Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
www.elsevier.com/locate/econbase
Optimal portfolios under a value-at-risk
constraint
K.F.C. Yiua; b;∗
a Department
of Economics, Birkbeck College, University of London, 7-15 Gresse Street,
London W1P 2LL, UK
b Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong,
Pokfulam, Hong Kong
Abstract
This paper looks at the optimal portfolio problem when a value-at-risk constraint is imposed.
This provides a way to control risks in the optimal portfolio and to ful-l the requirement of
regulators on market risks. The value-at-risk constraint is derived for n risky assets plus a risk-free
asset and is imposed continuously over time. The problem is formulated as a constrained utility
maximization problem over a period of time. The dynamic programming technique is applied
to derive the Hamilton–Jacobi–Bellman equation and the method of Lagrange multiplier is used
to tackle the constraint. A numerical method is proposed to solve the HJB-equation and hence
the optimal constrained portfolio allocation. Under this formulation, we -nd that investments in
risky assets are optimally reduced by the imposed value-at-risk constraint.
? 2003 Elsevier B.V. All rights reserved.
JEL classi,cation: G11
Keywords: Optimal portfolio; Value-at-risk; Dynamic programming
1. Introduction
One of the frequent questions in -nance is how to allocate a certain amount of
money in di;erent assets and at what time instant. The earliest approach to consider
the optimal portfolio problem is the so-called mean–variance approach. It was pioneered
by Markowitz (1952) and is basically a single-period model which makes an one-o;
decision at the beginning of the period and holds on until the end of the period.
∗
Corresponding author. Industrial and Manufacturing Systems Engineering, The University of Hong Kong,
Pokfulam, Hong Kong. Tel.: +852 2859 2586; fax: +852 2858 6535.
E-mail address: [email protected] (K.F.C. Yiu).
0165-1889/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0165-1889(03)00116-7
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K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
Gradually, researchers extended this single-period model to continuous-time models
(Merton 1969, 1971). By applying results from stochastic control theory to the optimal
portfolio problem, explicit solutions have been obtained for some special cases.
With the rapid development of the derivatives markets, together with margin tradings
on certain -nancial products, the exposure to losses of an investment can be many times
more than the initial capital allocation for that investment. Without a careful analysis
of the potential danger, the investment could cause catastrophical consequence when
a shock occurs. With the experience of recent failure of large -nancial institutions
such as the Barings Bank, suHcient risks control measures are clearly essential and
the regulators have started to set restrictions on limiting the exposure to market risks.
Following the Basle accord of 1988 which tightened the management of credit risk, the
G30 meeting in 1993 and the Basle Committee on Banking Supervision in 1995 made
guidelines and recommendations toward the market risk management, and suggested
models for measuring market risks.
In market risk management, it is widely accepted that value-at-risk (VaR) is a useful
summary measure of market risks and an option to be used by regulators and large
banks to set the requirement on capital reserves (Jorion 1997; Dowd, 1998). To be
precise, VaR is the maximum expected loss over a given horizon period at a given level
of con-dence. In order to ful-l the requirement, a portfolio must be able to control
the level of VaR. In Merton’s original formulation, the expected utility of wealth or
consumption was maximized over a certain period of time which yielded the optimal
allocation to the risky asset. However, VaR was not considered and the -nal portfolio
might fall short of the regulatory requirement. In particular, if a power utility function
is applied, the optimal allocation is a constant proportion of the wealth (Merton, 1971)
which will clearly violate the VaR restriction at some points.
In the literature, several approaches have been proposed to study the mean-VaR optimization and compare with the mean–variance approach (see, for example, Kluppelberg
and Korn, 1998; Alexander and Baptista, 1999; Kast et al., 1999). However, these
studies are basically static in settings. Formulation in a dynamic setting has started
to draw more attentions recently. This includes the approach presented by Luciano
(1998) and Basak and Shapiro (2001). They both focus on optimal portfolio policies
of a utility-maximizing agent and use the VaR as a constraint. In particular, Luciano
(1998) analyzes deviations from the constraint rather than explicitly applying the constraint to the optimization problem. Basak and Shapiro (2001) focus on imposing the
VaR constraint at one point in time in order to study trading between recalculated
VaRs. Apparently, the imposed constraint is to allow back-testing 1 at the horizon to
verify that losses indeed complied with the stated VaR.
