Technology and Investment, 2013, 4, 36-41
Published Online Febr uary 2013 (http://www.SciRP.org/journal/ti)
A New Class of Time-Consistent Dynamic Risk Measures
and its Application
Rui Gao, Zhiping Chen
School of Mathematics and statistics, Xi’an Jiaotong University, Xi’an, China
Received 2012
ABSTRACT
We construct a new time consistent dynamic convex cash-subadditive risk measure in this paper. Different from exist-
ing measures, both potential loss and volatility of risky objects are considered. Based on a one-period measure that dis-
torts financial values, punishes downside risk yet rewards upside potential, a dynamic time consistent version is con-
structed recursivel y through a modified translation property. We then establish a portfolio selection model and give its
optimal condition.
Keywords: Dynamic Risk Measure; Time-Consistent; Cash-Subadditive; Portfolio Optimization; Stochastic
Programming
1. Introduction
Financial activity is teemed with risk, therefore it is
crucial to construct reasonable risk measures and utilize
them on the optimal portfolio selection. One popular
definition of risk is volatility of random return of
portfolio, originated from Markowitz’s prominent
mean-variance model. Following him, hundreds of
momen t-based risk measures were proposed, such as
Mean Absolute Deviation [1] and Lower Partial Moment
[2]. Another common notion of risk is potential downside
loss below a certain target. Correspondingly, a number of
downside risk measures has been suggested in the
literature, such widely used measures as Value-at-Risk
(VaR) and Conditional Value-at-Risk (CVaR). In all the
above financial risk measures, the attention is put on
either the volatility of random return or the potential loss.
Nevertheless, this is insufficient. Both these two aspects
should be considered simultaneously. Only concentrating
on the volatility of return ignores the information on the
degree of potential loss; while merely emphasizing
potential loss not only neglects the dispersion of future
return but also throws away upside data. What’s more,
when considering the volatility of random return, many
researchers punish both downside risk and upside
potential. However, the volatility of random return above
certain target implies the potential of gaining much more
than expected. A higher upside variability generally
indicates a higher possibility to acquire good upside
performance, which is desirable for each rational investor.
Hence, upside potential should be rewarded. Based on
the analysis above, we believe that when defining risk
and its measure, we should consider both the potential
loss and volatility into consideration and distinguish
between downside risk and upside potential.
Generally speaking, an ideal risk measure should
satisfy some properties. In their seminal paper [3],
Artzner, Delbaen, Eber, Heath established an axiomatic
notion of coherent risk measures. They proposed that an
ideal risk measure should satisfy four properties:
monotonicity, subadditivity, positive homogeneity and
translation -invariance. Though it has been accepted by
many scholars, it is not perfect. For example, positive
homogeneity sometimes does not hold because a
financial position’s risk increases in a nonlinear way with
its volume due to liquidity risk. Hence, Follmer and
Schied [4] replaced subadditivity and positive
homogeneity by convexity and established a more
general concept of convex risk measures. In addition,
translation -invariance is questioned in [5] since the
ambiguity on interest rates and is suggested to be
replaced with cash-subadditivity, which implies that
additional loss of some amount of money is covered by
an additional reserve of the same amount. Hence, we
believe that for one-period risk measures it is reasonable
to assume monotonicity, convexity and
cash-subadditivity. From the perspective of economics
and finance, an ideal risk measure should reflect
inves to r ’s risk-averse attitude because risk is always a
subjective notion [6].
During the recent decade, dynamic risk measure has
attracted many researchers and practitioners, of which
the most important feature is time consistency describing
how risk assessments at different times are interrelated.

R. GAO, Z. CHEN
By certain translation property, which corresponds to
translation -invariance in one-period setting, time
consistent risk measures can be completely defined by
conditional risk measures recursively [7]. Thus, an usual
way to construct time-consistent risk measures is
establishing a static risk measure first and extending it to
dynamic setting by translation property. As illustrated
later, existed translation property cannot reflect risk
aversion and we will modified it into another version.
Bearing in mind the above limitation in existed risk
measures, a new class of time consistent dynamic
cash-subadditive convex risk measure is constructed.
Comparing with existing measures, our new risk measure
has the following advantages: we take into account the
potential loss and volatility of both downside risk and
upside potential, and thus the whole domain distribution
is utilized, which makes the new measure superior for
finding robust and stable investment decisions; by
suitably selecting the parameters in the model, our risk
measure can explicitly reflect the investor’s risk attitude;
when the risk measure is applied to portfolio selection
model, we give its optimality condition, which is useful
in determining the stochastic dual dynamic programming
method to solve the risk-averse multistage problem.
