Optimal Mean-Variance Investment Strategy Under

8/13/2019 Optimal Mean-Variance Investment Strategy Under

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arXiv:1011.4991v1[q-fin.PM]23Nov2010

Optimal mean-variance investment strategy under

value-at-risk constraints

Jun Yea,, Tiantian Lia,1

aDepartment of Mathematical Sciences, Tsinghua University Beijing, 100084, China.

Abstract

This paper is devoted to study the effects arising from imposing a value-

at-risk (VaR) constraint in mean-variance portfolio selection problem for aninvestor who receives a stochastic cash flow which he/she must then investin a continuous-time financial market. For simplicity, we assume that thereis only one investment opportunity available for the investor, a risky stock.Using techniques of stochastic linear-quadratic (LQ) control, the optimalmean-variance investment strategy with and without VaR constraint are de-rived explicitly in closed forms, based on solution of corresponding Hamilton-Jacobi-Bellman (HJB) equation. Furthermore, some numerical examples areproposed to show how the addition of the VaR constraint affects the optimalstrategy.

Keywords: Value-at-risk, Mean-variance portfolio,Hamilton-Jacobi-Bellman equation, Optimal investment strategy.2000 MSC: C02, C61, IM01.

1. Introduction

The mean-variance model ofMarkowitz(1952,1959) is a cornerstone ofmodern portfolio theory. The most important contribution of this model isthat it enables an investor to optimally select mean-variance efficient port-folios for seeking the highest return after specifying his acceptable risk level.

Corresponding author. Tel.:+86-10-62788974.Email addresses: [email protected](Jun Ye),

[email protected](Tiantian Li )1Tel.:+86-10-66931968

Preprint submitted to Insurance: Mathematics and Economics November 24, 2010
http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1http://arxiv.org/abs/1011.4991v1


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SinceMarkowitzs pioneering work, the mean-variance model was extended

from single-period case to multi-period discrete-time case (see Hakansson,1971; Pliska, 1997; Samuelson, 1969, etc.) and continuous time case dur-ing the last decades (seeCox and Huang,1989;Duffie and Richardson,1991;Karatzas et al.,1987;Schweizer,1996, etc.). However, when studying thesetwo kinds of dynamic portfolio selection models, most research works havebeen dominated by those of maximizing expected utility functions of the ter-minal wealth. Nevertheless, when using this approach, the tradeoff informa-tion between the risk and the expected return is implicit, which makes the in-vestment decision less intuitive. In2000,Zhou and Liintroduce the stochas-tic linear-quadratic (LQ) control as a general framework to study the mean-variance optimization problem. Within this framework they have established

a natural connection of the portfolio selection problems and standard stochas-tic control models and attained some elegant results for a continuous-timemean-variance model with determined coefficients.

When using stochastic LQ control approach to deal with continuous timemean-variance problem, the terminal wealth is a random variable with a dis-tribution that is often extremely skewed and shows considerable probabilityin regions of small values of the terminal wealth. This means that the op-timal terminal wealth may exhibit large shortfall risks. In order to preventinvestors from extremely dangerous positions in the market, it is thus morereasonable to consider asymmetric risk measures, e.g. value-at-risk (VaR),

to limit the exposure to market risks.In market risk management, it is widely accepted that VaR is a usefulsummary measure of market risks which regulatory authorities sometimesenforced investors to use. VaR is actually the maximum expected loss overa given horizon period at a given level of confidence. For comprehensiveintroduction to risk management using VaR, we refer the reader to Jorion(1997).