In this article, we impose the VaR as a dynamic constraint. For tractability, the constraints are calculated abstracting from within-interval trading and from considerations
of backtesting. At each instant, the VaR is estimated and is applied to inKuence the
investment decision. The model applies the VaR constraint over time and emphasizes
the repeated recalculations of VaRs in practice. This is clearly a feature not captured
1 This is recommended by the Basle committee as a means to verify the accuracy of VaR -gures. See,
for example, Jorion (1997).
K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
1319
by other models. We hope our model will provide another perspective on the optimal
behaviour and the economic implications of imposing the VaR constraint.
By applying the VaR constraint continuously over time, assuming that portfolio allocations do not change over a short horizon period, the agent cares about this calculated
VaR beyond his modelled horizon and constant investment opportunity set throughout,
we -nd that investments in risky assets are reduced whenever the VaR constraint
becomes active. The intuition behind this is very clear because there is a mapping
between VaRs and portfolio holdings at each instant, and if VaR is to be limited,
portfolio holdings in risky assets must be limited as well at all times. We believe that
any risk controls imposed in a way similar to here will achieve similar results. Moreover, we show numerically that when VaR is the risk control, consumption and total
expected utility are not considerably a;ected for certain choices of parameters. This
raises the question whether some continuously applied risk controls are more preferable
than others in a;ecting consumption and total expected utility apart from achieving risk
reduction.
The rest of the article is organized as follows. The VaR constraint is -rst derived
for n risky assets plus a risk-free asset to model market risks. The optimal portfolio
problem is then formulated as a constrained maximization of the expected utility, with
the constraint being the VaR level. Dynamic programming is applied to reduce the
whole problem to solving the Hamilton–Jacobi–Bellman equation (HJB-equation) coupled with the VaR constraint, and the method of Lagrange multiplier is then applied
to handle the constraint. Finally, a numerical method is then proposed to solve the
HJB-equation and hence the constrained optimal portfolios.
2. Continuous-time optimal portfolios
The problem is often formulated as maximizing the total expected utility of consumption or wealth over a certain time interval [0; T ]. At time t = 0, the agent is
endowed with initial wealth x0 and his/her problem is how to allocate investment
and/or consumption over the given time horizon. We assume that the agent’s investment opportunities are the following:
(i) The agent can invest money in the risk-free asset B at the deterministic short
rate of interest r. This deterministic process can be written as
dB(t) = rB(t) dt:
(1)
(ii) The agent can invest in n risky assets with the price process as (S1 (t); : : : ; Sn (t)).
We assume that the vector process S(t) follows the dynamics 2
dS(t) = D(S(t)) dt + D(S(t)) dW(t);
(2)
where W(t) is a k-dimensional standard Wiener process, is an n-vector, is an n×k
matrix, and D(S(t)) is the diagonal matrix diag[S1 ; : : : ; Sn ]. Let the amount allocated
2 For simplicity, we assume here that r; and are constants. However, the -nal results will also hold
if they are functions of time.
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K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
to S(t) be the n-vector !(t) and de-ne e to be the n-vector of 1, the dynamic of a
portfolio consisting of B(t) and S(t) with consumption c(t) is therefore
X (t) − !(t) e
dB(t) + !(t) D(S(t))−1 dS(t) − c(t) dt
B(t)
(3)
= (X (t) − !(t) e)r dt + !(t) ( dt + dW(t)) − c(t) dt
(4)
= (!(t) ( − re) + rX (t) − c(t)) dt + !(t) dW(t):
(5)
dX (t) =
Assuming the agent needs continuous consumption over the given period of time,
then the unconstrained optimal portfolio problem is given by
T
U (t; c(t)) dt
(6)
max E
!(t);c(t)
0
subject to
dX (t) = (!(t) ( − re) + rX (t) − c(t)) dt + !(t) dW(t):
(7)
3. VaR constraint
To formulate the VaR constraint, rewrite the dynamics as
dX (t) = ((t) − X (t)) dt + !(t) dW(t);
(8)
where
= −r;
(t) =
!(t) ( − re) − c(t)
:
−r
(9)
De-ning
Y (t) = et X (t);
(10)
we have
dY (t) = (t)et dt + et !(t) dW(t):
Let Pt = (s − t) be the horizon period, integrating gives
s
s
()e d +
e !() dW():
Y (s) − Y (t) = t
t
(11)
(12)
Assuming the portfolio is adjusted frequently so that the interval (s − t) is small, we
can approximate !() by !(t) and c() by c(t). 3 This means that there is no trading
in between constraint re-evaluation, and the consumption is roughly constant in the
given horizon period. This is a reasonable approximation since the portfolio can only
3 If r; and are functions of time, approximations will be required for these parameters as well over
the given horizon period.