This paper is organized as follows. Section 2.1 gives
the definition and property of the new one-period risk
measure, which is then extended to dynamic setting in
Section 2.2. We apply the risk measure to portfolio
selection model in Section 3 and presents our conclusion
in Section 4.
2. The New Risk Measure and its Properties
2.1. One-Period Setting
We first consider a one-period framework. Given a
probability space(,,)ΩPF , denote the random cost,
discounted by certain numéraire, of at time
T
by essen-
tially bounded random variable
X
in
()L∞F
. A
one-period risk measure is a mapping
: ()L ρ
∞→F
. A
larger value of
ρ
implies a riskier cost
X
. For
a∈
,
we denote
[]a
+
by
max{0, }a
. The notation “
:=
” means
“equal by definition”. For a random variable
X
,
[]XE
denotes its expectation;
x
α
is the
α
quantile of
X
. We
denote the indicator function of set
A
by
A
1.
For reasons demonstrated in the introduction, we pro-
pose here a new type of risk measure that takes into ac-
count the potential loss, downside risk, upside potential
and risk aversion. In order to illustrate the derivation of
our new risk measure, we look back on the notable
downside risk measure VaR, which is defined
as
1
VaR() :Xx
αα −
=
. When the random cost
X
is reduced
by
1
x
α −
amount of money, it becomes acceptable in the
sense that the random cost
Xx
α
−
is less than zero up to a
loss with probability
α
. However, as pointed out before,
the volatility of such random cost should also be meas-
ured. We use absolute deviation to measure the upside
potential and downside risk separately. The average
downside deviation is
11
[| ]Xx Xx
αα −−
≥−E, which
equals to
1
1
[]Xx
α
α
−+
−
−E
. Similarly, the upside potential
is
1
1
[(1 ]) Xx
α
α
−+
−
−−E
. Since we punish downside risk
and reward upside potential, the volatility of
1
Xx
α −
−
is
11
11
[ ](1)(1)[ ]XXx x
αα
λαλ α
−+ −+
−−
−− −−−EE
, where
λ
reflects the asymmetry between downside and upside.
To combine the potential loss and volatility together, a
simple way is by linear weight. It is not difficult to show
that to make the weighted sum be monotone, they have to
have equal weight. Moreover, taking into risk aversion
into consideration, all the financial values af or eme n-
tioned, including
X
and
1
Xx
α −
−, should be distorted by a
monotonically decreasing convex function
()
·w. Such
property of
w
indicates that risk-averse investors em-
phasize more on undesirable situation. For normalization
condition, we require(0) 0w=. Finally, we define our
new risk measure as follows.
Definition 1. Give n
[0,1] λ ∈
,
(0,1) α ∈
, the new
one-period risk measure
,
: ()L
λα
ρ
∞
→F
is defined
as
1
,1 1
1
()( )[()( )]
(1)(1)[ ()() ]
XwxwX wx
wx wX
λα αα
α
ρ λα
λα
−+
−−
−+
=+−
−− −−
E
E
(1)
whe re
inf{: []}xPX xx
α
α = ∈≤≥
and
()
·w is a
monot onically increasing convex continuous function
satisfying normalized condition
(0) 0w=
.
Rewriting the expectation in Equation (1) in integral
form, the measure has an equivalent form which facili-
tates us to study its property, demonstrating by the fol-
lowing proposition:
Proposition 1. For any
0 1 αλ <≤≤
, the risk measure
, λα
ρ
defined by (1) can be equivalently written as
1
1
,
1
() [()]()
1(1 )
u
XwXw xdu
λα α
λ αλ
ρα αα
−
−−
= +
−−
∫
E
(2)
Obviously, if
1 λ =
,
1
1,
1
( ())
u
wXx du
αα
ρ
−
=
∫
is
α
time
of
TNT ()X
α
defined in [8]; if
()wx x=
,
,
() (1)[]CVaR()XX X
λα α
ρ ββ =−+E
, where
1
(1 )(1) β λα
−
=−−
, is appeared in [9]; further if
1/2 α =
,
,1/ 21/ 21/ 2
[]2[]2(1)[]XXxx X
λ
ρλ λ
++
=+− −−−EE E
is
the deviation measure suggested in [10]. Therefore, our
new risk measure can be regarded as extensions to all
these risk measures.
