1. Introduction
We consider optimal investment for the one-step financial model with a finite number of assets. The classical Markowitz optimization problems are looking for portfolios that either maximize the expected return for a given variance threshold, or minimize the variance for a given expected return. However, using variance as a measure of risk has a serious drawback: high profits are penalized in the same way as high losses. Instead, in what follows we shall use coherent measures of risk (cf. [1]), that provide a much better quantification of risk.
In our set-up, the space of financial positions is a vector space with vector ordering
. Besides the origin 0 in E, we distinguish a (strictly positive) reference cash stream denoted by 1. In the space of linear price systems
, i.e., the algebraic dual of
, we fix a total subspace 
(i.e., if 
for all
, then
) and consider the weak*-topology on 
associated to the dual pair
.
A coherent measure of risk (see [2]) is a real valued mapping 
defined on 
which is subadditive:
for
, positive homogeneous: 
for
, monotonic:
if
, and translation invariant:
for 
and any real
.
The following property will be needed in our results. A coherent measure of risk 
is called continuous from below (cf. [1,3]) if 
for any sequence
in 
satisfying
. Note that, if 
has a strong order unit, continuity from below is equivalent to the more familiar condition: 
provided 
(see [3,4]).
2. Main Results
Our first result formulates and solves in our set-up the first Markowitz problem, i.e., the so-called “agentindependent optimization problem”: find portfolios that maximize the expected return for a given (measure of) risk. Particular cases have been considered in [5-8].
Theorem 1. Let 
be an ordered locally convex vector space, and 
a total Banach subspace of
. Let
and 
be fixed; if 
is a coherent measure of risk continuous from below, then the following optimization problem:
(1)
admits optimal solutions.
Proof. According to the structure theorem for coherent measures of risk (see e.g. [3], Theorem 2.1), 
admits the following representation
(2)
for some weak*-closed convex set
, in which all 
are positive (i.e., 
for
) and normalized (i.e.,
). Note that continuity from below of 
implies continuity in the order convergence topology of all 
in formula (2), see [3].
Let us define
where the bar denotes closure. By the continuity from below of 
and the Krein-Šmulian theorem (see e.g. [9]), the set 
is weak*-compact, hence 
is compact. Therefore, using (2), the definition of
, continuity from below of 
and James’ theorem (see [9]), we obtain for any
:
In particular the sup in (2) is achieved, and for any 
one has
(3)
where the threshold 
is given in formula (1).
As 
for
, from (2) it follows that, for some
, one has 
for
. Take 
in the latter and, using linearity, obtain
, i.e.,
. Similarly obtain
. This means
, hence the following is well defined:
(4)
Then the max value in (1) equals 
and is achieved at every 
satisfying:
(5)
Indeed, if 
for some
, take 
satisfying
Condition (3) and definition (4) imply that
and the max is achieved at every 
satisfying condition (5). □
Problem (1) is for investing a sum of money in securities; it is possible that the investor already possesses a capital with terminal value
, in which case minimizing the risk leads to the second Markowitz optimization problem, or “single-agent optimization problem”. Alternatively, one can seek the minimum price which allows us to sell a payment order, and then compile a hedging portfolio of assets such that the risk of the entire operation will be negative or zero. Our second result formulates and solves the second Markowitz problem in our set-up.
Theorem 2. Let 
be an ordered locally convex vector space, and 
a total Banach subspace of
. Let 
be fixed; if 
is a coherent measure of risk continuous from below, then the following optimization problem
(6)
admits optimal solutions.
Proof. Let us denote
Using a similar argument as in the proof of Theorem 1, we obtain for any
:
In particular, for all 
and any 
one has
(7)
As 
for
, using a similar argument as in the proof of Theorem 1, we obtain that the following is well defined
Then the min value in (6) is given by 
and is achieved at every 
satisfying:
(8)
Indeed, take 
satisfying
for some
.
Condition (7) and definition (8) imply that
and the min is achieved at every 
satisfying condition (9). □
3. Applications
Examples. 1) We can solve explicitly problems (1) and (6) in the important case of Tail VaR (short for Tail Value at Risk). More precisely, consider 
and define the Tail VaR of order 
as the coherent measure of risk with the representation (2) in which
, cf. [1,3,10]. One can easily check that Tail VaR is continuous from below. More, Tail VaR is one of the best coherent risk measures, because is the smallest law invariant coherent risk measure that dominates the Value of Risk (cf. [3,11]). In the context of Theorem 1, we have that
has the optimal solution equal to
. Indeed, one can easily check that 
and any positive constant multiple of 
is an optimal solution of (1). In the context of Theorem 2, we have that
has the optimal solution equal to Tail VaR
. Indeed, one can check that 
Tail VaR
because 
is an optimal solution of (6). This situation occurs in problems (1) and (6) for complete models, such as Black-Scholes and Cox-Ross-Rubinstein.
2) Recall that a coherent measure of risk identifies unacceptable positions, i.e. with strictly positive risk
. A good measure of the latter are the so-called relevant measures of risk: given
, a coherent measure of risk 
is called g-relevant (cf. [1,3,10]) if 
and 
imply
.
Let us consider
; we have 
, the Banach space of bounded finitely additive measures on F and absolutely continuous with respect to P. In this case, all functionals 
(given by formula (2) above) describing a coherent measure of risk continuous from below and 
-relevant are genuine (i.e., 
-additive) probability measures equivalent to
. The particular case
, i.e., g represents integration with respect to
, has been treated in [2], Theorem 3.4, and the associated optimization problems (1) and (6) have been completely solved in [8,12] (see also [4]).
The research of George Stoica was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.