Recognizing that risk management is typically not an investors primaryobjective, the investors would like to limit their risks while maximizing ex-pected utility. This leads to stochastic control problems under restrictions onsuch risk measures. There has been considerable interest in the study of port-

folio selection models subject to a VaR constraint. Kluppelberg and Korn(1998), Alexander and Baptista (2004, 2007) investigate the optimal port-folio choices subject to a VaR or conditional VaR (CVaR) constraint in astatic (one-period) setting. The similar problems in a dynamic setting hasstarted to draw more attentions recently, Basak and Shapiro(2001) focus on

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the optimal portfolio policies of a utility-maximizing agent by imposing the

VaR constraint at one point in time. Cuoco et al.(2008) developed a realis-tic dynamically consistent model of the optimal behavior of a trader subjectto risk constraints. They assume that the risk of the trading portfolio isre-evaluated dynamically by using the conditioning information, and hencethe trader must satisfy the risk limit continuously. Yiu (2004) explicitlyderived the standard VaR constraint on total wealth and obtained optimaltrading strategy(without consideration of re-insurance). Pirvu(2005) startedwith the model ofCuoco et al. (2008) and found the optimal growth port-folio subject to these risk measures. Pirvu(2007) extended those results byextensively studying the optimal investment and consumption strategies forboth logarithmic utility and non-logarithmic CRRA utilities.

Motivated byZhou and Li (2000) andYiu(2004), this paper addressesthe problem of an investor who receives an uncontrollable stochastic cash flowwhich he must then invest in a complete continuous-time financial market inorder to maximize the weighted average of the expectation and the varianceof his terminal wealth at a horizon time. The main focus in this paper is onthe mean-variance optimization problem of the investor subject to a risk limitspecified in terms of VaR on his future net worth. To our knowledge, thisproblem has not yet received a complete treatment in the existing literature.In this paper, we derive the optimal mean-variance investment strategy undera standard VaR constraint by solving the corresponding Hamilton-Jacobi-

Bellman (HJB) equation and explore how the addition of a risk constraintaffects the optimal solution.The rest of the paper is organized as follows. Section 2 describes the

model, including the definition of the VaR on the future net worth process.Section 3 contains the main characterization result of the VaR constraintand formulates the portfolio optimization problem that can be eventuallydiscussed as a stochastic LQ problem. Section 4 gives the explicit solution ofthe optimal mean-variance strategy without VaR constraint by solving thecorresponding HJB equation. Section 5 discusses the optimal mean-variancestrategy with VaR constraint. Finally, Section 6 provides some numericalexamples to show how the addition of the VaR constraint affects the optimal

strategy.

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2. The Model

2.1. Continuous-time investment in stochastic cash flow

All stochastic processes introduced below are supposed to be adaptedin a filtered probability space (, F, Ft, P), whereFt, t 0 is a filtrationsatisfying the usual conditions. Moreover, it is assumed throughout thispaper that all inequalities as well as equalities hold P-almost surely.

Following the framework of Browne (1995), for simplicity, and withoutany loss of generality, we assume that there is only one risky stock availablefor investment, whose price at time t will be denoted by Pt which satisfiesthe following stochastic differential equation

dPt= P

t(dt + dW

(1)

t ), (1)

where > 0 is the appreciation rate and > 0 is the volatility or thedispersion of the stock. W

(1)t is a standard Brownian motion.

Since we are concerned with investment behavior in the presence of astochastic cash flow, or an external risk process denoted by Yt, which isanother Browian motion with drift and diffusion parameter >0, that is

dYt= dt + dW(2)

t , (2)

where W(1)

t and W(2)

t are possibly correlated with correlation coefficient .In case there would only be one source of randomness left in the model, we

also assume that 2


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2.2. Value-at-risk

Now we want to introduce the definition of value-at-risk. Here we startby rewriting (3) into integration form

Xft =x +

t0

(fs + )ds +

t0

(fs)dW(1)

s +

t0

dW(2)s , (4)

where x >0 denotes the initial value of the portfolio. Notice that (4) leadsto

Xft+ =Xt+

t+t

(fs + )ds +

t+t

(fs)dW(1)

s +

t+t

dW(2)s , (5)

for any >0.If we assume that the investment strategy were kept constant during the

time period (t, t+ ], i.e. fs f, for any s (t, t+ ], then it followsimmediately from (5) that, given the strategy fand the associated wealthvalueXt= Xat timet, the random variableXt+(X, f) would be the futurevalue of the wealth at time t +

Xt+(X, f) =Xt+ (f + )+ f (W(1)t+ W(1)t ) + (W(2)t+ W(2)t ). (6)

Therefore we define the future net worth of the wealth process in horizonperiod (t, t+ ] by

Xt+(X, f)

Xt.

