Chaos, Solitons and Fractals 91 (2016) 516–521

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier.com/locate/chaos

Optimal consumption—portfolio problem with CVaR constraints

Qingye Zhang ∗, Yan Gao School of Management, University of Shanghai for Science and Technology, Shanghai 20 0 093, China

a r t i c l e i n f o

Article history:

Received 8 May 2016

Revised 25 July 2016

Accepted 29 July 2016

Keywords:

Dynamic portfolio selection

Conditional value-at-risk (CVaR)

Hamilton–Jacobi–Bellman (HJB) equation

Logarithmic utility function

a b s t r a c t

The optimal portfolio selection is a fundamental issue in finance, and its two most important ingredi-

ents are risk and return. Merton’s pioneering work in dynamic portfolio selection emphasized only the

expected utility of the consumption and the terminal wealth. To make the optimal portfolio strategy be

achievable, risk control over bankruptcy during the investment horizon is an indispensable ingredient. So,

in this paper, we consider the consumption-portfolio problem coupled with a dynamic risk control. More

specifically, different from the existing literature, we impose a dynamic relative CVaR constraint on it. By

the stochastic dynamic programming techniques, we derive the corresponding Hamilton–Jacobi–Bellman

(HJB) equation. Moreover, by the Lagrange multiplier method, the closed form solution is provided when

the utility function is a logarithmic one. At last, an illustrative empirical study is given. The results show

the distinct difference of the portfolio strategies with and without the CVaR constraints: the proportion

invested in the risky assets is reduced over time with CVaR constraint instead of being constant without

CVaR constraints. This can provide a good decision-making reference for the investors.

© 2016 Elsevier Ltd. All rights reserved.

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1. Introduction

Portfolio selection is an interesting and important issue in fi-

nance. It studies how to allocate an investor’s wealth among a bas-

ket of securities to maximize the return and minimize the risk.

In 1952, Markowitz firstly used variance to measure the risk and

proposed the so-called mean-variance (MV) model for the static

portfolio selection problem [1] , which laid the foundation of mod-

ern portfolio theory and inspired a great number of extensions and

applications. However, the MV model was criticized for its inappli-

cability. On the one hand, besides the difficulty of the covariance

matrix’s computation, variance, which takes the deviation both up

and down from the mean without discrimination as the risk, is in-

compatible with the actual portfolio situations. On the other hand,

the static buy-and-hold portfolio strategy which makes a one-off

decision at the beginning of the period and holds on until the

end of the period is usually inappropriate for a long investment

horizon. To overcome the classical MV model’s shortcomings, mea-

sures measuring the downside risk were proposed, such as semi-

variance, value-at-risk (VaR) [2] , conditional value-at-risk (CVaR)

[3] , etc. Meanwhile, the extension to the dynamic case is an issue

which has been extensively studied as well.

Among all downside risk measures, VaR is popular in practice

of risk management by virtue of its simplicity. It is defined as the

∗ Corresponding author. E-mail address: zhangqingye123@163.com (Q. Zhang).

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http://dx.doi.org/10.1016/j.chaos.2016.07.015

0960-0779/© 2016 Elsevier Ltd. All rights reserved.

uantile of the loss at a certain confidence level. But it has been

riticized for its lose of subadditivity. And as its modification, CVaR,

hich is defined as the average value of the loss greater than VaR,

as attracted increasing attention in recent years on the fact that

t is a coherent risk measure [4] and mean-risk model based on it

an be solved easily by linear programming method.

As for extensions from single-period portfolio to dynamic cases,

ulti-period setting and continuous-time circumstances are in-

luded. Merton [5] and Samuelson [6] extended the static model

o a continuous-time setting by utility functions and stochastic

ontrol theory respectively. Since then, the literature on dynamic

ortfolio selection has been dominated by expected utility maxi-

ization model. Moreover, multi-period and continuous-time MV

odel have been tackled by embedding technique [7] and linear

uadratic approach [8] respectively in 20 0 0. Afterwards, dynamic

V model became another hot research topic. From existing lit-

rature, it is easy to find that the optimal consumption-portfolio

roblem in continuous time is an interesting and appealing issue

9–13] . And in this paper, we intend to study this problem further.