The choice of depends on the investor's attitude toward
risk controls the heavy tails of loss distribution. Typical
decreasing convex functions are
12
[][ ]XX ββ
++
−−
37

R. GAO, Z. CHEN
(
12
0 ββ >>
),
exp() 1x β −
(
0 β >
),
x
β
( 1)
β ≥
. As for
concrete selection of and corresponding parameters, one
can refer to [8] for a detailed discussion.
The following proposition shows that under certain
mild specification, the new risk measure satisfies several
desirable mathematical properties.
Proposition 2. For any
0 1 αλ <≤≤
and
()
·w that is
differentiable and satisfies
0 1'w≤≤
, then the risk
measure defined in Equation (1) is a law-invariant,
convex, cash-subadditive risk measure.
Proof. The first two are direct corollaries of Theorem 1.
To prove the cash-subadditivity, we show that
,,
( )()XmX m
λα λα
ρρ +≤ +
. Rewriting the new risk
measure as
1
,0( )() u
w xu du
λα
ρφ
=
∫
, where
[0, )[ ,1]
():/() ()/(1)()uu u
αα
φλαλ αα =+− −11
, by
()
uu
xm xm+= +
, we have
1
,0
1
0
1
0
1
0
,
[( )
()(() )()
( )()
( )()
( )()
(
]
),
u
u
uu
u
Xmwxmudu
wxmu
w
du
wmudu
wxu um
X
x
d
m
λα
λα
ρφ
φ
ξφ
φ
ρ
+
= +
+= +
=
′
≤+
= +
∫
∫
∫
∫
where
[ ,]
uu u
x mx ξ ∈−
is determined by the mean
value theorem.
Besides, Equation (1) contains two parameters
λ
and
α
, which can be flexibly reflect investor's attitude
toward risk.
λ
is a factor linearly adjusting the balance
between downside risk and upside potential; is the
confidence level that the investors can accept. More
specifically, we have the following theorem.
Proposition 3. T he coherent risk measure
, λα
ρ
is
increasing with respect to
λ
, decreasing with respect to
α
, and continuous with respect to
α
and
λ
.
The monotonicity property of
, λα
ρ
with respect to
/ λα
can be used to reflect the investor’s attitude
toward risk. Concretely, the increasing property of
, λα
ρ
with respect to
λ
indicate that the greater the
λ
, the
larger the
, λα
ρ
. Investors who adopt a larger
λ
treat
X
riskier than those who choose a smaller
λ
. They
have a stronger tendency to risk aversion because the
concentrate more on downside risk than on upside
potential. When
1 λ =
, investors only consider downside
risk. On the other hand, the decreasing property of
, λα
ρ
with respect to
α
means that
, λα
ρ
with large
α
should be connected with the less risk-averse investor.
They are less conservative in the sense that they bear a
larger probability of loss.
Stochastic dominance rules are often utilized to
judge a new risk measure. From the point of view of
utility theory, it is desirable for a risk measure to
preserve second order stochastic dominance (SSD). An
equivalent definition of SSD in terms of quantile
function is the following:
SSD
XY
if and only if
0
()0, 01,
p
uu
xy dup−≤∀≤≤
∫
and there is a strict inequality of at least one
0
p
.
Proposition 4. The risk measure
, λα
ρ
preserves second
order stochastic dominance.
Proof. It suffices to prove the case when two random
variables
SSD
XY
whose quantile functions satisfy
uu
x y≡
/
in any interval ([0,1],)ab ⊂. Since the quantile
function is right-continuous, there exists only countable
intersection of u
x
and u
y
. If there is no intersection, it
follows that
uu
x y≤
and thus
,,
( )()XY
λα λα
ρρ ≤
due
to the monotonicity of
()
·
w
. Otherwise denote all the
intersection point ofu
x
and u
y
by
1
{}
N
nn
u
=
,
1N≤≤∞
,
and
0
::0,1, ()
Nk
uuk
+
+
== ∈
. Then we have
1
21
2
22
21
21
2
22
21
,,
0
[ /2]
0
[ /2]
0
'( )()
'( )(
()()[()()]( )
[()()]( )
[()()]()
()
())
0,
n
n
n
n
n
n
n
n
n
n
Nu
uu
u
n
Nu
uu
u
n
u
uu
u
Nu
nuu
u
n
u
n uu
u
XYw xwyu du
w xwyudu
w ywxu du
yu du
y xud
w
u
x
w
λα λα
ρρ φ
φ
φ
ξφ
ηφ
+
+
+
+
+
+
+
=
=
=
=
=
−
−= −
−
−−
−
−
≤
∑∫
∑∫
∫
∑∫
∫
where
2 21
[, ]
n nn
uu
ξ
+
∈ and
21 22
[, ]
n nn
uu
η
++
∈ are
determined by the mean value theorem. The last
inequality is deduced by the convexity of
()
·w and the
monotonicity of
()
u φ .
