Liability Concentration and Losses in Financial Networks

Liability Concentration and Losses in Financial Networks:
Comparisons via Majorization
Agostino Capponi
∗
Peng-Chu Chen
„
David D. Yao
…
March 18, 2015
Abstract
The objective of this study is to develop a majorization-based tool to compare
financial networks with a focus on the implications of liability concentration. Specifically, we quantify liability concentration by applying the majorization order to the
liability matrix that captures the interconnectedness of banks in a financial network.
We develop notions of balancing and unbalancing networks to bring out the qualitatively different implications of liability concentration on the system’s loss profile.
Using both analytical and numerical examples, we illustrate how to identify networks
that are balancing or unbalancing, and make connections to interbanking structures
identified by empirical research, such as perfect and imperfect tiering schemes. We
also discuss how our findings can inform policy and regulatory decisions.
Keywords: systemic risk, financial network, interbank liabilities, majorization.
1 Introduction
The financial industry is fraught with cases of bank failures due to large exposures to
certain counterparties. One such example is Johnson Matthey Bankers in the United
Kingdom in 1984. Bank assets more than doubled between 1980 and 1984, and loans
became concentrated only to a few borrowers, including Mahmoud Sipra, Rajendra
Sethia and ESAL Commodities, and Abdul Shamji. The quality of some of these loans
turned out to be worse than expected, and Bank of England had to intervene to prevent
a financial crisis. Another example is the Korean banking system during the crisis in
the late 1990s, when the country’s bank assets were concentrated on five largest banks.
∗
Email: [email protected], Industrial Engineering and Operations Research Department,
Columbia University, New York, NY10027.
„
Email: [email protected], School of Industrial Engineering, Purdue University, West Lafayette,
IN47906.
…
Email: [email protected], Industrial Engineering and Operations Research Department, Columbia
University, New York, NY10027.
1
These cases have prompted regulatory authorities in recent years to impose limits on
banks’ exposures. For example, the Core Principles for Effective Banking Supervision
(Core Principle 19) sets prudent limits on large exposures to a single borrower.
The goal of our study is to develop an analytical framework to assess how concentration of liabilities affects the net exposures towards individual entities, and in turn
induces losses in a financial system.
Our starting point is the basic model of Eisenberg and Noe (2001), which captures
the interlinking exposures among financial institutions. The model yields the loss contributed by each node (bank) in the network, computed as the difference between its
total liability and payment. We then use the majorization order (Arnold et al. (2011))
among vectors to express preferences between losses, i.e. one loss profile (vector) is
preferred to another only if it is majorized by the latter. This allows us to capture
the desired preference via a broad class of functions known as Schur convex functions
(Arnold et al. (2011)), which preserve the majorization order. This includes functions
such as summation and max; thus, a loss profile is preferred to another if it results in a
smaller total loss or a smaller worst-case loss.
More importantly, we use matrix majorization to compare relative liability matrices
in terms of concentration of liabilities. When a relative liability matrix is majorized by
another one, it means that in the first matrix the nodes tend to spread their liabilities
more evenly across the network than they do in the second matrix. (Refer to Figure 1,
where the relative liability matrix of the network on the left is majorized by the one on
the right, in both upper and lower panels.)
In addition, we develop two new notions of financial networks, referred to as balancing
and unbalancing. The distinguishing feature between the two is that the total asset values
of banks before clearing are more evenly distributed than their liabilities if the network
is balancing, whereas the former are less evenly distributed than the latter if the network
is unbalancing. (Again, refer to Figure 1, where the two networks in the upper panel are
balancing, as the asset values, both before and after clearing, are more even than the
liabilities; the two networks in the lower panel are unbalancing, as the asset values are
less even than the liabilities.) Moreover, in balancing systems losses occur at nodes with
larger liabilities, and larger losses are generated if the interbanking liabilities are less
concentrated (i.e., smaller under majorization); whereas in unbalancing systems losses
occur at nodes with smaller liabilities, and more concentrated liabilities will result in
larger losses. (Refer to Figure 3.) A good example of a system in a balancing state is the
interbanking network during the 2007/08 financial crisis, when entities such as Lehman
Brothers, Bear Stearns and Merrill Lynch were in financial distress. The interbanking
network during the Secondary Banking Crisis of 1973-75 in UK, caused by failures of
more than thirty small, highly leveraged and unregulated banks, serves as a concrete
example of a system in an unbalancing state.
The common denominator for both systems is that lowering the size of net exposures to banks with lower net worth leads to smaller losses. Our findings thus support
2
regulatory policies of the Basel Committee (BCBS (2014)) aiming at limiting the size of
large net exposures to individual counterparties. We find that for balancing networks, a
higher concentration of liabilities tends to reduce net exposures to banks with lower net
worth, which, in turn, leads to a more desirable loss profile; whereas in an unbalancing
network, this has the opposite effect, leading to a less desirable loss profile.
Moreover, our results add to the understanding of preventive policies in bringing
out their consequences and implications on the network. The regulator will monitor
the interbanking system and limit net exposures towards banks that fail to meet their
presribed capital requirements. In a balancing system, such actions will push banks with
large liabilities to lend more to banks with smaller liabilities in order to reduce their net
exposures to the latter, and thereby drive liabilities to a more concentrated level. When
the system is unbalancing, reducing net exposures will take the form of banks with
higher liabilities lending less to the ones with smaller liabilities, and consequently drive
the network of interbank liabilities towards a less concentrated state.
A brief overview of some related literature is in order. Most studies on interbanking
networks have focused on understanding the impact of shocks, originating in a specific
part of the network, on the overall financial system. Allen and Gale (2001) employ an
equilibrium approach to model the propagation of financial distress in a credit network.
Gai and Kapadia (2010) model how contagion spreads in a random network, and analyze
the knock-on effects of distress. Battiston et al. (2012a) and Battiston et al. (2012b)
characterize feedback effects arising from changing financial conditions of the network
nodes. Rogers and Veraart (2012) improve the realism of the Eisenberg and Noe (2001)
model by including liquidation costs at default. Elsinger et al. (2006) distinguish between
fundamental and contagious defaults in the Eisenberg and Noe (2001) framework, and
analyze feedback and domino effects via an empirical analysis. Glasserman and Young
(2014) impose distributional assumptions on the shocks and find that contagion effects
via network spillovers are usually low for realistic interbanking networks. Furfine (2003)
provides an empirical analysis quantifying contagion risk resulting from interbank federal
funds exposures data. Haldane and May (2001) draw analogies with ecosystems and
analyze how growth in interbanking claims leads to instability. Other studies have
explored the relation between the topological structure of the network and the magnitude
of defaults it experiences. Gai et al. (2011) analyze the degree to which networks with a
smaller number of key highly interconnected players is affected by target shocks. Elliott
et al. (2013) discuss the dependence of the probability of default cascades on integration
and diversification. Acemoglu et al. (2014) develop a theoretical framework to explain
the robust-yet-fragile tendency of financial networks. On the empirical side, Cont et al.
(2013) and Angelini et al. (1996) analyze, respectively, Brazilian and Italian interbank
systems, and show how contagion through the payment system can originate systemic
crisis.
The rest of the paper is organized as follows. We start with preliminaries on both
the Eisenberg-Noe model and the majorization order in §2, and formalize the notion of
a loss profile and loss preference in a financial network. We then spell out in §3 the tech-
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nical details in modeling a) the concentration of liabilities using matrix majorization,
and b) the notion of balancing versus unbalancing networks and their implications on
loss preference. This section is concluded with results (in §3.3) that identify networks
which belong to these special categories first from principles and then from primitive
data (such as the relative liability matrix, total liabilities and cash-flow vectors). The
next section, §4, then follows up with concrete examples, both analytical and numerical,
to illustrate and enhance these notions. In particular, we make connections to the studies on German and Italian banks respectively presented in Craig and Von Peter (2014)
and in Fricke and Lux (2013), where a tiering (or, core-periphery) structure has been
identified as the primary configuration of interbanking liabilities. We apply the balancing/unbalancing notions to the tiering structure and bring out the distinction between
perfect and imperfect tiering schemes. Concluding remarks and policy implications are
summarized in §5.
2 Loss Preferences
We describe the majorization method used to express loss preferences in Section 2.1.
We recall the Eisenberg-Noe framework in Section 2.2. We describe the objective of the
study in Section 2.3.
2.1 Loss Comparison Using Majorization
We start by providing basic notations and definitions related to majorization and refer
to Arnold et al. (2011) for a complete treatment of the subject.
Majorization is a preorder on vectors of real numbers, which measures the dispersion
among the elements in a vector. For any vector x ∈ Rn , we use x[1] , . . . , x[n] to denote the
ordered entries of x from largest to smallest (x[1] being the largest and x[n] the smallest).
Moreover, we use x(1) , . . . , x(n) to denote the ordered entries of x from smallest to largest
(x(1) being the smallest and x(n) the largest).
Definition 2.1. x is majorized by y, denoted by x ≺ y, if
k
k
n
n
∑ x[i] ≤ ∑ y[i] for k = 1, . . . , n − 1,
∑ x[i] = ∑ y[i] ,
i=1
i=1
i=1
i=1
k
k
n
n
(1)
or equivalently,
∑ x(i) ≥ ∑ y(i) for k = 1, . . . , n − 1,
∑ x(i) = ∑ y(i) .
i=1
i=1
i=1
(2)
i=1
x ≺ y indicates that the vector x is more evenly distributed than y. Replacing
the equality in Eq. (1) and (2) with ≤ and ≥ respectively leads to the notion of weak
submajorization and weak supermajorization.
4
Definition 2.2. For x, y ∈ Rn , x is weakly submajorized by y, denoted by x ≺w y, if
k
k
∑ x[i] ≤ ∑ y[i] for k = 1, . . . , n.
i=1
i=1
x is weakly supermajorized by y, denoted by x ≺w y, if
k
k
∑ x(i) ≥ ∑ y(i) for k = 1, . . . , n.
i=1
i=1
Interchangeably, we denote x ≻w y if y ≺w x and x ≻w y if y ≺w x.
