1. finance
2. bank

Systemic risk measures on general probability spaces
Eduard Kromer
Department of Mathematics, University of Gießen,
35392 Gießen, Germany;
email: [email protected]
Ludger Overbeck
Department of Mathematics, University of Gießen,
35392 Gießen, Germany;
email: [email protected]
Katrin A. Zilch∗
Department of Mathematics, University of Gießen,
35392 Gießen, Germany;
email: [email protected]
First version: May 23, 2013
This version: October 09, 2014
Abstract
In view of the recent financial crisis systemic risk has become a very important research object.
It is of significant importance to understand what can be done from a regulatory point of view
to make the financial system more resilient to global crises. Systemic risk measures can provide
more insight on this aspect. The study of systemic risk measures should support central banks
and financial regulators with information that allows for better decision making and better risk
management. For this reason this paper studies systemic risk measures on general probability
spaces. In our work we extend the axiomatic approach to systemic risk, as introduced in Chen
et al. (2013), in different directions. One direction is the introduction of systemic risk measures
that do not have to be positively homogeneous. The other direction is that we allow for a general
probability space whereas in Chen et al. (2013) only a finite probability space is considered. This
extends the scope of possible loss distributions of the components of a financial system to a great
extent and introduces more flexibility for the choice of suitable systemic risk measures.
Keywords systemic risk measure, general probability space, dual representation, risk attribution
1
Introduction
The recent financial crisis has posed several open questions of how to better understand and control
systemic risk. These open questions lead to a noticeable increase in the number of publications related
∗
Address correspondence to Katrin Zilch, University of Gießen, Department of Mathematics, Arndtstr. 2, 35392
Gießen, Germany; e-mail: [email protected]
1
to this topic. Systemic risk has been studied from different viewpoints and different approaches to the
various aspects of systemic risk have been chosen.
Several authors study systemic risk from the network modelling viewpoint and consider different
types or forms of contagion that spreads through the financial system, see for instance Nier et al.
(2007), Gai and Kapadia (2010), Amini et al. (2013), Cont et al. (2013) and Hurd and Gleeson (2011).
Another viewpoint is to consider structural models of clearing and to analyse clearing vectors as in
Eisenberg and Noe (2001) and the extensions of this paper in Cifuentes et al. (2005) and Rogers and
Veraart (2013). A good overview of the different approaches to systemic risk and the broad literature
that is concerned with this important research object is given in Staum (2013).
In this paper we study systemic risk from the viewpoint of central banks or financial regulators
that are interested in measurement and control of systemic risk and, in particular, in the attribution of
systemic risk of the whole system to different components of the financial system or financial network.
We believe that a thorough study of systemic risk measures is of major importance for maintaining
stability of the financial systems. In view of the effects of the crisis systemic risk has been dramatically
underestimated (Bartram et al. (2007)). Systemic risk measures that were introduced axiomatically
in Chen et al. (2013) represent a possible and reasonable way to better handle systemic risk from a
regulatory point of view. They provide a tool for the measurement and regulation of systemic risk
of an economy or a financial market and extend the previous traditional approach, where the entire
economy is treated as a portfolio consisting of single firms and the regulator is treated as a portfolio
manager. As outlined in Chen et al. (2013) the traditional portfolio approach has several drawbacks
in a systemic risk framework. For instance, it implicitly allows the netting of profits and losses across
the portfolio components. This may be undesirable from the perspective of a systemic regulator,
who is not able to directly cross-subsidize different firms with distinct ownership interests. For a
more detailed account on the shortcomings of the portfolio approach in a systemic risk measurement
framework we refer the reader to Chen et al. (2013). These shortcomings underline the importance
of the extension of the portfolio approach to a framework that allows for a more realistic and more
flexible measurement of systemic risk.
The approach to systemic risk in Chen et al. (2013) is general, since it is axiomatic and does not
rely on one specific systemic risk measure. Instead it covers a broad class of systemic risk measures.
For instance, the systemic risk measures introduced in Acharya et al. (2012), Gauthier et al. (2012)
and Tarashev et al. (2010) are covered by their approach. Nevertheless it is based on a finite probability space which reduces the scope of possible loss distributions of the different nodes in a financial
network dramatically. For instance the normal distribution is excluded by their approach. For this
reason the extension of the approach in Chen et al. (2013) to a general probability space is very important. Another aspect of the approach in Chen et al. (2013) is that it requires the systemic risk
measures to be positively homogeneous. This is a property that may not be desirable in every risk
measurement framework. We extend the approach of Chen et al. (2013) in two directions that cover
the criticism above. One direction is the extension to a general probability space. This requires a
study of systemic risk measures on infinite dimensional vector spaces which makes the proofs of the
different statements more technical but, on the other hand, allows for arbitrary loss distributions of
the different components of the financial system. The other direction is the extension of their approach
to systemic risk measures without requiring these risk measures to be positively homogeneous. This
introduces more flexibility for the choice of suitable systemic risk measures.
Another important aspect of the understanding of systemic risk is the possibility to explain what
2
fraction of systemic risk is caused by each firm in the financial network, see Drehmann and Tarashev
(2013) and Staum and Liu (2012). Our approach enables us to study such systemic risk attribution
methods in conjunction with systemic risk measures.
The outline of the paper is the following. In Section 2 we introduce our notation and define
systemic risk measures according to several different axioms. In Section 3 we provide a structural
decomposition result for systemic risk measures which states that every systemic risk measure can
be decomposed into an aggregation function and a single-firm risk measure. Since the examples of
systemic risk measures from Chen et al. (2013) can be carried over to our setting with a general
probability space we repeat some of their examples in Subsection 3.1 and add new examples that are
not covered by their approach. In Section 4 we introduce acceptance sets of systemic risk measures and
provide a representation of systemic risk measures with respect to these acceptance sets. This is the
primal representation of systemic risk measures. Furthermore we use the concept of convex duality
to provide a dual representation of systemic risk measures. In Section 5 we study an important
application of the dual representation of systemic risk measures. Under technical assumptions on the
underlying systemic risk measures it immediately provides a risk attribution method such that the
risk of the financial system can be allocated to the components of the system. This method has several
desirable properties such as the full allocation property and the no-undercut property.
2
Model and Notation
Throughout the paper we work with a general probability space (Ω, F, P). We consider a one-period
model consisting of a finite set of n firms that represent n nodes in a financial network. The losses
¯ = (X
¯1, . . . , X
¯ n ) ∈ (Lp )n for 1 ≤ p ≤ ∞ with
of these nodes are given by a random vector X
¯ i is the random loss of firm i. In the following equalities and inequalities
Lp := Lp (Ω, F, P), where X
¯
between random vectors X, Y¯ ∈ (Lp (Ω, F, P))n are interpreted componentwise P-a.s.
For 1 ≤ p < ∞, the space Lp (Ω, F, P) is equipped with the usual Lp -norm, i.e., kXkp :=
R
( |X|p dP)1/p = E[|X|p ]1/p for X ∈ Lp . For 1 ≤ p < ∞ the dual space (Lp (Ω, F, P))∗ is the space
Lq (Ω, F, P) where q ∈ (1, ∞] is such that 1/p + 1/q = 1. For p = ∞, the space L∞ (Ω, F, P) is
equipped with the norm kXk∞ = ess sup |X| and the dual space (L∞ (Ω, F, P))∗ is the Banach space
ba(Ω, F, P) of all bounded, finitely additive measures µ on (Ω, F) with the property that P(A) = 0
implies µ(A) = 0. To facilitate the notation and to keep it consistent, we use
X, ξ = E[Xξ]
for the
on (Lp (Ω, F, P), (Lp (Ω, F, P))∗ ), also for p = ∞ (in fact for µ ∈ ba(Ω, F, P), we
dual
pairing
R
have X, µ = Ω X(ω)dµ(ω) ). Analogously, the dual space of (Lp )n is given by ((Lp (Ω, F, P))n )∗ =
(Lq (Ω, F, P))n for 1 ≤ p < ∞ and ((L∞ (Ω, F, P))n )∗ = (ba(Ω, F, P))n . For p ∈ {1, ∞}, we consider the
weak* topologies on the dual spaces. Otherwise, the dual spaces are equipped with the corresponding
norm topologies. On (Lp )n we use the norm
¯ p,n :=
kXk
n
X
i=1
3
¯ i kp
kX
¯ ∈ (Lp )n . As before, we denote the natural pairing induced between (Lp )n and ((Lp )n )∗ by
for any X
n
X
¯
¯ i Ξi ]
X, Ξ =
E[X
i=1
¯ = (X
¯1, . . . , X
¯ n ) ∈ (Lp )n and Ξ = (Ξ1 , . . . , Ξn ) ∈ ((Lp )n )∗ . Moreover, we use the notation
for X
1n := (1, . . . , 1) and 0n := (0, . . . , 0).
In the following, we need the concept of the subdifferential of a systemic risk measure on (Lp )n .
¯ ∈ (Lp )n |ρ(X)
¯ < +∞}.
The domain of a convex functional ρ : (Lp )n → R∪{+∞} is the set domρ = {X
¯
The subdifferential of a systemic risk measure at X ∈ domρ is the set
(
)
n
X
p
n
∗
p
n
¯ = Ξ ∈ ((L ) ) ¯ i )Ξi ] ≤ ρ(Y¯ ) − ρ(X)
¯
∂ρ(X)
E[(Y¯i − X
∀Y¯ ∈ (L )
i=1
(
)
n
X
p n ∗
p
n
¯ ≤ ρ(X
¯ + Y¯ ) ∀Y¯ ∈ (L )
= Ξ ∈ ((L ) ) E[Y¯i Ξi ] + ρ(X)
i=1
¯ if ∂ρ(X)
¯ is nonempty. It is known that ∂ρ(X)
¯ is a singleton if and only
We call ρ subdifferentiable at X
¯ = {Dρ(X)},
¯
if ρ is Gˆateaux differentiable, see for instance Zalinescu (2002). This means that ∂ρ(X)
¯
¯
where Dρ(X) is the Gˆ
ateaux derivative of ρ at X.
A systemic risk measure ρ : (Lp )n → R ∪ {+∞} quantifies the risk associated with an economy
¯
¯1, . . . , X
¯ n . We will see that systemic
X that is represented by the random losses of the n nodes X
risk measures have a connection to coherent (single-firm) and convex (single-firm) risk measures that
were introduced and studied in Artzner et al. (1999), Delbaen (2002), F¨ollmer and Schied (2002) and
Frittelli and Rosazza Gianin (2002). In fact, later on we will show that a systemic risk measure can be
decomposed into a single-firm risk measure and an aggregation function. The differences between the
terms ”systemic risk measure” and ”single-firm risk measure” are explained in the definitions below.