K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
1321
be adjusted in discrete time and the decision made is based on the holdings at time t.
Thus, (12) becomes
s
Y (s) − Y (t) = (t)(es − et ) +
e !(t) dW();
(13)
t
which implies
X (s) = e−(s−t) (X (t) − (t)) + (t) +
t
s
e−(s−) !(t) dW():
(14)
This is an Ornstein–Uhlenbeck process except that the speed-of-adjustment parameter
is negative instead. The conditional mean on time t is given by
Et (X (s)) = (t) + e−(s−t) (X (t) − (t));
(15)
while the conditional covariance is given by
!(t) !(t) −|s−u|
Covt [X (s); X (u)] =
(e
− e−(s+u−2t) );
2
where = . The conditional variance is therefore given by
!(t) !(t)
Vt (X (s)) =
(1 − e−2(s−t) ):
2
In order to eliminate X (t) from the VaR constraint, de-ne the losses by 4
PX (t) = X (s) − er(s−t) X (t);
(16)
(17)
(18)
the VaR de-nition
Pr(PX (t) 6 − VaRt ) = k
(19)
implies
VaRt = −((t) − (t)erPt ) − "−1 (k)
e2rPt − 1
!(t) !(t):
2r
Substituting (t), the constraint of restricting the VaR at level R is
a1 !(t) !(t) + a2 !(t) + bc(t) 6 R;
where
a1 = −"−1 (k)
e2rPt − 1
;
2r
a2 = −
− re rPt
(e − 1);
r
1
b = (erPt − 1):
r
(20)
(21)
(22)
To interpret this constraint, we note that in one dimension, the constraint reduces to
a1 $|!(t)| + a2 !(t) + bc(t) 6 R:
(23)
Rearranging gives
− R + bc(t) 6 (a1 $ − a2 )!(t)
(24)
4 Note that if the conventional losses X (s) − X (t) are used as the measure, the derivation will not be
a;ected except that the VaR constraint will be more complicated.
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K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
and
(a1 $ + a2 )!(t) + bc(t) 6 R:
(25)
For reasonable parameters, (a1 $ + a2 ) is greater than zero. Therefore, constraint (25)
imposes an upper bound on !(t) to constrain the investment in the risky asset. In the
extreme case when !(t) ≡ 0; bc(t) ≈ c(t)Pt is the only factor to decrease the portfolio
value. Therefore, if R is chosen to be small, this will constrain the consumption as
well. Similarly, (a1 $ − a2 ) is greater than zero for reasonable parameters and (24)
imposes a lower bound on !(t) to constrain the short-selling of the risky asset.
The -nal optimal portfolio problem with VaR constraint is
T
U (t; c(t)) dt
(26)
max E
!(t); c(t)
0
subject to
dX (t) = (!(t) ( − re) + rX (t) − c(t)) dt + !(t) dW(t);
a1 !(t) !(t) + a2 !(t) + bc(t) 6 R:
(27)
(28)
4. Optimality conditions
In solving the optimal portfolio problem, the dynamic programming methodology
developed by Bellman will be applied. 5 The optimal portfolio problem is shown to
be equivalent to the problem of -nding a solution to the HJB-equation (Kamien and
Schwartz, 1991; Bjork, 1998). De-ne the optimal value function as
T
U (s; c(x; s)) ds ;
(29)
V (x; t) = sup E
!(x; t);c(x; t)
t
where x is a possible state of Xt . Denote
G(x; !(x; t); c(x; t)) ≡ (!(x; t) ( − re) + rx − c(x; t))
and
H (!(x; t)) ≡ !(x; t) !(x; t);
the corresponding HJB-equation is given by
9V
9V
U (t; c(x; t)) + G(x; !(x; t); c(x; t))
+ sup
9t
9x
!(x; t);c(x; t)
1
92 V
+ H (!(x; t)) 2 = 0
2
9x
5
(30)
The basic principle of dynamic programming, called the principle of optimality, can be put roughly as
follows. An optimal path has the property that whatever the initial conditions and control values over some
initial period, the control (or decision variables) over the remaining period must be optimal for the remaining
problem, with the state resulting from the early decisions considered as the initial condition. By exploiting
this principle a partial di;erential equation, known as the HJB-equation, can be derived.
