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On a Class of Portfolio Diversification Measures Induced by Risk Measures

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Abstract

We define an optimization problem and prove that its objective function verifies desirable theoretical properties for a portfolio diversification measure. We show that the unique solution of the optimization problem coincides to the Geometric Diversified Portfolio, supporting the validity of the intuition behind geometric diversification. As an important corollary result, since geometric diversification is defined on the base of given risk measures, we obtain that each risk measure induces a specific portfolio diversification measure that incorporates its risk information content. The appealing of the proposal mainly relates to the possibility to automatically define a portfolio diversification measure starting from any given risk measure, exploiting the number of theoretical proposals in one field, risk measurement, to develop proposals in the other field, portfolio diversification.
On a Class of Portfolio Diversification Measures
Induced by Risk Measures
Maria-Laura Torrente, Pierpaolo Uberti
Department of Economics, University of Genoa
Via Vivaldi 5, 16126 - Genoa, Italy
marialaura.torrente@economia.unige.it, pierpaolo.uberti@unige.it
Abstract
We define an optimization problem and prove that its objective func-
tion verifies desirable theoretical properties for a portfolio diversification
measure. We show that the unique solution of the optimization problem
coincides to the Geometric Diversified Portfolio, supporting the validity
of the intuition behind geometric diversification. As an important corol-
lary result, since geometric diversification is defined on the base of given
risk measures, we obtain that each risk measure induces a specific portfo-
lio diversification measure that incorporates its risk information content.
The appealing of the proposal mainly relates to the possibility to au-
tomatically define a portfolio diversification measure starting from any
given risk measure, exploiting the number of theoretical proposals in one
field, risk measurement, to develop proposals in the other field, portfolio
diversification.
Keywords: Portfolio Diversification Measures, Risk Measures, Coherent Risk
Measures, Deviation Risk Measures, Convex Risk Measures
JEL: G10, G11, D81
1 Introduction
The idea of diversification in portfolio theory has been first introduced by H.
Markowitz [15]. Despite its simplicity, no generally accepted unique definition
of diversification is available in the literature, giving the rise to the production
of many contributions on the topic. Several different papers deal with the asset
allocation problem from the point of view of diversification; among the others,
we enumerate the following contributions: in [5] and [6] the authors propose an
allocation rule based on the maximization of the so called diversification ratio; in
[8] the authors refer to the Equally Weighted Portfolio as naive diversification
and compare its out-of-sample performance to alternative approaches; in [7],
[14], [17] and [20] the Equally Risk Contribution is proposed as the strategy
1
that balances the risk exposure among the assets; in [16] the author proposes
to use principal component analysis to extract uncorrelated risk factors and
diversify the portfolio.
On the other hand, the axiomatic approach to risk measures dates back to
the paper of [4]. After this first attempt, many axiomatic approaches to risk
measurement have been proposed, see for example [18]. Among the others, we
recall the deviations risk measures defined in [19], the convex risk measures
proposed in [9], the spectral risk measures presented in [2], the downside risk
measures introduced in [21] and the dynamic risk measures discussed in [1].
Each axiomatic class contains an infinite number of risk measures. As a result,
the universe of possible risk measures proposed in the literature is deeply intri-
cate and risk measures are strongly interrelated each other, see [10]. Since risk
measures have been effectively used before the attempts of a theoretical axiom-
atization, a branch of literature has been subsequently involved in the problem
of classifying many existing risk measures within the different axiomatic classes:
for instance, [3] shows that CVaR is a coherent measure of risk, whereas in [12]
the coherency of expected shortfall is investigated.
The relation between portfolio diversification and risk measures is intimate.
In particular, the sub-additivity property, usually required for risk measures,
translates the economic idea of diversification, ensuring that any portfolio is
better in terms of risk than holding its single constituents.