Consider the following properties of a function ρ0 : Lp → R ∪ {+∞}:
(R1) Monotonicity: If X ≥ Y , then ρ0 (X) ≥ ρ0 (Y ) for any X, Y ∈ Lp .
(R2) Convexity: ρ0 (aX + (1 − a) Y ) ≤ aρ0 (X) + (1 − a) ρ0 (Y ) for any X, Y ∈ Lp and any a ∈ [0, 1].
(R3) Translation property: ρ0 (X + a) = ρ0 (X) + a for any X ∈ Lp and any a ∈ R.
(R4) Positive homogeneity: ρ0 (aX) = aρ0 (X) for any X ∈ Lp and any a ∈ R+ .
(R5) Constancy on R ⊂ R: ρ0 (a) = a for any a ∈ R.
(R6) Normalization: ρ0 (1) = 1.
The properties (R1)-(R4) are very well known and have been motivated in detail in the study of
coherent and convex (single-firm) risk measures, see for instance F¨ollmer and Schied (2002). A special
case of constancy on R ⊂ R is constancy on {1} which is the normalization property from above. The
constancy property was introduced and studied in Frittelli and Rosazza Gianin (2002). This property
follows from the translation property together with the positive homogeneity or together with the
property ρ0 (0) = 0.
4
Definition 2.1 A convex single-firm risk measure is a function ρ0 : Lp → R ∪ {+∞} that satisfies
the properties (R1) and (R2). A positively homogeneous single-firm risk measure is a convex singlefirm risk measure that additionally satisfies the properties (R4) and (R6). A coherent single-firm risk
measure is a positively homogeneous single-firm risk measure that additionally satisfies the property
(R3).
Now, consider the following properties of a function ρ : (Lp )n → R ∪ {+∞}:
¯ ≥ Y¯ , then ρ(X)
¯ ≥ ρ(Y¯ ) for any X,
¯ Y¯ ∈ (Lp )n .
(S1) Monotonicity: If X
¯
¯ ≥ ρ(Y¯ ).
(S2) Preference consistency: If ρ(X(ω))
≥ ρ(Y¯ (ω)) for almost all ω ∈ Ω then ρ(X)
(S3) fρ -constancy: Either Imρ|Rn = R and there exists a surjective map fρ : R → R such that
ρ(a1n ) = fρ (a) for any a ∈ R or Imρ|Rn = R+ and there exists a map fρ : R → R+ and b ∈ R+
such that fρ is surjective and strictly increasing on [b, ∞), fρ (a) = 0 for a ≤ b and ρ(a1n ) = fρ (a)
for any a ∈ R.
(S4) Convexity:
¯ + (1 − a) Y¯ ) ≤ aρ(X)
¯ + (1 − a) ρ(Y¯ ) for any X,
¯ Y¯ ∈ (Lp )n and
(S4a) Outcome convexity: ρ(aX
any a ∈ [0, 1].
¯
(S4b) Risk convexity: Suppose ρ(Z¯ (ω)) = aρ(X(ω))
+ (1 − a) ρ(Y¯ (ω)) for a given scalar a ∈ [0, 1]
¯ ≤ aρ(X)
¯ + (1 − a) ρ(Y¯ ).
and for almost all ω ∈ Ω. Then ρ(Z)
¯ = aρ(X)
¯ for any X
¯ ∈ (Lp )n and any scalar a ∈ R+ .
(S5) Positive homogeneity: ρ(aX)
(S6) Normalization: ρ (1n ) = n.
Note that the properties (S2) and (S4b) are understood in the following way: (S2) If P({ω ∈
¯
¯ ≥ ρ(Y¯ ). (S4b) If P({ω ∈ Ω|ρ(z) = aρ(¯
Ω|ρ(¯
x) ≥ ρ(¯
y ), (¯
x, y¯) = (X(ω),
Y¯ (ω))}) = 1 then ρ(X)
x) + (1 −
¯
¯
¯
¯
¯
¯
a)ρ(¯
y ), (¯
z, x
¯, y¯) = (Z(ω), X(ω), Y (ω))}) = 1 then ρ(Z) ≤ aρ(X) + (1 − a) ρ(Y ).
Moreover, the properties (S1), (S3) and (S4a) lead to the existence of the inverse function fρ−1
of fρ . In case of Imρ|Rn = R, the function fρ−1 is a map from R to R. In case of Imρ|Rn = R+ , the
function fρ is surjective on [b, ∞) and the inverse function is meant to be a map from R+ to [b, ∞).
In the following, we do not distinguish between these two types of inverse functions.
Definition 2.2 A positively homogeneous systemic risk measure is a function ρ : (Lp )n → R ∪ {+∞}
that satisfies the properties (S1), (S2), (S4), (S5) and (S6). If ρ satisfies the properties (S1)-(S4)
and the function fρ from property (S3) satisfies kfρ (Z)kp < ∞ and kfρ−1 (Z)kp < ∞ for any Z ∈ Lp ,
1 ≤ p < ∞, we will call ρ a convex systemic risk measure.
The properties (S1), (S4a), (S5) and (S6) can be interpreted in the same way as in the definition
of convex single-firm risk measures. The preference consistency and the risk convexity property were
¯
introduced in Chen et al. (2013). Preference consistency means that if the risk of economy X(ω)
∈ R is
¯
greater than the risk of economy Y (ω) ∈ R for almost all ω ∈ Ω, then the risk of the random economy
¯ ∈ (Lp )n should be greater than the risk of the random economy Y¯ ∈ (Lp )n . For the formulation of
X
¯
the risk convexity property we start with the risk of the economy Z(ω)
that is the convex combination
¯
¯
of the risk of the economies X(ω) and Y (ω) for almost all ω ∈ Ω. Then the risk convexity property
requires that the risk of the random economy Z¯ ∈ (Lp )n is at most the risk of the convex combination
5
¯ Y¯ ∈ (Lp )n . This means that the introduction of ”randomness”
of the risks of the random economies X,
¯ and Y¯ .
should not increase the risk of Z¯ above the risk of the convex combination of the risks of X
As already mentioned in Chen et al. (2013), if the economy consists of a single firm, then the risk
convexity property is implied by the outcome convexity, the translation property and ρ(0) = 0. For
additional information on these two properties we refer to Chen et al. (2013).
The fρ -constancy property is new. It simply tells us that if each component in the financial system
has the same constant loss a ∈ R, then the systemic risk measure is a function in a ∈ R that maps
from R to R. Surjectivity of fρ , in conjunction with the monotonicity and the convexity property of
ρ, implies in particular that the function fρ has to be strictly increasing, convex and unbounded. The
property that fρ is strictly increasing means that if each node has the same strictly higher constant
loss c ∈ R, c > a, then this should lead to a strictly higher risk ρ(c1n ) > ρ(a1n ). The unboundedness
of fρ ensures that the risk tends to infinity if the loss tends to infinity. This means that we preclude
systemic risk measures ρ that admit an upper bound. The fρ -constancy property is strongly connected
to the fΛ -constancy property that we introduce to define convex aggregation functions.
Consider the following properties of a function Λ : Rn → R:
(A1) Monotonicity: If x
¯ ≥ y¯, then Λ(¯
x) ≥ Λ(¯
y ) for any x
¯, y¯ ∈ Rn .
(A2) Convexity: Λ(a¯
x + (1 − a) y¯) ≤ aΛ(¯
x) + (1 − a) Λ(¯
y ) for any x
¯, y¯ ∈ Rn and any a ∈ [0, 1].
(A3) fΛ -constancy: Either ImΛ = R and there exists a surjective map fΛ : R → R such that Λ(a1n ) =
fΛ (a) for any a ∈ R or ImΛ = R+ and there exists a map fΛ : R → R+ and b ∈ R+ such that fΛ
is surjective and strictly increasing on [b, ∞), fΛ (a) = 0 for a ≤ b and Λ(a1n ) = fΛ (a) for any
a ∈ R.
(A4) Positive homogeneity: Λ(a¯
x) = aΛ(¯
x) for any x
¯ ∈ Rn and any a ∈ R+ .
(A5) Normalization: Λ (1n ) = n.
As before, the properties (A1)-(A3) imply the existence of the inverse function fΛ−1 .
Definition 2.3 A positively homogeneous aggregation function is a function Λ : Rn → R that satisfies
the properties (A1), (A2), (A4) and (A5). If Λ satisfies the properties (A1)-(A3) and the function fΛ
from (A3) satisfies kfΛ (Z)kp < ∞ and kfΛ−1 (Z)kp < ∞ for any Z ∈ Lp , 1 ≤ p < ∞, we will call Λ a
convex aggregation function.
Here, as in the definition of systemic risk measures, we have introduced the fΛ -constancy property.
The motivation behind this property is similar to the motivation of the fρ -constancy property. It tells
us that in case of the same constant loss a ∈ R of each component in the economy the aggregation
function is a function in a that maps from R to R. We need the constancy properties (R5), (S3) and
(A3) for the decomposition result in Theorem 3.1. These new properties allow us to consider convex
systemic risk measures respectively convex aggregation functions that are not positively homogeneous.
Although the constancy property is an additional technical condition that is not required in the
standard single-firm risk measures approach, it does not preclude systemic risk measures that are
based on standard single-firm risk measures. The reason for this is that the constancy property (R5)
of ρ0 follows from the translation property together with the positive homogeneity or together with the
property ρ0 (0) = 0. Both properties, in general, are satisfied by standard single-firm risk measures.
6
Before we move on to the decomposition theorem, the next lemma reveals the implications of the
fΛ -constancy property for the aggregation function Λ. We will see that this lemma is needed at a
critical point in the proof of the decomposition theorem.
Lemma 2.4 Let Λ : Rn → R be a convex aggregation function. Then Λ((Lp )n ) = Lp .
¯ = (X
¯1, . . . , X
¯ n ) ∈ (Lp )n , we can define a random variable Z ¯ by
Proof. For any X
X
(
¯ 1 (ω), . . . , X
¯ n (ω)} for ω ∈ A
max{X
ZX¯ (ω) =
¯ 1 (ω), . . . , X
¯ n (ω)} for ω ∈ Ω \ A
min{X
¯
where A = {ω ∈ Ω|Λ(X(ω))
≥ 0}. Since the maximum and the minimum of Lp -integrable random
p
¯ ∈ (Lp )n . From the monotonicity property
variables is L -integrable, we know that ZX¯ ∈ Lp for any X
¯
¯
of Λ we get 0 ≤ Λ(X(ω))
≤ Λ(ZX¯ (ω)1n ) for any ω ∈ A and 0 > Λ(X(ω))
≥ Λ(ZX¯ (ω)1n ) for any
ω ∈ Ω \ A. This leads to
¯
|Λ(X(ω))|
≤ |Λ(ZX¯ (ω)1n )| = |fΛ (ZX¯ (ω))|
¯ ∈ (Lp )n , there exists a random variable Z ¯ ∈ Lp such that
for any ω ∈ Ω. It follows that for any X
X
¯ ] ≤ E[|fΛ (Z ¯ )| ].