Definition 1 (Value-at-risk). Given a probability levelp(0, 1) and a hori-zon >0, the value-at-risk of the future net worth of the wealth process withinvestment strategyfat timet, denoted byV aRp,ft , is defined as

V aRp,ft = (Qp,ft )

, (7)

whereQp,ft =sup{L R :P(Xt+ XtL| Ft)< p}, (8)

andx = max{0, x}.Consequently,Qp,ft is thepquantile of the projected portfolio gain over

the time interval (t, t+]. In other words,V aRp,ft is the greatest loss over thenext period of lengthwhich would be exceeded only with a small conditionalprobabilityp if the current portfolio ft were kept the same.

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Proposition 1 (Computation of value-at-risk). We have

V aRp,ft = (Qp,ft ) = ((f + ) + f22 + 2 + 2f1(p)), (9)

where() and1() denote the standard normal distribution and inversedistribution functions respectively.

Proof of Proposition1. We have

P(Xt+ XtL| Ft)= P((f + )+ f (W

(1)t+ W(1)t ) + (W(2)t+ W(2)t )L| Ft)

= P(f (W(1)

t+ W(1)t ) + (W(2)t+ W(2)t )L (f + )| Ft)= P(Z

L (f + )

f22 + 2 + 2f | Ft)= (

L (f + )

f22 + 2 + 2f),

where Z = f (W

(1)t+W

(1)t )+(W

(2)t+W

(2)t )

f22+2+2ffollows standard normal distribution

conditionally. Thus, from

P(Xt+ XtL| Ft)< p,we know that

( L (f + )

f22 + 2 + 2f)< p,

which gives rise to

L < (f + ) +

f22 + 2 + 2f1(p).

And therefore

Qp,ft = sup{L R :L < (f + ) +

f22 + 2 + 2f1(p)}= (f + ) +

f22 + 2 + 2f1(p),

which immediately gives (9).

Note that even when f

0, which means there is no investment in risky

stock, V aRp,0t =( + 1(p)/) is also positive if/


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3. Statement of the Problem

We consider the optimal control problem of the investor who starts withan initial wealth X0 = x and must select a strategy fso as to maximize theweighted average of the expectation and the variance of his terminal wealth,subjected to the constraint that the VaR with the chosen portfolio is nolarger than a given level V aRat any timet[0, T]. In mathematical terms,this problem can be described as

supf

E[XfT (XfT)2]s.t. dXt= (ft + )dt + ftdW

(1)t + dW

(2)t , X0= x,

V aRp,ft

V aR,

t

[0, T].

(10)

where the parameter (representing the weight) is positive. We denote theoptimal solution of problem (10) byfV aR if it exists. Note that the upperbound V aR can be dependent on Xft and t, however in this paper, we setV aRto be a constant in order to obtain the explicit solution.

Proposition 2 (Computation of the value-at-risk constraint). The explicitform of the VaR constraint in problem(10) is

[ M

2N222(NM) , +), ifN= ,N < M,

[f1, +

), ifN < ,

[f2, f1], if0< N22 2 (M)2(12)2 , < M,

, otherwise,(11)

whereN= 1(1 p)/ >0, M=+ V aR/ >0, and

f1,2=2(M N2)

2(N22 2) , (12)

if = 4N2((1 2)2(2 N22) + (M )2)0 andN22 =2. Wealso assume thatf1 is always larger thanf2.