As far as we know, a first attempt to rigorously define portfolio diversifi-
cation can be found in [13] where, based on the theoretical structure used for
the axiomatization of risk measures, the coherent portfolio diversification mea-
sures are introduced providing a list of axioms and the corresponding economic
interpretations.
Risk measures constitute the key ingredient for the definition of alternative
asset allocation strategies. In this paper we consider the novel approach of the
geometric diversification strategy introduced in [22]: based on the intuitive idea
to underweight the allocation on the riskier asset classes while overweighting the
allocation on the less risky ones, such allocation strategy invests in the port-
folio that is equally distant, according to the so-called Risk Adjusted Distance
(RAD), from the single asset portfolios, that is the maximum concentrated ones.
Note that the RADs, conceived to take into account the single asset risk contri-
butions, allow to compute the distance between investment portfolios not only
in terms of difference in the allocation but also according to the underlying risk
measure.
In this framework, we first rewrite the conditions defining the Geometric
Diversified Portfolio as an equivalent optimization problem, show that its ob-
jective function, which we call the Geometric Portfolio Diversification Measure
(GPDM), verifies desirable properties for a portfolio diversification measure and
discuss each of them in relation to the concurrent economic relevance. Our ap-
proach differs from the one in [13] for a few substantial aspects: first, we take
into consideration the risk, not reducing the analysis of the diversification to
a mere evaluation of the vector of portfolio weights; second, we generally refer
to a long-short framework, without restricting to the long-only case; finally, in
2
most of the cases the properties we obtain imply the axioms introduced in [13].
Considering that the GPDM is defined on the basis of a RAD that introduces
in the approach an arbitrary risk measure, we may as well refer to it as a port-
folio diversification measure induced by a given risk measures. In other words,
each risk measure can be used to define a portfolio diversification measure that
contains its information about the risk. Note that such approach takes advan-
tage of the huge number of risk measures proposed in the literature. Interesting
portfolio diversification measures may be naturally developed in this way and
proved to verify desirable theoretical properties. Our approach is so general
that permits to directly use the risk measures starting from their axiomatic
definition without specifying the functional form. In particular, deviations risk
measures and coherent risk measures, see [19] and [4], induce GPDMs that dif-
fer exclusively for the behavior toward deterministic translations, reflecting the
specific features of the original risk measures.
The paper is organized as follows. Section 2 contains the notation together
with preliminary and background results useful for the comprehension of the
paper. In Section 3 the GPDM is introduced, its main theoretical properties are
proved and analyzed by providing their useful economic interpretation; further,
as a corollary result, it is shown that the GPDMs induced by the deviation risk
measures fulfill all the stated properties, and the ones induced by the coherent
and the convex risk measures do satisfy all but one of them. Finally, Section 4
contains some final remarks and concludes the paper.
2 Background material
In this section we introduce the notation used in the rest of the paper, see
Section 2.1, we enumerate some general properties that are required for the
definition of the axiomatic risk measures, see Section 2.2, and we recall, in a
formal way, the concept of geometric diversified portfolios, see Section 2.3.
2.1 Preliminaries and notation
Let mn2, let Matm×n(R) be the set of m×nreal matrices and Mm×n(R)
be the subset of Matm×n(R) containing all the full-rank matrices, i.e. rank(A) =
nfor each A∈ Mm×n(R). We indicate with 0m×nand 1m×nthe m×nmatrices
whose elements are all equal to zero and one respectively.
Throughout the paper, we will interpret the entries of the column vectors Aj,
with j= 1, . . . , n, of any matrix AMatm×n(R) as the historical returns of the
j-th asset of the portfolio, so that Ais the usual matrix of portfolio returns with
mobservations and nrisky assets. Further, we let Γn={w= (w1, . . . , wn)
Rn|Pn
j=1 wj= 1}and Wn={w= (w1, . . . , wn)Rn
0|Pn
j=1 wj= 1}be
respectively the set of long-short portfolios and long-only portfolios; a portfolio
is uniquely identified by the vector wwhere the j-th entry wjis the weight of
the j-th asset. The subset of Wnand Γncontaining the portfolios with at least
one null component is denoted by Wn, whereas the single asset portfolio with
3
the allocation concentrated on the j-th asset is denoted by ejWn, where
e1, . . . , endenote the standard basis of Rn.