E[|Λ(X)|
X
p
p
From kfΛ (Z)kp < ∞ for any Z ∈ Lp we get
¯ p ≤ kfΛ (Z ¯ )kp < ∞
kΛ(X)k
X
¯ ∈ (Lp )n . This means that Λ((Lp )n ) ⊂ Lp .
for any X
Now suppose that we have a random variable X ∈ Lp . We know from the properties (A1)-(A3) of
Λ that fΛ is a bijective function. Thus, the inverse function fΛ−1 exists and we can define a random
variable Y by
Y (ω) = fΛ−1 (X(ω)).
From kfΛ−1 (Z)kp < ∞ for any Z ∈ Lp we know that Y ∈ Lp . It follows that fΛ (Y ) = X such that the
fΛ -constancy property of Λ implies that for any X ∈ Lp there exists a random vector Y 1n ∈ (Lp )n
such that
Λ(Y 1n ) = fΛ (Y ) = X.
This means that Lp ⊂ Λ((Lp )n ) and we finally arrive at Λ((Lp )n ) = Lp .
2
Note that if p = ∞ and we have a convex aggregation function Λ : Rn → R, then the properties
kfΛ (Z)k∞ < ∞ and kfΛ−1 (Z)k∞ < ∞ for any Z ∈ L∞ are automatically satisfied.
Remark 2.5 Let R ∈ {R, R+ } and consider a positively homogeneous aggregation function Λ : Rn →
R. In this case, we do not need to claim the fΛ -constancy property. It follows immediately from the
properties of Λ that we have a function fΛ : R → R such that Λ(a1n ) = fΛ (a) and fΛ is given by
(
an,
a≥0
fΛ (a) :=
a(−Λ(−1n )), a < 0
for R = R and fΛ (a) = na+ for R = R+ . Here, we immediately have kfΛ (Z)kp < ∞ and kfΛ−1 (Z)kp <
∞ for any Z ∈ Lp . In conclusion, this means that each positively homogeneous aggregation function
is also a convex aggregation function. Similarly, each positively homogeneous systemic risk measure is
also a convex systemic risk measure.
7
3
Structural Decomposition
In this section we provide a structural decomposition result which states that any convex systemic
risk measure can be decomposed into a convex single-firm risk measure and a convex aggregation
function. This result does not require the systemic risk measure to be positively homogeneous and
thus extends the result from Chen et al. (2013) to a general probability space and to convex, not
necessarily positively homogeneous, systemic risk measures.
The proof of the decomposition result in the following theorem is similar to the proof of Theorem 1
in Chen et al. (2013) in several aspects but it also requires new arguments since it requires the fρ - and
the fΛ -constancy property that we have introduced before. Further on, in Corollary 3.3, we provide
an analogous decomposition result to Theorem 1 in Chen et al. (2013) for positively homogeneous
systemic risk measures on a general probability space.
Theorem 3.1 (a) A function ρ : (Lp )n → R ∪ {+∞} with ρ(Rn ) = R is a convex systemic risk
measure if and only if there exists a convex aggregation function Λ : Rn → R with Λ(Rn ) = R
and a single-firm risk measure ρ0 : Lp → R ∪ {+∞} that satisfies the constancy property on R
such that ρ is the composition of ρ0 and Λ, i.e.,
¯ = (ρ0 ◦ Λ)(X)
¯ for all X
¯ ∈ (Lp )n .
ρ(X)
(b) A function ρ : (Lp )n → R ∪ {+∞} with ρ(Rn ) = R+ is a convex systemic risk measure if and
only if there exists a convex aggregation function Λ : Rn → R with Λ(Rn ) = R+ and a single-firm
risk measure ρ0 : Lp → R ∪ {+∞} that satisfies the constancy property on R+ such that ρ is the
composition of ρ0 and Λ, i.e.,
¯ = (ρ0 ◦ Λ)(X)
¯ for all X
¯ ∈ (Lp )n .
ρ(X)
Proof. Define R = S = R in case of (a) and R = R+ and S = [b, ∞) for b ∈ R+ in case of (b). Suppose
that ρ is a convex systemic risk measure with fρ : R → R that is surjective and strictly increasing on
S. Define
Λ(¯
x) := ρ(¯
x)
(3.1)
for any x
¯ ∈ Rn . Since ρ is convex, it follows
Λ(a¯
x + (1 − a) y¯) = ρ(a¯
x + (1 − a) y¯)
≤ aρ(¯
x) + (1 − a) ρ(¯
y ) = aΛ(¯
x) + (1 − a) Λ(¯
y ).
for any x
¯, y¯ ∈ Rn and any a ∈ [0, 1]. Thus, Λ satisfies the convexity property (A2). Similarly, the
monotonicity (A1) of Λ holds due the monotonicity of ρ.
Since ρ satisfies the fρ -constancy, we have
Λ(a1n ) = ρ(a1n ) = fρ (a)
(3.2)
for any a ∈ R. This gives the fΛ -constancy (A3) of Λ for fΛ := fρ and Λ(Rn ) = R. From Lemma 2.4
we have Λ((Lp )n ) = Lp for (a) and it follows analogously that Λ((Lp )n ) = Lp+ for (b).
Now, consider ρ˜0 : Λ((Lp )n ) → R ∪ {+∞} given by
¯
ρ˜0 (X) := ρ(X)
¯ ∈ (Lp )n satisfies Λ(X)
¯ = X.
where X
8
(3.3)
We define ρ0 : Lp → R ∪ {+∞} by
(
ρ˜0 (X)
if Λ((Lp )n ) = Lp
ρ0 (X) :=
.
ρ˜0 (X + ) if Λ((Lp )n ) = Lp+
(3.4)
¯ Y¯ ∈ (Lp )n with Λ(X)
¯ = Λ(Y¯ ). Then we have
To show that ρ˜0 is well-defined, consider X,
¯
¯
ρ(X(ω))
= Λ(X)(ω)
≥ Λ(Y¯ )(ω) = ρ(Y¯ (ω)) and
¯
¯
ρ(X(ω))
= Λ(X)(ω)
≤ Λ(Y¯ )(ω) = ρ(Y¯ (ω))
¯ = ρ(Y¯ ), and thus ρ˜0
for almost all ω ∈ Ω. From the preference consistency of ρ it follows that ρ(X)
is well-defined. If ρ˜0 defined by (3.3) satisfies monotonicity, convexity and constancy on R, then ρ0
given in (3.4) satisfies the properties (R1), (R2) and (R5). Suppose that X, Y ∈ Λ((Lp )n ) are such
¯ = X, Λ(Y¯ ) = Y and X ≤ Y . The preference consistency of ρ and
that Λ(X)
¯
¯
ρ(X(ω))
= Λ(X)(ω)
≤ Λ(Y¯ )(ω) = ρ(Y¯ (ω))
for almost all ω ∈ Ω imply ρ˜0 (X) ≤ ρ˜0 (Y ). Now, let X, Y ∈ Λ((Lp )n ) and a ∈ [0, 1] and consider
¯ Y¯ , Z¯ ∈ (Lp )n satisfy
Z := aX + (1 − a)Y . Suppose that X,
¯
ρ˜0 (X) = ρ(X),
ρ˜0 (Y ) = ρ(Y¯ ),
¯
ρ˜0 (Z) = ρ(Z)
¯ = X, Λ(Y¯ ) = Y and Λ(Z)
¯ = Z. It follows
with Λ(X)
¯
¯
ρ(Z(ω))
= Λ(Z)(ω)
= Z (ω) = aX (ω) + (1 − a) Y (ω)
¯
¯
= aΛ(X)(ω)
+ (1 − a) Λ(Y¯ )(ω) = aρ(X(ω))
+ (1 − a) ρ(Y¯ (ω))
for almost all ω ∈ Ω. An application of the risk convexity of ρ yields
¯ ≤ aρ(X)
¯ + (1 − a) ρ(Y¯ ) = a˜
ρ˜0 (Z) = ρ(Z)
ρ0 (X) + (1 − a) ρ˜0 (Y ) ,
i.e., ρ˜0 is convex. Since for any real a ∈ R, there exists x
¯ ∈ Rn such that Λ(¯
x) = a and
ρ˜0 (a) = ρ(¯
x),
a = Λ(¯
x) = ρ(¯
x) implies ρ˜0 (a) = a for any a ∈ R. Thus, ρ˜0 admits the constancy property on R.
From the definition of Λ and ρ0 we get ρ = ρ0 ◦ Λ.
For the other part of the proof suppose that Λ is a convex aggregation function with fΛ : R → R
that is surjective and strictly increasing on S. Let ρ0 be a convex single-firm risk measure with
ρ0 (a) = a for any a ∈ R. Since Λ and ρ0 are monotone and convex, it is clear that ρ is monotone and
¯ Y¯ ∈ (Lp )n
satisfies the outcome convexity property ((S1) and (S4a)). Moreover, if two economies X,
satisfy
¯ (ω)) = ρ(X
¯ (ω)) ≥ ρ(Y¯ (ω)) = (ρ0 ◦ Λ)(Y¯ (ω))
(ρ0 ◦ Λ)(X
for almost all ω ∈ Ω, then the monotonicity of ρ0 implies
¯ (ω)) ≥ Λ(Y¯ (ω))
Λ(X
9
for almost all ω ∈ Ω. By the monotonicity of ρ0 , it follows that
¯ = (ρ0 ◦ Λ)(X)
¯ ≥ (ρ0 ◦ Λ)(Y¯ ) = ρ(Y¯ ),
ρ(X)
which means that ρ satisfies the preference consistency (S2).