As we know, in continuous-time setting, the movement of risky

ssets is always assumed to follow some stochastic process, and

ased on this assumption, stochastic control theory as well as the

artingale method is the main solving method. Among all stochas-

ic processes, geometric Brownian motion is used widely. But as

hown by empirical analysis, the actual return distribution of the

isky assets has the properties of aiguilles and fat tails. Scholars

ngaged in econophysics have carried out many studies on this

ssue and proposed several more appropriate models, e.g. [14–19] .

http://dx.doi.org/10.1016/j.chaos.2016.07.015http://www.ScienceDirect.comhttp://www.elsevier.com/locate/chaoshttp://crossmark.crossref.org/dialog/?doi=10.1016/j.chaos.2016.07.015&domain=pdfmailto:zhangqingye123@163.comhttp://dx.doi.org/10.1016/j.chaos.2016.07.015

Q. Zhang, Y. Gao / Chaos, Solitons and Fractals 91 (2016) 516–521 517

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he martingale method decomposes the dynamic optimization

roblem into two sub-problems [20,21] , while stochastic control

ethod [22–24] obtains the optimal control by introducing an

ptimal value function and is more intuitively. It should be noticed

hat a bankruptcy occurs when the wealth falls below a predefined

isaster level at any moment during the investment horizon. And

hen an investor is in bankruptcy, he is unable to pursue further

nvestment due to his high liability and low credit. To control the

isk of bankruptcy and execute the investment strategy, imposing

dynamic risk constraint on the instantaneous wealth throughout

he investment is obviously needed. Fortunately, the optimal

ortfolio model coupled with a dynamic VaR constraint, which

as proposed by Yiu recently [25] , provides a new perspective on

he risk management and gives us some inspiration. But, to our

nowledge, literature on the dynamic risk constraint is still scarce

24,26,27] . So, in this paper, we study the optimal consumption

nd portfolio problem further. More specifically, we study the

roblem of maximization the expected discounted utility of both

onsumption and the terminal wealth by imposing a dynamic rel-

tive CVaR constraint on it, which is more appropriate in practice.

The rest of this paper is organized as follows. In Section 2 , we

resent the market setting and problem formulation, where in-

lude the analytical expression of CVaR. In Section 3 , we derive the

amilton–Jacobi–Bellman (HJB) equation for general utility func-

ions using the dynamic programming technique. In Section 4 , we

mploy the Lagrange multiplier method to tackle the CVaR con-

traint and present the closed-form solutions for logarithmic util-

ty function. In Section 5 , we give an illustrative empirical study.

t last, we summarize the paper.

. Market setting and problem formulation

Consider a financial market with n + 1 assets: one risk-free as-et and n risky assets. The price process S 0 ( t ) of the risk-free asset

ollows the following ordinary differential equation

d S 0 (t) = S 0 (t ) rdt , t ∈ [0 , T ] S 0 (0) = s 0 here r > 0 is the risk-free rate, T > 0 is the terminal time of the

nvestment. The price processes of the n risky assets satisfy the

ollowing stochastic differential equations

d S i (t) = S i (t) [ μi d t +

n ∑ j=1

σi j d B j (t)

]

S i (0) = s i , i = 1 , . . . , n here μ = ( μ1 , . . . , μn ) ′ is the appreciation rate of returns, σ = σi j , i, j = 1 , . . . , n } is the volatility rate of returns satisfies the non-egenerate condition, B (t) = ( B 1 (t ) , . . . , B n (t )) ′ is an n dimensionaltandard Brownian motion with B i (t) and B j (t) mutually indepen-

ent for i � = j . Let ( �, F, P , { F t } t ≥ 0 ) be a filtered complete proba-ility space, where F = { F t ; t ≥ 0 } is the natural filtration generatedy the n dimensional Brownian motion B ( t ), F t = σ { B (s ) ; 0 ≤ s ≤ t}

s a σ -field representing the information available up to time t .et L 2 F (0 , T ; R n ) be the set of all R n valued, { F t ; t ≥ 0}-adapted andquare integrable stochastic processes.