2.2 Axiomatic risk measures
We recall the setting to provide a general axiomatic characterization of risk
measures, see among the others [4], [3], [18], [19]. We consider the usual prob-
ability space (Ω,A, P ) and the set of functions X: Ω R, treated as random
variables, that belong to the linear space L2(Ω,A, P ). Note that such space
obviously contains all constant random variables Xα, with αR. In par-
ticular, in the following, by abuse of notation we will use the real number αto
signify the constant random variable with value αalmost surely. Analogously,
inequalities of type XYor XY, with X, Y ∈ L2(Ω,A, P ), mean that they
hold almost surely.
Definition 2.1. A function ρ:L2(Ω,A, P )R∪ {+∞} is said to be:
-normalized if ρ(0) = 0;
-strictly positive if ρ(X)>0 for all non-constant Xand ρ(X) = 0 for
constant X;
-monotone if for each X1, X2with X1X2it follows that ρ(X1)ρ(X2);
-sub-additive if ρ(X1+X2)ρ(X1) + ρ(X2) for each X1, X2;
-shift-invariant if ρ(X+α) = ρ(X) for each Xand αR;
-translation-invariant if ρ(X+α) = ρ(X)αfor each Xand αR;
-positively homogeneous if ρ(αX) = αρ(X) for each Xand α0;
-convex if ρ(αX1+ (1 α)X2)αρ(X1) + (1 α)ρ(X2) for each X1, X2
and α[0,1].
Definition 2.2. Acoherent risk measure is a mapping ρ:L2(Ω,A, P )
R{+∞} which is monotone, sub-additive, translation-invariant and positively
homogeneous (see [4]). If ρis normalized, strictly positive, sub-additive, shift-
invariant and positively homogeneous then it is called a deviation risk measure
(see [18]), while, if the conditions of subadditivity and positive homogeneity are
replaced by the weaker property of convexity, the risk measure is said convex
(see [9]).
A more detailed list of the axiomatic definitions of risk measures proposed
in the literature is behind the scope of the present paper.
Remark 2.3. Note that the properties of translation-invariance and shift-
invariance given in Definition 2.1 are mutually incompatible, such that axiomatic
risk measures verify at most one of the two alternative requirements. Moreover,
if the defined risk measure is absolute, in the sense that its value has an intrinsic
4
economic meaning (for instance, as in the case of CVaR, whose value represents
the expected average loss at given confidence level), the property of translation-
invariance has an effective impact. On the opposite, in a relative context, where
the interest is only in the order among the alternatives with respect to the
risk measures, there is no actual difference between translation-invariance and
shift-invariance, as both properties do not impact the original order of the risks.
The previous remark is useful to anticipate the particular behavior toward
deterministic translations of the induced portfolio diversification measures pro-
posed in this paper (see Section 3) that strictly depends on the properties of
the considered risk measure.
2.3 Geometric Diversified Portfolio
In this section we recall the notions of Risk Adjusted Distance and Geometric
Diversified Portfolio, introduced in [22].
Let n2 be the number of assets, let Xj, for each j= 1, . . . , n, be the
return of the the j-th asset for a fixed period, and consider the tuple of n
possibly correlated random variables X= (X1, . . . , Xn)t. Using the notations
defined above, we assume that each Xj∈ L2(Ω,A, P ) and let ρ:L2(Ω,A, P )
R∪{+∞} be a chosen risk measure. Throughout the paper we make the standing
assumption that ρ(Xj)>0 for each j= 1, . . . , n and, in order to simplify the
notation, we denote by ρ(X)=(ρ(X1), . . . , ρ(Xn))t.
