x2 f 00n þ

1

yð1þ x� mÞ2

� ��fnþ1s þ

40.

Using the fact that f 0nþ1 ¼ jsof 0n at fnþ1s yields

0ob3x2 g00 � f 00n þ

1

yð1þ x� mÞ2

� ��fnþ1s þ

¼ b3x2 f 00nþ1 þ

1

yð1þ x� mÞ2

� ��fnþ1s þ

¼ ðSf nþ1Þ0jfnþ1

s þ.

Using similar arguments it can also be shown that ðBf nþ1Þ0jfnþ1

b�o0.

5. fns is clearly an increasing sequence bounded above, hence converges. fn

b being adecreasing sequence bounded below, also converges. We define f�s and f�b as thelimits of the sequences fn

s and fnb, respectively.

6. Since by definition, f is C2 in ðm� 1;f�s Þ; ðf�s ;f�bÞ and ðf

�b;1Þ, to show that

f 2 C2ðm� 1;1Þ, it is sufficient to show that

f 00ðf�s�Þ ¼ f 00ðf�sþÞ (66)

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and

f 00ðf�b�Þ ¼ f 00ðf�bþÞ. (67)

Since fns ! f�s , that is, no further improvement of fs is possible using (56),

ðSf Þ0jf�sþ ¼1

b3x2log

f 0

js

� �� b2ðjs � f 0Þ

� ��f�sþ

.

Boundary conditions yield that f 0ðf�sþÞ ¼ js, hence,

ðSf Þ0jf�sþ ¼ 0, (68)

that is,

f 00ðf�sþÞ ¼1

yð1þ f�s � mÞ2. (69)

Differentiating Sf ¼ 0 and rearranging to evaluate f 00ðf�s�Þ gives

f 00ðf�s�Þ ¼1

yð1þ f�s � mÞ2. (70)

Hence f is twice differentiable at f�s . Similar arguments show (67), that is, f is C2 atf�b.

7. We divide the proof into three parts. (a) Shows that Sfp0 in ðf�b;1Þ, (b)shows that Sfp0 in ðf�s ;f

�bÞ and (c) shows that Lf þ ðmaxcðuðcÞ � cf 0ÞÞp0 in

ðm� 1;f�s Þ. To complete the proof, we also need to show that ða0Þ Bfp0 inðm� 1;f�s Þ, ðb

0Þ Bfp0 in ðf�s ;f

�bÞ and ðc

0Þ Lf þ ðmaxcðuðcÞ � cf 0ÞÞp0 in ðf�b;1Þ.Since proof of ða0Þ; ðb0Þ and ðc0Þ follow the exactly the same arguments as (a), (b) and(c), respectively, we only show (a), (b) and (c) below.

(a) In ðf�b;1Þ Bf ¼ 0, that is,

f 0 ¼1

yð1þ xþ lÞ.

Therefore, in ðf�b;1Þ

Sf ¼ f 0 �1

yð1þ x� mÞ

¼1

yð1þ xþ lÞ�

1

yð1þ x� mÞo0.

(b) Say for some x 2 ðf�s ;f�bÞ, Sf jx40. We have Sf jf�s ¼ 0 and Sf jf�bo0 with

Sf jx40, therefore, there exists a x0 such that Sf jx040 and ðSf Þ0jx0

¼ 0. Using thesame arguments as in the Proof of 1.1 we can see that there exists another fnþ1

s asdefined in (56). This contradicts f�s being the converged value. Hence, Sf jxp0 forall x 2 ðf�s ;f

�bÞ.

(c) Let K �Lf þ ðmaxcðuðcÞ � cf 0ÞÞ. Note that, (50) can be written in O� as

Kf ¼ 0.

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Smoothness of f implies that

Kf jf�s ¼ 0.

From (49), at any x 2 ðm� 1;f�s Þ we have

f 0 ¼1

yð1þ x� mÞ(71)

and

f 00 ¼�1

yð1þ x� mÞ2. (72)

Eqs. (71) and (72) also hold at f�s since at f�s , Sf ¼ ðSf Þ0 ¼ 0.Substituting the values of f 0 and f 00 in ðKf

��x� Kf

��f�sÞ, we have

Kf jx �Kf jf�s ¼b3y�

x2

ð1þ x� mÞ2þ

f�sð1þ f�s � mÞ2

!