In what follows, we consider an investor entering the market

ith initial wealth w 0 . The investor allocates his wealth in these

+ 1 assets continuously and withdraws some funds out of theortfolio for consumption within the time horizon [0, T ], where

he investor’s objective is to maximize his expected discounted

tility of consumption and terminal wealth. Denote the consump-

ion process as { c ( t )} and the control process as { π ′ (t) , c(t) } = π1 (t) , . . . , πn (t) , c ( t )}, where the components of π ( t ) are propor-ions of the investor’s wealth invested in the risky assets, c ( t ) is

he consumption rate. A control strategy { π ′ ( t ), c ( t )} is admissiblef { π ′ (t) , c(t) } ∈ L 2

F (0 , T ; R n +1 ) is F t progressively measurable. Let

π , c ( t ) be the investor’s wealth at time t . Then the wealth pro-

ess satisfies the following stochastic differentiable equation

W π,c (t) = W π,c (t) {

n ∑ i =1

πi (t)

[ μi d t +

n ∑ j=1

σi j d B j (t)

]

+ [

1 −n ∑

i =1 πi (t)

] rd t − c(t) d t

}

= W π,c (t) {

[ π ′ (t) ̄μ + r − c(t)] dt + π ′ (t) σdB (t) }

(2.1)

here μ̄ = ( μ1 − r, . . . , μn − r) ′ is the excess rate of return. By Itoemma, we can obtain the unique solution of ( 2.1 ) as

π,c (t) = w 0 exp {∫ t

0

Q(s, π ′ (s ) , c(s )) ds + ∫ t

0

π ′ (s ) σdB (s ) }(2.2)

here t ∈ [0, T ] and Q(t, π ′ (t) , c(t)) = π ′ (t) ̄μ + r − c(t) − 1 / 2 ‖ π ′ (t) σ‖ 2 . For sufficiently small τ > 0, we may deem that the control

rocess { π ′ (s ) , c(s ) } s ∈ [ t ,t + τ ] is unchanged and stays at the presentalue all the time interval [ t, t + τ ] , which is reasonable in practice.hen,

π,c (t + τ ) = W π,c (t) exp { Q(t, π ′ (t) , c(t)) τ + π ′ (t) σ [ B (t + τ ) − B (t)] } . So the loss of wealth on the time interval [ t, t + τ ] can be

xpressed as

(t) = W π,c (t) − W π,c (t + τ ) = W π,c (t) { 1 − exp [ Q(t , π ′ (t ) , c(t )) τ

+ π ′ (t) σ (B (t + τ ) − B (t))] } . For the given time t , the random variable π ′ (t) σ (B (t + τ ) −

(t)) follows a normal distribution with mean zero and standard

eviation ‖ π ′ (t) σ‖ √ τ . Hence, for the given confidence level β ∈0.5, 1], the CVaR of the loss can be written as

V a R β (L (t))

= W π,c (t) {

1 − c 2 (t ) exp [

Q(t , π ′ (t ) , c(t )) τ + 1 2 ‖ π ′ (t ) σ‖ 2 τ

] }(2.3)

here c 2 (t) = �[ c 1 − ‖ π ′ (t) σ‖ √ τ ] / (1 − β) , c 1 = �−1 (1 − β) , �nd �-1 are the cumulative distribution function and its inverse

unction of the standard normal distribution, respectively. The

erivation of (2.3) is given in the appendix. By introducing an-

ther parameter β̄ ∈ (0 , 1) to be a benchmark of CVaR constraint,e have the relative CVaR constraint as

− c 2 (t) exp (

Q(t, π ′ (t) , c(t)) τ + 1 2 ‖ π ′ (t) σ‖ 2 τ

)≤ β̄. (2.4)

We call it relative CVaR since here CVaR β ( L ( t ))/ W π , c ( t ) is used

nstead of CVaR.