¯
¯ Y¯ , Z¯ ∈ (Lp )n we have ρ(Z(ω))
Now, suppose that for a given scalar a ∈ [0, 1] and economies X,
=
¯
aρ(X(ω))
+ (1 − a) ρ(Y¯ (ω)) for almost all ω ∈ Ω. This means
¯
¯
¯
ρ(Z(ω))
= ρ0 (Λ(Z(ω)))
= aρ0 (Λ(X(ω)))
+ (1 − a)ρ0 (Λ(Y¯ (ω)))
¯
= aρ(X(ω))
+ (1 − a)ρ(Y¯ (ω))
(3.5)
for almost all ω ∈ Ω. Since we have Λ(Rn ) = R and ρ0 (a) = a for all a ∈ R, it follows from (3.5) that
¯ (ω)) + (1 − a) Λ(Y¯ (ω))
Λ(Z¯ (ω)) = aΛ(X
for almost all ω ∈ Ω. From the convexity of ρ0 we get
¯ = ρ0 (Λ(Z))
¯ ≤ aρ0 (Λ(X))
¯ + (1 − a) ρ0 (Λ(Y¯ )) = aρ(X)
¯ + (1 − a) ρ(Y¯ ),
ρ(Z)
which means that ρ has the risk convexity property (S4b).
By the fΛ -constancy of Λ and ρ0 (c) = c for all c ∈ R, we obtain
ρ(a1n ) = ρ0 (Λ(a1n )) = fΛ (a)
for any a ∈ R. This is the fρ -constancy property (S3) with fρ := fΛ .
2
Remark 3.2 Note that in our proof of Theorem 3.1 the fΛ -constancy and the resulting Lemma 2.4
guarantee that ρ˜0 is defined for all X ∈ Λ((Lp )n ) in equation (3.3). Chen et al. (2013) use the positive
homogeneity property in this part of the proof.
Now, we can deduce the positively homogeneous case as a special case of Theorem 3.1.
Corollary 3.3 (a) A function ρ : (Lp )n → R ∪ {+∞} is a positively homogeneous systemic risk
measure with ρ(Rn ) = R if and only if there exists a positively homogeneous aggregation function
Λ : Rn → R with Λ(Rn ) = R and a coherent single-firm risk measure ρ0 : Lp → R ∪ {+∞} such
that ρ is the composition of ρ0 and Λ, i.e.,
¯ = (ρ0 ◦ Λ)(X)
¯ for all X
¯ ∈ (Lp )n .
ρ(X)
(b) A function ρ : (Lp )n → R ∪ {+∞} is a positively homogeneous systemic risk measure with
ρ(Rn ) = R+ if and only if there exists a positively homogeneous aggregation function Λ : Rn → R
with Λ(Rn ) = R+ and a positively homogeneous single-firm risk measure ρ0 : Lp → R ∪ {+∞}
such that ρ is the composition of ρ0 and Λ, i.e.,
¯ = (ρ0 ◦ Λ)(X)
¯ for all X
¯ ∈ (Lp )n .
ρ(X)
10
Proof. Note that every positively homogeneous systemic risk measure characterized by a positively
homogeneous aggregation function Λ and a positively homogeneous single-firm risk measure ρ0 is also
a convex systemic risk measure characterized by the convex aggregation function Λ and the single-firm
risk measure ρ0 . Hence, we can apply Theorem 3.1.
To prove the only if part, it remains to show that Λ defined by (3.1) and ρ˜0 defined by (3.3) are
positively homogeneous and normalized and additionally that ρ0 = ρ˜0 satisfies the translation property
in part (a). It is obvious that Λ inherits the positive homogeneity (A4) and normalization (A5) from ρ
and similarly ρ˜0 is positively homogenous since ρ and Λ satisfy the corresponding property. Moreover,
we have ρ0 (a) = a for all a ∈ R and ρ0 is sub-additive since ρ0 satisfies the convexity and the positive
homogeneity property. From these two properties it follows
ρ0 (X + a) ≤ ρ0 (X) + ρ0 (a) = ρ0 (X) + a and
ρ0 (X + a) = ρ0 (X − (−a)) ≥ ρ0 (X) − ρ0 (−a) = ρ0 (X) + a
for any a ∈ R such that the translation property (R3) is satisfied.
For the if part it remains to show that ρ is normalized (S6) and satisfies positive homogeneity
(S5). Again, ρ inherits positive homogeneity (S5) from Λ and ρ0 . Since in case of (a), ρ0 has the
positive homogeneity and the translation property and in case of (b), ρ0 is positively homogeneous
and normalized, we get ρ0 (a) = a for all a ∈ R. From this and from the normalization of Λ it follows
that ρ is normalized (S6).
2
3.1
Examples of systemic risk measures
In Chen et al. (2013) the authors have introduced several examples of positively homogeneous systemic
risk measures that are based on coherent single-firm risk measures and positively homogenous aggregation functions. These examples can still be applied on a general probability space. Additionally, we
introduce systemic risk measures that are not positively homogeneous, and thus are not covered by
the approach of Chen et al. (2013).
Example 3.4 If we sum up the losses/ profits of each firm i, then we get the linear aggregation
function
n
X
Λsum (¯
x) =
x
¯i .
(3.6)
i=1
Convex generalizations of the form
Λaff (¯
x) = ¯bt x
¯+c
for ¯b ∈ Rn with ¯b > 0 and c ∈ R are also possible.
If we denote by Qλ for λ ∈ (0, 1] the set of all probability measures Q P where the density
ϕQ := dQ/dP satisfies ϕQ ≤ 1/λ P-a.s., then we can write the Average Value at Risk as
AVaRλ (X) = max EQ [X]
Q∈Qλ
11
for any X ∈ Lp , see for instance F¨
ollmer and Schied (2002). This leads to the positively homogeneous
systemic risk measure (systemic expected shortfall from Acharya et al. (2010))
!
n
X
¯ = AVaRλ
¯i .
ρSEM (X)
X
i=1
As already pointed out in Chen et al. (2013), this might be an unrealistic choice for the aggregation
function since usually we do not want to subsidize losses of one firm with gains of other firms in
a systemic risk framework. In this context, the aggregation function in (3.6) covers the traditional
portfolio approach that we have mentioned in the Introduction.
The following example shows an aggregation function without this feature. Furthermore, it introduces
a systemic risk measure that is not positively homogeneous although the aggregation function itself is
positively homogeneous.
Example 3.5 If we do not want the compensation of losses by profits of other firms, then we can
define the positively homogeneous aggregation function
Λloss (¯
x) =
n
X
x
¯+
i .
(3.7)
i=1
This aggregation function sets all profits equal to 0. Thus, only losses are aggregated. The aggregation
function in (3.7) can be easily modified to an aggregation function
x)
Λbloss (¯
=
n
X
(¯
xi − b)+ ,
b ∈ R+
i=1
such that another lower bound than zero is introduced and only losses above b are aggregated. This
type of aggregation function is covered by part (b) of Theorem 3.1. Now, suppose that we have a
non-decreasing function g : [0, 1] → [0, 1] satisfying g(0) = 0 and g(1) = 1. This function induces the
distorted probability Pg (A) := g ◦ P(A) which can be used to define the distorted expectation Eg as a
Choquet integral:
Z 0
Z ∞
Pg (X > t)dt +
(Pg (X > t) − 1)dt, X ∈ L∞ .
Eg [X] :=
−∞
0
The distorted expectation can be used to define the (single-firm) distortion entropic risk measure by
ρg,γ
0 (X) =
1
log Eg [exp(γX)],
γ
γ > 0.
Now, we can use the aggregation function from (3.7) to define the distortion entropic systemic risk
measure by
"
!#
n
X
1
+
g,γ ¯
¯
ρ (X) = log Eg exp γ
X
, γ > 0.
(3.8)
i
γ
i=1
Although the aggregation function is positively homogeneous, the distortion exponential systemic risk
measure from (3.8) is a systemic risk measure that is not positively homogeneous since the underlying
g,γ satisfies the f constancy
single-firm risk measure ρg,γ
ρ
0 is not positively homogeneous. Nevertheless, ρ
with the function fρ (a) = na+ .
12
Chen et al. (2013) pointed out that both Λsum and Λloss are indifferent of how a large loss is spread in
the economy. Other choices of aggregation functions that take this feature into account are presented
in the following example.
Example 3.6 If the regulator strongly prefers several smaller losses against one large loss, we can
use the convex aggregation function
Λexp (¯
x) =
n
X
(exp(γ x
¯+
i ) − 1),
γ > 0.
(3.9)
i=1
Again, it is not allowed to subsidize losses with gains. Furthermore, Λexp is not positively homogeneous
and the regulator penalizes large losses exponentially. The function fΛ from the fΛ -constancy property
+
of Λ in this case is given by fΛ (a) = n(eγa −1). Note that we can use this convex aggregation function
with random vectors from (L∞ )n .
Another possible choice is the following convex aggregation function
Λplin (¯
x) =
n
X
λ(¯
xi )
i=1
with


0
λ(x) = ax


b(x − c) + ac
for x < 0
for 0 ≤ x < c
for x ≥ c
where 0 < a < b and c > 0. In this case, we penalize losses above a barrier c by increasing the slope
of the piecewise linear aggregation function.
Example 3.7 Consider a network of n nodes where each node represents a specific firm. Assume
that each firm i has liabilities to other firms in the network represented by a matrix Π = (Πij )i,j=1,...,n .
This means that firm i has to pay the proportion Πij of its total liabilities to firm j. Moreover, an
external regulator has the possibility to inject capital in the system. If we interpret x
¯ ∈ Rn as the
realized loss, then each firm has two possibilities to cover the losses: Either it receives money from the
regulator by an amount ¯bi or it reduces its payments y¯i to other firms. Note that if firm i decides to
reduce their payments to firm j, then firm j faces new losses Πij y¯i . Chen et al. (2013) introduced the
˜ CM that is motivated by the structural contagion model of Eisenberg and Noe (2001) and
function Λ
that measures the “net systemic cost of the contagion” by
( n
!)
X
ΛCM (¯
x) :=
min
y¯i + γ¯bi
.
(3.10)
P
¯bi +¯
¯j
yi ≥¯
xi + n
j=1 Πji y
∀i=1,...,n, ¯b,¯
y ∈Rn
+
i=1
This function can be normalized to
ΛCM (¯
x) =
˜ CM (¯
Λ
x) · n
˜ CM (1n )
Λ
such that ΛCM is a positively homogeneous aggregation function. γ > 1 in (3.10) represents the
possibility P
for the regulator to balance
Pn ¯ between aggregate shortfalls across the economy on interfirm
n
obligation i=1 y¯i and the cost i=1 bi of injecting new capital to support the economy.
13
4
Representations of systemic risk measures
In this section we study acceptance sets of systemic risk measures and use these sets to provide a
primal and a dual representation of convex and positively homogeneous systemic risk measures.
