Multi-asset portfolio selection problem with transaction costs

~ i MATHEMATICS AND COMPUTERS N SIMULATION ELSEVIER

Mathematics and Computers in Simulation 38 (1995) 163-172

Multi-asset portfolio selection problem with transaction costs M a r i a n n e A ki a n a, Jose Luis Menal di b,1, A g n r s S u l e m a a INRIA, Domaine de Voluceau Rocquencourt, BPI05, 78153 Le Chesnay Cedex, France b Wayne State University, Dept. of Mathematics, Detroit, MI 48202, USA

Abstract This paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate r and n risky assets whose prices are log-normal diffusions. We suppose that transactions between the assets incur a cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption. Dynamic Programming leads to a Variational Inequality for the value function which is solved by using a numerical algorithm based on policies iterations and multigrid methods. Numerical results are displayed for n = 1 and n = 2. Keywords: Portfolio selection; Transaction costs; Viscosity solution; Variational inequality; Multigrid methods

1. Statement of the problem Consider an investor who has available one riskless bank account paying a fixed rate of interest r and n risky assets modeled by log-normal diffusions with expected rates of return ai > r and rates of return variation 0-2. The investor consumes at rate c(t) from the bank account. Any movement of money between the assets incurs a transaction cost proportional to the size of the transaction, paid from the bank account. The investor's objective is to maximize over an infinite horizon an expected discounted utility o f consumption. We refer, a m o n g others, to [13,4,6,17,16,18,7,9] and [8, chapter 8.7] for related w o r k s and references herein. Let us detail the m a t h e m a t i c a l formulation o f the problem. Let (s2, ,T', P ) be a fixed c o m p l e t e probability space and (~t),>~o a given filtration. We denote by So(t) (resp. s i ( t ) for i = 1 . . . . . n) the a m o u n t o f m o n e y in the bank account (resp. in the ith risky asset) at time t and refer as s ( t ) = (si(t))i=o ...... the investor position at time t. We suppose that the evolution equations o f the investor holdings are l Supported in part under NSF grant DMS-9101360. 0378-4754/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 7 5 4 ( 9 3 ) E0079-K

164

M. Akian et al./Mathematics and Computers in Simulation 38 (1995) 163-172

( - ( 1 + / l i ) d £ i ( t ) + (1 - I ~ i ) d J D l i ( t ) ) ,

dso(t) = ( r s o ( t ) - c ( t ) ) d t + ,

_

(1)

d s i ( t ) = a i s i ( t ) d t + ~risi(t)dWi(t) + d E i ( t ) - d.a/li(t),

i = 1 . . . . . n,

with initial values si(O-) = xi, i = 0 . . . . . n, where W i ( t ) , i = 1 . . . . . n, are independent Wiener processes, ~.i(t) and ./~i(t) represent cumulative purchase and sale of stock i on [0, t] respectively and s ( t - ) denotes the left hand limit of the process s at time t. The coefficients ki and /,; represent the proportional transaction costs. A policy for investment and consumption is a set ( c ( t ) , ( E l ( t ) , J~/li(t) )i=l....,,) of adapted processes such that t

• c(t,w)>.O,

fora.e. (t,o~),

/c(s,w)ds
o

• £ i ( t ) and ~/[i(t) are right-continuous, non-decreasing and £ i ( 0 - ) = .A,4i(O-) = O. We define the solvency region as: S = { x = (Xo, x~ . . . . . x , ) E ~"+', W ( x )

/> O}

where IV(x) = x0 + ~i"2-1min((1 - t ~ i ) x i , (1 + ,~i)xi) represents the net wealth, that is the amount of money in the bank account resulting from setting the risky assets to zero. Given an initial endowment x in S, a policy is admissible if the bankruptcy time ~ defined as = inf{t >~ 0, s ( t ) ~ S }

(2)

is infinite. We denote by/.4(x) the set of admissible policies. The investor's objective is to maximize over all policies 79 in g/(x) the discounted utility of consumption OO

(3)

Jx(79) = Ex / e - a t u ( c ( t ) )dr LI

o

where E~ denotes expectation given the initial endowment x, S is a positive discount factor and u ( c ) is a utility function defined by C'

u(c) =--, 3/

0<3,<

1.