Let U 1 and U 2 be utility functions of the consumption and fi-

al wealth respectively. Both U 1 and U 2 are twice differentiable,

ncreasing, concave functions and satisfying U ′ 1 ( 0 + ) = U ′

2 ( 0 + )

+ ∞ and U ′ 1 (+ ∞ ) = U ′ 2 (+ ∞ ) = 0 , where U ′ ( 0 + ) = lim x → 0 + U ′ (x ) ,

′ (+ ∞ ) = lim x → + ∞ U ′ (x ) . Let ρ > 0 be the discount factor. Thenhe expected discounted utility is

[α

∫ T 0

e −ρt U 1 (c(t)) dt + (1 − α) e −ρT U 2 ( W π,c (T )) ], (2.5)

here α ∈ [0, 1] is a trade-off factor indicating the investor’smphasis on consumption and terminal wealth. If T = + ∞ , then

518 Q. Zhang, Y. Gao / Chaos, Solitons and Fractals 91 (2016) 516–521

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(2.5) is an infinite time horizon problem (i.e. lifetime consumption

problem) and in this case it is sufficient to think about only con-

sumption utility, see [9] . But it is more realistic to analyze a finite

time horizon. So in this paper, we let T be a finite number. Sum

up, we can obtain our model as follows:

max { π(·) ,c(·) }∈ u [0 ,T ]

E

[α

∫ T 0

e −ρt U 1 (c(t)) dt + (1 − α) e −ρT U 2 ( W π,c (T )) ]

s . t . (2 . 1) , (2 . 4) (2.6)

3. Hamilton–Jacobi–Bellman (HJB) equation

In order to employ the optimal control techniques of dynamic

programming, we define the optimal function as

(t, w ) = max { π(·) ,c(·) }∈ u ([ t,T ])

E t

×[α

∫ T 0

e −ρs U 1 (c(s )) ds + (1 − α) e −ρT U 2 ( W π,c (T )) ]

(3.1)

where t ∈ [0, T ], u ([ t, T ]) represents all the admissible strategiesin the time interval [ t, T ], E t stands for the conditional expectation

and W π,c (t) = w is known. Obviously, we obtain the whole port-folio strategy on condition that t = 0 . By Ito lemma, we can obtainthe HJB equation of (3.1) as

t + max { π(t) ,c(t) } { α exp (−ρt) U 1 (c(t)) + V W [ π ′ (t) ̄μ(t) + r − c(t)] w

+ 1 2

V W W ∥∥π ′ (t) σ∥∥2 w 2 } = 0 (3.2)

with the boundary condition V (T , w ) = (1 − α) e −ρT U 2 (w ) . Theo-retically, once the utility function is given, the optimal control pro-

cess can be obtained by solving the HJB equation.

There are many risk-averse utility functions, such as exponen-

tial function, logarithmic function, power function and HARA util-

ity function, etc. In [5] , Merton chose a power function and de-

rived the analytical solution for the value function V ( t, w ) by a trial

function in the form of separable variables. And most existing lit-

erature employs power utility functions, see [26,27] . In the next

section, we choose the logarithmic utility function, which is fre-

quently used in economics, as our utility function and derive the

closed form solution.

4. Optimal control process under logarithmic utility function

For simplicity, here we let α = 0 . 5 . That is, the consumptionutility and the terminal wealth utility have equal importance. Then

we may delete α and (1 − α) in Problem (2.6) . Let U 1 (·) = U 2 (·) =log (·) . Problem (2.6) can be reduced to

max { π(·) ,c(·) }∈ u [0 ,T ]

(1 − e −ρT

ρ+ e −ρT

)log w 0

+ E {∫ T

0

e −ρt [

log c(t) + 1 ρ

(1 − (1 − ρ) e −ρ(T −t) )

× Q(t, π ′ (t) , c(t)) ]

dt

}

s . t . 1 − c 2 (t) exp (

Qτ + 1 2 ‖ π ′ (t) σ‖ 2 τ

)≤ β̄.