Define the acceptance sets of the systemic risk measure ρ = ρ0 ◦ Λ with single firm risk measure
ρ0 and aggregation function Λ by
(4.11)
Aρ0 = (m, X) ∈ R × Lp m ≥ ρ0 (X) ,
p
p n
¯
¯
(4.12)
AΛ = (Y, Z) ∈ L × (L ) Y ≥ Λ(Z) .
The following properties of subsets of R × Lp , R × Rn and Lp × (Lp )n are needed in the subsequent
study.
Definition 4.1 Let V × W be two linear spaces. A set S ⊂ V × W satisfies the monotonicity property
if (v, w1 ) ∈ S, w2 ∈ W and w1 ≥ w2 implies (v, w2 ) ∈ S. A set S ⊂ V × W satisfies the epigraph
property if (v1 , w) ∈ S, v2 ∈ V and v2 ≥ v1 implies (v2 , w) ∈ S.
The next proposition provides the primal representation of systemic risk measures and connects
properties of single-firm risk measures and aggregation functions to the corresponding acceptance sets.
Proposition 4.2 Suppose that ρ = ρ0 ◦ Λ is a convex systemic risk measure with convex single-firm
risk measure ρ0 : Lp → R ∪ {+∞} and convex aggregation function Λ : Rn → R. Let Aρ0 and AΛ be
the corresponding acceptance sets.
(i) Aρ0 and AΛ satisfy the following properties:
(a) Aρ0 and AΛ satisfy the monotonicity property.
(b) Aρ0 and AΛ satisfy the epigraph property.
(c) Aρ0 and AΛ are convex sets.
(d) (a, a) ∈ Aρ0 with inf{r ∈ R|(r, a) ∈ Aρ0 } = a for all a ∈ ImΛ and (fΛ (a), a1n ) ∈ AΛ with
ess inf{Y ∈ Lp |(Y, a1n ) ∈ AΛ } = fΛ (a) for all a ∈ R.
If ρ is a positively homogeneous systemic risk measure, then the following properties are additionally satisfied:
(e) Aρ0 and AΛ are cones.
(f ) (n, 1n ) ∈ AΛ with ess inf{Y ∈ Lp |(Y, 1n ) ∈ AΛ } = n.
¯ ∈ (Lp )n , ρ admits the primal representation of
(ii) For any X
¯ = inf{m ∈ R(m, Y ) ∈ Aρ , (Y, X)
¯ ∈ AΛ }
ρ(X)
0
where we set inf ∅ := +∞.
14
(4.13)
Proof. Part (a)-(f) hold due to the claimed properties of ρ0 and Λ. Since ρ = ρ0 ◦ Λ, we immediately
get the following representation
¯ = inf m ∈ Rm ≥ (ρ0 ◦ Λ)(Z)
¯
ρ(Z)
for all Z¯ ∈ (Lp )n . By using
ρ0 (X) = inf {m ∈ R |m ≥ ρ0 (X) } = inf m ∈ R(m, X) ∈ Aρ0
and
¯ ∈ AΛ
¯ = ess inf Y ∈ Lp (Y, Z)
¯ = ess inf Y ∈ Lp Y ≥ Λ(Z)
Λ(Z)
for X ∈ Lp and Z¯ ∈ (Lp )n , it follows
¯ = inf m ∈ R(m, Λ(Z))
¯ ∈ Aρ = inf m ∈ R(m, ess inf Y ∈ Lp (Y, Z)
¯ ∈ AΛ ) ∈ Aρ
ρ(Z)
0
0
for all Z¯ ∈ (Lp )n . Finally, the monotonicity property of Aρ0 implies (4.13).
Lp
2
Rn
The next result shows that we can use subsets of R ×
and R ×
with specific properties to
define systemic risk measures. In contrast to Proposition 4.2, we start with specific subsets of R × Lp
and R × Rn as primal objects whereas in Proposition 4.2 we have used systemic risk measures to define
the corresponding acceptance sets.
Proposition 4.3 Assume that ∅ =
6 B ⊂ R × Lp and ∅ =
6 C ⊂ R × Rn and define
ρB
0 (X) := inf{m ∈ R |(m, X) ∈ B },
ΛC (¯
x) := inf{m ∈ R(m, x
¯) ∈ C}
for all x
¯ ∈ Rn and X ∈ Lp . Suppose that inf{m ∈ R(m, Y ) ∈ B} > −∞ for all Y ∈ Lp and
¯) ∈ C} > −∞ for all x
¯ ∈ Rn . Moreover, assume that C is such that for any x
¯ ∈ Rn ,
inf{m ∈ R(m, x
there exists m ∈ R with (m, x
¯) ∈ C. Then we have the following properties:
C
(a) If B and C satisfy the monotonicity property, then ρB
0 and Λ are monotone.
C
(b) If B and C are convex, then ρB
0 and Λ are convex.
C
(c) If B and C are cones, then ρB
0 and Λ are positively homogeneous.
(d) If (1, 1) ∈ B with inf{m ∈ R|(m, 1) ∈ B} = 1 and (n, 1n ) ∈ C with inf{m ∈ R|(m, 1n ) ∈ C} = n,
C
then ρB
0 and Λ are normalized.
(e) If (a, a) ∈ B and inf{m ∈ R|(m, a) ∈ B} = a for any a ∈ R, then ρB
0 (a) = a.
(f ) Define fC (a) := inf{m ∈ R(m, a1n ) ∈ C} for any a ∈ R and suppose that the function fC : R → R
is surjective. Then ΛC satisfies the fΛ -constancy with fΛC = fC .
In particular, if B and C satisfy all of the properties from (a)-(e), then ρB,C : (Lp )n → R ∪ {+∞}
defined by
¯ := inf m ∈ R(m, Y ) ∈ B, (Y, X)
¯ ∈ AΛC
¯ ∈ (Lp )n
ρB,C (X)
for X
(4.14)
(where AΛC is the acceptance set of ΛC ) is a positively homogeneous systemic risk measure with ρB,C =
−1
C
ρB
0 ◦Λ . If B and C satisfy the properties from (a), (b), (e) and (f ) and kfC (Z)kp < ∞ and kfC (Z)kp <
p
B,C
B,C
C
∞ for any Z ∈ L , then ρ
defined in (4.14) is a convex systemic risk measure with ρ
= ρB
0 ◦Λ .
Furthermore, B is a subset of AρB and C is a subset of AΛC .
0
15
Proof. If B is monotone, it follows for X, Y ∈ Lp with X ≥ Y that
{m ∈ R |(m, X) ∈ B } ⊂ {m ∈ R |(m, Y ) ∈ B } ,
B
which yields ρB
0 (X) ≥ ρ0 (Y ). Suppose B is convex, then convexity of ρ0 follows from
ρB
0 (aX + (1 − a)Y ) = inf {ax + (1 − a)y ∈ R |x, y ∈ R, a (x, X) + (1 − a) (y, Y ) ∈ B }
B
≤ inf {ax + (1 − a)y ∈ R |(x, X) , (y, Y ) ∈ B } = aρB
0 (X) + (1 − a)ρ0 (Y )
for any a ∈ [0, 1] and any X, Y ∈ Lp . Moreover, if B is a cone, we have
B
ρB
0 (aX) = inf {am ∈ R |(am, aX) ∈ B } ≤ inf {am ∈ R |(m, X) ∈ B } = aρ0 (X)
for a > 0 and X ∈ Lp . Let x < ρB
/ B. Hence,
0 (X) = inf {m ∈ R |(m, X) ∈ B }. It follows that (x, X) ∈
B
B
B
(ax, aX) ∈
/ B and ax < ρ0 (aX). Together, we arrive at ρ0 (aX) = aρ0 (X) for any a > 0. As a
B
B
B
consequence, we have ρB
0 (0) = ρ0 (a · 0) = aρ0 (0) for all a > 0, but this also implies ρ0 (0) = 0.
Similarly, ΛC inherits monotonicity, convexity and positive homogeneity from the corresponding
properties of C.
Moreover, (d), (e) and (f) are obvious. From the assumptions on C it follows for x
¯ ∈ Rn
ΛC (¯
x) = inf m ∈ R(m, x
¯) ∈ C ∈ R.
If all of the properties from (a)-(e) are satisfied, then ΛC considered as function on (Lp )n maps
into Lp and we obtain the composition
C ¯
C ¯
ρB
0 ◦ Λ (X) = inf m ∈ R (m, Λ (X)) ∈ B
¯ ∈ AΛC ) ∈ B
= inf m ∈ R(m, ess inf Y ∈ Lp (Y, X)
¯ ∈ AΛC
= inf m ∈ R(m, Y ) ∈ B, (Y, X)
(4.15)
and it follows from Corollary 3.3 that ρB,C is a positively homogeneous systemic risk measure. If
the properties from (a), (b), (e) and (f) and kfC (Z)kp < ∞ and kfC−1 (Z)kp < ∞ for any Z ∈ Lp
are satisfied, then again equation (4.15) holds and we know from Theorem 3.1 that ρB,C is a convex
systemic risk measure.
Finally, (x, X) ∈ B implies ρB
=
0 (X) = inf {m ∈ R |(m, X) ∈ B } ≤ x, hence (x, X) ∈ AρB
0
p
B
{(m, X) ∈ R × L |m ≥ ρ0 (X)}, which means that B ⊂ AρB . Similarly, we get C ⊂ AΛC : (x, x
¯) ∈ C
0
C
p
p
¯
implies Λ (¯
x) = inf m ∈ R (m, x
¯) ∈ C ≤ x, and therefore (x, x
¯) ∈ AΛC = {(Y, X) ∈ L × (L )n |Y ≥
¯
ΛC (X)}.
2
For the dual representation of convex systemic risk measures we need the lower semi-continuity of
the convex single-firm risk measure ρ0 . This property is introduced in the following definition.
Definition 4.4 Consider the function ρ0 : Lp → R ∪ {+∞}:
(i) ρ0 is called lower semi-continuous (l.s.c.) at X0 if for all sequences (Xm ) ⊂ Lp with kXm −
X0 kp → 0 and for all a < ρ0 (X0 ), there exists ma > 0 such that a ≤ ρ0 (Xm ) for all m ≥ ma .
(ii) ρ0 is called lower semi-continuous if ρ0 is lower semi-continuous for all X ∈ Lp .
16
The following theorem provides the dual representation of convex systemic risk measures. The
positively homogeneous case is covered by Theorem 4.8.
Theorem 4.5 Suppose that ρ = ρ0 ◦ Λ is a convex systemic risk measure characterized by a convex
aggregation function Λ that is continuous on (Lp )n and a l.s.c. convex single-firm risk measure ρ0 .