Proof of Proposition2. The VaR constraint can be written as

(f + +1(p)

f22 + 2 + 2f)V aR,

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which is equivalent to (

1(p)

)2(f22 + 2 + 2f)(f + + V aR

)2,

f + + V aR 0,

that is, after some simplifications, N2(f22 + 2 + 2f)(f + M)2,f + M0,

where N = 1(1p)/ > 0, M = + V aR/ > 0. Note that whenf =M/, N2(f22 +2 + 2f) > 0 = (f +M)2 always holds. Thisobservation will help us in the second case whenN

2

2

2

0.

Therefore we have a group of inequalities (N22 2) f2 + 2(N2 M) f+ (N22 M2)0,f M/. (13)

First we study the degeneration case: N22 2 = 0,i.e. N = .In this case, if 2(N2 M) > 0, i.e. N > M, then the first

inequality of (13) would imply

f M2

N2

2

2(N2 M) = M2

N2

2

2(N M) ,

however

M

> M2 N222(N M) ,

which, together with the second inequality of (13), leads to f .If 2(N2 M) = 0, i.e. N = M, then the first inequality of (13)

would implyN22 M2 =N22(1 2)0,

which leads to a contradiction. Thus there is no solution.If 2(N2 M) < 0, i.e. N < M, then the first inequality of (13)

would imply

f M2 N22

2(N M) ,

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so the constraint becomes

f[max{M

, M2 N22

2(N M)}, +) = [ M2 N222(N M) , +).

Secondly, we study the case: N22 2 0. f2 < f1

are two points of intersection of the parabola and the lateral axis. Using theobservation we have mentioned just now, we know that the first inequality of(13) will not hold iff=M/, which leads to f2 0.In this case, if

0, i.e.

N22 2 (M )2

(1 2)2 .

The constraint becomes f [f2, f1] [M/, +). Also by using the ob-servation, we haveM/ < f2f1 or f2f1

M

N2

N22 2 ,

where (M N2)/(N22 2) = 12(f1+ f2) is the symmetry axis of theparabola.

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To proceed, let V(t, x) = supf

E[u(XfT)|Xft = x] be the optimal valuefunction attainable by the investor starting from the state x at time t. Andwe will give the explicit form of the optimal strategy f in the followingtheorem.

Therom 1 (The optimal mean-variance strategy without VaR constraints).The optimal strategy to maximize expected utility at terminal time T is toinvest, at each timetT,

f=2

(x 12

) ( /)(T t)2(T t) + 2

, (15)

and then the optimal value function is

V(t, x) =(x 12

)2ek1(Tt) + k2(T t)(x 12

) + k3(T t) + 14

, (16)

where

k1 = 2B,

k2 = 2Ae2B(Tt)

2B(Tt)1 ,

k3 = 2A2e2B(Tt)[ Tt

2B(Tt)1 B( Tt2B(Tt)1)2 CA2 ],(17)

andA= /, B=12(/)2, C= 122(1 2).Proof. From Fleming and Rishel (1975), the corresponding HJB equation is

given by sup

f

{Vt+ [f + ]Vx+ 12 [f22 + 2 + 2f]Vxx}= 0,V(T, x) =x x2.

(18)

Assume that the HJB equation (18) has a classic solution V, which satisfiesVxx< 0. Then differentiating with respect to fgives the optimizer

f=2

VxVxx

. (19)

Substituting (19) back into (18), the HJB equation becomes, after some

simplification, equivalent to the following nonlinear Cauchy problem for thevalue function V

Vt+ AVx+ B V2xVxx

+ CVxx= 0,

V(T, x) =x x2 =(x 12)2 + 14, (20)

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where the constants A= /,B =12(/)2, C= 122(1 2).In order to simplify the boundary condition, let

y= x 12

, (21)

and rewrite the HJB equation (20) into the form Vt+ AVy+ B

V2yVyy

+ CVyy = 0,

V(T, y) =y2 + 14

.(22)

To solve this partial differential equation (22), we try to fit a solution ofthe form

V(t, y) =y2ek1(Tt) + k2(T t)y+ k3(T t) + 14

, (23)

wherek1,k1and k1are suitable coefficient, and note that by the form of (23)we have

Vt = k1y2ek1(Tt)k2yk3 ,

Vy = 2ek1(Tt)y+ k2(T t), (24)Vyy = 2ek1(Tt).

