A Mean-Variance Portfolio Optimal Under Utility Pricing

Abstract

An expected utility model of asset choice, which takes into account asset pricing, isconsidered. The obtained portfolio selection problem under utility pricing is solved under severalassumptions including quadratic utility, exponential utility and multivariate symmetric ellipticalreturns. The obtained unique solution, called optimal utility portfolio, is shown mean-variance efficientin the classical sense. Various questions, including conditions for complete diversification and thebehavior of the optimal portfolio under univariate and multivariate ordering of risks as well as risk-adjustedperformance measurement, are discussed.

Full text

Page 1

Journal of Mathematics and Statistics 2 (4): 445-452, 2006
ISSN 1549-3644
© 2006 Science Publications

A Mean-Variance Portfolio Optimal Under Utility Pricing
Corresponding Author: Hürlimann Werner, Feldstrasse 145, CH-8004 Zürich, Switzerland
445

Hürlimann Werner
Feldstrasse 145, CH-8004 Zürich, Switzerland

Abstract: An expected utility model of asset choice, which takes into account asset pricing, is
considered. The obtained portfolio selection problem under utility pricing is solved under several
assumptions including quadratic utility, exponential utility and multivariate symmetric elliptical
returns. The obtained unique solution, called optimal utility portfolio, is shown mean-variance efficient
in the classical sense. Various questions, including conditions for complete diversification and the
behavior of the optimal portfolio under univariate and multivariate ordering of risks as well as risk-
adjusted performance measurement, are discussed.

Key words: Portfolio selection, utility pricing, mean-variance efficiency, elliptical distributions

INTRODUCTION


Markowitz[1] is one of the cornerstone of modern
finance. Despite its shortcomings and restrictions
(multivariate normal returns for arbitrary preferences or
quadratic utility for arbitrary distributions) this
fundamental model remains popular, mainly due to its
analytical tractability and
implications.
In the present study, we consider a model of
portfolio selection, which takes into account asset
pricing. As usual, the goal is maximization of expected
utility of the investor’s terminal random wealth.
However, as side constraint, we suppose that asset
prices are determined by the utility principle, which
expresses the indifference between the future random
value and its certainty equivalent. The analysis is
restricted to the case where all assets are risky. This
assumption is most useful for strategic asset allocation,
where even money markets are considered risky
investments.
Our portfolio selection problem under utility
pricing is defined in Section 2 and solved under several
alternative assumptions. In Section 2.1, we assume
quadratic utility functions but arbitrary returns and
obtain the optimal quadratic utility portfolio. Assuming
multivariate normally distributed returns and an
exponential utility function in Section 2.2, the solution
is called optimal exponential utility portfolio. The
solution in Section 2.3, obtained for multivariate
symmetric elliptical returns but arbitrary left truncated
utility functions, is called optimal elliptical portfolio.
As a remarkable main result, we show in Theorem 3.1
that the three derived optimal utility portfolios define a
unique optimal utility portfolio, which is mean-variance
efficient in the classical sense. In Section 4, we study
the behavior of this portfolio with respect to some
typical situations. The conditions for complete
The mean-variance model of asset choice by
its rich empirical
diversification are stated. Then, we discuss the behavior
of the optimal portfolio under univariate and
multivariate ordering of risks as well as under risk-
adjusted performance measurement.

2. Portfolio selection under utility pricing: Consider
the portfolio selection over a static one-period in an
economy with a finite set n of risky assets, which has
the following characteristics:

S : initial investment amount
Rj : random return on the j-th risky asset, j 1,...,n
Wj : proportion of initial amount invested in the j-th
risky asset, j 1,...,n
=

Pj : initial price of the j-th risky asset, j 1,...,n
Let
jj
S w S
=⋅ be the initial amount invested in the j-th
∑
=

=

risky asset such that
n
j
j 1
=
SS
=
. Define the proportion
jλ of shares of the j-th risky asset by the condition
PS , j 1,...,n.
λ==

We suppose that the initial prices of the risky assets
are utility prices. For this, let be given non-decreasing
and concave utility functions
each risky asset. The utility principle determines prices
in such a way that there is indifference between
receiving the future random value
corresponding certainty equivalent Pj, that is prices
solve the system of equations
( )
jjj
u PE u P (1 R ),
=⋅+

Using these prices, the one-period increase in
wealth or random gain of a portfolio choice
( ,...,)
λλλ
=
is given by
jjj
(2.1)
ju (x), j 1,...,n
=
, one for
jj
P (1 R )
⋅+
and its
()
j 1,...,n.
=