þb2y

x

ð1þ x� mÞ�

f�sð1þ f�s � mÞ

� �

þ b1ðf ðxÞ � f ðf�s ÞÞ � log1þ f�s � m1þ x� m

� �� . ð73Þ

Solving for f in ðm� 1;f�s Þ gives

f ðxÞ ¼ f ðf�s Þ �1

ylog

1þ f�s � m1þ x� m

� �.

Therefore,

�yðf ðxÞ � f ðf�s ÞÞ ¼ log1þ f�s � m1þ x� m

� �.

This makes Eq. (73)

Kf jx �Kf jf�s ¼b3y�

x2

ð1þ x� mÞ2þ

f�sð1þ f�s � mÞ2

!

þb2y

x

ð1þ x� mÞ�

f�sð1þ f�s � mÞ

� �. ð74Þ

From Theorem 11.2 of Shreve and Soner (1994)

f�spð1� mÞs2

a� r� 1

� �. (75)

Rearranging the above yeilds,

2b3f�s

1þ f�s � m� b2p0. (76)

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Since x=ð1þ x� mÞ is increasing in x and xof�s

b3x

1þ x� � mþ

f�s1þ f�s � m

� �� b2o2b3

f�s1þ f�s � m

� �� b2p0. (77)

Multiplying by the positive quantity, ðf�s=ð1þ f�s � mÞ � x=ð1þ x� � mÞÞ andsubstituting in (74) gives

Kf jx �Kf jf�so0.

Since Kf jf�s ¼ 0 we have for x 2 ðm� 1;1Þ

Kf jxo0.

8. The verification theorem which shows that a smooth (C2ðm� 1;1Þ) solution tothe HJB equation is the optimal value function V defined by (10), can be found inDavis and Norman (1990). &

6. Results

The primary objective of this section is to compare and contrast the numericalresults obtained using our scheme with three well-known works in this area. First, weillustrate a typical boundary update sequence while using it to corroborate with theresults obtained by Davis and Norman (1990). Second, we compare our results toasymptotic results obtained by Janecek and Shreve (2004) and the we evaluate the‘simple policy’ approximation suggested by Constantinides (1986). Finally, we shedsome light on the convergence rates of the boundaries. During the discussions in thissection we seek to provide the readers with a greater intuition for the nature andstructure of the optimal policies.

6.1. Discussion 1: boundary iteration

As in Davis and Norman (1990), we choose the model parameters shown in Table1. We seek the optimal policy for a risk-averse investor who has the power utility ofconsumption (uðcÞ ¼ cg=g). We will use this case to illustrate the boundary update

Table 1

Table of parameters (Discussion 1)

Parameter Symbol Value

Stock’s expected rate of return a 12%

Volatility of stock s 0.4

Bank interest rate r 7%

Objective discounting rate y 10%

Risk aversion coefficient g �1

Transaction costs for buying l 5%

Transaction costs for selling m 5%

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Table 2

Intervals of inaction generated (Discussion 1)

Iteration Interval of inaction (% of wealth in stock)

0 [3.00, 50.00]

1 [3.22, 40.26]

2 [3.80, 29.12]

3 [5.25, 23.84]

4 [5.70, 22.23]

5 [5.77, 21.52]

6 [5.77, 21.52]


sequence. We begin with a transaction policy that buys/sells to maintain the fractionof wealth in stock between 3% and 50%. Table 2 shows the intervals of inactionobtained in the iteration sequence. When the boundary convergence tolerance value�f ¼ 10�2, convergence occurs in six iterations. The converged values are 5.77% and21.52%. Table 3 shows the value function obtained in each iteration.

We calculate the regions for various transaction levels and plot them as the solidline in Fig. 7. In the case of zero transaction costs, the optimal region is a point,known as the Merton point and is at 15.6% for the particular parameter choice. Thisplot agrees with the plot presented in Davis and Norman (1990, p. 706), Table 3.