(4)

We define the value function V as V(x) =

sup J~(79).

(5)

"PEU(x)

R e m a r k 1. When the process s ( t ) reaches the boundary 0S at time t, i.e. s ( t - ) E 3 $ , the only admissible policy is to jump immediately to the origin and remain there (see [ 16] ) with a null consumption. Consequently, if the initial endowment x is on the boundary, then V ( x ) = O. We make the assumptions (A.1)

r+2( 1-

T)

i=1

O'i

M. Ak&n et al./Mathematics and Computers in Simulation 38 (1995) 163-172

165

and (A.2)

0 ~
/~i ~ 0,

/~i +/xi > 0,

Vi = 1. . . . . n.

R e m a r k 2. When the transaction costs are equal to zero (Merton's problem), the value function V is finite iff assumption (A.1) is satisfied (see [6] for n = 1 and Section 5 below). Theorem 3. Under assumptions (A.1) and (A.2), (i) the value function V defined in (5) is y-Hrlder continuous and concave in ,_q and non-decreasing with respect to xi for i = 0 . . . . . n. (ii) V is the unique viscosity solution 2 o f the variational inequality (VI):

max{AV+u*(OV)

maxLiV, m a x M i V } = 0

OX 0 ' I
1<~i<~n

o

in S,

(6) (7)

v/as = o

where

1 S-" o-?x? O~V A V = -~ z__, , ' OXi 2 i=l

q-

OV

i =1

OV OV -OliXi-"~- rxo -OXo 69Xi

~g

(8)

OV

--

--7

(9)

L i V = --(1 q- /~i) Ox 0 q - O x i

c?V

OV

Mi V = ( 1 - ~L&i)OXo

Oxi,

(10)

and u* is the convex Legendre transform of u defined by u*(v)

C max(-cv+--) ~)o y

=

(

1 y

--

--

1)

.._L_ /)~,-I

(11)

.

The proof of this theorem is given in [2]. The solvency region S is divided as follows:

Bi={xE$,

Ly(x)

=0},

Si={xES,

My(x)=O},

NTi=S\(B;USi).

No transaction take place in N T = Ai"=~NTi. Outside NT, an instantaneous transaction brings the position to the boundary of NT: buy stock i in Bi, sell stock i in Si. After the initial transaction, the agent position remains in N T = {x E S, AV + u* (oV)Txo = 0 } , and further transactions occur only at the boundary (see [6]). We are concerned with the numerical solution of the variational inequality satisfied by the value function. An adequate change of variables is performed in Section 2 which reduces the dimension of the problem and simplifies the numerical study. Then, the variational inequality is discretized by finite difference schemes and solved by using an algorithm based on the "Howard algorithm" (policy iteration) and the multigrid method (Section 3). Numerical results for the value function and the optimal policy in the case of one bank account and one or two risky asset(s) are given in Section 4. 2 By viscosity solution of F(x, v, Dr, D2v)

= 0, we mean viscosity solution of - F ( ) = 0 in the sense of [5].

166

M. Akian et al./Mathematics and Computers in Simulation 38 (1995) 163-172

Finally, in Section 5, a brief theoretical study of the optimal strategy is done by using properties of the variational inequality; this analysis corroborates the numerical results.

2. Change of variables The value function V defined by (5) has the homothetic property [6] V p > O, V ( p x )

(12)

= pW(x).