(2.2)
1n
n
jjj
j 1
=
G( )PR .
λλ
=∑
(2.3)

Page 2

J. Math. & Stat. 2 (4): 445-452, 2006
446

concave utility function u(x). Then, the portfolio
selection problem of the investor consists to maximize
the expected utility of the terminal random
wealth
(
( )E u S G( )
ϕ λλ=+

(2.2).
In the next three Subsections three typical
situations are considered. We begin with the two well-
known mean-variance legitimated cases of quadratic
utility and multivariate normal returns and extend the
latter to the case of multivariate symmetric elliptical
returns.

2.1. Quadratic utility functions: Consider first
quadratic utility
2
1
j
2
u (x)u(x) xx , j 1,...,n
γ==−=
utility u'(x)1x
γ
= −
is non-negative provided
hence satiation is reached when
one assumes that rates of return and initial wealth are
such that satiation will not be attained[2].
Solving (2.2) yields the quadratic utility prices
11
P, j 1,...,n,
1(1 k )
γµ
++
where
jj
, k
µ
are the means and coefficients of
variation of
j
R , j 1,...,n
=
suffices to solve the first order conditions
(
ii
i
λ
∂
Under the assumption of a quadratic utility
function, one obtains the equivalent conditions
(
jjij
j 1
1S
γ
=
−
Let
1n
e(,...,)
µµ=
be the vector of expected returns
and let
(
ij
V Cov R ,R

matrix of second order moments, whose inverse is
assumed to exist. Then (2.6) has the solution
(
jj
j
1S
γ
−
It follows that the optimal weight vector
o
1n
w (w ,...,w )
=
, with
P
w,i 1,...,n
P
λ
∑
satisfies the simple relationship
1
o
V e
w,
A
%
Where 1(1,...,1)
=
denotes a vector of ones. The
solution (2.9) defines the optimal quadratic utility
portfolio. Furthermore, one has using (2.7) that

S
PA
1S1S
γγ
−−
Suppose an investor uses a non-decreasing and
)

 under the conditions
functions
. The marginal
1
x
γ−
≤
,
1
x
γ−
≥
. To avoid this,
j
2
j
1
2
j
=⋅=
(2.4)
. To maximize
( )
ϕ λ
it
)
( )
P E R u' S
⋅

G( ) 0,i 1,...,n.
ϕ λ∂
λ=⋅+  =

=
(2.5)
)
n
iji
PCov R ,R


,i 1,...,n
γ
λ µ µµ


⋅⋅+==
∑
(2.6)
) ()
ijij
E R R


µ µ


=+=
%
be the
)
1
PV e ,
%
j 1,...,n.
=
γ
λ
−
⋅=
(2.7)
o
i
i i
jj
λ
==
, (2.8)
−
=
%
with
()
T1
A
%
1V e
%
−
=⋅
, (2.9)
jj
γγ
λ
=⋅=
∑
%, that is
A
+
1
S
1 A
γ =⋅
%
%
(2.10)

utility prices
Inserting (2.10) into (2.4) one obtains the quadratic
j
2
j
1
2
j
1
+
1 A
A
%
PS, j 1,...,n.
=
1
The optimal quadratic utility proportions defined
in (2.1) are then given by
(
j
j
P
1 A
+

2.2. Multivariate normal returns and exponential
utility: Another important situation, which allows for
explicit analytical formulas, is a multivariate normal
distribution of returns and exponential utility functions
(
u(x)1 e, j 1,...,n
γ
==−=
coefficient of risk aversion. Again, denote by
means and coefficients of variation of
Solving (2.2) yields the exponential utility prices
11
P, j 1,...,n.
k
γµ
Since ()
i
R ,SG( )
λ+
is bivariate normal, the
application of Stein’s
()
()
[
i
E u'' SG( )Cov R ,G( ) ,i
λ=+⋅

(as (A.6) in Appendix). It follows that (2.5) is
equivalent to the conditions
n
P Cov R ,R,
γλµ
=
(
V Cov R ,R
matrix, whose inverse is assumed to exist. Then (2.14)
has the solution
(
jj
j

The optimal weight vector
by
1
o
V e
w,
A
and defines the optimal exponential utility portfolio.
With (2.15) one has
S
γγ=⋅
∑
Inserted into (2.13) yields the exponential utility prices
1S
P, j 1,...,n.
kA
µ
The optimal exponential utility proportions (2.1)
are then given by
(
jjj
2
j
j
P