Definition 2.4. (Risk Adjusted Distance) The Risk Adjusted Distance
(RAD) dρ(X):Rn×Rn7→ Ris defined by
dρ(X)(x, y) =
n
X
j=1
1
ρ(Xj)(xjyj)2
1
2
,x, y Rn.
Definition 2.5. (Geometric Diversified Portfolio) Let dρ(X)be a RAD;
the Geometric Diversified Portfolio (GDP) with respect to dρ(X)is represented
by the portfolio w= (w
1, . . . , w
n)Rnsuch that
n
X
j=1
w
j= 1 and dρ(X)(w, ei) = dρ(X)(w, ek),i, k ∈ {1, . . . , n}.
Equivalently the coordinates of the GDP satisfy
w
j=1
21n2
nMρ(X)
ρ(Xj), j = 1, . . . , n, (1)
where Mρ(X)is the arithmetic mean of ρ(X1), . . . , ρ(Xn).
Remark 2.6. The GDP wis a long-only portfolio, that is wWnif and
only if n3 or n > 3 and Mρ(X)n2
nρmax, where ρmax is the maximum
of ρ(X1), . . . , ρ(Xn) (see [22, Proposition 3.3]). Further, notice that the more
restrictive condition w
j>0 for each j= 1, . . . , n is equivalent to the cases n3
or n > 3 and Mρ(X)>n2
nρmax.
5
3 Geometric portfolio diversification measures
In this section we introduce the Geometric Portfolio Diversification Measures
(GPDMs), a class of functions that, starting from suitable risk measures, aim to
quantify the portfolio diversification. Note that, by construction, the GPDMs
are induced by risk measures and rely on the geometric diversification strategy,
recalled in Section 2.3, as a pure technical instrument for their definition. We
use the notations and assumptions as provided in Section 2; further, we let Aj
be the realizations of the random variable Xj, i.e. the mreturns of the j-th
asset, for each j= 1, . . . , n, and denote by A= (A1, . . . , Am)Matm×n(R) the
usual matrix of portfolio returns.
Definition 3.1. (Geometric Portfolio Diversification Measure) The Ge-
ometric Portfolio Diversification Measure (GPDM) with respect to the risk mea-
sure ρis the map Φρ(X): Γn×Matm×n(R)\ {0m×n} → Rdefined by
Φρ(X)(w, A) := (rank(A)1) 1fρ(X)(w)
maxj=1,...,n fρ(X)(ej)(2)
for each wΓnand AMatm×n\ {0m×n}, where fρ(X): ΓnR0is:
fρ(X)(w) :=
n
X
j=1
n
X
k=j+1 d2
ρ(X)(w, ej)d2
ρ(X)(w, ek)2(3)
for each wΓn.
In the following we enumerate and prove some important theoretical prop-
erties that characterize Φρ(X)as a diversification measure and, afterwards, we
discuss about their economic interpretation. It is useful to precede all these
results with Lemma 3.2, containing some technical issues on the function fρ(X).
Lemma 3.2. Let fρ(X)be the map introduced in Definition 3.1 and wbe the
GDP with respect to dρ(X). Then the following properties hold true:
(i) fρ(X)(ei)>0, for each i= 1, . . . , n;
(ii) fρ(X)(ei)fρ(X)(ej)if and only if ρ(Xi)ρ(Xj)for each i, j = 1, . . . , n;
(iii) fρ(X)is strictly convex;
(iv) for each permutation matrix ΠMatn×n(R)we have fρ(XΠ)tw) =
fρ(X)(w)for each wΓn;
(v) wis the unique global minimum of fρ(X)over Γnand fρ(X)(w)=0.
If ρis shift-invariant then
(vi) fρ(X+α)=fρ(X)for each αR.
If ρis positively homogeneous then
6
(vii) fρ(αX)=1
α2fρ(X)for each α > 0.
Proof. We let ρj=ρ(Xj)>0 for each j= 1, . . . , n.