¯ ∈ (Lp )n , ρ(X)
¯ can be expressed as
Then for any X
( n
)
X
¯ =
¯ i Ξi ] − α(ξ, Ξ)
ρ(X)
sup
E[X
(4.16)
(ξ,Ξ)∈(Lp )∗ ×((Lp )∗ )n
i=1
with α : (Lp )∗ × ((Lp )∗ )n → R ∪ {+∞} that is given by
(
α(ξ, Ξ) =
{−m + E[Y ξ]} +
sup
(m,Y )∈Aρ0
−E[V ξ] +
sup
¯
(V,Z)∈A
Λ
sup
¯
(m,Y )∈Aρ0 ,(V,Z)∈A
Λ
)
E[Z¯i Ξi ]
(4.17)
i=1
(
=
n
X
−m + E[(Y − V )ξ] +
n
X
)
E[Z¯i Ξi ] .
i=1
In addition, a solution (ξ, Ξ) ∈ (Lp )∗ × ((Lp )∗ )n of the optimization problem (4.16) satisfies
ξ ≥ 0,
E[ξ] ≤ 1,
Ξ ≥ 0.
Proof. By the primal representation of ρ in Proposition 4.2, we have
¯ = inf m ∈ R(m, Y ) ∈ Aρ , (Y, X)
¯ ∈ AΛ
ρ(X)
0
¯
=
inf
m + ιAρ (m, Y ) + ιA (Y, X)
Λ
0
(m,Y )∈R×Lp
where an indicator function of a set D ⊂ E ∈ {R × Lp , Lp × (Lp )n }, ιD : E → R ∪ {+∞}, is defined by
(
0
if x ∈ D,
ιD (x) :=
∞ otherwise.
If we define the support function sAρ0 : E ∗ → R ∪ {∞} for the convex set Aρ0 by
sAρ0 (−x, ξ) :=
{−mx + E[Y ξ]} ,
sup
(m,Y )∈Aρ0
then we have ι∗Aρ (−x, ξ) = sAρ0 (−x, ξ) . The duality theorem for conjugate functions (see for instance
0
Theorem 5 in Rockafellar (1974)) leads to
ιAρ0 (m, Y ) = ι∗∗
sup
mx + E[Y ξ] − ι∗Aρ (x, ξ)
Aρ (m, Y ) =
0
=
sup
(x,ξ)∈R×(Lp )∗
−mx + E[Y ξ] − sAρ0 (−x, ξ)
(
=
sup
(x,ξ)∈R×(Lp )∗
0
(x,ξ)∈R×(Lp )∗
−mx + E[Y ξ] −
sup
(m,
ˆ Yˆ )∈Aρ0
17
)
n
o
−mx
ˆ + E[Yˆ ξ]
.
(4.18)
Similarly, we get for the indicator function of the set AΛ
(
)
n
X
∗
¯ =
¯ i Ξi ] − ι (−ψ, Ξ)
ιAΛ (Y, X)
sup
−E[Y ψ] +
E[X
AΛ
(ψ,Ξ)∈(Lp )∗ ×((Lp )∗ )n
(
=
−E[Y ψ] +
sup
(ψ,Ξ)∈(Lp )∗ ×((Lp )∗ )n
i=1
n
X
(4.19)
(
¯ i Ξi ] −
E[X
sup
−E[Yˆ ψ] +
¯
(Yˆ ,Z)∈A
Λ
i=1
inf
(m,Y )∈R×Lp
m − mx + E[Y ξ] − E[Y ψ] +
sup
(x,ξ)∈R×(Lp )∗ ,
))
E[Z¯i Ξi ]
.
i=1
¯
Finally, we obtain for ρ(X)
¯ =
¯
ρ(X)
inf
m + ιAρ0 (m, Y ) + ιAΛ (Y, X)
(m,Y )∈R×Lp
(
=
n
X
n
X
¯ i Ξi ]
E[X
i=1
(ψ,Ξ)∈(Lp )∗ ×((Lp )∗ )n
)
−
ι∗Aρ
0
(−x, ξ) −
ι∗AΛ (−ψ, Ξ)
(
=
sup
inf
p
(m,Y )∈R×L
(x,ξ)∈R×(Lp )∗ ,
(ψ,Ξ)∈(Lp )∗ ×((Lp )∗ )n
m − mx + E[Y ξ] − E[Y ψ] +
n
X
¯ i Ξi ]
E[X
(4.20)
i=1
)
− ι∗Aρ (−x, ξ) − ι∗AΛ (−ψ, Ξ)
0
=
( n
X
sup
)
¯ i Ξi ] − ι∗ (−1, ξ) − ι∗ (−ξ, Ξ) .
E[X
Aρ
AΛ
0
(1,ξ)∈R×(Lp )∗ ,
(ξ,Ξ)∈(Lp )∗ ×((Lp )∗ )n
i=1
Since the dual indicator functions ι∗Aρ and ι∗AΛ are closed and convex (see Rockafellar (1974), Theorem
0
5), we can interchange the infimum with the supremum in (4.20) by using Theorem 6 and 7 in
Rockafellar (1974). With
(
)
n
X
α(ξ, Ξ) =
sup
−m + E[(Y − V )ξ] +
E[Z¯i Ξi ]
¯
(m,Y )∈Aρ0 , (V,Z)∈A
Λ
i=1
it immediately follows that
¯ =
ρ(X)
( n
X
sup
(ξ,Ξ)∈(Lp )∗ ×((Lp )∗ )n
)
¯ i Ξi ] − α(ξ, Ξ) .
E[X
i=1
It remains to show the claimed properties of the solutions to the optimization problem (4.16). First,
consider the case p < ∞. Suppose that ξ < 0 on a set A ∈ F with P[A] > 0. For each (m, Y ) ∈ Aρ0 ,
define Z(Y,k) for k ∈ N by Z(Y,k) := (−|Y | − k)IA + Y IAc . Then Z(Y,k) ∈ Lp and Y ≥ Z(Y,k) .
Moreover, E[Z(Y,k) ξ] = E[(−|Y | − k)ξIA ] + E[Y ξIAc ] = E[(−|Y | − k)ξ|A]P[A] + E[Y ξIAc ]. Since
E[(−|Y | − k)ξ|A]P[A] > 0 and E[Y ξIAc ] ∈ R, we can further increase k (respectively decrease Z(Y,k) )
18
such that E[Z(Y,k) ξ] > 0. From the monotonicity of Aρ0 it follows that (m, Z(Y,k) ) ∈ Aρ0 . Since we can
further increase k (respectively decrease Z(Y,k) ) such that E[Z(Y,k) ξ] tends to infinity, it follows that
αn (ξ, Ξ) = ∞. As a consequence, we only have to consider ξ ≥ 0. The same argumentation applies if
we suppose that Ξ < 0. Then again the monotonicity of the acceptance set AΛ implies αn (ξ, Ξ) = ∞
such that we only have to consider Ξ ≥ 0.
Now, let p = ∞. Suppose that ξ ∈ ba satisfies ξ(A) < 0 on a set A ∈ F. For each (m,
R Y ) ∈ Aρ0 ,
∞
define Z(z,Y ) := zIA − kY k∞ IAc ∈ L with R 3 z < −kY k∞ . Then Y ≥ Z(z,Y ) and Z(z,Y ) dξ =
c
c
−zξ(A)−kY k∞ ξ(Ac ). Since ξ is bounded, we
R have |ξ(A )| ≤ M(ξ) ∈ R, hence −kY k∞ ξ(A ) ≥ −M(ξ,Y ) .
It follows that we can decrease z such that Z(z,Y ) dξ > 0. From the monotonicity of Aρ0 it follows that
(m, Z(z,Y ) ) ∈ Aρ0 . Since we can further decrease z such that E[Z(z,Y ) ξ] tends to infinity, it follows that
αn (ξ, Ξ) = ∞. As a consequence, we only have to consider ξ ≥ 0. The same argumentation applies if
we suppose that Ξ < 0. Then again the monotonicity of the acceptance set AΛ implies αn (ξ, Ξ) = ∞,
such that we only have to consider Ξ ≥ 0.
Since ρ0 satisfies the constancy property, we have ρ0 (1) = 1. This means that (1, 1) ∈ Aρ0 .
Now, suppose that ξ ∈ (Lp )∗ is such that −1 + E[ξ] > 0. Since (λ, λ) ∈ Aρ0 for any λ > 0 and
limλ→∞ (−λ + E[λξ]) = limλ→∞ (λ(−1 + E[ξ])) = ∞, it follows that αn (ξ, Ξ) = ∞. This yields that we
only have to consider ξ ∈ (Lp )∗ such that E[ξ] ≤ 1.
2
Remark 4.6 Note that Theorem 4.5 is satisfied for convex aggregation functions Λ which are Lp continuous. Another possibility is to require 1-Lipschitz continuity for Λ as a real valued function on
Rn . Indeed, 1-Lipschitz continuity automatically yields Lp -continuity.
This can be seen as follows: If the convex aggregation
function Λ : Rn :→ R is 1-Lipschitz continPn
uous, then it satisfies |Λ(¯
x) − Λ(¯
y )| ≤ k¯
x − y¯k = i=1 |x¯i − y¯i | for all x
¯, y¯ ∈ Rn . Furthermore, let
¯ m ) ⊂ (Lp )n be a sequence such that X
¯m → X
¯ in Lp . Now, consider p < ∞. The measurablilty of
(X
Λ together with the 1-Lipschitz continuity and the H¨
older inequality imply
!1/p
n
n
X
X
(p−1)/p
p
¯ m ) − Λ(X)|
¯ ≤
¯ m )i − X
¯i| ≤ n
¯ m )i − X
¯i|
|Λ(X
|(X
|(X
P-a.s.,
i=1
¯ m ) − Λ(X)|
¯ p≤
which means that |Λ(X
leads to
i=1
¯ m )i − X
¯ i |p P-a.s. This
¯
¯ p ≤ np−1 Pn |(X
i=1
i=1 |(Xm )i − Xi |
Pn
¯ m ) − Λ(X)k
¯ pp ≤ np−1
kΛ(X
n
X
¯ m )i − X
¯ i kpp .
k(X
i=1
Pn
¯
¯
¯ m ) − Λ(X)|
¯ ≤
In case of p = ∞, we have |Λ(X
i=1 |(Xm )i − Xi | P-a.s. and kXk∞ = inf{r ∈
∞
¯ m ) − Λ(X)k
¯ ∞ ≤
R||X|
for X ∈ L . As a consequence, we immediately obtain kΛ(X
Pn ≤¯r P-a.s.}
¯
¯
¯
¯
¯
i=1 k(Xm )i − Xi k∞ . From the previous arguments, kXm − Xkp → 0 yields kΛ(Xm ) − Λ(X)kp → 0
p
for all p ∈ [1, ∞], which means that Λ is L -continuous.