The boundary condition is naturally satisfied by the solution form of (23).By substituting (24) into (22), we have

(k1y2ek1(Tt) k2y k3) + A(2ek1(Tt)y+ k2(T t))

+B[2ek1(Tt)y+ k2(T t)]2

2ek1(Tt) + C(2ek1(Tt)) = 0,

(25)

which requires k1,k2 and k3 to satisfy

k1ek1(Tt) + B(2ek1(Tt)) = 0,

k2 2Aek1(Tt) + 2Bk2(T t) = 0,k3+ Ak2(T t) + B k

22(Tt)2

2ek1(Tt) + C(2ek1(Tt)) = 0.(26)

Solving equations (26), we derive (17), and hence (16) after replacing y by(x 12). Since we have the value function in explicit form, it becomes easyfor us to obtain the optimal control of (15) by substituting the value for VxandVxx from (24) into (19).

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5. Optimal Mean-variance Strategy under VaR Constraints

In this section, we come back to the problem (10). Our objective is togive the optimal control fV aR as well as the corresponding value functionV in explicit form. Since in Theorem1we have already found the optimalcontrol f which optimize the problem of (14) without the VaR constraintTherefore, we obtain fV aR = f

as long as f satisfies the VaR constraint.However, the global optimizer can not be the local optimizer when f failsthe VaR constraint.

Remind ourselves of the proof in Theorem 1, by applying the dynamicprogramming approach we are tackling with a static optimization problem(18). Rewrite the problem and add the VaR constraint to it and we have

supf

{12Vxx

2 f2 + (Vx+ Vxx) f+ (Vt+ Vx+ 122Vxx)},s.t. fI

(27)

where the intervalsIhave the same forms of (11) in Proposition2,therefore,the constraint f I is equivalent to the VaR constraint in problem (10).Since we know that 1

2Vxx

2 < 0, it becomes a simple problem to find thepeak of the parabola, which opens down, on each interval I. Specifically,

Case 1 when N =, N < M,

fV aR =

f, f[ M2

N22

2(NM) , +),M2N222(NM) , otherwise.

(28)

Case 2 when N < ,

fV aR =

f, f[f1, +),f1, otherwise.

(29)

Case 3 when 0< N22 2 (M)2(12)2 , < M,

fV aR =

f

, f[f2, f1],f2, f

(, f2),f1, f

(f1, +).(30)

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Except these three cases listed above, there is no solution which can

satisfies the constraint. Heref1 and f2 have the same expressions as shownin Proposition2, andf has the expression of (15) in Theorem1.Before giving the optimal control as well as the corresponding value func-

tion in explicit form, we have to admit that we are going to omit the firstcase, i.e. when N = happens. On the one hand, this case could hardlyhappen so that we can benefit little from studying them in practice, on theother hand, the procedures of studying the first case is quite similar with theother two, it will be therefore a mere repetition.

Therom 2 (The optimal mean-variance strategy under VaR constraints).The optimal strategy to maximize expected utility at terminal timeTsubjected

to the VaR constraint is to investfV aR.WhenN < ,

fV aR = max{f1, f}, (31)and the optimal value function is

V(t, x) =

(x 12

)2ek1(Tt) + k2(T t)(x 12

)

+k3(T t) + 14

,ff1,

1

4 [(x 1

2)2 + 2D1(x 1

2)(T t)

+D21(T t)2 + 2E1(T t)],f< f1;

(32)

When0< N22 2 (M)2(12)2 , < M,

fV aR= max{f2, min{f1, f}}, (33)

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and the optimal value function is

V(t, x) =

(x 12

)2ek1(Tt) + k2(T t)(x 12

)

+k3(T t) + 14

,f[f2, f1],

1

4 [(x 1

2)2 + 2D2(x 1

2)(T t)

+D22(T t)2 + 2E2(T t)],f(, f2),

1

4 [(x 1

2)2 + 2D1(x 1

2)(T t)

+D21(T t)2 + 2E1(T t)],f(f1, +)

(34)where k1, k2 and k3 have the same expressions of (17) in Theorem 1, f1,2have the expressions of (12), andDi = fi + , Ei=

12

(f2i2 + 2 + 2fi),

fori= 1, 2.