(i) By some computations, using the expression of fρ(X), see formula (3), and
Definition 2.4, we get
fρ(X)(w) =
n
X
j=1
n
X
k=j+1  1
ρj
1
ρk2wj
ρj
wk
ρk2
.(4)
By evaluating the above expression at w=eiwe have:
fρ(X)(ei) =
i1
X
j=1
n
X
k=j+1,k6=i1
ρj
1
ρk2
+
n
X
j=i+1
n
X
k=j+1 1
ρj
1
ρk2
+
n
X
j=1,j6=i1
ρj
+1
ρi2
>0.(5)
(ii) We consider the nontrivial case i6=j; by some computations, using ex-
pression (5), we get:
fρ(X)(ej)fρ(X)(ei) = 1
ρj
1
ρin
X
k=1,k6=i,k6=j
4
ρk
from which the result immediately follows.
(iii) Let w, z Γn, with w6=zand α(0,1). Recalling that g(x) = x2is a
strictly convex function we obtain
fρ(X)(αw + (1 α)z) =
=
n
X
j=1
n
X
k=j+1  1
ρj
1
ρk2αwj
ρj
wk
ρk2(1 α)zj
ρj
zk
ρk2
=
n
X
j=1
n
X
k=j+1 α 1
ρj
1
ρk2wj
ρj
wk
ρk
+(1 α) 1
ρj
1
ρk2zj
ρj
zk
ρk2
< α
n
X
j=1
n
X
k=j+1  1
ρj
1
ρk2wj
ρj
wk
ρk2
+ (1 α)
n
X
j=1
n
X
k=j+1  1
ρj
1
ρk2zk
ρj
zk
ρk2
=αfρ(X)(w) + (1 α)fρ(X)(z),
which proves that fρ(X)is strictly convex.
7
(iv) Let πbe the permutation of {1, . . . , n}associated to Π, that is π(j) satisfies
eπ(j)= Πtej, for each j= 1, . . . , n. Note that fρ(X)(w) can also be
expressed as follows:
fρ(X)(w) = 1
2
n
X
j=1
n
X
k=1 d2
ρ(X)(w, ej)d2
ρ(X)(w, ek2
.
Using formula (4) and the fact that πis a bijection we get
fρ(XΠ)tw) = 1
2
n
X
j=1
n
X
k=1  1
ρπ(j)
1
ρπ(k)2wπ(j)
ρπ(j)
wπ(k)
ρπ(k)2
=fρ(X)(w),
for each wΓn, so that the item is proved.
(v) The uniqueness of the global minimum of fρ(X)over Γndirectly follows
from item (i). Further, using formula (1) and (4), we get fρ(X)(w) = 0;
consequently, since fρ(X)(w)0 for each wΓn, the result follows.
(vi) By the shift-invariance of ρit immediately follows that fρ(X+α)(w) =
fρ(X)(w), for each wΓn, so that the item is proved.
(vii) Using formula (4) and the positive homogeneity of ρwe get
fρ(αX)(w) =
n
X
j=1
n
X
k=j+1  1
αρj
1
αρk2wj
αρj
wk
αρk2
=1
α2fρ(X)(w),
for each wΓn, so that the item is proved.
Remark 3.3. Note that the function Φρ(X)is well-defined for any choice of the
risk measure ρ: this is an immediate consequence of Lemma 3.2, item (i), which
implies that maxj=1,...,n fρ(X)(ej)>0. Further, Lemma 3.2, item (ii), implies
that there is an ordering on the evaluations fρ(X)(ej) which depends on the
values ρ(Xj), where j= 1, . . . , n. In particular, assuming that ρ(X1). . .
ρ(Xn), which is always the case through a feasible reordering, then fρ(X)(en)
. . . fρ(X)(e1)> fρ(X)(w) = 0, where the last inequality follows from Lemma
3.2, item (v).