K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
1323
with the boundary conditions
V (x; T ) = 0;
V (0; t) = 0:
(31)
In solving the HJB-equation, the static optimization problem
1
92 V
9V
max
+ H (!(x; t)) 2
U (t; c(x; t)) + G(x; !(x; t); c(x; t))
!(x; t); c(x; t)
9x
2
9x
(32)
subject to the VaR constraint
a1 H (!(x; t)) + a2 !(x; t) + bc(x; t) 6 R
(33)
can be tackled separately to reduce the HJB-equation to a non-linear partial di;erential
equation of V only. Introducing the Lagrange function as
L(!(x; t); c(x; t); '(x; t)) = U (t; c(x; t)) + G(x; !(x; t); c(x; t))
+
9V
9x
92 V
1
H (!(x; t)) 2 − '(x; t)(R − a1 H (!(x; t))
2
9x
− a2 !(x; t) − bc(x; t));
(34)
the -rst-order necessary conditions of the static optimization problem are given by
9V
!(x; t)
92 V
( − re)
(35a)
+ a2 = 0;
+ !(x; t) 2 + '(x; t) a1 9x
9x
H (!(x; t))
9U
9V
=
− '(x; t)b;
9c
9x
'(x; t)(R − a1 H (!(x; t)) − a2 !(x; t) − bc(x; t)) = 0;
'(x; t) 6 0:
(35b)
(35c)
(35d)
Rearranging (35a) gives
!opt (x; t) =
−Vx −1 ( − re) − 'opt (x; t)−1 a2
;
Vxx + 'opt (x; t)a1 = H (!opt (x; t))
(36)
where Vx ≡ 9V=9x and Vxx ≡ 92 V=9x2 . In addition, (35b) is used to solve for copt (x; t)
while (35c) is applied to solve for 'opt (x; t) whenever 'opt (x; t) = 0. Substituting
!opt (x; t); copt (x; t) and 'opt (x; t) into (30) gives
9V
9V
92 V
1
+ U (t; copt (x; t)) + G(x; !opt (x; t); copt (x; t))
+ H (!opt (x; t)) 2 = 0;
9t
9x
2
9x
(37)
which can then be solved for the optimal value function Vopt (x; t). Because of the nonlinearity in copt (x; t) and !opt (x; t), the -rst-order conditions together with the HJBequation are a highly non-linear system and numerical methods are required to solve
for !opt (x; t), copt (x; t); 'opt (x; t) and Vopt (x; t) iteratively.
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K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
5. Numerical methods and results
In many cases, the utility function is de-ned to be a power function of consumption,
such as
U (t; c(x; t)) = e−)t c(x; t)* ;
) ¿ 0; 0 ¡ * ¡ 1:
(38)
The economic reasoning behind this is that we now have an in-nite marginal utility at
c = 0. This will force the optimal consumption plan to be positive throughout the planning period. Merton (1971) has derived the analytical solution for the value function
V (x; t) under this type of utility function. He used a trial function of the form
V (x; t) = e−)t h(t)x* ;
) ¿ 0; 0 ¡ * ¡ 1;
(39)
which separates the x and t variables. Substituting the trial function into the HJBequation, it reduces to a Bernoulli equation for h(t) which is an ordinary di;erential
equation. Here, this approach is extended for the VaR-constrained optimal portfolio
problem. It is derived for one risky asset plus a risk-free asset, but can be extended
easily for n risky assets.