The derivation is given in the appendix. It is not difficult to find

that the above problem is equivalent to the following one

max { π(t) ,c(t) }

log c(t) + 1 ρ

(1 − (1 − ρ) e −ρ(T −t) ) Q(t, π ′ (t) , c(t))

s . t . 1 − c 2 (t) exp (

Qτ + 1 ∥∥π ′ (t) σ∥∥2 τ) ≤ β̄ (4.1)

2

heorem 1. The solution of the optimization problem (4.1) is

∗(t)

=

⎧ ⎪ ⎨ ⎪ ⎩

(σσ ′ ) −1 μ̄, i f 1 − c 2 (t) exp (

Qτ + 1 2

∥∥π ′ (t) σ∥∥2 τ)< β̄k ∗1 (σσ

′ ) −1 μ̄, i f 1 − c 2 (t) exp (

Qτ + 1 2

∥∥π ′ (t) σ∥∥2 τ)= β̄

∗(t)

=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

ρ

1 − (1 − ρ) e −ρ(T−t) , i f 1 − c 2 (t) exp (

Qτ + 1 2

∥∥π ′ (t) σ∥∥2 τ) < β̄k ∗2 ρ

1 − (1 − ρ) e −ρ(T−t) , i f 1 − c 2 (t) exp (

Qτ + 1 2 ‖ π ′ (t) σ‖ 2 τ

)= β̄

,

here k ∗1

is the root of Eq. (4.4) in the variable x , k ∗2

is defined by

4.3) .

roof. Obviously, the objective function is concave with respect

o π ( t ) and c ( t ). The feasible region is a convex set. So, Problem4.1) is a convex programming and its KKT point is the same as

he optimal point. Define its Lagrange function as

(t, π(t) , c(t) , λ(t)) = log c(t) + 1

ρ(1 − (1 − ρ) e −ρ(T −t) ) Q(t , π ′ (t ) , c(t ))

−λ(t) [

1 − c 2 (t) exp (

Qτ + 1 2

∥∥π ′ σ∥∥2 τ)− β̄] , here λ( t ) is the Lagrange multiplier. Let ( π∗, c ∗, λ∗) be its KKT

oint, then

∂

∂πL (t, π ∗, c ∗, λ∗) = 0

∂

∂c L (t, π ∗, c ∗, λ∗) = 0 [

1 − c 2 (t) exp (

Qτ + 1 2

∥∥π ′ σ∥∥2 τ)− β̄] ( π ′ ∗, c ∗, λ∗)

≤ 0 , λ∗ ≥ 0 { λ[

1 − c 2 (t) exp (

Qτ + 1 2

∥∥π ′ σ∥∥2 τ)− β̄] } ( π ′ ∗, c ∗, λ∗)

= 0

.

In what follows, we discuss Problem (4.1) in two different cases.

Case 1. If 1 − c 2 (t) exp (Qτ + 1 / 2 ‖ π ′ σ‖ 2 τ ) < β̄ , then the rela-ive CVaR constraint is inactive at the optimal solution. In this case,

roblem ( 4.1 ) is reduced to the following unconstrained problem

max (t) ,c(t)

log c(t) + 1 ρ

(1 − (1 − ρ) e −ρ(T −t) ) Q(t, π ′ (t) , c(t)) .