In Proposition 3.4 we look for the maximum of the GPDM and show that
the optimization problem admits a unique maximum attained at the GDP as
defined in Definition 2.5.
Proposition 3.4. Let Φρ(X)be the GPDM with respect to dρ(X)and A
Matm×n(R)be a nonzero matrix. The optimization problem maxωΓnΦρ(X)(w, A)
admits a unique solution that coincides with w, the GDP with respect to dρ(X),
and satisfies Φρ(X)(w, A) = rank(A)1.
8
Proof. By Proposition 3.2, item (i), it follows that maxj=1,...,n fρ(X)(ej)>0,
consequently
max
wΓn
Φρ(X)(w, A) = (rank(A)1) 1minwΓnfρ(X)(w)
maxj=1,...,n fρ(X)(ej).
Since wis the unique global minimum of fρ(X)over Γnand fρ(X)(w) = 0 (see
Proposition 3.2, item (v)), the result immediately follows.
As a consequence of Proposition 3.4 and by formula (1), we note that the
point wwhich realizes the maximum of the GPDM never coincides with any
element of the standard basis ej,j= 1, . . . , n, meaning that the diversification
is always better than full concentration in any of the single asset portfolios.
The main properties of Φρ(X)are shown in Propositions 3.5, 3.6 and 3.7.
Proposition 3.5. The GPDM Φρ(X)satisfies the following properties.
P1. (Quasi-concavity) Let AMatm×n(R)be a non-zero matrix; for any
w, z Γnand α[0,1] it holds that
Φρ(X)(αw + (1 α)z, A)min{Φρ(X)(w, A),Φρ(X)(z , A)},
with strict inequality for at least one value of α.
P2. (Generalized Size Degeneracy) Let AMatm×n(R)be such that rank(A)>
1; for each i, j = 1, . . . , n it holds Φρ(X)(ei, A)Φρ(X)(ej, A)0if and
only if ρ(Xi)ρ(Xj). Further, there exists k∈ {1, . . . , n}such that
Φρ(X)(ek, A)=0.
P3. (Generalized Risk Degeneracy) Let AMatm×n(R)be a non-zero matrix
such that Aiand Ajare linearly dependent for each i, j = 1, . . . , n; then
Φρ(X)(w, A)=0for each wΓn.
P4. (Reverse Risk Degeneracy) For each wWn\Wnthe equation Φρ(X)(w, A) =
0in the variable AMatm×n(R)admits a solution Awhich is lower
comonotonic.
P5. (Symmetry) Let AMatm×n(R)be a non-zero matrix and ΠMatn×n(R)
be a permutation matrix; then Φρ(X)tw, AΠ) = Φρ(X)(w, A)for each
wΓn.
Proof. P1. We start considering some trivial cases: if rank(A) = 1 the property
is obviously verified, so we assume rank(A) >1. Further, in the cases
α= 0 and α= 1, the property immediately holds, so we let α(0,1).
Denote ξ(α) = αw+(1α)zΓn; by Lemma 3.2, item (iii), fρ(X)(ξ(α)) <
9
αfρ(X)(w) + (1 α)fρ(X)(z). Consequently
Φρ(X)(ξ(α), A) = (rank(A)1)) 1fρ(X)(ξ(α))
maxj=1,...,n fρ(X)(ej)
>(rank(A)1) 1αfρ(X)(w) + (1 α)fρ(X)(z)
maxj=1,...,n fρ(X)(ej)
= (rank(A)1) α1fρ(X)(w)
maxj=1,...,n fρ(X)(ej)
+ (1 α)1fρ(X)(z)
maxj=1,...,n fρ(X)(ej)
min{Φρ(X)(w, A),Φρ(X)(z, A)},
thus property P1 is proved.
P2. The first result follows by simply combining the result of Lemma 3.2,
item (ii), with the definition of Φρ(X). To prove the second result it
is enough to consider an index k∈ {1, . . . , n}such that fρ(X)(ek) =
maxj=1,...,n fρ(X)(ej); by definition of Φρ(X)it follows that Φρ(X)(ek, A) =
0, so property P2 is proved.