Proof. Since the explicit form (31) and (33) of the optimal control fV aR inthis theorem are only the rescript of (29) and (30), we only have to work outwith the corresponding value function V.

When N < happens, substituting (29) in (18) and we have

0 = Vt+ ( /)Vx 12(/)2 V2xVxx

+ 12

2(1 2)Vxx, ff1,Vt+ (f1 + )Vx+ 12(f21 2 + 2 + 2f1)Vxx, f< f1,

(35)with the same terminal condition V(T, x) =x x2.

Since we have already solved the first equation of (35) in Theorem1, wewill focus on the second equation which appears to be much more easier tohandle for its linearity.

Using the notation ofD1 and E1, the second equation of (35) with theterminal condition can be written as another Cauchy problem

Vt+D1Vx+ E1Vxx= 0, for t < T,V(T, x) =x

x2.

(36)

Applying the same trick by letting y = x 12, then Vt+ D1Vy+E1Vyy = 0, for t < T,V(T, y) =y2 + 14.

. (37)

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In order to apply the Fourier transform, change the terminal condition into

initial condition by letting U(t, y) = V(T t, y) Ut D1Uy E1Uyy = 0, fort >0,U(0, y) =y2 + 14.

(38)

Then we have

U(t, y) = ( 1

42 y2 2D1yt D21t2 2E1t), (39)

which immediately gives the second part of (32).

Similarly when 0< N222 (M)2(12)2 , < M, the corresponding

Cauchy problem is described as

0 =

Vt+ ( /)Vx 12(/)2 V2x

Vxx+ 12

2(1 2)Vxx, f[f2, f1],Vt+ (f2 + )Vx+

12

(f22 2 + 2 + 2f2)Vxx, f

(, f2),Vt+ (f1 + )Vx+

12(f

21

2 + 2 + 2f1)Vxx, f(f1, +),

(40)with terminal condition V(T, x) =x x2.

Almost the same steps can be taken if we replace the notation f1,D1andE1 with f2,D2 andE2 respectively, and (34) will be obtained.

6. Illustration of the Solutions

In this section we will illustrate the result of Theorem1and Theorem2.Without loss of generality, we set the initial wealth level between [0 , 1]. TheVaR horizon period is chosen to be 1 trading day, nearly 1/260 calendaryear, while the terminal year is set to be 10 calendar year. Confidence levelis 1 p= 99%, and the upper VaR limit is V aR= 0.02, which is 2% of theinitial wealth. For the stochastic cash flow, we use = 0.01, = 0.14. Inthe market, the risk-free interest r = 0, as we always assumed, and for thestock, = 0.05, = 0.3. the correlation coefficient ofW

(1)t and W

(2)t is set

to be = 0.2, and the parameter in the quadratic utility function is = 1.We summarize these parameters below in the Table1.

The setting of the parameters in Table1 satisfies the three conditions ofcase 3:

0< N22 2 (M )2

(1 2)2 , < M,

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Table 1: Parameters

x T p V aR1 0.05 0.3 0.01 0.14 0.2 1 10 1/260 0.01 0.02

whereN=N1(1p)/= 37.74,M= +V aR/= 5.273. Therefore theoptimal control has the expression fV aR = max{f2, min{f1, f}}. We showthe optimal strategy without the VaR limit as well as the constrained one on(x, t)[0, 1] [0, 10] in the Figure1 below.