When the VaR constraint is imposed, although the variation of V (x; t) in x is still
well modelled by the term x* , the function h will depend on x as well because of
! and c. However, we shall show later numerically that for reasonable parameters, h
is a slow varying function of x and its derivative with respect to x is therefore very
small. Let the utility function be de-ned by (38), then the HJB-equation for the value
function is given by
9V
9V
*
+ e−)t copt
(x; t) + (!opt (x; t)(. − r) + rx − copt (x; t))
9x
9t
1 2
92 V
+ !opt
(x; t)$2 2 = 0:
2
9x
(40)
Neglecting the derivatives of h with respect to x, we have
9V
= *e−)t h(x; t)x*−1 ;
9x
92 V
= *(* − 1)e−)t h(x; t)x*−2 ;
9x2
9V
= e−)t h (x; t)x* − )e−)t h(x; t)x* :
9t
(41)
(42)
Substituting into (40), dividing by e−)t x* and rearranging gives
h (x; t) + A(!opt (x; t); x)h(x; t) + B(copt (x; t); h(x; t)) = 0
(43)
with the terminal condition
h(x; T ) = 0;
where
(44)
A(!opt (x; t); x) = *
!opt (x; t)
(. − r) + r
x
+
2
(x; t) 2
1 !opt
$ *(* − 1) − );
2
x2
(45)
K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
B(copt (x; t); h(x; t)) =
*
(x; t) *h(x; t)copt (x; t)
copt
:
−
x*
x
1325
(46)
To avoid the singularity in calculating negative powers of h(x; t) near to the terminal
time T , (43) is transformed into
g (x; t) + (1 − 0)A(!opt (x; t); x)g(x; t) + (1 − 0)B(copt (x; t); g(x; t))g(x; t)* = 0
(47)
with the terminal condition
g(x; T ) = 0;
(48)
where
g(x; t) = h(x; t)1−0 ;
0=−
*
:
1−*
(49)
In the unconstrained case, (47) reduces to
g (t) + (1 − 0)Ag(t) + (1 − 0)B = 0;
where
A=*
(. − r)2
1 (. − r)2
+
r
+
$2 (1 − *)
2 $2 (1 − *)
(50)
− );
B = 1 − *:
(51)
This unconstrained solution will be used as an initial guess to the iterative algorithm.
Dividing the computational domain into a grid of Nx ×Nt mesh points and omitting (x; t)
in all variables for the simplicity of notation, the -nal algorithm can be summarized
as follows:
(0)
(0)
(0)
(1) 'opt
= xh−1=(1−*) and solve (50) for the
= 0; !opt
= −(. − r)x=($2 (* − 1)); copt
unconstrained solution. Set k = 0.
(k+1)
(2) For x = [0; Px; : : : ; Nx Px] and t = [(Nt − 1)Pt; : : : ; Pt; 0], calculate 'opt
from
(k+1)
(k)
(k)
(R − (a1 $ + a2 )!opt
− bcopt
) = 0;
'opt
(k+1)
!opt
=
(k+1)
(a1 $ + a2 )
−Vx(k) (. − r) − 'opt
$2 Vxx(k)
(52)
;
(53)
(k+1)
and copt
from
(k+1) *−1
(k+1)
)
= (*x*−1 h(k) − e)t 'opt
b):
*(copt
(54)
(3) For x = [0; Px; : : : ; Nx Px] and n = Nt − 1; : : : ; 0, solve
(k+1)
(k+1)
(k+1)
(k+1) (k)
+ Pt(1 − 0)A(!opt
; x)gn+1
+ (1 − 0)B(copt
; gn )(gn(k) )* :
gn(k+1) = gn+1
(55)
(4) Return to 2 with k = k + 1 until convergence.
A Fortran program has been written to implement the above procedure and is executed on a Digital alpha workstation. The terminal year is chosen to be 20 and Nt is
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K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
Fig. 1. Asset allocations with and without the VaR constraint. In this case, the parameters are
1
, the maximum loss allowed
. = 0:2; $ = 0:5; r = 0:1; ) = 0:2; * = 0:5, the horizon period is Pt = 50
is R = 100 with probability k = 0:01. In the -gure, kinks are produced whenever the VaR constraint becomes
active to reduce investments in the risky asset.
1
-xed at 1000, which corresponds to a VaR horizon period Pt = 50
≈ 7 days. The
stochastic process is chosen arbitrarily with . = 0:2; $ = 0:5 and the risk-free rate is
r = 0:1. The parameters in the utility function are taken to be ) = 0:2; * = 0:5. For
the VaR constraint, the maximum loss is limited to R = 100 with probability k = 0:01.