According to first-order necessary conditions, the optimal solu-

ion { πM ( t ), c M ( t )} must satisfy

μ̄ − σσ ′ π(t) | { πM (t) , c M (t) } = 0 1

c(t) − 1 − (1 − ρ) e

−ρ(T −t)

ρ

∣∣∣∣{ πM (t) , c M (t) }

= 0 .

ence,

M (t) = (σσ ′ ) −1 μ̄, c M (t) = ρ1 − (1 − ρ) e −ρ(T −t) . (4.2)

Case 2 If 1 − c 2 (t) exp (Qτ + 1 / 2 ‖ π ′ σ‖ 2 τ ) = β̄ , then the rela-ive CVaR constraint is active. Let ∂

∂πL (t, π, c, λ) = 0 , that is

1 − (1 − ρ) e −ρ(T −t) ρ

[ ̄μ − σσ ′ π(t)]

+ λ{[

∂ c 2 (t)

∂π+ c 2 (t) ̄μτ

]exp

[(π ′ (t) ̄μ + r − c(t)) τ

]}= 0 .

Q. Zhang, Y. Gao / Chaos, Solitons and Fractals 91 (2016) 516–521 519

Fig. 1. The risk-free asset’s portfolio versus time.

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Fig. 2. The optimal consumption rate versus time.

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π

By calculating the above equation, we find that the optimal

ortfolio π∗( t ) is parallel to πM ( t ). So, we let π ∗(t) = k 1 πM ,

∗(t) = k 2 c M . Then, there must exist the unique (k ∗1 , k ∗2 ) be the op-imal solution of the following problem

ax k 1 , k 2

log k 2 c M (t) + 1 ρ

(1 − (1 − ρ) e −ρ(T −t) ) Q(t, k 1 π ′ M (t) , k 2 c M (t))

. t . 1 − c 2 (t) exp (

Qτ + 1 2

∥∥k 1 π ′ M (t) σ∥∥2 τ) = β̄. By first-order necessary conditions, there must exist a unique

agrange multiplier λ∗ such that

{ log k 2 c M (t) + 1 ρ

(1 − (1 − ρ) e −ρ(T −t) ) Q(t, k 1 π ′ M (t) , k 2 c M (t)) }

= λ∗∇ {

1 − c 2 (t) exp (

Qτ + 1 2 ‖ k 1 π ′ M (t) σ‖ 2 τ

)− β̄

} .

By eliminating λ∗, we obtain

∗2 = f (k ∗1 ) = 1 +

k ∗1 ‖ π ′ M σ‖ 2 − π ′ M μ̄ϕ( c 1 −k ∗1 ‖ π ′ M (t) σ‖

√ τ )

�( c 1 −k ∗1 ‖ π ′ M (t) σ‖ √

τ ) ‖ π ′ M σ‖ √

τ− k ∗

1 ‖ π ′ M σ‖ 2 , (4.3)

here k ∗1

is the root of the equation

(1 − β̄ )(1 − β) = �( c 1 − x ‖ π ′ M (t) σ‖ √ τ ) × exp [(x π ′ M (t) ̄μ + r − f (x ) c M (t)) τ ] (4.4)

n the variable of x .

. Empirical analysis

In this section, we choose four stocks from Chinese financial

arket with codes 60 0611,60 0796,60 0 054 and 60 0118, and use

heir daily closing prices from 01/04/2010 to 01/04/2016. By com-

utation, we have μ = (0 . 0316 , 0 . 1013 , 0 . 0485 , 0 . 1291) ′ and

=

⎡ ⎢ ⎣

0 . 5812 0 0 0 0 . 3184 0 . 4915 0 0 0 . 2445 0 . 1258 0 . 3631 0 0 . 3392 0 . 1769 0 . 0967 0 . 6337

⎤ ⎥ ⎦ .

Let r = ρ = 0 . 017 , β= 0 . 95 , β̄ = 0 . 06 , T = 5 (year), τ = 0 . 02 ≈ aeek.

Fig. 1 presents the optimal portfolio of the risk-free asset at dif-

erent time. When there is no relative CVaR constraint, the propor-

ion of the wealth invested in the risk-free asset is constant. How-

ver, as the value of the Lagrange multiplier turns from zero to

ositive, the constraint becomes binding. And the investor will in-

rease the proportion of the wealth invested in the risk-free asset,

.e., the investor will reduce the proportion invested in the risky

ssets. Fig. 2 presents the optimal consumption rate at different

ime. Since k ∗2 is very close to 1, the optimal consumption rate withelative CVaR constraint and without it is almost the same.

