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
The concept of stochastic dominance is related to an axiomatic model of risk-averse preferences Fishburn [52]. It originated from the theory of majorization Hardy,
Litltewood and Polya [70], Marshall and Olkin [109] for the discrete case and was later extended to general distributions Quirk and Saposnik [146], Hadar and Russell [66],
Hanoch and Levy [68], Rothschild and Stiglitz [155]. It is nowadays widely used in economics and finance Bawa[7], Levy [99].
In the stochastic dominance approach, random returns are compared by a point-wise comparison of some performance functions constructed from their distribution functions.
Figure 3.3. Mean–risk analysis. Portfolio x is better than portfolio y in the mean–risk sense, but none of them is efficient.
For a real random variable V, its first performance function is defined as the rightcontinuous cumulative distribution function of V :
R P
∈ ≤
= η
η η
for V
F
V
}, {
. A random return V is said Lehmann[94], Quirk and Saposnik [144] to stochastically
dominate another random return S to the first order, denoted
S V
FSD
f
, if
R ∈
= η
η η
all for
F F
S V
,
. The second performance function
2
F
is given by areas below the distribution function F,
∫
∞ −
=
η
α α
η
d V
F V
F ;
;
2
for
R ∈
η
3.14
46
and defines the weak relation of the second-order stochastic dominance SSD. That is, 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
R ∈
η
. 3.15 see Hadar and Russell [66], Hanoch and Levy [99]. The corresponding strict
dominance relations
FSD
f
and
SSD
p
are defined in the usual way
S V
FSD
f ⇔
S V
SSD
p
. and
S V
SSD
f
3.16 For portfolios, the random variables in question are the returns defined by 3.11. To void
placing the decision vector, x, in a subscript expression, we shall simply write It will not lead to any confusion, we believe. Thus, we say that portfolio x dominates portfolio y
under the FSD rules, if
; ;
y F
x F
η η =
for all
R ∈
η
, where at least one strict inequality holds. Similarly, we say that x dominates y under the SSD rules
y R
x R
SSD
f
, if
; ;
2 2
y F
x F
η η =
for all
R ∈
η
, with at least one inequality strict.
; η
η
x
F x
F
R
=
and
;
2 2
η η
x
F x
F
R
=
.
Figure 3.4. The expected shortfall function. Stochastic dominance relations are of crucial importance for decision theory. It is known
that
y R
x R
FSD
f
if and only if
] [
] [
y R
U x
R U
E E
≥
for any nondecreasing function U· for which these expected values are finite. Also,
y R
x R
SSD
f
if and only if
] [
] [
y R
U x
R U
E E
≥
for every nondecreasing and concave U· for which these expected values are finite see, e.g., Levy [97].
47
For a set P of portfolios, a portfolio
P ∈
x
is called SSD-efficient or FSD-efficient in P if there is no
P ∈
y
such that
x R
y R
SSD
f
or
x R
y R
FSD
f
. We shall focus our attention on the SSD relation, because of its consistency with risk-
averse preferences: if
y R
x R
SSD
f
, then portfolio x is preferred to y by all risk- averse decision makers. By changing the order of integration we can express the function
;
2
x F
⋅
as the expected shortfall Orgyczak ang Rusczynski [129]: for each target value
η
we have
] ,
[max ;
2
x R
E x
F −
=
η η
. 3.17
The function
;
2
x F
⋅
is continuous, convex, nonnegative and nondecreasing. Its graph is illustrated in Figure 3.4.
Following [42, 43], we introduce the following definition.
Definition 2.1 Ruszczy´nski and Vanderbei [156] The mean-risk model
, ρ
μ
is consistent with SSD with coefficient
α
, if the following relation is true
y y
x x
y R
x R
SSD
λρ μ
λρ μ
− ≥
− ⇒
f
for all
α λ
≤ ≤
. for all
α λ
≤ ≤
In fact, as we shall see in the proof below, it is sufficient to have the above inequality satisfied for
α
; its validity for all
α λ
≤ ≤
follows from that. The concept of consistency turns out to be fruitful. In [129] we have proved the following
result.
Theorem 3.1. The mean–risk model in which the risk is defined as the absolute
semideviation,
} ,
{max x
R x
x −
= μ
δ E
, 3.18
is consistent with the second-order stochastic dominance relation with coefficient 1. We provide an easy alternative proof here.