Finally, Px = 2 and Nx = 500 are used which has the range x ∈ [0; 1000]. In this case,
the unconstrained solution suggests that ! = 0:8x.
A total of -ve cycles have been executed where changes in h(x; t) became negligible.
Fig. 1 compares the asset distribution for di;erent portfolio values with and without the
VaR constraint at di;erent times. From the -gure, a good control over the investment
in the risky asset has been achieved and the proportions invested in the risky asset have
been reduced in order to ful-l the VaR constraint. In particular, when the constraint
is not active, the optimal portfolio follows the unconstrained solution; as the portfolio
value increases, the VaR constraint becomes active and allocates less to the risky
asset. This is particularly so when t is close to the maturity time T . The points where
the constraint becomes active produces kinks in the curves. Figs. 2 and 3 show the
Lagrange multipliers and the VaR values along x at two di;erent times, respectively.
They show the exact portfolio values when the VaR constraint becomes active where
the Lagrange multipliers are negative instead of zeroes.
The consumption is depicted in Fig. 4 along x at two di;erent times. Clearly, very
similar consumption patterns are obtained for both unconstrained and constrained optimization. Thus, for this set of parameters, the consumption pattern has not been a;ected
greatly by the risk control. Fig. 5 depicts the optimal value function Vopt (x; 0) with
and without the VaR constraint. Since the utility function is de-ned solely by the discounted consumption and because the consumption pattern is not changed very much,
K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
1327
Fig. 2. The Lagrange multipliers show that the VaR constraint becomes active whenever ' becomes negative.
Fig. 3. The VaR for di;erent portfolio values.
therefore, only a small decrease in the optimal total expected utility is observed by
imposing the VaR constraint in this example.
In order to demonstrate the slow varying property of h in x, Fig. 6 depicts the
h(x; t) function over time. When x is small, the VaR constraint is not active which
implies h is a function of t only. When x is large, the VaR constraint becomes active
and h depends on x as well. Therefore, in the -gure, two values of x are plotted,
namely x = 1000 where the VaR constraint is active and x = 500 where the VaR
constraint is mostly inactive. Clearly, from the -gure, only a very little variation in
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K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
Fig. 4. The consumption pattern for di;erent portfolio values at two di;erent times show that the consumption
pattern is not a;ected very much by the imposed risk constraint in the -rst example.
Fig. 5. The optimal total expected utility for di;erent portfolio values shows that it has not been decreased
very much by the imposed risk constraint in the -rst example.
x is observed. Therefore, neglecting the derivatives of h(x; t) with respect to x will
not a;ect the -nal results very much in this example. Furthermore, the accuracy of
the solution can be assessed by calculating the residual value of the HJB-equation
as
9Vopt
9Vopt
92 Vopt
1
+ Uopt + Gopt
+ Hopt
2i; j =
;
(56)
2
9t
9x
9x2 i; j
K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
1329
Fig. 6. The plot of the h(x; t) function over time shows that it is a slow varying function in x in the -rst
example.
where the derivatives of Vopt are calculated via the -nite di;erence approximations.
The discrete errors can then be measured by the norm
Nx N t
”AV = 2i;2 j =(Nt × Nx ):
(57)
i=1 j=1
In this example, ”AV = 4:804e−2 has been achieved.
In the second example, the set of parameters are chosen so that the risky asset
resembles the behaviour of a stock index. Therefore, the stochastic process is chosen
as . = 0:12; $ = 0:2 and the risk-free rate is r = 0:05. The parameters in the utility
function are taken to be ) = 0:1; * = 0:3. The rest of the parameters are same as in the
-rst example. In this case, because the $ is relatively small, the unconstrained solution
suggests a signi-cant borrowing to invest in the risky asset which is equal to ! = 2:5x.
However, when the VaR constraint is imposed, the amount of borrowing is reduced,
as shown in Fig. 7. Thus, risk control has been achieved. The portfolio values where
the constraint becomes active can be seen from the Lagrange multipliers and the VaR
plot as shown in Figs. 8 and 9, respectively. Again, the h(x; t) function over time is
plotted in Fig. 10 in order to demonstrate the slow varying property of h in x. The
discrete error measure of the HJB-equation is given by ”AV = 2:792e−2 , which shows
that the solution obtained is reasonably accurate.