We next explain how we express preference between losses using majorization. Let
x ∈ Rn be a vector, whose i-th component xi is interpreted as the loss generated by entity
i in a financial network. We say that the loss vector x is preferred to the loss vector y
if x ≺w y. Our choice is driven by the following consideration. Consider two networks
a and b consisting of the same set of entities. We now think of x as the loss vector
associated with network a and of y as the loss vector associated with network b. The
difference between a and b lies in the interbanking structure and their outside assets. If,
for any k ∈ {1, . . . , n}, the sum of the k largest losses generated by entities in network a
never exceeds the corresponding quantity in network b, then we prefer the interbanking
network a to b. In particular, this means that the maximum loss generated by a node in
the network a does not exceed the maximum loss generated by a node in the network b
(when k = 1). Further, it also implies that the total loss in the network a never exceeds
the corresponding quantity in the network b (when k = n).
Since our objective is to measure losses, we would want them to be invariant with
respect to the entities which generate them. In other words, it should not matter in
which specific nodes the losses occurred, but only their sizes should play a role. This
property is preserved when weak submajorization is used to express preferences. Recall
that weak submajorization is a preorder, i.e. x ≺w y and y ≺w x together imply that
x = yP for some permutation matrix P, but not that x = y. Hence, a permutation of the
loss vector is equally preferred to the original loss vector.
2.2 The Eisenberg-Noe Model
We define the loss vector associated with a financial system using the framework proposed
by Eisenberg and Noe (2001). They consider a network of interbank liabilities consisting
of n nodes, where each node represents a financial institution. Let L ∈ Rn×n
≥0 be the
interbank liability matrix with lij denoting the amount of liabilities owed by i to j, and
c ∈ R1×n
≥0 be the outside assets vector, in which each component ci represents the value
of outside assets held by node i. As in Glasserman and Young (2014), we allow each
node to also have liabilities outside the interbanking network. More specifically, we let
e ∈ R1×n
≥0 be the outside liability vector, where each element ei denotes the amount of
liabilities of node i towards entities outside the network. These liabilities are assumed
to have equal priority to the interbank liabilities.
5
n
The total liability vector is denoted by ` ∈ R1×n
≥0 , with `i = ∑j=1 lij + ei being the total
amount of obligations from node i to all other nodes and the outside network. Further,
we denote by
⎧
lij
⎪
⎪ `i
πij = ⎨
⎪
⎪
⎩0
if `i > 0
if `i = 0,
the relative size of liabilities owed by i to j. Here, Π is the interbanking relative liability
matrix. Because i may also owe to entities outside the network, ∑nj=1 πij ≤ 1 for each i,
i.e. Π is row substochastic.
It is well known from the work of Eisenberg and Noe (2001) that, subject to the
bankruptcy laws, the clearing payment vector p∗ ∈ R1×n
≥0 is a solution to the system
p∗ = ` ∧ (p∗ Π + c) ,
where each component
p∗i
⎫
⎧
n
⎪
⎪
⎪
⎪
∗
= min ⎨`i , ∑ πji pj + ci ⎬ .
⎪
⎪
⎪
⎪
⎭
⎩ j=1
For any two vectors, x, y ∈ Rn ,
x ∧ y ∶= (min{x1 , y1 }, min{x2 , y2 }, . . . , min{xn , yn }) .
When ei = 0 for all i, the clearing payment is uniquely determined provided that
the financial system satisfies the regularity condition introduced in Eisenberg and Noe
(2001). Such a condition states that a financial system is regular if the risk orbit of each
node i, consisting of all nodes j reachable from i via a directed path, is a surplus set.
This means that every node in the set is not liable to any node outside it and the outside
asset value of each node in the set is positive. A simple sufficient condition guaranteeing
this is that the outside asset value of each node is strictly positive at all times. When
ei > 0 for some i, the clearing payment vector is also uniquely determined if the financial
system satisfies the condition introduced in Glasserman and Young (2014), i.e. from
every node i there exists a chain of positive obligations to some node j that has positive
obligations to the outside network, or equivalently to say, Π has spectral radius less than
1. Clearly, p∗ depends on Π, `, and c. For convenience, we use hereafter p∗ (Π, `, c) to
denote the clearing payment vector associated with the financial system (Π, `, c).
For future purposes, we distinguish between the asset value of a node before clearing
and after clearing. The vector of asset values before clearing is given by `Π + c as in
Glasserman and Young (2014). The vector of asset values after clearing is given by
p∗ (Π, `, c)Π + c.
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2.3 Objective of the Study
We aim at understanding how losses are affected by concentration of liabilities. The
loss vector s corresponding to the financial system (Π, `, c) is defined as the difference
between the total liability vector and the clearing payment vector, i.e.
s(Π, `, c) ∶= ` − p∗ (Π, `, c).
(3)
The i-th component of the above vector denotes the amount of loss generated by node
i. We illustrate in Figure 1 the behavior which we aim at capturing. The top graphs
give an interbanking network where the node with the highest outstanding liabilities
(node 3) has the lowest net worth. The bottom graphs give an interbanking network
where the node with the lowest oustanding amount of liabilities (node 1) has the lowest
net worth. Both top and bottom panels have in common that the node with lowest net
worth generates the highest loss in the system. Then the network with the lowest net
exposure to this node is always the most preferred in terms of losses.
However, there is a distinguishing feature between the interbanking network structures. In the top panels, the undesired system is the network whose liabilities are less
concentrated. In the bottom panels, instead, the network with higher concentration of
liabilities is the undesired one.
Our objective is to capture this behavior quantitatively and analyze the impact of
policies aiming at reducing net exposures. Such policies are currently being considered
by the Basel Committee, see BCBS (2014), which is proposing to limit the size of large
net exposures to single counterparties. We discuss policy implications in Section 5.
The next section formally defines balancing systems which capture the network behavior reported in the top panels of Figure 1, and unbalancing systems to characterize
the network behavior in the bottom panels.
3 Concentration of Liabilities
The objective of this section is to quantitatively analyze how concentration of liabilities
affects the loss profile of a financial system. We introduce definitions which will be
extensively used in the derivation of the main results in Section 3.1. We give the main
theorems comparing loss preferences in balancing and unbalancing systems in Section
3.2. We give a characterization of these systems in terms of the structural network
parameters in Section 3.3.
3.1 Preliminary Definitions and Results
We start recalling the definition of similarly ordered vectors. Two vectors x and y are
similarly ordered if (xi − xj )(yi − yj ) ≥ 0 for all i, j (see also Arnold et al. (2011) Ch6
A.1.a).
7
(a) Evenly distributed liabilities,
` = (4, 8, 20), p∗ = (4, 8, 12), s = (0, 0, 8),
asset values before clearing = (15, 14, 12),
asset values after clearing = (11, 10, 12),
net exposures to node 3 = (8, 6, 0).
(b) Unevenly distributed liabilities,
` = (4, 8, 20), p∗ = (4, 8, 18), s = (0, 0, 2),
asset values before clearing = (11, 12, 18),
asset values after clearing = (10, 11, 18),
net exposures to node 3 = (6, 2, 0).
(c) Evenly distributed liabilities,
` = (12, 16, 20), p∗ = (10, 16, 20), s = (2, 0, 0),
asset values before clearing = (10, 28, 36),
asset values after clearing = (10, 27, 35),
net exposures to node 1 = (0, 0, 3).
(d) Unevenly distributed liabilities,
` = (12, 16, 20), p∗ = (4, 16, 20), s = (8, 0, 0),
asset values before clearing = (4, 22, 48),
asset values after clearing = (4, 22, 40),
net exposures to node 1 = (0, 0, 9).
Figure 1: Top panels: Networks where nodes with large outstanding liabilities have low
net worth. Bottom panels: Networks where nodes with small outstanding liabilities
have low net worth. Here, ` denotes the liability vector, p∗ is the clearing payment, s is
the loss vector defined as in Eq. (3). The asset value before clearing and after clearing
are computed using the formulae in Section 2.2. The net exposure node i has to j is
computed as lji − lij .
8
Now, we define what it means for a matrix to be order preserving.
Definition 3.1. Let D be an n × n matrix and A ⊂ Rn be a subset of the space of n
dimensional real-valued vectors. D is order preserving w.r.t A if for x ∈ A, xD and x
are similarly ordered.
Further, we define the class of matrices which preserve the order of weak majorization
relations between vectors.
Definition 3.2. Let D be an n × n matrix and A ⊂ Rn be a subset of the space of n
dimensional real-valued vectors.
(I) D is weak submajorization preserving w.r.t A if for x, y ∈ A,
x ≺w y implies xD ≺w yD.
(II) D is weak supermajorization preserving w.r.t A if for x, y ∈ A,
x ≺w y implies xD ≺w yD.
Suppose that a relative liability matrix is order preserving w.r.t. A. If a node x
borrows more than y, x also lends more than y. This is because `Π is similarly ordered
to ` and each component in the former is the amount each node lends. If such a matrix
is also weak submajorization preserving w.r.t. A, as we will prove later, this implies the
following. If all the nodes in a set of nodes O borrow more than a node x and x borrows
more than y, then x must borrow more from O than y borrows from O. Instead, suppose
that such a matrix is also weak supermajorization preserving w.r.t. A. If all the nodes
in O borrow less than a node x and x borrows less than y, then x must borrow more
from O than y borrows from O.
For any vector, x ∈ Rn , define the operation
∆x ∶= (0, x(2) − x(1) , x(3) − x(2) , . . . x(n) − x(n−1) ) .
Such an operation computes the increment from one component to the next rank ordered
component in a vector.
Last, in order to compare two network topologies with different degrees of homogeneity, the notion of majorization for vectors is generalized to majorization for matrices.
Definition 3.3. Let X and Y be m × n matrices. X is said to be majorized by Y,
X ≺ Y, if there exists a doubly stochastic matrix S such that X = YS.
If X ≺ Y, it means that each row in X is more evenly distributed than the corresponding row in Y. Indeed, when m = 1, it is well known that this definition is equivalent
to Definition 2.1 (see Arnold et al. (2011) Ch.2 Theorem B.2).
We next use matrix majorization to compare financial systems in terms of concentration of liabilities. More specifically, we give the following
Definition 3.4. Given two financial systems (Πa , `, c) and (Πb , `, c), we say that b is
more concentrated than a if Πa ≺ Πb .
9
3.2 Loss preferences in Balancing and Unbalancing Systems
The characterization of balancing and unbalancing systems hinges upon the order preserving properties of the relative liability matrix. The presence of zero elements on the
diagonal of the matrix Π, i.e. πii = 0, restricts drastically the set of matrices Π which
are order preserving. However, we can define relaxed versions of these matrices which
are equivalent to the original matrix Π in terms of produced clearing payments. We first
give the definition and then prove the equivalence.