At the end of this section, we consider the computation
and minimization of
, λα
ρ
. Let
:(, )X gx ω =
be the
random cost associated with the decision vector
x∈G
,
representing element of feasible set
G
, and the random
vector
ω
, standing for the uncertainties in the market
which affects the random cost, such as capital gains and
dividends. Similar to the arguments in [11], we have the
following proposition.
Proposition 5. Intr oducing the following auxiliary
function
38

R. GAO, Z. CHEN
1
,
1
( ,)[((,))]{[(((,))) ]},
11
GxEwgxE wgx
λα
λ λα
ηωηαω η
αα
−+
−−
=++ −
−−
then
, λα
ρ
has an equivalent form
,,
( )min( ,).x Gx
λαη λα
ρη
∈
=
Minimization of
, λα
ρ
with respect to
X∈X
is
equivalent to
,(,) ,
min()min( ,)
xx
x Gx
λαη λα
ρη
∈ ∈×
=
GG
.
Moreover, we have
(,)
,,
(,)arg min
arg min(),arg min(,).
x
x
x
xx Gx
η
λα λα
η
ξ
ρη η
∗∗
∈×
∗∗
∈∈
∈
∈∈
⇔


G
G
2.2. Multi-period setting
In this section we extend our new one-period risk
measure
, λα
ρ
to the dynamic setting. Consider a
filtered probability space
1
, ,( )(),
T
tt=
ΩFF P
, where
1},{=∅ ΩF
and
T
= ΩF, and an adapted stochastic
process
t
X
,
, ,1t T= …
, representing discounted random
return process. Define the space
(, ,)
tt
∞
=Ω PLL F
and
,tT tT
= ×LL L
. The one-period risk measure naturally
induces a sequence of one-period conditional risk
measure
1
1
{} :
T
ttt
ρ
−
=
→L
:
1
1
()[()| ]
1
inf[(()},){
|]
t
tt
t
tt
tt
t
X wX
wX
η
λ
ρα
λα ηα η
α
−+
∈
−
=−
−
++−

E
E
F
F
A dynamic risk measure for stochastic process
,,
tT
X X…
is a sequence of conditional risk measures
1
,1
{}
T
tT t
ρ
−
= , where
,,
:
tTtTt
ρ →LL
assesses the risk of the
sequence
,,
tT
X X…
from the perspective of time
t
. Our
aim is to construct the dynamic risk measure
1
,1
{}
T
tT t
ρ
−
=
based on the conditional risk measure
1
1
{}
T
tt
ρ
−
=
.
The key to constructing dynamic risk measures from
one-period ones is the translation property, which arises
from translation -invariance property of one-period risk
measure. A static risk measure
ρ
satisfies
translation -invariance property if for all
m∈
,
() ()XmX m ρρ += +
, which implies cash-invariance,
i.e., ()
mm
ρ =. This suggests the risk of a riskless cost,
in terms of potential loss, can be described as the its
present value. Then the corresponding translation
property in existed papers is for all
1,
{}
tT
X∈L
and
,11, Tt …−=
,
, 1,1
(,,, )(0,,, )
tTt tTttTtT
XXXX XX ρρ
++
…=+ …
(3)
Condition (3) indicates that the risk of a stochastic
process ),( ,
tT
X X…from the perspective of time
t
is the
aggregation of its riskless component
t
X
and its risky
component 1,
()
,
tT
X X
+…, and by translation-invariance
property, the risk of the riskless component
t
X
is its op-
posite of its value. Nevertheless, when taking into ac-
count the interest rate [5] and more importantly, the in-
vesto r ’s risk-ave rs e behavior, it is better to measure risk-
less object’s risk by a distortion of its value instead of
itself, i.e ., the cash-invariance and corresponding transla-
tion-invariance should be replaced with
() (),() ()(),mwmXmXwm m ρ ρρ =+= +∈ (4)
where ()·wis a monotonically decreasing convex conti-
nuous function. The monotonicity of
()
·w guarantees
that the smaller the
m
, the riskier it is, and the convexity
entails risk-aversion. Intuitively speaking, the dynamic
risk measure
1
,1
{}
T
tT t
ρ
−
=
induced by conditional risk meas-
ure sequence
1
1
{}
T
tt
ρ
−
=
should satisfies
,1 11
(0,)(),1, ,1,
tttt t
XXtT ρρ
++ +
= =…−
(5)
which identifies the conditional risk measure t ρ with
the dynamic risk measure for two-period process
,1tt
ρ
+
.