Consequently, the (n + 1)-dimensional VI (6) satisfied by V can be reduced to a n-dimensional VI by considering the new state variables:

{

n (net wealth),

p = xo + ~_,( 1 - IXi)xi,

i=l (1

Yi-

-

(13)

].)~i)xi

P

(fraction of net wealth invested in stock i)

i = 1 . . . . . n,

,

and the new control variable C = c / p (fraction of net wealth dedicated to consumption). The function V ( x ) can be written as V(x)

= V(p(l - ~Yi), i=~

PY~

PY"

)

(1 - / x l ) . . . . . (1 - / x . )

(14)

= p~'W(y)

where the function W(y)

yl yn = V(1 -)_.~Yi, ) i=~ (1 - ~ l ) . . . . . (1 - #n)

(15)

is defined in g = { Y = ( Y l . . . . . y.) EIR", 1 -

i=l

Ai +/xi {y;}_ >/0} ]--/zi

with {y}- = m a x ( 0 , - y ) . The function W is bounded (see [2]) and satisfies max(,4W + u * ( B W ) ,

max LiW, max ~ l i W ) = 0

l <~i<~n

(16)

l <~i~n

o in $ with the boundary condition W = 0 on 0,~, where

Z

• - :- w +

j,k=l a:k oYJ Oy~

aw- w, bj j=~

(17)

tl

B W = T W - ~-.~yjOW j=l Oyj '

(18)

M. Akian et al./Mathematics and Computers in Simulation 38 (1995) 163-172

aw LiW = - -

A i .qt_ /'£i

(19)

aW

l~l i W . . . .

167

(20)

c~yi '

and

a jk -- --Y~ k ~-~ ~r2( ~3~i-

Y i ) ( (~ji -

Yi ) ,

(21)

i=1 n

(22)

bj = yj ~-~[ ('y - 1 )o'2yi + ol i - r] ( (~ij - - Yi), i=1

fl=B-

y ( r + ~_.[ ( o t i - r ) y i +

o-~y~]).

(23)

i=1

R e m a r k 4. Eq. (16) only depends on v = ( v i ) i = l , . . . , n with vi = (hi + ~i)//(1 - - ] , £ i ) , and so does the function W. Consequently, it is sufficient to compute the value function V when the transaction costs on sale/z~ are equal to zero. Using the properties of V and (16), we deduce that W is concave, non-negative and non-decreasing with respect to each coordinate Yi. In order to simplify the numerical computation, we restrict the admissible region S to n

S + = { X E A n+l , X 1 . . . . .

X n ~ 0 , X0 " ~ - Z ( I

--]LLi)X i ~ O)

i=l

that is, we suppose that the amounts of money allocated in the risky assets are non-negative, while the amount of money in the bank account can be negative as long as the net wealth remains non-negative. This is not restrictive since, when cei > r, the no-transaction cone is inside S + and a trajectory which starts in S ÷ remains in S÷(see [6] for n = 1). This leads to the study of the VI (16) in the domain (/~-)n. No boundary condition has to be specified since the VI degenerates on {Yi = 0}, i = 1. . . . . n. Note that the function W has bounded derivatives in (11~+)n. For numerical purpose, a second change of variables is performed in order to bring (I~÷) n to [0, 1] ", namely: ?1

~P(z) = ~'I(1 - z i ) W ( y ) ,

Y___.L__, with z i - 1 + Y i '

i= 1,...,n.

i=1

The function 0 is bounded and concave with respect to zi, i = 1. . . . . n, and has bounded derivatives.

3. Numerical methods We consider equations of the form:

168

M. Akian et al./Mathematics and Computers in Simulation 38 (1995) 163-172

max(APW+u(P)) =0

in ~ = (0,1) m,

W=O

on F

P E "Pad

(24)

where A e is a second order degenerated elliptic operator

APW(x)

+

bi(x,P)

(x) - B ( x , P ) W ( x )

i=1

i,j=l

with n!