Remark 4.7 Note that if the function fΛ : R → R from the fΛ -constancy property is positively
homogeneous (which does not imply that Λ is positively homogeneous), then a solution (ξ, Ξ) ∈ (Lp )∗ ×
((Lp )∗ )n to optimization problem (4.16) additionally satisfies
n
X
E[Ξi ] ≤ fΛ (1)E[ξ].
i=1
19
(4.21)
As noted in Remark 2.5, positively homogeneous aggregation functions induce a specific function fΛ
such that (4.21), in this case, becomes
n
X
E[Ξi ] ≤ nE[ξ].
i=1
This is exactly the additional property that Ξ from a solution (ξ, Ξ) to optimization problem (4.22) in
Theorem 4.8 will satisfy.
For the dual representation of positively homogeneous systemic risk measures we need the following
sets:
A∗ρ0 := { (x, ψ) ∈ R × (Lp )∗ | mx − E[Y ψ] ≥ 0 ∀ (m, Y ) ∈ Aρ0 } ,
¯ ∈ AΛ .
A∗Λ := (ξ, Ξ) ∈ (Lp )∗ × ((Lp )n )∗ E[Y ξ] − E[Z¯ t Ξ] ≥ 0 ∀(Y, Z)
Note that A∗ρ0 and A∗Λ are equal to the dual cones to Aρ0 and AΛ if we neglect the sign change.
Theorem 4.8 Suppose that ρ = ρ0 ◦Λ is a positively homogeneous systemic risk measure characterized
by a positively homogeneous aggregation function Λ that is continuous on (Lp )n and a l.s.c. positively
¯ ∈ (Lp )n , ρ(X)
¯ can be expressed
homogeneous single-firm risk measure ρ0 . Then for all economies X
as
n
X
¯ =
¯ i Ξi ].
ρ(X)
sup
E[X
(4.22)
(1,ξ)∈A∗ρ0 ,(ξ,Ξ)∈A∗Λ i=1
In addition, feasible points (ξ, Ξ) for this problem must satisfy
ξ ≥ 0,
E[ξ] ≤ 1,
Ξ ≥ 0,
n
X
E[Ξi ] ≤ nE[ξ].
i=1
Proof. Since every positively homogeneous systemic risk measure characterized by a positively homogeneous aggregation function Λ and a positively homogeneous single-firm risk measure ρ0 is also a
convex systemic risk measure characterized by the convex aggregation function Λ and the single-firm
¯ ∈ (Lp )n , ρ(X)
¯ can be expressed as
risk measure ρ0 , we can apply Theorem 4.5. Hence, for any X
( n
)
X
¯ i Ξi ] − α(ξ, Ξ)
¯ =
ρ(X)
sup
E[X
(ξ,Ξ)∈(Lp )∗ ×((Lp )∗ )n
i=1
with α : (Lp )∗ × ((Lp )∗ )n → R ∪ {+∞} that is given by
(
α(ξ, Ξ) =
sup
¯
(m,Y )∈Aρ0 ,(V,Z)∈A
Λ
−m + E[(Y − V )ξ] +
n
X
)
E[Z¯i Ξi ] .
i=1
Since ρ0 and Λ are positively homogeneous, we know that Aρ0 and AΛ are cones and we get
(
0
if (1, ξ) ∈ A∗ρ0 and (ξ, Ξ) ∈ A∗Λ
α(ξ, Ξ) =
∞ otherwise.
20
The
is verified as follows: If (1, ξ) ∈ A∗ρ0 and (ξ, Ξ) ∈ A∗Λ , then −m + E[(Y − V )ξ] +
Pn equality
¯
¯
¯
i=1 E[Zi Ξi ] ≤ 0 for all (m, Y ) ∈ Aρ0 and (V, Z) ∈ AΛ . Since ρ0 (0) = 0 and Λ(0) = 0, we get
∗
α(ξ, Ξ) = 0. If (1, ξ) ∈
/ Aρ0 , then there exists (m, Y ) ∈ Aρ0 with −m + E[Y ξ] > 0. Since Aρ0 is a
cone, λ(m, Y ) ∈ Aρ0 for λ > 0. It follows α(ξ, Ξ) = ∞. Similarly, we can deduce α(ξ, Ξ) = ∞ in case
of (ξ, Ξ) ∈
/ A∗Λ .
Finally, we get representation (4.22) and by Theorem 4.5 feasible
Pn points (ξ, Ξ) of this problem
must satisfy ξ ≥ 0, E[ξ] ≤ 1 and Ξ ≥ 0. The additional property i=1 E[Ξi ] ≤ nE[ξ] follows as in
Remark 4.7.
2
Remark 4.9 Note that proper convex single-firm risk measures on L∞ that satisfy the translation
property (R3) are finite and continuous on L∞ (see Corollary 2.3 in Kaina and R¨
uschendorf (2009)).
Thus, in case of convex systemic risk measures on (L∞ )n that are based on such convex single-firm
risk measures, we do not require the lower semi-continuity of the convex single-firm risk measure ρ0
in Theorem 4.5 and Theorem 4.8.
Remark 4.10 In case of ρ(Rn ) = R and p < ∞, the first component of feasible (ξ, Ξ) ∈ (Lp )∗ ×
((Lp )∗ )n additionally satisfies E[ξ] = 1. Indeed, since ρ0 satisfies the constancy property, we have
ρ0 (−1) = −1, and therefore (−1, −1) ∈ Aρ0 . If we suppose that ξ ∈ (Lp )∗ is such that 1 − E[ξ] > 0,
then (λ, λ) ∈ Aρ0 for any λ > 0 and limλ→∞ (λ − E[λξ]) = limλ→∞ (λ(1 − E[ξ])) = ∞ implies
α(ξ, Ξ) = ∞. It follows that E[ξ] ≥ 1. Together with the properties of feasible solutions in Theorem
4.5 and Theorem 4.8, this yields E[ξ] = 1 for feasible (ξ, Ξ) ∈ (Lp )∗ × ((Lp )∗ )n . Hence, ξ represents a
density function. For p = ∞, this corresponds to ξ ∈ M1,f (P), the set of finitely additive set functions
which are normalized to ξ(Ω) = 1.
The dual representation (4.22) enables us to state the following corollary which connects the Ξcomponent of a solution (ξ, Ξ) to (4.22) to the subdifferential of ρ.
¯ ∈ (Lp )n . If ρ = ρ0 ◦ Λ is a positively homogeneous systemic risk
Corollary 4.11 Fix an economy X
measure with dual representation (4.22), then for every optimal solution (ξ o , Ξo ) to (4.22), Ξo is a
¯ i.e., Ξo ∈ ∂ρ(X).
¯
subgradient of ρ at X,
Furthermore, if Λ, considered as a function from (Lp )n to Lp , is Gˆ
ateaux differentiable on a
¯
¯
¯
¯
¯
neighborhood U of X, the mapping Y 7→ DΛ(Y ) is continuous at X and ρ0 is continuous at Λ(X),
o
∗
∗
p
∗
p
n
∗
¯ and Ξ ∈ (DΛ(X))
¯ ∂ρ0 (Λ(X))
¯ where (DΛ(X))
¯
then ρ is continuous at X
: (L ) → ((L ) ) denotes
¯
the adjoint operator of DΛ(X).
Proof. Consider an optimal solution (ξ o , Ξo ) to (4.22). Then
n
X
i=1
¯ i )Ξo ] =
E[(Y¯i − X
i
n
X
i=1
E[Y¯i Ξoi ] −
n
X
¯ i Ξo ] =
E[X
i
i=1
n
X
¯ ≤ ρ(Y¯ ) − ρ(X)
¯
E[Y¯i Ξoi ] − ρ(X)
i=1
for every Y¯ ∈ (Lp )n , which implies the first statement. Moreover, from Corollary 4.1.1 in Kurdila and
¯
Zabarankin (2005) it follows that the requirements on Λ yield that Λ is Fr´echet differentiable at X.
Now, the last statement follows from Theorem 2 in Section 4.4.2 of Ioffe and Tihomirov (1979).
2
21
¯ ∈ Lp . With
Remark 4.12 Consider a positively homogeneous systemic risk measure ρ = ρ0 ◦Λ and X
the assumptions of Corollary 4.11 on Λ and ρ0 , the corollary implies that for any optimal solution
(ξ o , Ξo ) to (4.22), Ξo can be represented as
¯ ∗·θ
Ξo = (DΛ(X))
¯
for θ ∈ ∂ρ0 (Λ(X)).
5
Risk Attribution
In this section we study methods that provide a possible answer to the question what fraction of
systemic risk should be attributed to the individual components of the financial system. This is an
important question since central banks and regulators should know the influence of the individual
components on the systemic risk of the whole system to be able to control the risk of the whole
system.
Suppose that the dual problem (4.22) has a solution (ξ o , Ξo ). Then the dual representation (4.22)
enables us to define (as in Chen et al. (2013)) a systemic risk attribution in conjunction with Ξo to
firm i (or node i in a financial network) by
¯ Ξo ) = E[X
¯ i Ξoi ],
ki (X,
i = 1, . . . , n.
(5.23)
If the dual optimal solution is unique, this implies that the systemic risk attribution
¯ Ξo ) = (k1 (X,
¯ Ξo ), . . . , kn (X,
¯ Ξo ))
k(X,
is unique. Again, from (4.22) it immediately follows that
¯ =
ρ(X)
n
X
¯ Ξo ).
ki (X,
i=1
This is a desirable property of a risk attribution method. It is usually called full allocation property
and several different allocation methods with this property have been studied in the literature about
risk capital allocations in the traditional portfolio framework, see for instance Denault (2001), Tasche
(2004), Kalkbrener (2005), Cheridito and Kromer (2011) and Kromer and Overbeck (2014).
Now, consider the dual problem (4.16) and suppose that (ξ o , Ξo ) is an optimal solution to this
dual problem. If we are interested in a similar risk attribution method to (5.23) that satisfies the full
allocation property, we can simply define
¯ ξ o , Ξo ) = E[X
¯ i Ξo ] − γi α(ξ o , Ξo ), i = 1, . . . , n
ki (X,
(5.24)
i
Pn
where γi , i = 1 . . . , n, are chosen such that i=1 γi = 1. This guarantees the desired full allocation
property for the risk attribution method (5.24) in conjunction with systemic risk measures that are
not positively homogeneous.