6.2. Discussion 2: comparison with asymptotics

Janecek and Shreve (2004) provide an asymptotic expansion for the value functionaround the Merton point in powers of the transaction cost parameter l, for the one-stock case with power utility. In particular, they provide expansion in powers of l1=3.They assume that the transaction costs for both buying and selling are equal, that is,m ¼ l. In this subsection, we compare and contrast their results with those obtainedusing our computational scheme. Theorem 1 of Janecek and Shreve (2004) showsthat the value function at the Merton point can be expressed as

1

gcmp �

9

32ð1� gÞy4

mpð1� ympÞ4

� �1=3

cg�2mp s2l2=3 þOðlÞ, (78)

where ymp is the Merton point and cmp is the consumption fraction when notransaction costs are present. That is,

ymp ¼1

ð1� gÞa� r

s2, (79)

cmp ¼y� rg1� g

�1

2s2y2

mpg. (80)

We will denote the first two terms (other than the OðlÞ) of (78) by ~cmp andcmp will represent the value at the Merton point given by our solution. Thedifference between the two value functions at the Merton point is ce ¼ j

~cmp � cmpj.

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Table 3

Value functions for each boundary iteration. Illustrating policy improvement (Discussion 1)

g pm � pl sm � sl sl=sm sm=sm Y

�8.0 0.07636 5.8168 0.67094 1.1444 13.4508

�7.5 0.08195 6.1362 0.66951 1.1449 14.3926

�7.0 0.08853 6.5024 0.66669 1.1448 15.4921

�6.5 0.09594 6.8911 0.66485 1.1446 16.7120

�6.0 0.10405 7.2881 0.66228 1.1409 18.0234

�5.5 0.11473 7.7953 0.66004 1.1419 19.7256

�5.0 0.12734 8.3542 0.65572 1.1396 21.6871

�4.5 0.14412 9.0560 0.65112 1.1411 24.2270

�4.0 0.16355 9.7572 0.64647 1.1371 27.0118

�3.5 0.19081 10.6638 0.63865 1.1347 30.7179

�3.0 0.22886 11.7440 0.62969 1.1332 35.4366

�2.5 0.28360 12.9891 0.61669 1.1272 41.2906

�2.0 0.37498 14.5997 0.59649 1.1215 49.0691

�1.5 0.54712 16.5787 0.56355 1.1107 58.4276

�1.0 0.97829 19.0766 0.49658 1.0836 67.2705

-10 -9 -8 -7 -6 -5 -4-12

-11

-10

-9

-8

-7

-6

-5

-4

log(

ψe)

log(λ)

Fig. 4. ce vs l (Discussion 2).


If the accuracy of cmp was greater than or equal to OðlÞ then ce will be oforder OðlÞ. Fig. 4 plots the logðceÞ versus logðlÞ. A slope of k on the log – logplot indicates that ce ¼ Oðlk

Þ. The dashed line in Fig. 4 shows that logðceÞ vs logðlÞfor has slope close to 1. This confirms that c differs from the exact value by Oðlk

Þ

with kX1.

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-10 -9 -8 -7 -6 -5 -4-10

-9

-8

-7

-6

-5

-4

log(λ)

log(

ψe)

Fig. 5. pe vs l (Discussion 2).


The asymptotic expansion for the width of the interval of inaction is given byJanecek and Shreve (2004) as

12

1� gy2mpð1� ympÞ

2

� �1=3

l1=3 þOðl2=3Þ. (81)

Again denote the first term above by ~p, by p the width of interval of inactionobtained by our computational scheme and the difference by pe ¼ j ~p� pj. Fig. 5shows the plot of logðpeÞ vs logðlÞ. As can be seen the slope is 1, again agreeing withJanecek and Shreve (2004) and further suggesting that if further expanded thecoefficient of the l2=3 term would be zero.

When l is zero, that is, with no transaction costs, it is obvious that the valuefunction is maximized at and symmetric around Merton point. But for l40, it is notclear if the value function is maximized at and/or symmetric around the Mertonpoint. For the parameters listed in Table 1, Fig. 6 plots the value function obtainedby our computational scheme and the value function obtained by the asymptoticexpansion (Janecek and Shreve, 2004, Eq. (3.2)). We find that the value function isneither maximized nor is symmetric around the Merton point, which is an artifact ofthe form of asymptotic expansion. The intervals of inaction for various levels oftransaction costs are plotted in Fig. 7. Of course what the asymptotic expansionlooses in accuracy it more than makes up in simplicity and elegance.