~ ao( x,

P )Tli~Tj ~ O,

fl(x, P) >~O,

VX E f2 ~'I E R m, P E T~ad .

i,j=l

7~ad is a closed subset of N k and F is a part of the boundary 0£2, which consists of faces of the m-cube s2. In 0S2 \ F, the operator A p is degenerated for any P and no boundary condition is needed. Variational inequalities can be formulated in this form by using an additive discrete control which selects the equation which satisfies the maximum. The numerical study of Eq. (24) consists in two steps: first, discretize (24), and then solve the discrete equation by means of an iterative method.

3.1. Discretization Eq. (24) is discretized by using the finite difference method. Since A e is degenerated, first order derivatives ~ are approximated by one-sided difference approximation according to the sign of the drift term bi (x, P ) (see [ 12] ). Let h = 1/N (N E N*) denote the finite difference step in each coordinate direction. We obtain a system of Nh non-linear equations of Nh unknowns {Wh(x), x E ah):

max(Ae~Wh +u(P))(x) = 0 ,

p C79ad

"

Vx E f2h

(25)

where Nh = ~f2h "~ 1/h m. Because of the degeneracy of the operator A P at some points of the closed m-cube ~0 and the presence of mixed derivatives, Ahe does not satisfy the usual Discrete Maximum Principle (i.e. (APWh(X) <~ O, Vx E Oh) ~ ( W h ( x ) ) 0, Vx E Oh)). Consequently, Eq. (25) may not be stable, even for small step h. However A P can be written as the sum of a symmetric negative definite operator and an operator which satisfies the Discrete Maximum Principle; we thus infer the stability of A f which is confirmed by numerical experiments. Eq. (25) is solved by the (Full) Multigrid-Howard algorithm based on the "Howard algorithm" (policy iteration) and the multigrid method [ 1 ].

3.2. The (Full-)Multigrid-Howard algorithm The Howard algorithm [ 11,3] also named policy iteration consists in an iteration algorithm on the control and value functions (starting from p0 or IV°):

M. Akian et al./Mathematics and Computers in Simulation 38 (1995) 163-172

169

for n ~> 1

P" E Argmax(APW "-l + u ( P ) ) ,

(26)

for n >/0

W" is solution of A~"W + u(P") = 0.

(27)

When A~ satisfies the Discrete Maximum Principle, the sequence Wn decreases and converges to the solution of (25) and the convergence is in general superlinear (see [3,1]). The exact computation of step (27) is expensive in dimension m ~> 2 (the complexity of a direct method is O(N] -z/m)). We thus use the multigrid-Howard algorithm introduced in [ 1 ]: in (27), W" is computed by a multigrid method with initial value W n-1. The advantage is that each multigrid iteration takes a computing time of O(Nh) and contracts the error by a factor independent of the discretization step h. For a detailed description of the multigrid algorithm, see for example [ 14,10]. Let ,M P denote the operator of an iteration of the multigrid method associated to the equation APW ÷ u(P) = 0. Starting from W°, we proceed the following iteration: (26), for n/> 1

{ W"'0 = W "-I, for i = 1 to m. Wn = W~,m..

w.,i = / ~ p , , ( w n , i _ l )

'

(28)

This algorithm is converging to the solution W~ of (25) if W° is sufficiently close to W~ and m. is large enough (independently of the step h) [1]. In order to get rid of the constraint on W°, we introduce the FMGH algorithm which uses the idea of the Full-Multigrid method (FMG) (see [ 1] ). Let us consider the sequence of grids (S2k)k~>~ of steps hk = 2 -k and denote by Z~+~ the operator of the m-linear interpolation from grid s'2~ to grid s2k+l. Given an initial value W~ on the grid s2j, we perform, at each grid level k, h iterations of (28) and interpolate the resulting function W~. We then iterate this procedure with W~k+l = Z~+IW~ as initial value. Under appropriate assumptions (see [ 1 ] ), the error between W~ and the solution W; of (25) with h = hk is in the order of the discretization error, for any k. This property is realized for any initial value W~l, if the numbers mn and h are large enough (but independent of the level k). Consequently, this algorithm solves Eq. (25) (with an error in the order of the discretization error) with a computing time of O(Nh).