Definition 3.5. A 3-tuple (Πα , `, cα ), α ∈ [0, 1), is called the α-relaxed equivalent
version of a financial system (Π, `, c), if Πα = (1 − α)Π + αI and cα = (1 − α)c. Here I is
the identity matrix.
As shown below, multiplying a vector x by a relative liability matrix, Π, we can
obtain a vector xΠ with the opposite ordering of x. We avoid such a situation by
replacing Π with Πα . Clearly, when α is set to 0.5, the vector xΠ0.5 preserves the
ordering of x.
⎛ 0 0.5 0.5⎞
(1 2 3) ⎜0.5 0 0.5⎟ = (2.5 2 1.5)
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ⎝0.5 0.5 0 ⎠
x
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
⎛ 0.5 0.25 0.25⎞
(1 2 3) ⎜0.25 0.5 0.25⎟ = (1.75 2 2.25) .
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ⎝0.25 0.25 0.5 ⎠
x
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
Π0.5
Π
We next prove that the clearing payment is invariant with respect to relaxation, i.e.
the clearing payment associated with the relaxed equivalent version coincides with the
clearing payment of the original financial system. The proof of Lemma 3.6 is reported
in the Appendix.
Lemma 3.6. Let (Π, `, c) be a financial system, then it holds that p∗ (Π, `, c) = p∗ (Πα , `, cα )
for α ∈ [0, 1).
Next, we specify the set P of payment vectors which will be used to define balancing
and unbalancing systems. We choose
P = {p ∣ p is similarly ordered to `, (c ∧ `) ≤ p ≤ `} .
The set P includes all payment vectors where each node must use at the least the
value of his outside assets to repay his outstanding liabilities. It is possible, however,
that the equity value of a node i, given by [pΠ+c−p]i , is positive but the actual payment
does not cover liabilities, i.e. pi < `i and absolute priority can be violated. It can also
happen that limited liability is also violated, i.e. [pΠ + c]i < pi , and node i pays more
than the total value of his asset after receiving payments from the network. Moreover,
given two nodes in P one pays a higher amount than the other if it also has higher
outstanding liabilities.
10
We can then define balancing and unbalancing financial systems. Figure 2 illustrates
the structural properties of these systems.
Definition 3.7. Let (Π, `, c) be a financial system.
(I) (Π, `, c) is balancing if for each α-relaxed equivalent version of Π which is order
preserving w.r.t. P, ∆p ≤ ∆` implies that ∆(pΠα + cα ) ≤ ∆` for p ∈ P.
(II) (Π, `, c) is unbalancing if for each α-relaxed equivalent version of Π which is order
preserving w.r.t. P, ∆p ≥ ∆` implies that ∆(pΠα + cα ) ≥ ∆` for p ∈ P.
Intuitively, balancing and unbalancing financial systems may be understood as follows. Consider the setting where the matrix of relative liabilities distributes larger
payments to nodes with higher total liabilities. In a balancing system, if the payments
made by nodes are more evenly distributed than the corresponding liabilities, the value
of their assets after payments have been received will also be more evenly distributed
than the liabilities. Vice versa, in an unbalancing financial system, if the payments made
are less evenly distributed than the liabilities, their asset values after payments have been
received will also be less evenly distributed than the liabilities. This precisely captures
the behavior of the networks illustrated in Figure 1, where the asset values after clearing
are more evenly distributed than liabilities in the top two graphs (balancing system),
and are less evenly distributed than liabilities in the bottom two graphs (unbalancing
system).
We make the following assumption on the vectors of total liabilities and outside
assets:
Assumption 3.8. For any financial system (Π, `, c), ` and c are similarly ordered.
Our assumption is supported by empirical studies, see for instance figures 5, 6, and
7 in Laeven et al. (2014) showing that the bank with larger assets make more loans
(outside assets) to consumers while holding less capital, which thus results in higher
outstanding liabilities.
The next theorem compares the loss profiles in two balancing financial systems.
Theorem 3.9. Let (Πa , `, c), (Πb , `, c) be two balancing financial systems. Suppose there
exists α ∈ [0, 1) such that both Πaα and Πbα are order preserving w.r.t. P and
(I) Πaα or Πbα is weak submajorization preserving w.r.t. P,
(II) Πaα ≺ Πbα .
Then,
pa∗ (Πa , `, c) ≺w pb∗ (Πb , `, c) and s(Πa , `, c) ≻w s(Πb , `, c).
11
Figure 2: The x-axis indicates that the components of each plotted vector are sorted
from smallest to highest. Balancing system: slope of the payment vector smaller than
the slope of the liability vector implies that the slope of the vector of asset values after
the payments is smaller than the slope of the liability vector. Unbalancing system: slope
of the payment vector higher than the slope of liability vector implies that the slope
of the vector of asset values after the payments is higher than the slope of the liability
vector. In order to illustrate the salient features of two types of systems, each vector has
been shifted by subtracting the smallest component from each of its sorted entries.
Next, we give the basic intuition behind the result and refer to the appendix for the
formal proof.
(1) If a and b are balancing and Πaα is more evenly distributed than Πbα , then a distributes smaller payments to nodes with large total liabilities as compared to b.
If the payments are the same in both systems, and given by pb∗ , this leads to
∑ki=1 [pb∗ Πaα ][i] ≤ ∑ki=1 [pb∗ Πbα ][i] , for any k = 1, . . . , n.
(2) Suppose Πaα is weak submajorization preserving w.r.t. P. Then nodes with higher
liabilities have more interbanking claims on nodes with average liabilities rather than
on nodes with small liabilities. If two payments are made in system a, one more
evenly distributed than the other, Πaα will distribute smaller payments to nodes with
high liabilities if the payments are more homogeneous. This yields ∑ki=1 [pa∗ Πaα ][i] ≤
∑ki=1 [pb∗ Πaα ][i] for k = 1, . . . , n.
Combining (1) and (2) we obtain the result in Theorem 3.9. This result is directly related
to concentration of liabilities. When two systems are balancing, the more concentrated
system b is preferred to the less concentrated system a.
The next theorem compares the loss profiles in two unbalancing financial systems.
Theorem 3.10. Let (Πa , `, c), (Πb , `, c) be two unbalancing financial systems. Suppose
there exists α ∈ [0, 1) such that both Πaα and Πbα are order preserving w.r.t. P and
12
(I) Πaα or Πbα is weak supermajorization preserving w.r.t. P,
(II) Πaα ≺ Πbα .
Then
pa∗ (Πa , `, c) ≺w pb∗ (Πb , `, c) and s(Πa , `, c) ≺w s(Πb , `, c).
The proof of Theorem 3.10 is reported in the Appendix. Here, we give the main
intuition behind the result which can also be understood in two steps, as for the balancing
case.
(1) Because Πaα is more evenly distributed than Πbα , and both a and b are unbalancing,
if the same payments are made by nodes in both systems, system a will distribute
larger payments to nodes with smaller liabilities as compared to b. This in turn
leads to ∑ki=1 [pb∗ Πaα ](i) ≥ ∑ki=1 [pb∗ Πbα ](i) for k = 1, . . . , n.
(2) Since Πaα is weak supermajorization preserving, nodes with small liabilities have
higher claims on nodes with average outstanding liabilities rather than on nodes
with large liabilities. If two payments are made, and one is more evenly distributed
than the other, Πaα will distribute larger payments to nodes with small outstanding
liabilities if the payment is more homogeneous. This leads to ∑ki=1 [pa∗ Πaα ](i) ≥
∑ki=1 [pb∗ Πaα ](i) for k = 1, . . . , n.
Again, combining (1) and (2) above, we obtain the result in Theorem 3.10. This is again
directly related to concentration of liabilities. When two systems are unbalancing, the
less concentrated system a is preferred to the more concentrated system b.
We next discuss the financial implications of the above given theorems. Theorem
3.9 can be interpreted as follows. The proof shows that the clearing payment vector
is similarly ordered to the liability vector and, in a balancing system, more evenly distributed than it. This directly yields that the loss vector is also similarly ordered to the
liability vector, thus implying that losses are more likely to occur at nodes with higher
outstanding liabilities. This behavior is reflected in the left panel of Figure 1. A symmetric interpretation can be given for Theorem 3.10. The proof of this theorem shows
that the clearing payment vector is similarly ordered to the liability vector. However, in
unbalancing systems the former is less evenly distributed than the latter. This implies
that the loss vector is reversely ordered to the liability vector. Consequently, larger
losses will occur at nodes with smaller total liabilities, see also right panel of Figure 1.
Theorems 3.9 and 3.10 provide analytical support to the behavior illustrated, respectively, in the left and right panels of Figure 3.
3.3 Balancing and Unbalancing Systems: Identification
So far, we have focused on studying two classes of financial systems, namely balancing
and unbalancing networks, defined in terms of the relaxed equivalent version of the orig13
Figure 3: Two financial systems consisting of six nodes with liability vector satisfying
`6 > `5 > ⋅ ⋅ ⋅ > `1 . The graphs illustrate how the asset value of the nodes after clearing
vary. The left panel illustrates that, when both systems are balancing, losses occur at
the nodes with larger liabilities. Moreover, larger losses occur in the less concentrated
financial system (system a). The right panel shows that if both systems are unbalancing,
losses occur at the nodes with smaller liabilities. In this case, larger losses occur in the
more concentrated system (system b).
inal financial network. This brings the question of whether it is possible to recognize this
type of systems from structural properties of the network, i.e. the matrix of liabilities,
outside liabilities, and outside assets. This is the objective here.
In the sequel, the following notation will be useful. Given a financial system (Π, `, c),
we denote by µ a permutation vector defined by µi = j if `(i) = `j for i = 1, . . . , n. For
brevity, given any matrix X, we also use the abbreviated notation xµij ∶= xµi µj .
Here, we make the assumption that each node in the network has strictly positive
liabilities. Such an assumption is satisfied in all concrete financial systems, in particular
in the systems identified by recent empirical studies, see Craig and Von Peter (2014). The
next proposition gives necessary and sufficient conditions for the system to be balancing
and unbalancing.
Proposition 3.11.