6.3. Discussion 3: comparison with the approximation suggested by Constantinides

Constantinides (1986) considers the one-stock problem and restricts his search to aclass of ‘simple’ policies. These are policies for which consumption is proportional to

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0.05 0.1 0.15 0.2 0.25-134.5

-134.4

-134.3

-134.2

-134.1

-134

-133.9

Val

ue fu

nctio

n

y

Asymptotic

Computed

Merton point

Fig. 6. c vs y (Discussion 2).

0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

y

λ

AsymptoticComptational

Fig. 7. Intervals of inaction vs transaction costs (Discussion 1, 2).


wealth in bank, x, and the transaction policy is characterized by an interval ofinaction. The optimal transaction policy is indeed ‘simple’, but it is not clear whetherthe optimal consumption policy is ‘simple’ and if it is not, how good anapproximation it is. We consider the case that is frequently considered inConstantinides (1986), Table 5. Fig. 8 shows both c=x and c=ðxþ yÞ. Here,c is the optimal consumption. It is clear that the optimal consumption is not‘simple’, that is, consumption is not proportional to x. The c=ðxþ yÞ plot in Fig. 8suggests that consumption is proportional to the total wealth ðxþ yÞ. A closer look

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0.5 1 1.5 2 2.5 30.05

0.1

0.15

0.2

0.25

0.3

C/x and C/(x+y)

Money in Bank (with 1$ in stock)

C/xC/(x+y)

Fig. 8. c=x and c=ðxþ yÞ (Discussion 3).

0.5 1 1.5 2 2.5 30.104

0.1045

0.105

0.1055

0.106

0.1065

0.107

0.1075

0.108C/(x+y)

Money in Bank for 1$ in stock

C/(x+y)

Fig. 9. c=ðxþ yÞ (Discussion 3).


at c=ðxþ yÞ in Fig. 9 shows that c=ðxþ yÞ is not constant but is nevertheless a betterapproximation. Evidence of consumption proportional to wealth being a relativelygood approximation even in higher dimensional settings can be found in Muthura-man and Kumar (2006).

Constantinides (1986) also makes the observation that under ‘simple policies’ thewidth of the regions of inaction are insensitive to risk aversion and to the variance of

ARTICLE IN PRESS


the stock. The region of inaction being a cone in two dimensional space, thedefinition of width is not obvious. In Constantinides (1986) four different measuresfor width have been used. They are, the difference in fraction of wealth in stock atthe transaction boundaries (denoted by pm � pl), the difference in slopes of thetransaction boundaries (denoted by sm � sl) and the value of each of the slopes of thetransaction boundaries divided by the slope of the Merton line (denoted bysl=sm; sm=sm). For various g’s and s’s, Table 4 shows these measures of width. Theyalso show Y, the value of the angle subtended by the cone, which we consider as theprimary measure of the width of the cone. Under any of these measures, the size ofthe region of inaction is sensitive to both risk aversion and variance when therestrictions to ‘simple policies’ are relaxed. The primary result of Constantinides(1986), that transaction costs have only a second order effect on liquidity premium,is robust to the assumption of simple consumption policies, as clearly demonstratedin the appendix of Constantinides (1986) (Tables 4 andTables 5).

Table 4

Measures of width for various g and s (Discussion 2)

s pm � pl sm � sl sl=sm sm=sm Y

0.20 0.98702 19.1794 0.49685 1.0891 13.4508

0.25 0.39139 15.7043 0.59261 1.1797 14.3926

0.30 0.24432 13.3933 0.60835 1.2436 15.4921

0.35 0.17630 11.4816 0.60117 1.2887 16.7120

0.40 0.13649 9.8894 0.58620 1.3233 18.0234

0.45 0.11028 8.5757 0.56945 1.3524 19.7256

0.50 0.09247 7.5534 0.55061 1.3828 21.6871

0.55 0.07961 6.7392 0.53063 1.4143 24.2270

0.60 0.06927 6.0257 0.51125 1.4396 27.0118

0.65 0.06030 5.3559 0.49882 1.4576 30.7179

0.70 0.05448 4.9194 0.47409 1.4875 35.4366

0.75 0.04916 4.4980 0.45602 1.5131 41.2906

0.80 0.04447 4.1135 0.43878 1.5329 49.0691

0.85 0.04019 3.7525 0.42096 1.5424 58.4276

0.90 0.03719 3.4972 0.40523 1.5732 67.2705

Table 5

Table of parameters (Discussion 3)