P3. From the hypotesis it follows that rank(A) = 1, consequently Φρ(X)(w, A) =
0, so that property P3 is proved.
P4. Since by Lemma 3.2, item (iii), fρ(X)is a strictly convex function then
fρ(X)(w)<maxj=1,...,n fρ(ej) for each wWn\Wn; consequently, A
is a solution of the equation Φρ(X)(w, A) = 0 if and only if rank(A) = 1,
so that property P4 is proved.
P5. From Lemma 3.2, item (iv), and equality rank(AΠ) = rank(A), it imme-
diately follows that Φρ(X)tw, AΠ) = Φρ(X)(w, A), so that property P5
is proved.
We provide the economic interpretation of the properties in Proposition 3.5.
Quasi-concavity together with Generalized Size Degeneracy imply the preference
for diversification with respect to concentration. Quasi-concavity is implied by
concavity; even if constituting a less restrictive requirement, it translates the
economic intuition that a portfolio is always better diversified compared to the
situation of holding its single constituents. Generalized Size Degeneracy man-
ages the behavior of the measure in the extreme cases of single asset portfolios:
in particular, the diversification measure gradually increases starting from the
portfolios invested in one of the maximum risk assets, at which a diversification
equal to zero is attained, to the portfolios invested in one of the minimum risk
assets, at which the maximum diversification among the single asset portfolios
is reached. Further, as already pointed out as a consequence of Proposition 3.4,
10
none of the single assets can give rise to a portfolio with maximum diversifica-
tion. Generalized Risk Degeneracy prevents the possibility that the diversifica-
tion measure is sensitive to the mere number of assets in the portfolio without
considering the number of its effectively different assets; in particular, it as-
serts that adding redundant assets, eventually obtained by simply constructing
a combination of the original assets, is not an effective way to improve diversi-
fication. Reverse Risk Degeneracy avoids the occurrence of a specific unwanted
extreme case, namely that a portfolio of linear independent assets results to
have the same degree of the null diversification. Simmetry is very intuitive and
states that the degree of diversification of a portfolio does not depend on the
order the assets are considered.
Proposition 3.6. Let Φρ(X)be the GPDM with respect to the risk measure ρ.
If ρis shift-invariant then Φρ(X)satisfies the following property.
P6. (Shift-invariance) Let AMatm×n(R)be a non-zero matrix such that
1m×16∈ span(A),11×n6∈ span(At)and αR; then Φρ(X)(w, A +α) =
Φρ(X)(w, A)for each wΓn.
If ρis positively homogeneous then Φρ(X)satisfies the following property.
P7. (Homogeneity) Let AMatm×n(R)be a non-zero matrix and α > 0; then
Φρ(X)(w, αA) = Φρ(X)(w, A)for each wΓn.
Proof. P6. Since rank(α1m×n) = 1 the rank-sum inequality, see [11, Section
0.4.5], yields rank(A+α) = rank(A) which, combined with the result of
Proposition 3.2, item (vi), yields Φρ(X)(w, A +α)=Φρ(X)(w, A), so that
property P6 is proved.
P7. From equality rank(αA) = rank(A) and Proposition 3.2, item (vii), it
immediately follows that Φρ(X)(w, αA) = Φρ(X)(w, A), so that property
P7 is proved.
We provide the economic interpretation of the two properties in Proposi-
tion 3.6. Shift-invariance makes explicit the idea that no deterministic trans-
lation of the returns in matrix A, including, eventually, the one obtained by
adding the risk free asset to the portfolio, is able to positively impact the degree
of diversification. Homogeneity asserts that the diversification of a portfolio is
not affected by any scale transformation of the input data.