According to (4) and (5), translation property (3) is
modified into for all
1,
{}
tT
X∈L
,
,11, 1
(,,, )()(,(, )),
tTtt TttttTt T
XXXwXXX ρ ρρ
+ ++
…= +…
(6)
whe re
)·(
t
w
is a monotonically decreasing convex conti-
nuous fu nct ion.
Based on the above analysis, we define our new
dynamic risk measure as follows.
Definition 3. Let
1
{}
T
tt
w
=
is a sequence of monotonically
decreasing convex differentiable function satisfying for
all
1tT≤≤
,
(0) 0
t
w=
and (1
)0
wt−′
≤≤
. The new time
consistent dynamic risk measure
1
,1
{}
T
tT t
ρ
−
=
induced by
one-period risk measure (1) through modified translation
property (6) is recursively defined as
,1, 1
(()(,, )), ,11,
tTtttt TtT
wtX XTX ρ ρρ
++ == +……−
(7)
As a fo r e me nt ioned , time consistency is the most
important issue of dynamic risk measures. One of the
most commonly used versions is introduced in [12].
Definition 2. A dynamic risk measure1
,1
{}
T
tT t
ρ
−
=is time
consistent if for all
1T τθ ≤≤<
and all seque nces
1
{}
T
tt
X
=
,
1,
{}
T
t ttT
Y=∈L,
,,
,,,1 and
( ,,)(,,)
kk
TTT T
Yk
XX Y
X
Y
θ θθθ
τθ
ρρ
= =…−
…≤…
(8 )
39

R. GAO, Z. CHEN
imply
,,
(,, )(,,)
tTTT Tt
X XYY
ττ
ρρ …≤ …
.
This definition is intuitive since it indicates that if
the future subsequence of sequence
t
X
is at least as good
as the subsequence of another sequence
t
Y
and today’s
value of
t
X
is the same as that of
t
Y
, then
t
X
is at least
as good as
t
Y
from the perspective of today. We show that
our new risk measure satisfies time consistency.
Proposition 6. Suppose a dynamic risk measure
1
,1
{}
T
tT t
ρ
−
=
satisfies condition (6)-(7), then it is time consistent if and
only if for all
1T τθ ≤≤<
and all
1,
{}
tT
X∈L
,
,
1
, 1,
( ,,,,)
( ,,,( ,,),0,,0)
TT
T TT
XXX
X XwXX
ττ θ
τ τθθθ θ
ρ
ρρ
−
−
…… =
… ……
(9)
where ·
()
w
τ
satisfies properties stated in Proposition 2.
Proof. Suppose sequences
1
{}
T
tt
X=
,
1,
{}
T
t ttT
Y
=
∈L satisfy
the Equation (9), by the monotonicity of
1
,1
{}
T
tT t
ρ
−
=
and
w
θ
, it follows that
, 1,
, 1,
( ,,,(,,),0,,0)
(,, ,(,,),0,,0)
T TT
T TT
XX XX
YY YY
τ τθθ θ
ττθ θθ
ρρ
ρρ
−
−
… ……
≤ ………
.
If identity (10) holds, then
,,
( ,,)(,,)
TTTT
X XYY
τ τττ
ρρ …≤ …
.
3. Portfolio Selection Model and Optimal
Condition
Based on the new dynamic risk measure, we establish a
multistage portfolio selection model in this section.