Proof. First, it is clear from 3.17 that the line
x μ
η −
is the asymptote of
;
2
x F
⋅
for
∞ →
η
. Therefore
y R
x R
SSD
f
implies that
y x
μ μ ≥
. 3.19
48
Secondly, setting
x μ
η =
in 4 we obtain
} ,
{max y
R x
x −
≤ μ
δ E
. Since
≥ − y
x μ
μ
, we have
, max
, max
, max
y R
x y
x y
R y
y x
y R
x −
+ −
≤ −
+ −
= −
μ μ
μ μ
μ μ
μ
Taking the expected value of both sides and combining with the preceding inequality we get
y y
x x
δ μ
μ δ
+ −
=
, which can be rewritten as
y y
x x
δ μ
δ μ
+ ≥
−
. 3.20
Combining inequalities 3.19 and 3.20 with coefficients
λ
− 1
and
λ
, where
] 1
, [
∈ λ
, we obtain the required result. An identical result under the condition of finite second moments has been obtained in
Ogryczak and Ruszczynski [128] for the standard semideviation, and further extended in Ogryczak and Ruszczynski [129] to central semideviations of higher orders and
stochastic dominance relations of higher orders see also Gotoh and Konno [64]. Elementary calculations show that for any distribution
2 1
x x
δ δ
=
, where
x δ
is the mean absolute deviation from the mean:
x x
R x
μ δ
− = E
. 3.21
Thus,
x δ
is a consistent risk measure with the coefficient
2 1
= α
. The mean–absolute deviation model has been introduced as a convenient linear programming mean–risk
model by Konno and Yamazaki [86]. Another useful class of risk measures can be obtained by using quantiles of the
distribution of the return Rx. Let
x q
p
denote the p-th quantile In the financial literature, the quantity
x q
p
W, where W is the initial investment, is sometimes called the Value at Risk of the distribution of the return Rx, i.e.,
49 x
δ
.
We may define the risk measure
⎥ ⎦
⎤ ⎢
⎣ ⎡
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
− −
− =
z x
R x
R z
p p
x
p
, 1
max E
ρ
. 3.22
In the special case of
2 1
= p
the measure above represents the mean absolute deviation from the median. For small p, deviations to the left of the p-th quantile are penalized in a
much more severe way than deviations to the right. Although the p-th quantile
x q
p
might not be uniquely defined, the risk measure
x
p
ρ
is a well defined quantity. Indeed, it is the optimal value of a certain optimization problem:
⎥ ⎦
⎤ ⎢
⎣ ⎡
⎟⎟⎠ ⎞
⎜⎜⎝ ⎛
− −
− =
z x
R x
R z
p p
x
p
, 1
max min
E
X
ρ
. 3.23
It is well known that the optimizing z will be one of the p-th quantiles of Rx see, e.g., Bloomfield [17]. In Ogryczak and Ruszczynski [127] we have proved the following
result.
Theorem 2. The mean–risk model with the risk defined as
x
p
ρ
is consistent with the second-order stochastic dominance relation with coefficient 1, for all
1 ,
∈ p
. Again, we provide here an alternative proof.
Proof. Let us consider the composite objective in our mean–risk model scaled by p:
; x
p x
p x
p G
p
ρ μ −
=
. 3.24 If follows from 3.23 that we can represent it as an optimal value:
[ ]
[ ]
, 1
max sup
; z
x R
p x
R z
p p
x p
G −
− −
− =
E μ
X
. 3.25 Clearly, we have the identity
z x
R p
x R
z z
x R
p x
R z
p −
+ −
= −
− −
, max
, 1
max
. Using this in 3.25 we obtain
[ ]
, sup[
;
2
x z
F pz
x p
G −
=
X
. 3.26
50
Figure 3.6 The absolute Lorenz curve Therefore, the function
; x
G ⋅
is the Fenchel conjugate of
;
2
x F
⋅
see Fenchel [50] and Rockafellar[148]. Consequently, the second-order dominance
y R
x R
SSD
f
implies that
; ;
y p
G x
p G
≥
for all
] 1
, [
∈ p
. Recalling 13 we conclude that
y y
x x
p p
ρ μ
ρ μ
− ≥
−
. Since we also have 8, Definition 1 is satisfied with all
] 1
, [
∈ λ
. ■
Interestingly, the function
; x
G ⋅
can also be expressed as the integral:
∫
=
η α
α ,
d x
q x
p G
3.27 non-uniqueness of the quantile does not matter here. Indeed, it follows from 3.26 that
the quantile
x q
p
, which is the maximizer in 3.26, is a subgradient of
; x
G ⋅
at p see Fenchel [50] and Rockafellar[150]. The integral in 3.27 is called the absolute Lorenz
curve and is frequently used for nonnegative variables and in a normalized form in income inequality studies see Arnold [2], Gastwirth [59], Lorenz [106], Hardy,
Litlewood and Polya [70] and the references therein. It is illustrated in Figure 3.6.
51
C
HAPTER 4
THE DOMINANCE-CONSTRAINED PORTFOLIO PROBLEM
4.1 Introduction