As shown in this figure, the VaR constraint actually gives an upper anda lower bound surface to the strategy surface. From the expression of (12),we know that f1,2 are independent ofx and t, and strictly negative in this

case. Therefore the bound surfaces are horizontal and below level zero, whichactually constrain the behavior of short-selling.

We could also compare the optimal value function with and without theVaR constraint, which are illustrated in Figure2 for case 3. In both figureswe could observe that the VaR constraint is active during the time periodof approximately t [0, 3][6, 10]. The optimal function of constrainedproblem is identical to that of the unconstrained one duringt[3, 6], and itbecomes inferior when the constrain is active.

By changing the value of drift and diffusion parameter of the risky stock,investment in them becomes much more promising than in case 3. See the

table below, note that we do not change the values of other parameters.

Table 2: Parameters

x T p V aR

1 0.8 0.02 0.01 0.14 0.2 1 10 1/260 0.01 0.02

The setting of the parameters in Table2 satisfies the condition of case 2,i.e. N < , where N = 37.74. In Figure3, we observe that when time tis near 0, the investment in risky asset appears to be rather radical. Thisresults from the superiority of the stock, and the investor can hardly get

any loss under such circumstance, and therefore do not activate the VaRconstraint. When time goes to approximately t= 5, the constraint becomesactive and gives the optimal strategy a lower bound near level zero, whichis the only bound surface brought by the VaR constraint in case 2. Theoptimal value function surfaces are shown in Figure 4, where the surface

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of the constrained problem remains identical to that of the unconstrained

problem until t = 5. All as we expected, the constrained surface becomesinferior since then. Note that in this case, the optimal value without VaRconstraint appears almost horizontal, which seems unreasonable. This resultsfrom the uncommon condition that N < /, which directly leads to the hugeabsolute value ofB =0.5(/)2. Look at the expressions of (16) and (17),the factor e2B(Tt) almost annihilate the first three terms in V(t, x), whichmakesV(t, x) becomes a constant of 1/(4) approximately.

References

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Basak, S., Shapiro, A., 2001. Value-at-risk-based risk management: Optimalpolicies and asset prices. Review of Financial Studies 14, 371405.

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Jorion, P., 1997. Value at Risk: the New Benchmark for Controlling MarketRisk. Irwin, Chicago, IL.

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Karatzas, I., Lehoczky, J.P., Shreve, S.E., 1987. Optimal portfolio and con-

sumption decisions for a small investor on a finite time-horizon. SIAMJournal on Control and Optimization 25, 15571586.

Kluppelberg, C., Korn, R., 1998. Optimal portfolios with bounded value-at-risk. Working paper, Munich University of Technology.

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Pliska, S.R., 1997. Introduction to Mathematical Finance. Blackwell,Malden.

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Schweizer, M., 1996. Approximation pricing and the variance-optimal mar-tingale measure. The Annals of Probability 24, 206236.

Yiu, K.F.C., 2004. Optimal portfolios under a value-at-risk contraint. Journalof Economic Dynamics and Control 28, 13171334.

Zhou, X.Y., Li, D., 2000. Continuous-time mean-variance portfolio selection:a stochastic LQ framework. Applied Mathematics and Optimization 42,1933.

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0

2

4

6

8

10 0

0.2

0.4

0.6

0.8

10.4

0.3

0.2

0.1

0

0.1

0.2

0.3

wealthtime

without VaR

with VaR

Figure 1: Optimal strategies in case 3

0

2

4

6

8

10 0

0.2

0.4

0.6

0.8

1

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

wealthtime

with VaR

without VaR

Figure 2: Value functions in case 3

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0

2

4

6

8

10 0

0.2

0.4

0.6

0.8

1

1500

1000

500

0

500

1000

wealthtime

without VaRwith VaR

Figure 3: Optimal strategies in case 2

0

2

4

6

8

10 0

0.2

0.4

0.6

0.8

1

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

wealthtime

with VaR

without VaR

Figure 4: Value functions in case 2

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Optimal Mean-Variance Investment Strategy Under

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