(I) (Π, `, c) is balancing if and only for j = 1, . . . , n − 1,
n
µ
µ
∑ `(i) (πi(j+1) − πij ) + c(j+1) − c(j) ≤ `(j+1) − `(j) .
i=1
(II) (Π, `, c) is unbalancing if for j = 1, . . . , n − 1,
n
µ
µ
+
∑ [`(i) − (`(1) − c(1) ) ] (πi(j+1) − πij ) + c(j+1) − c(j) ≥ `(j+1) − `(j) .
i=1
14
Vice versa, if (Π, `, c) is unbalancing, then for j = 1, . . . , n − 1,
n
µ
µ
∑ `(i) (πi(j+1) − πij ) + c(j+1) − c(j) ≥ `(j+1) − `(j) .
i=1
+
Here, x denotes the positive part of x.
The formal proof is reported in the appendix. Here, we explain how the above
characterization reflects the description of balancing and unbalancing financial systems
given in the previous section. Proposition 3.11 indicates that it is enough to consider the
case when P only consists of a payment vector coinciding with the total liability vector.
Item (I) indicates that a system is balancing if and only if the total asset values of the
nodes before clearing are more evenly distributed than the liabilities. For unbalancing
systems, item (II) indicates that a necessary condition for the system to be unbalancing
is that the total asset values of the nodes before clearing are less evenly distributed than
the liabilities. However, for this to be sufficient the total asset values of nodes before
clearing must be even more heterogeneous than the liabilities.
4 Examples
We provide examples of balancing and unbalancing financial systems from real financial
networks. Section 4.1 describes the network structure of the financial system considered.
Section 4.2 develops equivalent characterizations of order preserving, weak submajorization and weak supermajorization preserving which can be easily verified on financial
networks. Section 4.3 gives numerical examples of tiered systems and apply the theoretical results developed earlier to express preferences between loss profiles.
4.1 Tiered Financial Systems
We consider tiered financial systems. Such a choice is motivated by empirical evidence
provided by Craig and Von Peter (2014) using historical data of the German banking
system from 1999 to 2007, and by Fricke and Lux (2013) using the overnight interbank
transactions in the Italian market from 1999 to 2010. They uncovered that the pattern
of interbanking liabilities follows a tiered core-periphery structure, i.e. the network is
centered around a set of core nodes which intermediate between numerous smaller nodes
in the periphery. In particular, such a network has following properties.
(I) The size of the core node is significantly larger than that of peripheral nodes.
Craig and Von Peter (2014) find that the average size of core banks is 51 times
that of peripheral banks.
(II) The peripheral nodes do not borrow from or lend to other peripheral nodes.
A financial system satisfying (I) and (II) is called perfectly tiered financial system, while
a financial system which only satisfies (I) is called imperfectly tiered financial system.
15
4.2 Order and Weak Majorization Preserving Characterizations
Our objective is to construct specific examples of tiered systems and verify that the
conditions given in theorems 3.9 and 3.10 are satisfied. Using Proposition 3.11 we can
determine if the system is balancing or unbalancing. However, for the theorems to be
applied, we need to verify that the relaxed equivalent version of a system is order preserving, weak submajorization and weak supermajorization preserving. This is usually
hard to verify using the definition. Here, we develop easily verifiable characterizations.
We start characterizing the α-relaxed equivalent version which is order preserving
w.r.t. P. This is given in the following lemma.
Lemma 4.1. Πα is order preserving w.r.t. P if and only if
n
n
i=k
i=k
µ
µ
∑ πα,ij ≤ ∑ πα,i(j+1) for k = 1, . . . , n, j = 1, . . . , n − 1.
The proof of this lemma can be obtained as a special case of Proposition E.1 in
Arnold et al. (2011) Ch.2, by restricting the set of vectors to those consisting only of
positive entries. For the sake of completeness, we report this proof in the appendix.
Such an α-relaxed equivalent version has the following implications, which are useful
to characterize balancing and unbalancing financial systems.
Lemma 4.2. If Πα is order preserving w.r.t. P, for any p ∈ P, it must hold that
(I) ∆p ≤ ∆` implies that ∆ (pΠα ) ≤ ∆ (`Πα ).
(II) ∆p ≥ ∆` implies that ∆ (pΠα ) ≥ ∆ ((` − (`(1) − p(1) )) Πα ) .
The proof of Lemma 4.2 is reported in the appendix. The asymmetry of (I) and (II)
comes from that we have ≤ sign both in p ≤ ` and ∆p ≤ ∆` in (I), but different signs in
p ≤ ` and ∆p ≥ ∆` in (II). The quantity, `(1) − p(1) , is the adjustment needed to reduce
` to another vector componentwise smaller than p. Indeed,
˜ α) ,
∆ ((` − (`(1) − p(1) ))Πα ) = ∆ (`Π
˜
where `˜(1) = p(1) , `˜(2) = p(1) + ∆`2 , . . . , ˜l(n) = p(1) + ∑ni=2 ∆`i . Clearly, ∆p ≥ ∆`˜ and p ≥ `.
For any α-relaxed equivalent version of Π which is order preserving w.r.t. P, the
following lemma identifies when it is also weak submajorization preserving or weak supermajorization preserving w.r.t. P.
Lemma 4.3. Let Πα be order preserving w.r.t. P. Πα is weak submajorization preserving w.r.t. P if and only if for k = 1, . . . , n, i = 1, . . . , n − 1,
n
n
j=k
j=k
µ
µ
∑ πα,ij ≤ ∑ πα,(i+1)j .
16
Lemma 4.4. Let Πα be order preserving w.r.t. P. Πα is weak supermajorization
preserving w.r.t. P if and only if for k = 1, . . . , n, i = 1, . . . , n − 1,
k
k
j=1
j=1
µ
µ
∑ πα,ij ≥ ∑ πα,(i+1)j .
Both proofs are reported in the appendix.
4.3 Numerical Examples
We formalize the classes of perfectly tiered and imperfectly tiered financial systems
which will be considered. For illustration purposes, we set the number of nodes n = 4,
and choose nodes 1, 2, 3 to be peripheral and node 4 to be core. For convenience, the
order of total liabilities in ` is set to `1 ≤ `2 ≤ `3 ≪ `4 , i.e. µi = i for i = 1, . . . , 4.
Let (Πa , `, c) be an imperfectly tiered financial system, i.e. lending and borrowing
are allowed between peripheral nodes. We set
Using Proposition 3.11 (I), (Πa , `, c) is balancing if it satisfies
4
a
a
∑ `i (πi(j+1) − πij ) + cj+1 − cj ≤ `j+1 − `j for j = 1, . . . , 3 .
(4)
i=1
Using the same proposition, (Πb , `, c) is balancing if it satisfies
b
b
⎧
) + c2 − c1 ≤ `2 − `1
− π41
`4 (π42
⎪
⎪
⎪
⎪
b
b
⎨`4 (π43 − π42 ) + c3 − c2 ≤ `3 − `2
⎪
⎪
⎪
b
b
b
b
⎪
⎩`1 π14 + `2 π24 + `3 π34 − `4 π43 + c4 − c3 ≤ `4 − `3 .
(5)
The above inequalities are obtained by specializing the condition in Proposition 3.11, (I)
to Πb .
For (Πa , `, c) to be unbalancing we require
4
a
a
∑ [`i − (`1 − c1 )] (πi(j+1) − πij ) + cj+1 − cj ≥ `j+1 − `j for j = 1, . . . , 3.
(6)
i=1
Such a condition follows directly from the condition in Proposition 3.11, (II). For (Πb , `, c)
to be unbalancing we require
b
b
⎧
[`4 − (`1 − c1 )] (π42
− π41
) + c2 − c1 ≥ `2 − `1
⎪
⎪
⎪
⎪
b
b
⎨[`4 − (`1 − c1 )] (π43 − π42 ) + c3 − c2 ≥ `3 − `2
⎪
⎪
⎪
b
b
b
b
b
b
b
b
⎪
⎩(`1 π14 + `2 π24 + `3 π34 − `4 π43 ) − (`1 − c1 )(π14 + π24 + π34 − π43 ) + c4 − c3 ≥ `4 − `3 ,
(7)
which is obtained by specializing the condition in Proposition 3.11 (II) to Πb .
17
4.3.1
Balancing Networks
˜ a , `, c), (Πb , `, c) given by
Consider the three financial systems, (Πa , `, c), (Π
0.26 0.32 0.32⎞
⎛ 0
0.24
0
0.33 0.33⎟
⎜
˜a =⎜
⎟,
Π
⎜0.24 0.33
0
0.33⎟
⎝0.33 0.33 0.33
0 ⎠
0.32 0.32 0.26⎞
⎛ 0
0.28
0
0.28 0.34⎟
⎜
⎟,
Πa = ⎜
⎜0.28 0.28
0
0.34⎟
⎝0.33 0.33 0.33
0 ⎠
0
0
0.9⎞
⎛ 0
0
0
0.9⎟
⎜ 0
⎟,
Πb = ⎜
⎜ 0
0
0
0.9⎟
⎝0.33 0.33 0.33 0 ⎠
` = (4g, 6g, 8g, 50g) and c = (3g, 4g, 5g, 25g), where g > 0. Choosing α = 1/4,
we obtain
⎛0.25
⎜0.21
Πa1/4 = ⎜
⎜0.21
⎝0.25
0.23
0.25
0.21
0.25
0.23
0.21
0.25
0.25
⎛0.25
0.17
˜a =⎜
Π
1/4 ⎜
⎜0.17
⎝0.25
0.21⎞
0.25⎟
⎟,
0.25⎟
0.25⎠
0.21
0.25
0.25
0.25
0.23
0.25
0.25
0.25
0.23⎞
0.25⎟
⎟,
0.25⎟
0.25⎠
0
0
0.67⎞
⎛0.25
0.25
0
0.67⎟
⎜ 0
⎟,
Πb1/4 = ⎜
⎜ 0
0
0.25 0.67⎟
⎝0.25 0.25 0.25 0.25⎠
˜ a are both order and weak submajorization preserving w.r.t. P.
where Πa1/4 and Π
1/4
This is because the conditions in lemmas 4.1 and 4.3 are satisfied, and Πb1/4 is order
preserving w.r.t. P given that it satisfies Lemma 4.1.
