41
Other stochastic optimization models involving general risk functional were considered by Dentcheva and Ruszczynski [42], Rockafellar and Uryasev [152]. Model 3.2–3.4
correspond to a new approach in stochastic optimization problem.
3.2 Stochastic dominance
In the stochastic dominance approach, random returns are compared by a point-wise comparison of some performance functions constructed from their distribution functions.
For a real random variable
V
, its first performance function is defined as the right- continuous cumulative distribution function of
V
:
} {
, η
η ≤
= V V
F P
for
R ∈
η
. A random return
V
is said stochastically dominate another random return
S
to the first order Dentcheva and Ruszczynski [42], Lehmann [96] and Quirk and Saposnik [146],
denoted
S V
FSD
f
, if
; ;
η η
S F
V F
≤
for all
R ∈
η
. Define the function
;
2
⋅ V
F
as
α α
η
η
d V
F V
F
∫
∞ −
= ;
;
2
for
R ∈
η
, 3.5
as an integral of a nondecreasing function, it is a convex function of
η
and defines the weak relation of the second-order stochastic dominance SSD. That is, the random return
V
stochastically dominates
S
to the second order, denoted
S V
SSD
f
, if
; ;
2 2
η η
S F
V F
≤
for all
R ∈
η
. The corresponding strict dominance relations for
FSD
f
and
SSD
f
are defined in the usual way:
S V f
if and only if
S V f
,
V S f
. Furthermore, for
, ,
P V
F L
m
Ω ∈
we can define recursively the functions
∫
∞ −
−
=
η
α α
η d
V F
V F
k k
, ;
1
for
1 ,
3 k
+ =
∈ m
R, η
. 3.6 Furthermore, for
, ,
P V
F L
m
Ω ∈
we can define recursively the functions
42
∫
∞ −
−
=
η
α α
η d
V F
V F
k k
, ;
1
for
1 ,
3 k
+ =
∈ m
R, η
. 3.6
Figure 3.3 First order dominance
R E
∈ −
= =
+ ∞
−
∫
η η
α α
η
η
for X
d X
F X
F ;
;
1 2
Figure 3.2 Scond-order dominnce They are also convex and nondecreasing functions of the second argument.
Definition 3.1. A random variable
, ,
1 -
k
P X
F L
Ω ∈
dominates in the kth order another random variable
, ,
1 -
k
P Y
F L
Ω ∈
if
; ;
η η
Y F
X F
k k
≤
for all
R ∈
η
. 3.7
We shall denote relation 3.7 as
Y X
k
f 3.8
and the set of X satisfying this relation as
} :
, ,
{
1
Y X
P X
Y A
k k
k
f F
L Ω
∈ =
−
. 3.9
43
By changing the order of integration we can express the function
;
2
⋅ V
F
as the expected shortfall Rockafellar and Uryasev [152]: for each target value
η
we have
[ ]
+
− =
;
2
V E
V F
η η
, 3.10
where
, max
V V
− =
−
+
η η
. The function
;
2
⋅ V
F
is continuous, convex, nonnegative and nondecreasing. It is well defined for all random variables V with finite
expected value.
3.3 The portfolio problem
Let
n
R R ,...,
1
be random returns of n assets. We assume that
∞ ]
[
j
R E
for all
n j
, 1
=
. Our aim is to invest our capital in these assets in order to obtain some desirable characteristics of the total return on the investment. Denoting by
n
x x ,...,
1
the fractions of the initial capital invested in assets
n ,...,
1
respectively we can easily derive the formula for the total return:
n n
x R
x R
x R
+ +
= ...
1 1
. 3.11 Clearly, the set of possible asset allocations can be defined as follows:
X
{ }
n j
x x
x x
j n
n
, 1
, ,
1 ...
:
1
= ≥
= +
+ ∈
= R
, where
} ,...,
{
1 n
x x
x =
. In some applications one may introduce the possibility of short positions, i.e., allow some
j
x
’s to become negative. Other restrictions may limit the exposure to particular assets or their groups, by imposing upper bounds on the
j
x
’s or on their partial sums. One can also limit the absolute differences between the
j
x
’s and some reference investments
j
x
, which may represent the existing portfolio, etc. Our analysis will not depend on the
detailed way this set is defined; we shall only use the fact that it is a convex polyhedron. All modifications discussed above define some convex polyhedral feasible sets, and are,
therefore, covered by our approach.
44
The main difficulty in formulating a meaningful portfolio optimization problem is the definition of the preference structure among feasible portfolios. If we use only the mean
return
[ ]
x R
x E
=
μ
, then the resulting optimization problem has a trivial and meaningless solution: invest
everything in assets that have the maximum expected return. For these reasons the practice of portfolio optimization resorts usually to two approaches.
In the first approach we associate with portfolio
x
some risk measure
x ρ
representing the variability of the return
x R
. In the classical Markowitz model
x ρ
is the variance of the return,
[ ]
x R
ar x
V =
ρ
, but many other measures are possible here as well.
The mean–risk portfolio optimization problem is formulated as follows:
] [
max x
x
x
λρ μ −
∈
X
3.12 Here,
λ
is a nonnegative parameter representing our desirable exchange rate of mean for risk. If
=
λ
, the risk has no value and the problem reduces to the problem of maximizing the mean. If
λ
we look for a compromise between the mean and the risk. The general question of constructing mean–risk models which are in harmony with
the stochastic dominance relations has been the subject of the analysis of the recent papers Dentcheva and Ruszczynski [41, 42], Rothschild and Stiglitz [155], Ogryczak and
Ruszczynski [127, 128]. We have identified there several primal risk measures, most notably central semi-
deviations, and dual risk measures, based on the Lorenz curve, which are consistent with the stochastic dominance relations.
The second approach is to select a certain utility function
R R
→ :
u
and to formulate the following optimization problem
[ ]
max x
R u
E
x
X
∈
3.13 It is usually required that the function u· is concave and nondecreasing, thus
representing preferences of a risk-averse decision maker. The challenge here is to select
45
the appropriate utility function that represents well our preferences and whose application leads to non-trivial and meaningful solutions of 3.13.
3.4 Consistency with stochastic dominance