2.3. Multivariate elliptical returns and left truncated
utility: Suppose that the random vector of returns
(R ,...,R ) follows a multivariate symmetric elliptical
density of returns with mean e and positive definite
covariance matrix V given by
(1 k )
µ
+
=⋅⋅
+
%
(2.11)

)
o
j
2
j
1
2
j
o1
j
w1(1 k )
+
S V e ,
%
j 1,...,n
=
µ
λ
−
+
=⋅ =⋅
%
(2.12)
)
x
1
ju (x)
γ
− ⋅
, with
γ
the
jj
, k
.
µ
the
j
R , j 1,...,n
=
j
2
j
1
2
j
=⋅=
(2.13)
Lemma yields
]
i
Cov R ,u' S


G( )
λ
1,...,n.
λ
+

=

jjiji
j 1
i 1,...,n.




⋅⋅==
∑
(2.14)
Denote by
)
ij




=
the covariance
)
1
PV e , j 1,...,n.
= γ λ⋅
−
=
(2.15)
o
1n
w(w ,...,w )
=
is given
−
=
with
()
T1
A1V e
−
=⋅
, (2.16)
jj
PA
λ=
, hence
1
A S
⋅
γ
−
=
.
j
2
j
1
2
j
=⋅=
(2.17)
)
o
j
o21
1
w
SkV e ,j 1,...,n
=λµ
−
=⋅ =⋅
(2.18)
1n

Page 3

J. Math. & Stat. 2 (4): 445-452, 2006
447
T1
1
f(x)g (x


e) V (xe)
det(V)
−


=⋅−−
(2.19)

generator
everywhere differentiable concave utility function of
the investor is assumed to take non-negative values
such that u(x)0
= for all
≤
x
such that the utility associated to undesirable negative
portfolio returns vanishes (see the examples below for
illustration). Applying Fang et al. [3], Theorem 2.16, p.
43 and Theorem 2.18, p. 45, one notes that
()
i
R ,SG( )
λ
+
is bivariate elliptical and satisfies the
linear regression property required in Theorem A.1 of
the Appendix. Since u(x)=0 for x ≤ l , the variant (A.1)
of Stein’s Lemma yields the relationship
()
()
[]
Var G( )
λ
It follows that (2.5) is equivalent to the system of
equations
()
[][]
n
PCov R ,R,i1,...,n
λµ
=
which has the solution
[]
(
()
Cov G( ),u' S G( )
λλ
+

The optimal weight vector (2.8) is given by
V e
w,
A
and defines the optimal elliptical portfolio. By (2.16)
this coincides with the optimal exponential utility
portfolio.
To illustrate, we assume in the examples below left
truncated exponential utility functions such that for
j 1,...,n
=
one sets
0,x P (1),
µ≤+
Where
[
)
[
The
)
g: 0,
function.
0,
∞ →∞
is
non-decreasing
some appropriate
almost
λ
. The constantl is chosen
[]
i
i
Cov R ,u' S




=
G( )
Cov G( ),u' SG( )
Cov R ,G( ) ,i1,...,n.
λ
+λλ
λ
+



⋅=
(2.20)
jjiji
j 1
Cov G( ),u' S
Var G( ) E u'(S
λ
∑
G( )
G( ))
+
λλ
λ


+

−⋅
⋅


 =

=
(2.21)
)
()
1
jj
j
Var G( ) E u' S G( )
λ⋅
PV e , j 1,...,n
λ
λ
−
+




= −⋅=
(2.22)
1
o
−
=
with
()
T1
A1V e
−
=⋅
, (2.23)
j u (x)


=
{}
()
jj
1
jjjj
1 exp
−
xP (1 ),xP (1),
γγµµ
−




⋅−−+≥+
(2.24)
and
u(x)


=
[]
{
is the mean expected return of the
portfolio. The special choice of the left truncation
models the fact that returns below average are
associated a zero utility. This might be appropriate for
the design of optimal portfolios, whose goal is to “beat
the market”. In this situation, the pricing equations (2.2)
are equivalent to

−⋅


}
()
1
0,xS(1 ),
1 exp
−
xS(1),xS(1),
µ
γγµµ
−
≤+
⋅−−+≥+
(2.25)
where
1
E S G( )


µλ
−

=
{}
jj
jj
R
jj
E exp(PR ) 1
γ
1
2
1exp(P),j 1,...,n
=
µ
γµ
≥
= −−
(2.26)
where use has been made of
margins).