˜ a , `, c) and (Πb , `, c) are balancing using the criteria
We next check if (Πa , `, c), (Π
identified in the previous section. Specializing Eq. (4) to (Πa , `, c), we obtain
4
a
a
a
a
∑ `i (πi2 − πi1 ) + c2 − c1 = 0.57g ≤ 2g = `2 − `1
i=1
4
∑ `i (πi3 − πi2 ) + c3 − c2 = 0.43g ≤ 2g = `3 − `2
i=1
4
a
a
∑ `i (πi4 − πi3 ) + c4 − c3 = 6.1g ≤ 42g = `4 − `3 ,
i=1
˜ a , `, c), we obtain
while for (Π
4
a
a
πi2 − π
˜i1 ) + c2 − c1 = 1.47g
∑ `i (˜
i=1
18
≤ 2g = `2 − `1
4
a
a
a
a
πi3 − π
˜i2 ) + c3 − c2 = 0.53g + g ≤ 2g = `3 − `2
∑ `i (˜
i=1
4
πi4 − π
˜i3 ) + c4 − c3 = 6g
∑ `i (˜
≤ 42g = `4 − `3 .
i=1
˜ a , `, c) are balancing.
These inequalities are always satisfied, hence both (Πa , `, c) and (Π
b
For (Π , `, c), specializing Eq. (5) we obtain
b
b
`4 (π42
− π41
) + c2 − c1 = g
≤ 2g = `2 − `1
b
b
`4 (π43
− π42
) + c3 − c2 = g
≤ 2g = `3 − `2
b
`1 π14
b
+ `2 π24
b
+ `3 π34
b
− `4 π43
+ c4 − c3 = 19.53g ≤ 42g = `4 − `3 .
Since the inequalities are all satisfied, (Πb , `, c) is balancing.
ˆ a , `, c) and (Π
˜ a , `, c), we
By taking convex combinations of the financial systems (Π
a
ˆ
can generate a large class of balancing systems. Concretely, let (Π , `, c) = λ(Πa , `, c) +
˜ a , `, c) for some λ ∈ [0, 1]. Then it is clear that (Π
ˆ a , `, c) is balancing. Moreover,
(1−λ)(Π
ˆ a must also be order and weak submajorization preserving w.r.t. P, and be majorized
Π
1/4
by Πb1/4 .
ˆ a , `, c) and (Πb , `, c) satisfy the assumptions of Theorem 3.9, it must hold
Since (Π
ˆ a , `, c) ≻w s(Πb , `, c). This means that the perfectly tiered network is preferred
that s(Π
to the imperfectly tiered network if both networks are balancing.
This can be understood as follows. When a network is balancing, we know from the
discussion in Section 3.2 that the losses will occur at the larger nodes, i.e. the core nodes,
see left panel of Figure 3. Moreover, from the result in Theorem 3.9 we deduce that in
the imperfectly tiered network, the clearing payments are more evenly distributed than
the clearing payments associated with the perfectly tiered network. This is because in
the former, more clearing payments are made to periphery and less payments to the
core nodes, while in the latter all payments from periphery are only made to the core
nodes. Hence, the core nodes are more likely to default and generate higher losses in the
imperfectly tiered network.
4.3.2
Unbalancing Networks
˜ a , `, c), (Πb , `, c) in which
We consider three financial systems, (Πa , `, c), (Π
0.35 0.39 0.26⎞
⎛ 0
0
0.36 0.35⎟
⎜0.29
⎟,
Π =⎜
⎜0.29 0.29
0
0.42⎟
⎝0.29 0.29 0.29
0 ⎠
0.38 0.37 0.25⎞
⎛ 0
0
0.34 0.37⎟
⎜0.29
⎟,
Π =⎜
⎜0.29 0.33
0
0.38⎟
⎝0.29 0.29 0.29
0 ⎠
˜a
a
19
0
0
⎛ 0
0
0
⎜ 0
Πb = ⎜
⎜ 0
0
0
⎝0.21 0.33 0.33
1⎞
1⎟
⎟,
1⎟
0⎠
` = (14g, 16g, 18g, 50g) and c = (3g, 8g, 13g, 56g), where g > 0. Choosing
α = 1/4, we obtain
⎛0.25
⎜0.22
Πa1/4 = ⎜
⎜0.22
⎝0.21
0.26
0.25
0.21
0.22
0.29
0.27
0.25
0.22
0.20⎞
0.26⎟
⎟,
0.32⎟
0.25⎠
⎛0.25
0.21
˜a =⎜
Π
1/4 ⎜
⎜0.21
⎝0.21
0.29
0.25
0.25
0.22
0.27
0.26
0.25
0.22
0.19⎞
0.28⎟
⎟,
0.29⎟
0.25⎠
0
0
0.75⎞
⎛0.25
0
0.25
0
0.75
⎟
⎜
⎟.
Πb1/4 = ⎜
⎜ 0
0
0.25 0.75⎟
⎝0.15 0.25 0.25 0.25⎠
˜ a are both order and weak supermajorization preserving w.r.t. P
Here, Πa1/4 and Π
1/4
because they satisfy the conditions in lemmas 4.1 and 4.4. Moreover, Πb1/4 is order
preserving w.r.t. P because it satisfies the conditions in Lemma 4.1. Moreover, the
doubly stochastic matrices
⎛0.34
⎜0.22
S=⎜
⎜0.22
⎝0.22
0.33
0.29
0.15
0.23
0 ⎞
⎛0.37 0.33 0.30
0.21 0.2 0.23 0.36⎟
˜=⎜
⎜
⎟
S
⎜0.21 0.2 0.20 0.39⎟
⎝0.21 0.27 0.27 0.25⎠
0.33
0 ⎞
0.23 0.26⎟
⎟,
0.15 0.48⎟
0.29 0.26⎠
˜=Π
˜a .
are such that Πb1/4 S = Πa1/4 and Πb1/4 S
1/4
˜ a , `, c) and (Πb , `, c) are all unbalancing using the
We next check if (Πa , `, c), (Π
criteria identified in section 4.2. Specializing Eq. (6) to (Πa , `, c), we obtain
4
a
a
∑ (`i − (`1 − c1 )) (πi2 − πi1 ) + c2 − c1 = 4.68g ≥ 2g = `2 − `1
i=1
4
a
a
∑ (`i − (`1 − c1 )) (πi3 − πi2 ) + c3 − c2 = 4.91g ≥ 2g = `3 − `2
i=1
4
a
a
∑ (`i − (`1 − c1 )) (πi4 − πi3 ) + c4 − c3 = 34.24g ≥ 32g = `4 − `3 ,
i=1
˜ a , `, c), we get
and specializing it to (Π
4
a
a
πi2 − π
˜i1 ) + c2 − c1
∑ (`i − (`1 − c1 )) (˜
i=1
20
= 5.13g ≥ 2g = `2 − `1
4
a
a
πi3
−π
˜i2
) + c3 − c2 = 4.54g =≥ 2g = `3 − `2
∑ (`i − (`1 − c1 )) (˜
i=1
4
a
a
πi4 − π
˜i3 ) + c4 − c3 = 34.03g ≥ 32g = `4 − `3 .
∑ (`i − (`1 − c1 )) (˜
i=1
Since the above inequalities are satisfied, both systems are unbalancing. Next, specializing Eq. (7) to (Πb , `, c), we obtain
b
b
[`4 − (`1 − c1 )] (π42
− π41
) + c2 − c1 = 9.7g
≥ 2g = `2 − `1
b
b
[`4 − (`1 − c1 )] (π43
− π42
) + c3 − c2 = 5g
b
b
b
b
b
(`1 π14
+ `2 π24
+ `3 π34
− `4 π43
) − (`1 − c1 )(π14
≥ 2g = `3 − `2
b
b
b
+ π24
+ π34
− π43
) + c4 − c3 = 45g ≥ 32g = `4 − `3 .
The above inequalities are satisfied, hence (Πb , `, c) is unbalancing.
ˆ a , `, c) = λ(Πa , `, c) + (1 − λ)(Π
˜ a , `, c), where λ ∈ [0, 1], i.e. any
Next, consider (Π
a
a
˜ , `, c). Clearly, (Π
ˆ a , `, c) must also be unbalconvex combination of (Π , `, c) and (Π
ˆ a must also be order and weak supermajorization preserving w.r.t. P
ancing, and Π
1/4
ˆb .
and be majorized by Π
1/4
ˆa
Since (Π , `, c) and (Πb , `, c) both satisfy the assumptions in Theorem 3.10, it must
ˆ a , `, c) ≺w s(Πb , `, c). Hence, the imperfectly tiered network is preferred to
hold that s(Π
the perfectly tiered network if both networks are unbalancing. This can be understood
as follows. When a network is unbalancing, the discussion in Section 3.2 indicates that
larger losses will arise at smaller nodes, i.e peripheral nodes, see right panel of Figure 3.
Moreover, from the result in Theorem 3.9 we obtain that in the perfectly tiered network,
the clearing payment is less evenly distributed than the clearing payments associated
with the imperfectly tiered network. This is because in the latter more clearing payments
are made to periphery and less to the core nodes, while in the perfectly tiered network
the peripheral nodes only receive payments from the core node, and hence they are easier
to default and generate higher loss.
5 Concluding Remarks and Policy Implications
This study is what we believe to be the first to use majorization as a basic tool to compare
financial networks with a focus on the implications of liability concentration. Specifically,
we quantify liability concentration by applying the majorization order to the liability
matrix that captures the interconnectedness of banks in a financial network. We then
develop notions of balancing and unbalancing networks to bring out the qualitatively
different implications of liability concentration on the system’s loss profile.
Our findings support current regulatory policies of the Basel Committee (BCBS
(2014)) aiming at limiting the size of large net exposures to individual counterparties.
Specifically, the findings of §3 indicate that, regardless of whether the system is balancing
21
or unbalancing, lowering net exposures to nodes with low net worth always improves the
loss profile.
Comparing the value of total assets (pre-clearing) with liabilities, a regulator can
determine whether a given system is balancing or unbalancing, and use the outcome
of our analysis to develop preventive policies. In a balancing system, the regulator will
monitor banks with high outstanding liabilities since they are the most likely to generate
losses. If these banks do not satisfy their capital requirements, i.e. their net worth is not
sufficiently high, then the regulator may cap other banks’ net exposures to them. Under
such policy, nodes with large liabilities will have to lend to nodes with smaller liabilities
in order to reduce their net exposures to them, and consequently shift the network of
interbanking liabilities towards a more concentrated state (e.g. from imperfectly tiered
to perfectly tiered). In contrast, in an unbalancing system, the regulator will monitor
banks with small liabilities as they are the most likely source of loss. By capping their
net exposures when their capital requirements are not met, banks with larger liabilities
will lend less to nodes with smaller liabilities so as to reduce their net exposures to them.