Finally, Figs. 11 and 12 depict the comparisons of consumption and the optimal total
expected utility, respectively. It shows that for this set of parameters, the consumption
is not greatly a;ected, and hence there is only a small drop in the optimal total expected
utility even after the VaR constraint is imposed. In order to understand the inKuence of
* on the optimal total expected utility, we increase * in the utility function to 0.5 so that
the agent is now less risk-aversed. The unconstrained solution (! = 3:5x) suggests that
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K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
Fig. 7. Asset allocations with and without the VaR constraint. In this case, the parameters are
1
, the maximum loss al. = 0:12; $ = 0:2; r = 0:05; ) = 0:1; * = 0:3, the horizon period is Pt = 50
lowed is R = 100 with probability k = 0:01. In this case, the agent is to borrow and invest into the risky
asset. Again, imposing the VaR constraint reduces the allocation to risky asset and produces kinks.
Fig. 8. The Lagrange multipliers show that the VaR constraint becomes active whenever ' becomes negative.
even more money should be borrowed to invest into the risky asset. Once risk control
is imposed, the investment in the risky asset is decreased signi-cantly. Under this
circumstance, the optimal total expected utility is decreased by a noticeable amount as
shown in Fig. 13. The decrease is caused by a wider variation in h(x; t) in x as shown in
Fig. 14. The intuition behind this is obvious because when the agent is less risk-aversed,
the risk constraint will certainly impose more restrictions on the agent’s decision on
investing heavily in the risky asset and the decrease in the optimal total expected
K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
1331
Fig. 9. The VaR for di;erent portfolio values.
Fig. 10. The plot of the h(x; t) function over time shows that it is a slow varying function in x again in the
second example.
utility will therefore be larger. Another problem has been raised here. Because of the
variation of h(x; t) in x, the discrete error measure of the HJB-equation has dropped
to ”AV = 0:353, showing that the numerical method in solving the HJB-equation is
not as accurate as in the other examples.
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K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
Fig. 11. The consumption pattern for di;erent portfolio values at two di;erent times show that the consumption pattern is not a;ected very much by the imposed risk constraint in the second example.
Fig. 12. The optimal total expected utility for di;erent portfolio values shows that it has not been decreased
very much by the imposed risk constraint in the second example.
6. Conclusion
The optimal portfolio problem together with a VaR constraint has been studied. For
the continuous-time model where the constraint is applied continuously, the dynamic
programming technique and the stochastic control theory have been applied to reduce
the problem to solving the HJB-equation coupling with the VaR constraint. The method
K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
1333
Fig. 13. The optimal total expected utility for di;erent portfolio values. In this case, the parameters are
1
. = 0:12; $ = 0:2; r = 0:05; ) = 0:1; * = 0:5, the horizon period is Pt = 50
, the maximum loss allowed is
R = 100 with probability k = 0:01. Since * is larger here, the agent is less risk-aversed and therefore prone
to make riskier decision when the risk constraint is not imposed. This accounts for the noticeable drop as a
result.
Fig. 14. The plot of the h(x; t) function over time shows that it has a higher variation in x in this example.
of Lagrange multiplier has been applied to handle the constraint. A numerical method
has been proposed to solve for the constrained optimal portfolios.
From the numerical results, we -nd that the constrained optimal portfolios invested
less in the risky asset. This is due to the fact that the VaR constraint is applied over
time so that there is a direct relationship between VaRs and portfolio holdings at each
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K.F.C. Yiu / Journal of Economic Dynamics & Control 28 (2004) 1317 – 1334
instant. We believe that any risk controls imposed in a way similar to here will achieve
similar results. Since the constraint is imposed di;erently from other models in the
literature, the method provides another insight into the problem of optimal portfolios
with regulatory controls over market risks. Although the results derived here will still
hold if r; and are functions of time, allowing for stochastic r; and will be
more complicated. It would certainly be of interest to look at this as an extension to
the present study.
Acknowledgements
The author would like to thank Professor William Perraudin, Dr Mike Orszag and
Professor Elias S.W. Shiu for encouragements and helpful discussions, and to thank
the three anonymous referees for their comments.
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