Suppose we have initial capital
1
X
in
n
assets at
stage 1, each of which has respective net expected return
rate
1
(, , )
t tnt
rrr= …
at stage
, ,2t T= …
, forming a
random process with a known distribution (for example,
can be determined by Vector Auto Regr e ssion model
()VAR p
). We assume this process is stagewise
independent, i.e., t r is independent of
11
,,
t
rr − …
for
, ,2t T= …
. This assumption is not very realistic but in
most cases, we can transform the across stage dependent
process into stagewise independent by adding state
variables to the model (cf. [13]). Suppose further a
self-balance model, that is, we reallocate our portfolio at
each stage
,11, Tt …−=
, but without investing
additional money during the time period. At each stage
t
,
we decide the amount of the
n
assets
1
(,, )
t tnt
xxx…=
,
satisfying the balance of wealth constraints
,1
11
(1 )
nn
ititi t
ii
x rx
−
= =
= +
∑∑
,
,11,Tt …−=
. The sequence of
decisions
t
x
satisfies nonanticipativity (implementable)
constraints, i.e. ,
t
x
is a function of information available
at the current stage, say the process 1),,(t
r r
…. We
assume there are no short sales or borrowing:
0
it
x≥
for
all
i
and
t
. Our goal is minimizing the risk of the whole
process over all the implementable and feasible policies,
measured by our new dynamic risk
measure
1, 1
(,, )
TT
XX ρ …
.
We write the dynamic programming equations for
the multistage problem. At the stage
11, ,tT=− …
, a
realization of
[] 1
: (,, )
ttrr r= …
is known. We solve the
problem
0
11
1
,, ,1
11
():()( )
s.t. (1
min
)
xt
n
t titttt
i
nn
ititit
ii
xw x
x rx
Vx
≥
−+
=
−
= =
= +
= +
∑
∑∑
V
(10)
whe re
11111
1
[( ):( )],()(1) ()
n
tttttTTTiTT
i
xV xxrx ρρ
++−−−
=
== +
∑
VV
By conjugate duality theory (c.f. [9]), we can prove the
following optimal condition of problem (11).
Proposition 7.
t
x
∗
is an optimal solution of (11) if and
only if there exists
1
()
tt
x π
∗
−
∈D
such that
(, )0tt t Lx π
∗∗
∈∂
, where
1
()
t
x
−
D is the set of optimal
solutions of the dual problem
1
11
1
max sup((1 )
),
{[( )]}
tt
nnn
tit ittittit
x
tt
ii
i
xx wx xr
π
ππ
+
= =
=
− −−+∑∑
∑V
and
(, )
tt t
L x π
is the Lagrangian
1 ,1
1 11
( ,)( )(1)( )(),
n nn
ttttittttititt
i ii
xw xxrxxL ππ
+−
===
=++ +−
∑ ∑∑
V
a
nd the subdifferential of
f
at
x
is denoted as
()fx∂
.
Further, the function
t
V
is differentiable at
1t
x−
and
11
() ()
tt tt
Vxx
−−
∂=11D
, where
11
is a
n
dimensional vector whose elements are 1.
Proof. First, the function
t
V
is convex for
, ,1t T= …
.
Indeed, since
T
w
and
T
ρ
are convex and increasing,
the convexity of
T
V
follows the fact that the
composition of increasing convex function is still convex
and that the minimum function preserves convexity. By
induction and increasing and convex property of t ρ ,
convexity of
, ,,11
t
Vt T=− …
is obtained. Next, since
t
V is finite and continuous with respect to
1t
x−
, by
conjugate duality theory (c.f. Theorem 7.8 in [9]), we
obtain the result.
The optimal condition and the subdifferential of
t
V
is
very useful practically. For example, it has been pointed
40

R. GAO, Z. CHEN
out in [13] that the complexity of Sample Average
Approximation (SAA) method for solving multistage
stochastic programming grows exponentially in the
number of scenarios and stages. An tractable way to
solve the SAA problem approximately is by stochastic
dual dynamic programming (SDDP) method. The
subdifferential of
t
V
is critical when deciding the SDDP
algorithm. Indeed, we can modify the SDDP algorithm
easily in [13] based on
t
V∂
.
4. Conclusion
By linearly combining the downside measure and
dispersion measure which punishes downside risk and
rewards upside potential together, meanwhile distorting
the financial value, this paper proposes a new class of
one-period risk measure. The new static measure satisfies
convexity, cash-subadditivity, preserves second order
stochastic dominance, and can reflect investor’s risk
attitude. Based on this static measure, we then construct a
dynamic time consistent risk measure using a modified
translation property. Under this new dynamic measure,
we establish a portfolio selection model whose goal is to
minimize the risk of the whole process. By conjugate
duality theory, we derive its optimal condition, which
facilitates us implement the SDDP algorithm for solving
the multistage stochastic program. We only consider
several theoretical property of our new risk measure,
nevertheless, whether such measure is practically ideal
needs some empirical research using realistic data. This
issue is left for future research.
5. Acknowledgemen ts
I am grateful to Professor Alexander Shapiro in Georgia
Institute of Technology for his sincere help on my work.
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