This will in turn make the interbanking matrix of liabilities less concentrated (e.g. move
from perfectly tiered to imperfectly tiered).
Acknowledge
We gratefully acknowledge J. George Shanthikumar for discussions on the concept of
majorization.
Appendix
The following lemma identifies sufficient conditions under which the minimum operation
preserves the weak majorization relation. This is important because the computation of
the clearing payment vector requires taking the minimum between two vectors.
Lemma .1. Let x, y, z ∈ Rn≥0 such that x and y are similarly ordered to z.
(I) If z[i] ≤ a[i] implies z[k] ≤ a[k] for k > i , a ∈ {x, y}, then x ≺w y implies (x ∧ z) ≺w
(y ∧ z).
(II) If z(i) ≤ a(i) implies z(k) ≤ a(k) for k > i , a ∈ {x, y}, then x ≺w y implies (x ∧ z) ≺w
(y ∧ z).
Proof. (I) Because x and y are similarly ordered to z, clearly, (x ∧ z) and (y ∧ z) are
similarly ordered to z. Hence, proving (x ∧ z) ≺w (y ∧ z) is equivalent to show
k
k
∑ min {x[i] , z[i] } ≤ ∑ min {y[i] , z[i] } for k = 1, . . . , n.
i=1
i=1
22
(8)
Let mx = min{i = 1, . . . , n∣z[i] ≤ x[i] }, my = min{i = 1, . . . , n∣z[i] ≤ y[i] }. It must hold that
for k = 1, . . . , my − 1,
k
k
k
k
∑ min {x[i] , z[i] } ≤ ∑ x[i] ≤ ∑ y[i] = ∑ min {y[i] , z[i] } ,
i=1
i=1
i=1
(9)
i=1
where the second inequality is due to x ≺w y and the last equality follows from the
assumption. Moreover, for k = my , . . . , n,
k
∑ min {x[i] , z[i] }
i=1
= 1{mx <my }
my −1
k
k
⎛my −1
⎞
⎛my −1
⎞
{x
}
min
,
z
+
z
+
1
∑
∑ [i]
∑ x[i] + ∑ min {x[i] , z[i] }
[i] [i]
{mx ≥my }
⎝ i=1
⎠
⎝ i=1
⎠
i=my
i=my
k
k
≤ ∑ y[i] + ∑ z[i] = ∑ min {y[i] , z[i] } ,
i=1
i=my
(10)
i=1
where we use 1 to denote the indicator function. Here, the first and last equalities follow
by the assumption, and the second equality because x ≺w y. Using Eq. (9) and (10), we
obtain the inequality in (8).
(II) Because x and y are similarly ordered to z, clearly, (x∧z) and (y∧z) are similarly
ordered to z. Hence, proving (x ∧ z) ≺w (y ∧ z) is equivalent to show
k
k
∑ min {x(i) , z(i) } ≥ ∑ min {y(i) , z(i) } for k = 1, . . . , n.
i=1
(11)
i=1
Let mx = min{i = 1, . . . , n∣z(i) ≤ x(i) }, my = min{i = 1, . . . , n∣z(i) ≤ y(i) }. For k =
1, . . . , mx − 1, we must have
k
k
k
k
∑ min {x(i) , z(i) } = ∑ x(i) ≥ ∑ y(i) ≥ ∑ min {y(i) , z(i) } ,
i=1
i=1
i=1
(12)
i=1
where the first equality follows from the assumption and the second inequality because
x ≺w y. Moreover, for k = mx , . . . , n,
mx −1
k
k
∑ min {x(i) , z(i) } = ∑ x(i) + ∑ z(i)
i=1
i=1
≥ 1{mx <my }
⎛mx −1
i=mx
k
⎞
⎛mx −1
⎞
{y
}
{y
}
y
+
min
,
z
+
1
min
,
z
+
∑ (i) ∑
∑
∑ z(i)
(i) (i)
{mx ≥my }
(i) (i)
⎝ i=1
⎠
⎝ i=1
⎠
i=mx
i=mx
k
k
= ∑ min {y(i) , z(i) } ,
(13)
i=1
where the first and last equalities follow from the assumption, and the second inequality
because x ≺w y. Using Eq. (12) and (13), we obtain the inequality (11).
23
Lemma .2. Let x, y ∈ Rn≥0 such that x and y are similarly ordered. Then, ∆x ≤ ∆y if
and only if ∆(x ∧ y) ≤ ∆y.
Proof. We first prove that ∆x ≤ ∆y ⇒ ∆(x ∧ y) ≤ ∆y. For i = 1, . . . , n − 1,
∆(x ∧ y)i+1 = (x ∧ y)(i+1) − (x ∧ y)(i)
= min {x(i+1) , y(i+1) } − min {x(i) , y(i) }
⎧
x(i+1) − x(i) if x(i+1) ≤ y(i+1) and x(i) ≤ y(i)
⎪
⎪
⎪
⎪
= ⎨x(i+1) − y(i) if x(i+1) ≤ y(i+1) and x(i) ≥ y(i)
⎪
⎪
⎪
⎪
⎩y(i+1) − y(i) if x(i+1) ≥ y(i+1) and x(i) ≥ y(i)
≤ y(i+1) − y(i)
= ∆yi+1 ,
where the second equality holds because x is similarly ordered to y by assumption.
Moreover, the third equality does not include the case x(i+1) ≥ y(i+1) and x(i) ≤ y(i) ,
because such a case violates the assumption that ∆x ≤ ∆y. The same assumption leads
to the last inequality. Hence, by definition, ∆(x ∧ y) ≤ ∆y.
Next, we prove the reverse implication by showing that ∆x ≥ ∆y ⇒ ∆(x ∧ y) ≥ ∆y.
∆(x ∧ y)i+1 = (x ∧ y)(i+1) − (x ∧ y)(i)
= min {x(i+1) , y(i+1) } − min {x(i) , y(i) }
⎧
x(i+1) − x(i) if x(i+1) ≤ y(i+1) and x(i) ≤ y(i)
⎪
⎪
⎪
⎪
= ⎨y(i+1) − x(i) if x(i+1) ≥ y(i+1) and x(i) ≤ y(i)
⎪
⎪
⎪
⎪
⎩y(i+1) − y(i) if x(i+1) ≥ y(i+1) and x(i) ≥ y(i)
≥ y(i+1) − y(i)
= ∆yi+1 ,
where the second equality follows from the assumption that x is similarly ordered to y.
The third equality does not include the case x(i+1) ≤ y(i+1) and x(i) ≥ y(i) because such
a case violates the assumption that ∆x ≥ ∆y. Using the same assumption we obtain the
the last inequality. Hence, we can conclude that ∆(x ∧ y) ≥ ∆y.
Proof of Lemma 3.6
Proof. As pointed out in Eisenberg and Noe (2001), the clearing payment vector p∗ is
obtained as the solution to the following optimization problem:
max f (x),
x
s.t. x(I − Π) ≤ c,
24
0 ≤ x ≤ `,
where the objective function f is any real valued increasing function of the vector x.
Multiplying both sides of the first constraint by (1 − α), with α ∈ [0, 1), will lead to an
equivalent optimization problem. But this leads to:
x[I − (1 − α)Π − αI] ≤ (1 − α)c.
˜ ∶= (1 − α)Π + αI and c by c˜ ∶= (1 − α)c, the
That is, if we replace Π by another matrix Π
clearing payment vector will stay the same.
Proof of Theorem 3.9
Proof. Recall from Eisenberg and Noe (2001) that the clearing payment is the fixed
point of a function. Let M be the set of matrices with nonnegative entries whose row
sums are less than or equal to 1. Define the vector function F ∶ [0, `] × M → [0, `] by
F (p, Πα ; `, cα ) ∶= ` ∧ (pΠα + cα ) .
The clearing payment vector, p∗ (Πα , `, cα ), is then given by the function f ∶ M → [0, `],
f (Πα ) = FIX(F (⋅, Πα )). Define a sequence of functions {fu (Πα )}∞
u=0 as
fu (Πα ) ∶= F (fu−1 (Πα ), Πα ) ,
f0 (Πα ) ∶= 0.
It has been proven in Eisenberg and Noe (2001), Lemma 5, that fu (Πα ) increasingly
converges to f (Πα ) componentwise for all Πα ∈ M. Hence, proving pa∗ (Πaα , `, cα ) ≺w
pb∗ (Πbα , `, cα ) is equivalent to prove that fu (Πaα ) ≺w fu (Πbα ) for u = 1, 2, . . . . For brevity,
we denote hereafter fu (Πzα ) by fuz and p∗ (Πα , `, cα ) by p∗ .
We first prove that ∆(fuz Πzα + cα ) ≤ ∆`, and fuz is similarly ordered to ` for z ∈ {a, b}
by induction. For u = 0, we know that f0z = 0 and ∆f0z = 0 ≤ ∆`. Clearly, f0z is similarly
ordered to `. Following the assumptions that (Πz , `, c) is balancing and Πzα is order
preserving w.r.t. P, it must hold that ∆(`Πzα + cα ) ≤ ∆` because ` ∈ A. This implies
that ∆cα ≤ ∆` because `Πzα and cα are similarly ordered. Hence,
∆(f0z Πzα + cα ) = ∆cα ≤ ∆`.
(14)
Assume that fuz is similarly ordered to ` and that the inequality (14) holds. Then we
want to prove the statement for u + 1. Because Πzα is order preserving w.r.t. P, fuz ∈ P,
and cα is similarly ordered to `, it must hold that fuz Πzα + cα is similarly ordered to `,
z
hence fu+1
= ` ∧ (fuz Πzα + cα ) is similarly ordered to `. Using the induction hypothesis,
z z
i.e. ∆(fu Πα + cα ) ≤ ∆`, and Lemma .2, we obtain
z
∆fu+1
= ∆ [` ∧ (fuz Πzα + cα )] ≤ ∆`.
z
This further leads to ∆(fu+1
Πzα + cα ) ≤ ∆` because (Πz , `, c) is balancing and Πzα is
order preserving w.r.t. P, which concludes the induction.