Example 2.1: Normal inverted gamma returns
It appears instructive to illustrate our results with a
non-trivial but tractable
distribution, which finds wide interest in both Insurance
and Finance. The mixture of a normal with inverted
gamma variance yields the Pearson type VII
distribution or generalised Student t[4-6]. It has been
proposed to model financial returns by Praetz[7],
Blattberg and Gonedes[8],
Hürlimann[11,12]. An actuarial application is found in
Hürlimann[13]. The multivariate density
1
2
jj
Pr(R )
µ
≤= (symmetric
multivariate elliptical
Kon[9], Taylor[10],
()
(
⋅Γ
)
()
n
2
n
2
T1
n
2
β
n
2
n
2
1 (x

+

e)
β
C(x e)
f(x),
B, det(C)
( )
+
B,, 0,
()
β
π
Γ
ββ
β
−+
−


−⋅⋅−
=
⋅
=>
(2.27)
has location-scaled transformed Pearson VII marginal
densities
()
i
2
ii
()
2
x
1,i 1,...,n
c



If ß>1 the variance
i
σ =
υ
βυ
=
1
2
1
1
i
ii
1
c
f (x)B,
β
β
µ
−
−+
=⋅








−
+=
(2.28)
[]
2
i
Var R
exists and one has
iii
c2( 1)
βσ=−⋅
. If
,1,2,3,...
2
=
, one recovers a
location-scale transformed Student t with υ degrees of
1
2
Bowers distribution. If
variable
()
ii
2( 1)Rc
βµ−⋅−⋅
normal random variable. On the other hand, any linear
combination
(
1
RwR ,...,R
=⋅
freedom. In particular
β =
is a Cauchy and ß=1 is a
β → ∞
converges to a standard
the random
1
ii
−
)
T
n
has density
()
1
2
()
2
1
1
2
R
1
c
x
f (x)B,1,
c
β
µ
β
−+
−








−






=⋅+
(2.29)
with
analytical formulas below use the special integral
function
∞
−−
=+
∫
T
we
µ =⋅ and
T
cw C w
⋅=⋅
[3], Theorem 2.16). The
2abx
0
I(a,b)(1 x ) edx
, (2.30)
whose computational evaluation is discussed in the
Appendix B.
Under the assumption ß>1 (variances are finite),
one obtains after some calculation the formulas
[]
[]
Cov G( ),u'(S G( ))
22
B( ; ) I(, Sc),
21
β
−

which inserted into (2.22) yield the relationships
[]
1
1
2
1
2
1
1
2
1
2
E u'(SG( ))
λ
λ

⋅

B( ; )
β
λ
I(
γ
, Sc),
γ
λ
Var G( )
β
β
ββγ
−
−
+=⋅+
+ = − ⋅⋅


−
⋅−
(2.31)

Page 4

J. Math. & Stat. 2 (4): 445-452, 2006
448
()
1
2
1
2
jj
1
j
I(
I(
, Sc)
, Sc)
γ
1
γ
2
2
j 1,...,n
=
1
2
P
V e ,
β
β
γ
β
β
λ
−






+
−
−
−
=⋅⋅
(2.32)

equations (2.26) are equivalent to
1
1
2
1
jj
2
1 exp(P ),
γµ
= −−
Using (2.32) and
A similar calculation shows that the utility pricing
1
2
jj
B( ; )
β
exp(P) I(
j 1,...,n
=
, Sc)
γ γ µ
−
β
−⋅⋅+
(2.33)
jj
SP
λ
=∑
one sees that
1
2
1
2
I(
I(
, Sc)
, Sc)
γ
2
2
1
2
SA
β
β
γ
β
β
γ






+
−
−
−
=⋅⋅
(2.34)

Inserting its numerical value into (2.33) it is possible to
determine numerically the corresponding truncated
exponential utility prices and the optimal proportions
defined in (2.1).