Parameter Symbol Value

Stock’s expected rate of return a 15%

Volatility of stock s 0.2

Bank interest rate r 10%

Objective discounting rate y 10%

Risk aversion coefficient g �1

Transaction costs for buying l 5%

Transaction costs for selling m 5%

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6.4. Discussion 4: boundary iteration convergence rates

In this subsection we seek to shed some light on how the rates of boundaryconvergence are affected by changes in transaction costs and risk-aversioncoefficients. Apart from providing insight into convergence rates, the various

0 1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

80

90

100

Iteration (n)

% o

f Ω0

No. of iterations w.r.t Transaction costs

0.5%

1%2%

5%

10%

Fig. 10. Boundary convergence w.r.t. transaction levels (Discussion 4).

0 1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

70

80

90

100

Iteration (n)

% o

f Ω0

No. of iterations w.r.t Gamma

0

-0.2

-0.5-0.7

-1-1.2

-1.5

Fig. 11. Boundary convergence w.r.t. g (Discussion 4).

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parameter instances considered in this subsection provide evidence of robustness ofthe computational scheme. The stopping criteria used in our implementation is thatthe maxðfn

s � fn�1s ;fn�1

b � fnbÞ be less than the tolerance value �f set to 10�4. Each

boundary iteration is begun with the initial boundary O0 ¼ ð3; 50Þ. We keep theparameter set in Table 1 as the base case while first varying the transaction costs andthen varying the risk-aversion parameter g.

For various transaction costs (l ¼ m ¼ 0:5%; 1%; 2%; 5% and 10%), Fig. 10 plotsthe size of the subsequent On’s (as a percentage of O0, that is 100On=O0) vs iterationnumber. Note that smaller transaction cost levels indicate smaller regions ofinaction. Hence it is natural that more iterations are required for the convergence ofthe boundaries when transaction costs become small. But it is interesting to note thatthe number of required iterations does not significantly increase even for substantialchanges in transaction levels. Fig. 11 similarly plots the size of On as a percentage ofO0 for various values of the risk-aversion parameter g ¼ �1:5;�1:2;�1;�0:7;�0:5;�0:2 and 0. The case g ¼ 0 corresponds to log utility. Again one cannotice that iteration numbers are not that sensitive to changes in the risk-aversioncoefficient.

7. Conclusions

We have described an efficient computational scheme to solve the portfoliooptimization problem with proportional transaction costs. We have shown that themethod converges to the optimal value function and that the converged solution issmooth. We have also compared the results obtained by our method to the resultsobtained by Davis and Norman (1990), asymptotic expansions obtained by Janecekand Shreve (2004) and approximations suggested by Constantinides (1986).

Free-boundary problems are very common in finance, economics, queueingnetwork analysis and other problems which are formulated as stochastic controlproblems that have singular controls (like transaction costs). The spirit of theboundary update sequence lies in solving free-boundary problems by transformingthem to a sequence of fixed-boundary problems. We believe that this is a powerfulidea and as demonstrated by results in Muthuraman and Kumar (2006), can beeasily extended to solving multi-dimensional free boundary problems. We do hopethat researchers would be able to modify the update conditions easily to solve otherproblems as well. Convergence proof for the scheme in multiple dimensions is quitechallenging and is the subject of future work.

Acknowledgments

I am indebted to Sunil Kumar for his guidance. I would also like to thankDarrell Duffie, Kenneth Judd, Steven Shreve, Karel Janecek and an anonymousreferee for their comments, discussions and feedback. All errors are myresponsibility.

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a computational scheme for optimal investment – consumption with proportional transaction costs

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