25
Next, we prove that fua ≺w fub by induction. Without loss of generality, we take
Πaα to be submajorization preserving w.r.t. P. If Πbα is submajorization preserving
w.r.t. P, we obtain the same result and the proof proceeds in a symmetric fashion by
interchanging the roles of a and b. For u = 0, by definition, f0a = 0 ≺w 0 = f0b . Assume
fua ≺w fub . Then we want to prove the statement for u + 1. First, we deduce that
(fua Πaα + cα ) ≺w (fub Πaα + cα ) ≺w (fub Πbα + cα ) ,
where the first inequality follows from the assumption that Πaα is weak submajorization
preserving and the fact (we just proved) that fua , fub and cα are all similarly ordered to
`; the second inequality is due to that Πaα ≺ Πbα , and Πaα and Πbα are order preserving
w.r.t. P. For z ∈ {a, b}, because ∆(fuz Πzα + cα ) ≤ ∆`, `[i] ≤ (fuz Πzα + cα )[i] must imply
that `[k] ≤ (fuz Πzα + cα )[k] for n ≥ k > i. Moreover, fuz Πzα + cα is similarly ordered to `.
Using Lemma .1 (I), we deduce
a
b
fu+1
= [` ∧ (fua Πaα + cα )] ≺w [` ∧ (fub Πbα + cα )] = fu+1
by taking x = fua Πaα +cα , y = fub Πbα +cα , and z = `. This concludes the proof that fua ≺w fub
for u = 1, 2, . . . .
By definition of weak submajorization, this means that
k
k
i=1
i=1
b
a
∑ fu,[i] − ∑ fu,[i] ≥ 0 for k = 1, . . . , n.
Letting u → ∞, the above leads to
k
k
i=1
i=1
b
a
∑ f[i] − ∑ f[i] ≥ 0 for k = 1, . . . , n,
hence,
pa∗ = f a ≺w f b = pb∗ .
(15)
Notice that we have proved that for u = 1, . . . , n, z ∈ {a, b}, ∆(fuz Πzα + cα ) ≤ ∆`. By
Lemma .2, this implies that
∆ [` ∧ (fuz Πzα + cα )] ≤ ∆` and ∆pz∗ ≤ ∆`,
The last inequality above is obtained by letting u → ∞, and further leads to
z∗
z∗
`[1] − p[1]
≥ `[2] − pz∗
[2] ≥ ⋅ ⋅ ⋅ ≥ `[n] − p[n] .
Together with Eq. (15), the above inequality leads to
k
k
k
k
i=1
i=1
i=1
i=1
a∗
a∗
b∗
b∗
∑[` − p ][i] = ∑ `[i] − p[i] ≥ ∑ `[i] − p[i] = ∑[` − p ][i]
for k = 1, . . . , n, or equivalently
s(Πa , `, c) ≻w s(Πb , `, c).
26
Proof of Theorem 3.10
Proof. Let M be the set of matrices with nonnegative entries whose row sums are less
than or equal to 1, and define the vector function F ∶ [0, `] × M → [0, `] by
F (p, Πα ; `, cα ) ∶= ` ∧ (pΠα + cα ) .
The clearing payment vector, p∗ (Πα , `, cα ), is then given by the function f ∶ M → [0, `],
f (Πα ) = FIX(F (⋅, Πα )). Define a sequence of functions {fu (Πα )}∞
u=0 as
fu (Πα ) ∶= F (fu−1 (Πα ), Πα ) ,
f0 (Πα ) ∶= `.
Iteration the above equation yields a monotonically decreasing sequence f0 (Πα ), f1 (Πα ), . . . .,
which is bounded below, and hence componentwise converges to f (Πα ) for all Πα ∈ M.
Then, proving that pa∗ (Πaα , `, cα ) ≺w pb∗ (Πbα , `, cα ) is equivalent to prove that fu (Πaα ) ≺w
fu (Πbα ) for u = 1, 2, . . . . For brevity, we denote hereafter fu (Πzα ) by fuz and p∗ (Πα , `, cα )
by p∗ .
We first prove that ∆(fuz Πzα + cα ) ≥ ∆`, and fuz is similarly ordered to ` for z ∈ {a, b}
by induction. For u = 0, we know that f0z = ` and ∆f0z = ∆` ≥ ∆`. Clearly, f0z is similarly
ordered to `. Following the assumptions that (Πz , `, c) is unbalancing and Πzα is order
preserving w.r.t. P, it must hold that
∆(f0z Πzα + cα ) ≥ ∆`.
(16)
Assume that fuz is similarly ordered to ` and that the inequality (16) holds. Then we
want to prove the statement for u + 1. Since Πzα is order preserving w.r.t. P, fuz ∈ P,
and cα is similarly ordered to `, it must hold that fuz Πzα + cα is similarly ordered to `,
z
= ` ∧ (fuz Πzα + cα ) is similarly ordered to `. Using the induction hypothesis,
hence fu+1
z z
i.e. ∆(fu Πα + cα ) ≥ ∆`, and Lemma .2, we obtain
z
= ∆ [` ∧ (fuz Πzα + cα )] ≥ ∆`.
∆fu+1
z
This further leads to ∆(fu+1
Πzα + cα ) ≥ ∆` because (Πz , `, c) is unbalancing and Πzα is
order preserving w.r.t. P, which concludes the induction.
Next, we prove fua ≺w fub by induction. Without loss of generality, we take Πaα to be
supermajorization preserving w.r.t. P. If Πbα is supermajorization preserving w.r.t. P,
we obtain the same result and the proof proceeds in a symmetric fashion by interchanging
the role of a and b. For u = 0, by definition, f0a = ` ≺w ` = f0b . Assume fua ≺w fub , then we
want to prove the statement for u + 1.
(fua Πaα + cα ) ≺w (fub Πaα + cα ) ≺w (fub Πbα + cα ) ,
where the first inequality follows from the assumption that Πaα is weak supermajorization
preserving w.r.t. P and the fact (we just proved) that fua , fub and cα are all similarly
ordered to `; the second inequality is due to that Πaα ≺ Πbα , and Πaα and Πbα are order
27
preserving w.r.t. P. For z ∈ {a, b}, because ∆(fuz Πzα + cα ) ≥ ∆`, `(i) ≤ (fuz Πzα + cα )(i)
must imply that `(k) ≤ (fuz Πzα + cα )(k) for n ≥ k > i. Moreover, fuz Πzα + cα is similarly
ordered to `. Using Lemma .1 (II), we deduce
a
b
fu+1
= [` ∧ (fua Πaα + cα )] ≺w [` ∧ (fub Πbα + cα )] = fu+1
,
by taking x = fua Πaα +cα , y = fub Πbα +cα , and z = `. This concludes the proof that fua ≺w fub
for u = 1, 2, . . . .
By definition of weak supermajorization,
k
k
i=1
i=1
b
a
∑ fu,(i) − ∑ fu,(i) ≤ 0 for k = 1, . . . , n.
Letting u → ∞, the above leads to
k
k
i=1
i=1
b
a
∑ f(i) − ∑ f(i) ≤ 0 for k = 1, . . . , n,
hence,
pa∗ = f a ≺w f b = pb∗ .
(17)
Notice that we have proved that for u = 1, . . . , n, z ∈ {a, b}, ∆(fuz Πzα + cα ) ≥ ∆`. By
Lemma .2, this implies that
∆ [` ∧ (fuz Πzα + cα )] ≥ ∆` and ∆pz∗ ≥ ∆`,
The last inequality above is obtained by taking the limit u → ∞, and further leads to
z∗
z∗
`(1) − pz∗
(1) ≥ `(2) − p(2) ≥ ⋅ ⋅ ⋅ ≥ `(n) − p(n) .
Together with Eq. (17), the above inequality leads to
k
k
k
k
i=1
i=1
i=1
i=1
a∗
a∗
b∗
b∗
∑[` − p ][i] = ∑ `(i) − p(i) ≤ ∑ `(i) − p(i) = ∑[` − p ][i]
for k = 1, . . . , n, or equivalently
s(Πa , `, c) ≺w s(Πb , `, c).
Proof of Proposition 3.11
28
Proof. (I) We first prove the necessity. Suppose for any α-relaxed equivalent version
which is order preserving w.r.t. P, we have ∆` ≥ ∆p implying that ∆` ≥ ∆(pΠα + cα )
for any p ∈ P. Choosing p = `, it must hold that ∆` ≥ ∆(`Πα + cα ), or componentwise,
for j = 2, . . . , n,
`(j) − `(j−1) = [∆`]j ≥ [∆(`Πα + cα )]j = [∆ (α` + (1 − α)(`Π + c))]j
= [α` + (1 − α)(`Π + c)](j) − [α` + (1 − α)(`Π + c)](j−1)
n
µ
µ
) + c(j) − c(j−1) ]
= α(`(j) − `(j−1) ) + (1 − α) [∑ `(i) (πij
− πi(j−1)
i=1
µ
µ
) + c(j) − c(j−1) holds.
which shows that the inequality `(j) − `(j−1) ≥ ∑ni=1 `(i) (πij
− πi(j−1)
Next, we prove the sufficiency. By Lemma 4.2, we know that for any Πα which is
order preserving w.r.t. P, it must hold that
∆(pΠα ) ≤ ∆(`Πα ) for p ∈ P.
(18)
So, it must hold that
[∆(pΠα + cα )]j ≤ [∆(`Πα + cα )]j = [∆ (α` + (1 − α)(`Π + c))]j
= [α` + (1 − α)(`Π + c)](j) − [α` + (1 − α)(`Π + c)](j−1)
n
µ
µ
= α(`(j) − `(j−1) ) + (1 − α) [∑ `(i) (πij
− πi(j−1)
) + c(j) − c(j−1) ]
i=1
≤ α(`(j) − `(j−1) ) + (1 − α)(`(j) − `(j−1) )
= [∆`]j
for j = 2, . . . , n, where the first inequality follows because Πα is order preserving with
respect to P, cα is similarly ordered to `, and from Eq. (18). The last inequality follows
from the assumption of the proposition. This concludes the proof.
(II) We first show the necessary condition for a system to be unbalancing. Because
(Π, `, c) is unbalancing, choosing p = `, we must have that ∆` ≤ ∆(`Πα + cα ), or componentwise, for j = 2, . . . , n,
`(j) − `(j−1) = [∆`]j ≤ [∆(`Πα + cα )]j = [∆ (α` + (1 − α)(`Π + c))]j
= [α` + (1 − α)(`Π + c)](j) − [α` + (1 − α)(`Π + c)](j−1)
n
µ
µ
= α(`(j) − `(j−1) ) + (1 − α) [∑ `(i) (πij
− πi(j−1)
) + c(j) − c(j−1) ] ,
i=1
µ
µ
which shows that `(j) − `(j−1) ≤ ∑ni=1 `(i) (πij
− πi(j−1)
) + c(j) − c(j−1) holds.