Example 2.2: Symmetric Kotz type returns
Another tractable family of symmetric elliptical
distributions has the density
[
exp)()(
) det(
>+>
nNsr

with the normalizing constant
( )
()
2s
π
⋅Γ
This family, introduced by Kotz[14], is called
symmetric Kotz type distribution in Fang et al.[3]. It
reduces to the multivariate normal distribution in case
1
2
Ns1,r
= ==
. The marginal densities take the form
Which is an implicit equation for the parameter γ .
][]
{}
,
, 02, 0,
)()()(
1
1
1
−−−⋅−−=
−
−
−
exCexrexCex
C
K
xf
s
T
N
T
n
(2.35)
2N n 2
2s
n
2
n
2
n
2N n 2
+ −
s
Kr
+ −
⋅Γ
=⋅
(2.36)
2(N 1) 2s
1ii
i
iii
K
c
xx
f (x) expr
cc
µµ
−






















−−
=⋅⋅−
(2.37)
and the portfolio return is again of symmetric Kotz type
such that
2(N 1)
1
R
f (x) exp
cc

Analytical evaluation requires the following special
integral function
1
J(a,b,r,s)x edx
b
whose computational evaluation is provided in the
Appendix B. A straightforward calculation yields the
formulas
[]
[]
Cov G( ),u'(S G( ))K (Sc)
1, Sc,r,s).
γ+
Proceeding now as in Example 2.1, it is possible to
determine numerically the corresponding truncated
exponential utility prices and the optimal proportions in
(2.1).
2s
K
xx
r
c
µµ
−










−−










=⋅⋅−
(2.38)
2s
arxbx
0
∞
−−
=⋅∫
(2.39)
1
λ
2
1
E u'(S G( ))
+
K Sc J(2(N 1), Sc,r,s),
γ
γ=
J(2(N 1)
λγ
⋅λ
=
+
⋅−
−
(2.40)
3. Mean-variance efficiency of the optimal utility
portfolios: It is worthwhile and instructive to compare
portfolio selection under utility pricing with the
classical mean-variance portfolio selection. Let the set


==



Recall that a portfolio
w ∈
variance boundary if and only if, for some µ , the
n
j
j 1
=
Www1
∑
describe all portfolio choices.
W
belongs to the mean-
portfolio w solves the problem
{}
2
w
T
w W
∈
min w Vw
σ=

subject to
to the mean-variance efficient frontier if and only if no
portfolio
Wv∈
exists such that
σσ
≤
, where at least one of the inequalities is strict.
From Merton[15] or Huang and Litzenberger[2], one
knows that a portfolio w belongs to the mean-variance
boundary if and only if
µ⋅+=
hgw
,
where g and h are the vectors defined by
(
(
hC V A V e
D
with
T1T
A1 (V e),Be(V e),
C1 (V 1),DBC A .
=⋅=−
The mean-variance efficient frontier consists of
T
w
we
µµ
=⋅ =
. A portfolio wW
∈
belongs
vw
µµ
≥
and
vw
(3.1)
)()
))(
1 T
−
1T
1T1 T
1
D
1
g B V eA V,
µ
µ
−
−−




=−




=−
(3.2)
1
T12
−−
−
=⋅=⋅
(3.3)
those weight vectors (3.1) for which
A
C
µ ≥
. It is
remarkable that the optimal portfolios (2.9), (2.16) and
(2.23) are all equal and belong to the mean-variance
efficient frontier.

Theorem 3.1: The optimal quadratic utility portfolio,
the optimal exponential utility portfolio and the optimal
elliptical portfolio coincide and are mean-variance
efficient.

Proof: The expected return of the optimal exponential
utility portfolio is
T1
oTo
e(V e)
A
B
A
ew
µ
−
⋅
=⋅==
.
Using (3.1) and (3.2) one obtains immediately
1
o
V e
gh
A
belongs to the mean-variance boundary. Since
BA
AC
is mean-variance efficient. Since (2.23) is identical to
(2.16), it remains to show that (2.9) and (2.16) yield the
same optimal portfolios. Let w % and w denote the
optimal portfolios defined by (2.9) and (2.16). To
obtain them, one solves the linear systems Vx
that
µ
−
+ ⋅=
, which by (3.1) shows that
o
w
o
µ =>
(because
2
DBC A0
=−>
) the portfolio
e
=
%%
,