4. Numerical results Eq. (16) is solved in ( ~ + ) " for n = 1 and n = 2, and with y = 0.3, ~ = 10%, r = 7%. F o r n = 1, we set al = 11%, Ol = 3 0 % and A=A1 = 2 , 4 , 6 , 8 or 10%. The regions BI and Sl are of the form (see Section 5): BI = [0,7r-] and Sl = [rr+,+cxD) with 0 < 7r- < 7r+. The functions rr + and zr- are displayed in Fig. 1 as functions of A. The initial point zr+(0) = z r - ( 0 ) has been set to 7r]', where 7r~ is the optimal proportion of risky asset in the Merton problem (defined below). For n = 2, we set ce = (11%, 15%) , o-= (30%, 35%) and A = ( 2 % , 5 % ) . The value function W, and the optimal consumption C are displayed in Fig. 2. Then the partition of the domain is displayed in Fig. 3.

M. Akian et al./Mathematics and Computers in Simulation 38 (1995) 163-172

170 0.80 0.70 0.60 0.50 0.40 0.30 0.20

I

0.oo

0.05

0.1o

~+

Fig. 1. Graph of ~-+ and 7r- for n = 1, y = 0.3, 8 = 10%, r = 7%, al = 11%, oq = 30% and/~ = 0.

1 7 . 6 6 ~

0.097 0.096 0.095

0.0~4,~

Y

0.v93 ~ 1.5

1.5 1

0.5

l

1

0.5

Fig. 2. Value function W (left) and optimal consumption C (right) for y = 0.3, 8 = 10%, r = 7%, a = (11%, 15%), o" = (30%, 35%), A = (2%, 5%) and/z = (0,0). N o t e that, at first sight, the boundaries o f the regions Bi and Si seem to be straight lines o f e q u a t i o n Yi = constant. This w o u l d m e a n that the investment policies are decoupled, a l t h o u g h the d y n a m i c s are correlated. In fact, w h e n the cost for purchase A2 grows, the region NT2 g r o w s as expected but the b o u n d a r i e s o f S~ and B~ are also perturbed. Moreover, a variation o f a2 and 02 affect both NT2 and NT1. O t h e r n u m e r i c a l tests can be f o u n d in [ 2 ] .

5. Theoretical analysis of the optimal strategy 5.1. No transaction costs: the Merton problem W h e n A = / x = O, the o p t i m a l investment strategy is to keep a constant fraction o f total wealth in each risky asset. U n d e r a s s u m p t i o n ( A . 1 ) , the optimal proportion, d e n o t e d by 7r* is given b y

M. Akian et al./Mathematics and Computers in Simulation 38 (1995) 163-172 1.6

171

y2

1.4

I

B1 N $2

NT1

N S~

$1 fq $2

I

1.2

I

1.0

/ I

B1~ NT2

o,8

NT

~

Sl fq NT~

f) 0.6

.

......

,

0.4

o.2 0.0

B1 f3 B2 ""'~'"1'''" 0,0

0.2

NT1 fq B2

Sx f) B2

"]"'"''"l'~"~'"~['~'r~'T"l"""~"l~'~''"'l 0.,4

0.6

0.8

1.0

'''~f~''' 1.2

1,4

1.6 yl

Fig. 3. B o u n d a r i e s o f the regions Bi, Si a n d N ~ for Y = 0.3, B = 10%, r = 7%, a = ( 1 1 % , 1 5 % ) ,

o" = ( 3 0 % , 3 5 % ) ,

A = (2%,5%) and ~ = (0,0). O' i --

r

zr* - o.2(1 -- y)" The optimal fraction of wealth dedicated to consumption is

l-y

2(l-y)

i=l \

~r,

,,

j/

and the value function W is equal to C*(~-l) W -

-

-

Y The regions "sell i" and "buy i" are characterized by Bi = {y E (IR+) ", y; ~< ~[} and Si = {y c (IR+) ", Yi >~ Ir~[}. 5.2. A general shape o f the transaction regions