The following theorem is a generalization of Theorem 4 in Chen et al. (2013) that provides a
type of ”no-undercut” property of the risk attribution method (5.23). The ”no-undercut” property
in conjunction with risk attribution or risk capital allocation methods was studied in Delbaen (2002)
and Denault (2001) and for additional information on this property we refer the interested reader to
these papers.
22
¯ ∈ (Lp )n .
Theorem 5.1 Let ρ = ρ0 ◦ Λ be a positively homogeneous systemic risk measure and fix X
¯ = (¯
¯1, . . . , a
¯ n ) ∈ (Lp )n . If (ξ o , Ξo ) is an optimal solution to (4.22),
For a
¯ ∈ Rn+ denote a
¯?X
a1 X
¯n X
then
n
X
¯ Ξo ) ≤ ρ(¯
¯
a
¯i ki (X,
a ? X).
i=1
Proof. From the dual representation we have
¯ =
ρ(¯
a ? X)
sup
n
X
(1,ξ)∈A∗ρ0 ,(ξ,Ξ)∈A∗Λ i=1
¯ i Ξi ].
a
¯i E[X
(5.25)
Suppose k is obtained by a dual optimal solution Ξo . Then Ξo is a feasible solution of the dual
¯ and hence a feasible solution of the dual representation of ρ(¯
¯ in
representation (4.22) of ρ(X),
a ? X)
n
(5.25) for any a
¯ ∈ R . It follows
¯ ≥
ρ(¯
a ? X)
n
X
¯ i Ξoi ] = a
¯ Ξo ).
a
¯i E[X
¯t k(X,
i=1
2
¯ ∈ (Lp )n is scaled componentwise by the vector a
The result states that if an economy X
¯ ∈ Rn+ such
¯ is at least as large
that we get a new economy a
¯ ? X , the systemic risk of this new economy ρ(¯
a ? X)
as the weighted sum of risk attributed to the firms in the original economy.
Now, we move on to differentiability results for positively homogeneous systemic risk measures
that rely on the dual representation from Theorem 4.8. For the statement of these results we need
the following sets:
V # := {(ξ, Ξ) ∈ (Lp )∗ × ((Lp )n )∗ |(1, ξ) ∈ A∗ρ0 , (ξ, Ξ) ∈ A∗Λ }
and
)
n
X
¯ =
¯i]
(ξ, Ξ) ∈ V ρ(X)
E[Ξi X
(
¯ :=
V (X)
#
#
i=1
Theorem 5.2 Let ρ = ρ0 ◦ Λ be a finite valued positively homogeneous systemic risk measure. Then
¯ ∈ (Lp )n in the direction of Y¯ ∈ (Lp )n that is given by
ρ has a directional derivative at X
¯ Y¯ ) =
D+ ρ(X;
max
¯
(ξ,Ξ)∈V # (X)
n
X
E[Ξi Y¯i ].
i=1
¯ ∈ Lp × (Lp )n . ρ is finite valued, hence ρ0 is finite valued. Thus, there
Proof. Take an arbitrary (U, Z)
exists mU ∈ R such that ρ0 (U ) ≤ mU , which means that (mU , U ) ∈ Aρ0 . Since (ξ, Ξ) ∈ V # , this
means that (1, ξ) ∈ A∗ρ0 and (ξ, Ξ) ∈ A∗Λ . From (1, ξ) ∈ A∗ρ0 it follows that E[U ξ] ≤ mU . Moreover,
¯ ≤ Y ¯ and since (ξ, Ξ) ∈ A∗ , we have E[Z¯ t Ξ] ≤ E[Y ¯ ξ]. There
there exists Y ∈ Lp such that Λ(Z)
Z
Z
Λ
exists mZ¯ such that ρ0 (YZ¯ ) ≤ mZ¯ . From (1, ξ) ∈ A∗ρ0 it again follows that E[YZ¯ ξ] ≤ mZ¯ . Thus, there
¯ such that
exists a constant M(U,Z)
¯ (depending on (U, Z))
E[U ξ] + E[Z¯ t Ξ] ≤ M(U,Z)
¯
23
¯ ∈ Lp × (Lp )n was chosen arbitrarily, we have the pointwise
for each (ξ, Ξ) ∈ V # . Since (U, Z)
boundedness of V # .
According to Corollary 6.16 in Aliprantis and Border (2006), this is equivalent to norm boundedness of V # . Now, since V # is convex, weak*-closed and norm bounded, we have from Alaoglu’s
Theorem (see P
Theorem 6.21 in Aliprantis and Border (2006)) that V # is weak*-compact. The function
¯ i ] is continuous in (ξ, Ξ) and thus the maximum on the set V # is attained
¯ ξ, Ξ) = n E[Ξi X
J(X,
i=1
according to James’ Theorem, see Theorem 6.36 in Aliprantis and Border (2006). Moreover, since the
¯ ξ, Ξ) satisfies the assumptions D1 from Bernhard and Rapaport (1995), the assertion
function J(X,
follows from Theorem D1 in Bernhard and Rapaport (1995).
2
The following corollary is an immediate consequence of the previous theorem.
Corollary 5.3 Let ρ = ρ0 ◦ Λ be a finite valued positively homogeneous systemic risk measure. If
¯ ∈ (Lp )n , then ρ is Gˆ
¯ with
(4.22) has a unique solution (ξ o , Ξo ) ∈ V # at X
ateaux-differentiable at X
¯
¯
Gˆ
ateaux-derivative at X in the direction Y that is given by
¯ Y¯ ) =
Dρ(X;
n
X
E[Ξoi Y¯i ].
i=1
References
Acharya, V., L. Pedersen, T. Philippon, and R. M. (2012). Measuring systemic risk. CEPR Discussion
Paper 8824. Available at http://www.cepr.org/pubs/dps/DP8824.asp.
Acharya, V., J. Santos, and T. Yorulmazer (2010). Systemic risk and deposit insurance premiums.
Federal Reserve Bank of New York Economic Policy Revision 16 (1), 89–99.
Aliprantis, C. and K. Border (2006). Infinite Dimensional Analysis - A Hitchhiker’s Guide. Springer,
Berlin.
Amini, H., R. Cont, and A. Minca (2013). Resilience to contagion in financial networks. Mathematical
Finance., Available at http://onlinelibrary.wiley.com/doi/10.1111/mafi.12051/full.
Artzner, P., F. Delbaen, J. Eber, and D. Heath (1999). Coherent measures of risk. Mathematical
Finance 9 (3), 203–228.
Bartram, S., G. Brown, and J. Hund (2007). Estimating systemic risk in the international financial
system. Journal of Financial Economics 86 (3), 835–869.
Bernhard, P. and A. Rapaport (1995). On a theorem of Danskin with an application to a theorem of
von Neumann-Sion. Nonlinear Analysis, Theory, Methods & Applications 24 (8), 1163–1181.
Chen, C., G. Iyengar, and C. Moallemi (2013). An axiomatic approach to systemic risk. Management
Science 59 (6), 1373–1388.
Cheridito, P. and E. Kromer (2011). Ordered contribution allocations: theoretical properties and
applications. The Journal of Risk 14 (1), 123–135.
24
Cifuentes, R., G. Ferrucci, and H. Shin (2005). Liquidity risk and contagion. Journal of the European
Economic Association 3 (2-3), 556–566.
Cont, R., E. Santos, and A. Moussa (2013). Network structure and systemic risk in banking systems.
In Handbook of Systemic Risk, pp. 327–368. Cambridge University Press.
Delbaen, F. (2002). Coherent risk measures on general probability spaces. In Advances in Finance
and Stochastics, Essays in Honour of Dieter Sondermann, pp. 1–37. Springer, Berlin.
Denault, M. (2001). Coherent allocation of risk capital. The Journal of Risk 4 (1), 1–34.
Drehmann, M. and N. Tarashev (2013). Measuring the systemic importance of interconnected banks.
Journal of Financial Intermediation 22 (4), 586–607.
Eisenberg, L. and T. Noe (2001). Systemic risk in financial systems. Management Science 47 (2),
236–249.
F¨ollmer, H. and A. Schied (2002). Stochastic Finance: An Introduction in Discrete Time. Walter de
Gruyter, Berlin.
Frittelli, M. and E. Rosazza Gianin (2002). Putting order in risk measures. Journal of Banking and
Finance 26 (7), 1473–1486.
Gai, P. and S. Kapadia (2010). Contagion in financial networks. Proceedings of the Royal Society A
466, 2401–2423.
Gauthier, C., M. Lehar, and M. Souissi (2012). Macroprudential capital requirements and systemic
risk. Journal of Financial Intermediation 21 (4), 594–618.
Hurd, T. and J. Gleeson (2011). A framework for analyzing contagion in banking networks. Working
Paper. Available at http://arxiv.org/abs/1110.4312.
Ioffe, A. and V. Tihomirov (1979). Theory of extremal problems. North-Holland, Amsterdam.
Kaina, M. and L. R¨
uschendorf (2009). On convex risk measures on Lp -spaces. Mathematical Methods
of Operations Research 69 (3), 475–495.
Kalkbrener, M. (2005). An axiomatic approach to capital allocation. Mathematical Finance 15 (3),
425–437.
Kromer, E. and L. Overbeck (2014). Suitability of capital allocations for performance measurement.
The Journal of Risk 16 (6), 31–58.
Kurdila, A. and M. Zabarankin (2005). Convex Functional Analysis. Birkh¨auser, Basel.
Nier, E., J. Yang, T. Yorulmazer, and A. Alentorn (2007). Network models and financial stability.
Journal of Economic Dynamics and Control 31, 2033–2060.
Rockafellar, R. (1974). Conjugate Duality and Optimization. CBMS-NSF Regional Conference Series
in Applied Mathematics. Society for Industrial and Applied Mathematics.
25
Rogers, L. and L. Veraart (2013). Failure and rescue in an interbank network. Management Science 59 (4), 882–898.
Staum, J. (2013). Handbook on Systemic Risk, Chapter 20: Counterparty Contagion in Context:
Contributions to Systemic Risk. Cambridge University Press.
Staum, J. and M. Liu (2012). Systemic risk components in a network model of contagion. Working
Paper. Available at http://ssrn.com/abstract=1726107.
Tarashev, N., C. Borio, and K. Tsatsaronis (2010). Attributing systemic risk to individual institutions.
Working Paper No. 308, Bank for International Settlements, Basel. Available at http://www.bis.
org/publ/work308.pdf.
Tasche, D. (2004). Allocating portfolio economic capital to sub-portfolios. In Economic Capital: A
Practitioner Guide, pp. 275–302. Risk Books.
Zalinescu, C. (2002). Convex Analysis on General Vector Spaces. World Scientific Publishing.
26