Page 5

J. Math. & Stat. 2 (4): 445-452, 2006
449
Vxe
=
and set
T
x
%
w
%
1x
)
=
(
⋅
%,
, one sees immediately
T
x
w
1x
=
⋅
. Since
V
%
VM
=+
with
ij
M
~
~
µ µ=
that the equation
(
Vx 1 e x
=−
%
constant.
exV =
. But
is equivalent to the equation
(
e xe V e
%
follows
)
T
e
⋅
%
)
TT1
B
%
−
==
%
is a
It that
()(
%
)
()()
1
1
T
T1
1 B V e
−
%
x
%
V e
A
w
%
w
1x
%
1 1 B V e
−
−
−
−
====
⋅
⋅
, which shows
the assertion.
In view of the present unification result, the
portfolio defined by the weight vector
1
o
V e
A
w
−
=
will
simply be called optimal utility portfolio. As an
additional property, it is important to observe that any
mean-variance efficient portfolio
optimal in the following sense. For any such w,
investing the amount S yields by construction a terminal
random wealth, whose corresponding expected utility
satisfies the
()
(
E u S G( ) E u S G(
λ
+≤+


optimal proportion vector defined in (2.1). In the very
narrow sense of maximizing this expected utility, the
optimal utility portfolio is the preferred mean-variance
efficient portfolio choice.

4. Some properties of the optimal utility portfolio:
o
ww ≠
is non-
inequality
λ is the
)
o
)
λ

, where
o
Given the optimal utility portfolio
A
eV
w
1
−
=
, which
maximizes the expected utility of the terminal random
wealth under various assumptions, it appears useful to
study the behavior of this particular solution with
respect to some typical situations.
In the special case n = 2, often used as illustration,
we use the notations
X
R , R for the random returns
instead of
12
R , R . Assume that the bivariate distributed
returns have means
deviations
XY
,
σσ
and correlation coefficient ρ . The
coefficients of variation
σ
µ

Question 1: When is complete diversification optimal?
In case some of each risky asset is purchased, that
is
i
w0
>
for all
{}
i 1,...,n
∈
diversification[16-18]. This situation occurs if and only if
1
V e0
>
(each vector component is strictly positive).
For example, if n=2 one has
2
1
Y
2
222
X
XY
(1)
σ µ
ρ σ σ
−

Y
XY
,
µµ
, standard
are denoted
Y
Y
Y
X
X
X
kk
µ
σ
==
,
.
, one speaks of complete
−
XXYY
YXYX
1
V e0
σ µ


ρσ σ µ
ρσ σ µ
−



−
−
=⋅>
(4.1)
if and only if
YX
XY
k
k
k
k
min,
ρ











<
or equivalently
XYXY
XYXY
R
µ
R
µ
R
µ
R
µ
Cov,min Var ,Var.























<
(4.2)

correlated returns, which are typically observed in
financial markets and whose financial risk cannot be
eliminated through diversification[19].

Question 2: How does the optimal portfolio behave
under univariate ordering of risks?
In general, let
1
R (R ,...,R ), i
=
vectors of multivariate normally distributed returns and
let
w , i1,2
=
, the corresponding optimal portfolios.
The random returns of the optimal portfolios are
T
(i)
i
Rw

=

their expected exponential utilities, one knows from
standard ordering of risks theory[20] that
[][]
12
E u(R ) E u(R )
≤

if and only if
,
21
RR
sl
≤

where
sl
≤ denotes the stop-loss order or equivalently
the increasing convex order. Under normally distributed
returns, the relation (4.4) holds if and only if the means
and standard deviations are ordered as follows:
,.
µµσσ≤≤

From the proof of Theorem 3.1, one knows
B
i
i µ
, where the constants
In particular, this situation allows for positively
(i)(i) (i)
n
1,2,
=
be two
(i)
denoted here
(i)
R , i1,2
⋅=
. Comparing
(4.3)
(4.4)
1212
(4.5)
that
2 , 1, =
i
=
A
i
ii
A , B are
defined as in (3.3). Using Huang and Litzenberger[2],
formula (3.11.2b), one obtains further
(
iiiii
i
D
These results allow for a numerical evaluation of
the criterion (4.5), which is left to the interested reader.

Question 3: How does the optimal portfolio behave
under multivariate ordering of risks?
Given some multivariate ordering of relative
riskiness between the two vectors R(1) and R(2), how do
the corresponding optimal weight vectors W(1) and W(2)
behave? This difficult question has only been scarcely
discussed in the literature. For similar but different
portfolio selection problems, one finds some results in
the studies by Fishburn and Porter[21], Landsberger and
Meilijson[22], Eeckhoudt and Gollier[24], Hürlimann[18].
To illustrate, let us restrict our attention to the bivariate
situation n=2 and let us assume that the normally
distributed pairs
XY
(R ,R ) and
margins, in particular equal means
)
22
i
2
i
i
B
A
1
C 2AB,i 1,2.
σµµ=⋅−⋅+==
(4.6)
(1)(1)(2)
X
(2)
Y
µ
(R ,R ) have equal
,
µ
, equal
XY