Proposition 3.7. Let Φρ(X)be the GPDM with respect to dρ(X). Let A
Matm×n(R)be a nonzero matrix, An+1 Matm×1(R)be the return of a further
asset and A+= (A|An+1)Matm×(n+1) (R). Let wand w
+be the unique solu-
tions of the optimization problems maxwΓnΦρ(X)(w, A)and maxw+Γn+1 Φρ(X)(w+, A+)
respectively. Then the GPDM Φρ(X)satisfies the following properties.
P8. (Generalized Duplication Invariance) If An+1 is linearly dependent on the
columns of Athen Φρ(X)(w
+, A+)=Φρ(X)(w, A).
11
P9. (Generalized Size Monotonicity) If An+1 is linearly independent of the
columns of Athen Φρ(X)(w
+, A+)>Φρ(X)(w, A).
Proof. Proposition 3.4 yields Φρ(X)(w, A) = rank(A)1 and Φρ(X)(w
+, A+) =
rank(A+)1. Since by the hypothesis of P8 and P9 we have either rank(A+) =
rank(A) or rank(A+)>rank(A), the results are immediately proved.
The two properties of Proposition 3.7 illustrate the behavior of GPDM when
the number nof portfolio assets increases. In particular, the Generalized Dupli-
cation Invariance states that the diversification remains unchanged by adding
to the portfolio a linearly redundant asset while, on the opposite, the General-
ized Size Monotonicity asserts that adding an effectively different asset strictly
improves the portfolio diversification.
There are many analogies between the properties of Proposition 3.5, 3.6 and
3.7 and the axioms characterizing a coherent portfolio diversification measure
introduced in [13]. The properties of Quasi-concavity,Reverse Risk Degener-
acy and Simmetry are the same as the analogous axioms in [13], just as the
properties of Shift-invariance and Homogeneity which in our case hold under
given assumptions on the risk measure ρ. The properties of Generalized Size
Degeneracy,Generalized Risk Degeneracy,Generalized Duplication Invariance
and Generalized Size Monotonicity are indeed a generalization of the analogous
axioms presented in [13] from which we have been inspired. In fact, the fol-
lowing situations occur: regarding Generalized Size Degeneracy, if we consider
assets with the same level of risk, which is equivalent to neglect the assets’ risk
information, the diversification measure is zero for all single asset portfolios;
regarding Generalized Risk Degeneracy, if we consider multiple copies of the
same assets we always get a null diversification; regarding Generalized Duplica-
tion Invariance and Generalized Size Monotonicity, adding one additional asset
which is either equal to one of the portfolio constituents or completely different
from all of them, yields a diversification which is either equal to or greater than
its value on the original portfolio.
Remark 3.8. We underline how the proposed approach is suitable to transform
a risk measure in a GPDM. This choice, as highlighted in the introduction,
permits to take advantage of the huge number of risk measures proposed in the
literature to define induced diversification measures.
In the following corollary, summing up the results of Propositions 3.5, 3.6
and 3.7, we state the properties of the GPDMs induced by some well-known
classes of risk measures, recalled in Definition 2.2.
Corollary 3.9. If ρis a:
deviation risk measure then Φρ(X)satisfies all the properties P1-P9;
coherent risk measure then Φρ(X)satisfies all the above properties except
P6;
12
a convex risk measure then Φρ(X)satisfies all the above properties except
P7.
4 Conclusions
It often happens that researchers focus their attention on infinitesimal small
aspects on one specific field taking a chance of loosing a comprehensive view of
the total problem. In this paper we highlight the intimate relation between risk
measures and portfolio diversification measures and outline an effective way to
construct a portfolio diversification measure starting from a given risk measure.
The instrument used to bridge risk and diversification is represented by the
geometric diversified portfolios. As a byproduct, this paper supports the validity
of the geometric diversification strategy in portfolio theory, showing that the
GDP can be equivalently found as the solution of a maximization problem where
the objective function is characterized as a portfolio diversification measure.
Conflict of interest
The authors declare that they have no conflict of interest.
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