. Conclusion

The optimal consumption-investment problem with a dynamic

VaR constraint has been studied. For general utility function, we

erived the HJB equation by the dynamic programming technique.

s for the logarithmic utility function, the method of Lagrange

ultiplier has been applied to tackle the constraint and the closed

orm solution is presented.

From the empirical study’s results, we find that the constrained

roblem invested less in the risky assets. This is due to the fact

he CVaR constraint is imposed all the time. How to extend the

dea of dynamic risk control to other portfolio problems, such as

he benchmark process, dynamic mean-variance model, etc, is our

uture interest.

ompeting interests

The authors declare that they have no competing interests.

uthors contributions

All authors contributed equally to the manuscript, read and ap-

roved the final manuscript.

cknowledgements

We would like to thank two anonymous reviewers for their

elpful comments and the support of the National Natural Science

oundation of China (Project No. 1171221 ).

ppendix

. The derivation of formula ( 2.3 )

We divide the derivation into three steps. Denote the loss ,

here = L (t) . For simplicity, we denote W π , c ( t ) by W, π ( t ) by, Q ( t, π ′ ( t ), c ( t )) by Q , B (t + τ ) − B (t) by B respectively.

(1) Derivation the probability density function (PDF) of the loss.

http://dx.doi.org/10.13039/501100001809

520 Q. Zhang, Y. Gao / Chaos, Solitons and Fractals 91 (2016) 516–521

C

C

∫

∫

H∫

E

E

{

s

R

Since the cumulative distribution function of the loss is

F (x ) = P ( ≤ x ) = P (W − W e Qτ e π ′ σB ≤ x )

= P {

1

‖ π ′ σ‖ √ τ log W − x W e Qτ

≤ B √ τ

}

= 1 − �(

1

‖ π ′ σ‖ √ τ log W − x W e Qτ

).

So, the PDF of the loss is

f (x ) = F ′ (x )

= −φ(

1

‖ π ′ σ‖ √ τ log W − x W e Qτ

)· 1 ‖ π ′ σ‖ √ τ ·

W e Qτ

W − x ·−1

W e Qτ

= φ(

1

‖ π ′ σ‖ √ τ log W − x W e Qτ

)· 1 (W − x ) ‖ π ′ σ‖ √ τ ;

(2) Derivation the VaR.

By definition,

β = P ( ≤ V aR ) = P (W − W e Qτ e π ′ σB ≤ V aR ) = P (

W − V aR W e Qτ

≤ e π ′ σB )

= P (

1

‖ π ′ σ‖ √ τ log W − V aR

W e Qτ≤ π

′ σB ‖ π ′ σ‖ √ τ

)So, 1 − β = �( 1 ‖ π ′ σ‖ √ τ log W −VaR W e Qτ ) ,

1 ‖ π ′ σ‖ √ τ log

W −VaR W e Qτ

= �−1 (1 − β) ˆ = c 1 .

Therefore, V aR = W − W e Qτ e ‖ π ′ σ‖ √

τ c 1 = W ( 1 −e Qτ+ ‖ π′ σ‖ √ τ c 1 ) ,

where c 1 = �−1 (1 − β) ; (3) Derivation the CVaR.

By definition,

V aR = E(| ≥ V aR ) = 1 1 − β

∫ + ∞ VaR

x f (x ) dx

= 1 1 − β

∫ + ∞ VaR

x

(W − x ) ‖ π ′ σ‖ √ τ· 1 √

2 πe

− 1 2 ( 1 ‖ π ′ σ‖ √ τ log W−x W e Qτ ) 2

dx (∗)