Next, we derive the sufficient condition. By Lemma 4.2, we know that for any Πα
which is order preserving w.r.t. P, it must hold that if ∆p ≥ ∆`, then
∆(pΠα ) ≥ ∆ ((` − (`(1) − p(1) )) Πα ) ,
29
(19)
for p ∈ P. Then, it follows that
[∆(pΠα + cα )]j ≥ [∆ ((` − (`(1) − p(1) )) Πα + cα )]j
= α(`(j) − `(j−1) ) + (1 − α) [∆ ((` − (`(1) − p(1) )) Π + c)](j)
≥ α(`(j) − `(j−1) ) + (1 − α) [∆ ((` − (`(1) − c(1) )+ ) Π + c)](j)
= α(`(j) − `(j−1) )
n
µ
µ
) + cj − c(j−1) ]
+ (1 − α) [∑ [`(j) − (`(1) − c(1) )+ ] (πij
− πi(j−1)
i=1
≥ α(`(j) − `(j−1) ) + (1 − α)(`(j) − `(j−1) )
= [∆`]j
for j = 2, . . . , n, where the first inequality follows because Πα is order preserving with
respect to P, cα being similarly ordered to `, and from Eq. (19); the second inequality
follows from c(1) ≤ p(1) ; the last inequality follows from the assumption of the proposition.
This concludes the proof.
Proof of Lemma 4.1
Proof. We first prove the sufficiency. Let p ∈ P. For j = 1, . . . , n − 1,
n
µ
µ
[pΠα ]µj+1 − [pΠα ]µj = ∑ p(i) (πα,i(j+1)
− πα,ij
)
i=1
n−1
µ
µ
µ
µ
= p(n) (πα,n(j+1)
− πα,nj
) + ∑ p(i) (πα,i(j+1)
− πα,ij
)
i=1
n
n−2
i=n−1
n
i=1
n−3
i=n−2
n
i=1
µ
µ
µ
µ
≥ p(n−1) ∑ (πα,i(j+1)
− πα,ij
) + ∑ p(i) (πα,i(j+1)
− πα,ij
)
µ
µ
µ
µ
≥ p(n−2) ∑ (πα,i(j+1)
− πα,ij
) + ∑ p(i) (πα,i(j+1)
− πα,ij
)
µ
µ
)≥0
≥ ⋅ ⋅ ⋅ ≥ p(1) ∑ (πα,i(j+1)
− πα,ij
i=1
µ
µ
where each inequality above follows from the inequality ∑ni=k πα,ij
≤ ∑ni=k πα,i(j+1)
for
k = 1, . . . , n. This shows that pΠα is similarly ordered to p and Πα is order preserving
with respect to P.
Next, we prove the necessity. Choose p ∈ P such that p(n) = z1 > 0 and p(i) = 0 for
i = 1, . . . , n − 1. Because pΠα is similarly ordered to p, it holds that for j = 2, . . . , n,
n
n
i=1
i=1
µ
µ
µ
µ
− ∑ p(i) πα,i(j−1)
= (πα,nj
− πα,n(j−1)
)z1 ,
0 ≤ [pΠα ]µj − [pΠα ]µj−1 = ∑ p(i) πα,ij
30
µ
µ
≥ 0 because z1 > 0. Choose p ∈ P such that p(n) =
− πα,n(j−1)
which implies that πα,nj
z1 > p(n−1) = z2 > 0 and p(i) = 0 for i = 1, . . . , n − 2. Because pΠα is similarly ordered to
p, it holds that for j = 2, . . . , n,
n
n
µ
µ
− ∑ p(i) πα,i(j−1)
0 ≤ [pΠα ]µj − [pΠα ]µj−1 = ∑ p(i) πα,ij
i=1
=
µ
(πα,nj
i=1
µ
− πα,n(j−1) ) z1
µ
µ
) z2
− πα,(n−1)(j−1)
+ (πα,(n−1)j
µ
µ
µ
µ
µ
µ
) z2
− πα,(n−1)(j−1)
+ πα,(n−1)j
− πα,n(j−1)
) (z1 − z2 ) + (πα,nj
− πα,n(j−1)
= (πα,nj
which implies that
µ
µ
µ
µ
− πα,(n−1)(j−1)
+ πα,(n−1)j
− πα,n(j−1)
πα,nj
µ
πα,nj
≥
µ
− πα,n(j−1)
z2 − z1
z1
=1− .
z2
z2
We can always choose z2 sufficiently close to z1 so to make the right hand side of the
above equation close to 0. Hence limz2 →z1 1 − zz21 = 0, which implies that
n
µ
µ
µ
µ
µ
µ
πα,nj
− πα,n(j−1)
+ πα,(n−1)j
− πα,(n−1)(j−1)
= ∑ πα,ij
− πα,i(j−1)
≥ 0.
i=n−1
Next, we choose p ∈ P such that p(n) = p(n−1) = z1 > p(n−2) = z2 > 0 and p(i) = 0
for i = 1, . . . , n − 3. Following similar arguments as above, we can derive the inequality
µ
µ
∑ni=n−2 πα,ij − πα,i(j−1) ≥ 0 for j = 2, . . . , n. Repeating this procedure, we can prove that
µ
µ
∑ni=k πα,ij − πα,i(j−1) ≥ 0 for k = 1, . . . , n, j = 2, . . . , n.
Proof of Lemma 4.2
Proof. For j = 1, . . . , n − 1, p ∈ P,
[∆(`Πα )]j+1 − [∆(pΠα )]j+1
n
µ
µ
= ∑(`(i) − p(i) ) (πα,i(j+1)
− πα,ij
)
i=1
µ
µ
µ
µ
= (`(1) − p(1) ) (πα,1(j+1)
− πα,1j
) + [(`(1) + ∆`2 ) − (p(1) + ∆p2 )] (πα,2(j+1)
− πα,2j
) + ...
n
n
k=2
n
k=2
µ
µ
+ [(`(1) + ∑ ∆`k ) − (p(1) + ∑ ∆pk )] (πα,n(j+1)
− πα,nj
)
n
µ
µ
µ
µ
= (`(1) − p(1) ) (∑ πα,i(j+1)
− πα,ij
) + (∆`2 − ∆p2 ) (∑ πα,i(j+1)
− πα,ij
) + ...
i=1
µ
+ (∆`n − ∆pn ) (πα,n(j+1)
i=2
µ
− πα,nj
)
.
31
µ
µ
≥0
−πα,ij
If ∆p ≤ ∆`, then [∆(`Πα )]i+1 −[∆(pΠα )]i+1 ≥ 0 because ` ≥ p and ∑ni=k πα,i(j+1)
for k = 1, . . . , n. This proves (I). Vice versa, if ∆p ≥ ∆`, then
n
µ
µ
)≤0
[∆(`Πα )]j+1 − [∆(pΠα )]j+1 − (`(1) − p(1) ) (∑ πα,i(j+1)
− πα,ij
i=1
µ
µ
≥ 0 for k = 1, . . . , n. This proves (II).
− πα,ij
because ∑ni=k πα,i(j+1)
Proof of Lemma 4.3
Proof. First, we prove the sufficiency. Let x, y ∈ P and x ≺w y. For k = 1, . . . n,
n
n
∑ [xΠα ]µj − ∑ [yΠα ]µj
j=k
j=k
n
n
n
µ
µ
= ∑ (∑ x(i) πα,ij
− ∑ y(i) πα,ij
)
j=k i=1
n
i=1
n
n
n
j=k
i=2
j=k
µ
µ
µ
= ∑ (x(i) − y(i) ) ∑ πα,1j
+ ∑ (x(i) − y(i) ) ∑ (πα,2j
− πα,1j
)
i=1
(20)
n
µ
µ
+ ⋅ ⋅ ⋅ + (x(n) − y(n) ) ∑ (πα,nj
− πα,(n−1)j
)
j=k
≤ 0,
where the first equality follows because Πα is order preserving w.r.t. P and last inequalµ
µ
ity is due to x ≺w y and to the assumption that ∑nj=k πα,(i−1)j
≤ ∑nj=k πα,ij
for i = 2, . . . , n.
Hence, using the definition of weak submajorization, xΠα ≺w yΠα .
We then prove the reverse implication. Choose x, y ∈ P such that 0 ≤ x(t) < y(t) ,
y(t) −x(t) = x(t−1) −y(t−1) for some t ∈ {2, . . . , n} and x(u) = y(u) for u = 1, . . . , n, u ∉ {t−1, t}.
Clearly,
n
n
n
n
∑ x(i) < ∑ y(i) ,
∑ x(i) = ∑ y(i) for u = 1, . . . , n, u =/ t,
i=t
i=u
i=t
(21)
i=u
and x ≺w y. Because Πα is order and weak submajorization preserving w.r.t. P, Eq. (20)
must hold for x and y, and by Eq. (21) the last inequality can be simplified to
n
n
µ
µ
∑ (x(i) − y(i) ) ∑ (πα,tj − πα,(t−1)j ) ≤ 0.
i=t
j=k
µ
µ
Because ∑ni=t (x(i) − y(i) ) < 0, we obtain that ∑nj=k (πα,tj
− πα,(t−1)j
) ≥ 0 for k = 1, . . . , n.
This concludes the proof.
Proof of Lemma 4.4
32
Proof. The proof follows similar arguments used in Lemma 4.3. To prove the sufficiency,
we obtain ∑kj=1 [xΠα ]µj − ∑kj=1 [yΠα ]µj ≥ 0 for k = 1, . . . , n, x, y ∈ P by expanding vectormatrix products and combining terms in a similar way as done for the inequality (20).
To prove the necessity of the condition, we choose x, y ∈ P such that x(t) > y(t) ≥ 0,
x(t) − y(t) = y(t+1) − x(t+1) for some t ∈ {1, . . . , n − 1} and x(u) = y(u) for u = 1, . . . , n, u ∉
{t, t + 1}. Then we obtain
t
k
µ
µ
∑ (x(i) − y(i) ) ∑ (πα,tj − πα,(t+1)j ) ≥ 0,
i=1
j=1
for k = 1, . . . , n. Then, we conclude the proof following similar arguments as in Lemma 4.3.
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