We can derive formally from VI (16) and the concavity of W, without numerical computation, the general shape of the transaction regions, given in Fig. 4. The region Si is of the form Si = {y C (~x+) n, Yi ~ 77"+(Y))

where 7ri+ is some mapping of ~ = (Yl . . . . . Yi-l, Yi+l . . . . . y,). Similarly, we get 1

Bi = {y E ( ~ + ) " , Yi ~< zr~-( 1 + Aiy/~)}" Suppose n = 2. The function W is constant with respect to Yl in Sl. Consequently the pieces of boundaries aB2 and 3S2 included in S~ are straight lines of equation Yz = constant. Similarly, the pieces of boundaries 3B2 and 0S2 included in B~ are straight lines of equation y2/(1 + A~y~) = constant. By symmetry, we get similar properties for the boundaries 3B~ and OS~ as displayed in Fig. 4.

172

M. Akian et aL /Mathematics and Computers in Simulation 38 (1995) 163-172 Y2

B1 0 $2

r~YT1 [~l $2

c~ B1 fiB2 /

1

VT1OB2 !

/

S1 N S2

,5"10 B2

Fig. 4. General shape of the transaction regions.

References [ 1] M. Akian, Analyse de l'algorithme multigrille FMGH de rrsolution d'rquations d'Hamilton-Jacobi-Bellman, Analysis and Optimization of Systems, Lecture Notes in Contr. and Inf. Sci. 144 (Springer, Berlin, 1990) 113-122. [2] M. Akian, J.L. Menaldi and A. Sulem, On an Investment-Consumption model with transaction costs, Rapport de Recherche Inria 1926 (1993). [ 3 ] R. Bellman, Introduction to the mathematical theory of control processes (Academic Press, New York, 1971). [4] G.M. Constantinides, Capital market equilibrium with transaction costs, J. Political Economy 94 (1986) 842-862. [5] M.G. Crandall, H. Ishii and EL. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992) 1-67. [6] M. Davis and A. Norman, Portfolio selection with transaction costs, Math. Oper. Res. 15 (1990) 676-713. [7] B. Fitzpatrick and W. Fleming, Numerical methods for an optimal investment-consumption model, Math. Oper. Res. 16 (1991) 823-841. [8] W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions (Springer, New York, 1993). [9] W.H. Fleming and T. Zariphopoulou, An optimal investment-consumption model with borrowing, Math. Oper. Res. 16 (1991) 802-822, [ 10] W. Hackbusch and U. Trottenberg, eds., Multigfid Methods, Lecture Notes in Mathematics 960 (Springer, Berlin, 1981 ). [ 11] R.A. Howard, Dynamic Programming and Markov Process (M1T Press, Cambridge, MA, 1960). [12] H.J. Kushner and P.G. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time (Springer, Berlin, 1993). [13] M.J.E Magill and G.M. Constantinides, Portfolio selection with transaction costs, J. Econom. Theory 13 (1976) 245-263. [ 14] S.E McCormick, ed., Multigrid methods, SIAM Frontiers Appl. Math. 5 (1987). [15] R.C. Merton, Optimum consumption and portfolio rules in a continuous time model. J. Econom. Theory 3 (1971) 373-413. [ 16] S.E. Shreve, H.M. Soner and V. Xu, Optimal investment and consumption with two bonds and transaction costs, Math. Finance 1 (1991) 53-84. [ 17] M. Taksar, M.J. Klass and D. Assaf, A diffusion model for optimal portfolio selection in the presence of Brokerage fees, Math. Oper. Res. 13 (1988) 277-294. [ 18 ] T. Zariphopoulou, Investment-consumption model with transaction fees and Markov chain parameters, SIAM J. Control Optim. 30 (1992) 613-636.