Let 1 ‖ π ′ σ‖ √ τ log W −x W e Qτ

= t , then x = W − W e Qτ+ ‖ π ′ σ‖ √

τ t , dx =−W e Qτ+ ‖ π ′ σ‖

√ τ t · ‖ π ′ σ‖ √ τdt . Therefore, ( ∗) can be rewritten as

follows

V aR = 1 1 − β

∫ −∞ c 1

−W (1 − e Qτ · e ‖ π ′ σ‖ √ τ t ) · 1 √ 2 π

e −t 2

2

dt

= W 1 − β

∫ c 1 −∞

(1 − e Qτ · e ‖ π ′ σ‖ √ τ t ) · 1 √ 2 π

e −t 2

2

dt

= W 1 − β [ �( c 1 ) − e

Qτ+ 1 2 ‖ π ′ σ‖ 2 τ · �( c 1 − ‖ π ′ σ‖ √ τ ) ]

= W [

1 − e Qτ+ 1 2 ‖ π ′ σ‖ 2 τ · �( c 1 − ‖ π′ σ‖ √ τ )

1 − β

]= W [ 1 − c 2 e Qτ+ 1 2 ‖ π ′ σ‖

2 τ ]

where c 2 = �( c 1 −‖ π′ σ‖ √ τ )

1 −β . �

2. The simplification of ( 2.6 )

Taking the log on both sides of ( 2.2 ), we have

log ( W π,c (t)) = log w 0 + ∫ t

0

Q(s, π ′ (s ) , c(s )) ds + ∫ t

0

π ′ (s ) σdB (s ).

Then the utility of the consumption is T

0

e −ρt log (c(t) W π,c (t)) dt

= ∫ T

0

e −ρt log c(t) dt + 1 − e −ρT

ρlog w 0

+ ∫ T

0

e −ρt dt ∫ t

0

Q(s, π ′ (s ) , c(s )) ds

+ ∫ T

0

e −ρt dt ∫ t

0

π ′ (s ) σdB (s )

Exchanging the order of integral by Fubini’s theorem, we have T

0

e −ρt dt ∫ t

0

Q(s, π ′ (s ) , c(s )) ds

= ∫ T

0

Q(s, π ′ (s ) , c(s )) ds ∫ T

s

e −ρt dt

= ∫ T

0

e −ρs − e −ρT ρ

Q(s, π ′ (s ) , c(s )) ds

ence, T

0

e −ρt log (c(t) W π,c (t)) dt

= 1 − e −ρT

ρlog w 0

+ ∫ T

0

e −ρt [

log c(t) + 1 ρ

(1 − e −ρ(T −t) ) Q(t, π ′ (t) , c(t)) ]

dt

+ ∫ T

0

e −ρt dt ∫ t

0

π ′ (s ) σdB (s ) .

By taking mathematical expectation on both sides, we have

∫ T 0

e −ρt log (c(t) W π,c (t)) dt

= 1 − e −ρT

ρlog w 0

+ E {∫ T

0

e −ρt [ log c(t) + 1

ρ(1 − e −ρ(T −t) ) Q(t, π ′ (t) , c(t))

] dt

}.

Similarly, we can get

[e −ρT U 2 ( W π,c (T ))

]= e −ρT log w 0 + e −ρT E

∫ T 0

Q(t, π ′ (t) , c(t)) dt .

Problem (2.6) can be reduced to

max π(·) ,c(·) }∈ u [0 ,T ]

(1 − e −ρT

ρ+ e −ρT

)log w 0

+ E {∫ T

0

e −ρt

×[

log c(t) + 1 ρ

(1 − (1 − ρ) e −ρ(T −t) ) Q(t, π ′ (t) , c(t)) ]

dt

} . t . 1 − c 2 (t) exp

(Qτ + 1

2

∥∥π ′ (t) σ∥∥2 τ) ≤ β̄. �eferences

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Optimal consumption-portfolio problem with CVaR constraints1 Introduction2 Market setting and problem formulation3 Hamilton-Jacobi-Bellman (HJB) equation4 Optimal control process under logarithmic utility function5 Empirical analysis6 Conclusion Competing interests Authors contributions Acknowledgements Appendix1 The derivation of formula (2.3)2 The simplification of (2.6)

References