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# The Systemic Events Hypothesis

Mathis Mourey

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# The Systemic Events Hypothesis

Mathis MOUREY<sup>a,1,1</sup>

<sup>a</sup>Univ. Grenoble Alpes, Grenoble INP*, CERAG, 38000 Grenoble, France

*Institute of Engineering and Management Univ. Grenoble Alpes

<sup>b</sup>The Hqgue University of Applied Sciences, CERAG, Den Haag, Netherlands

## Abstract

This article attempts to provide a theoretical framework for systemic risk assessment. We start from the fundamental definitions and build our way up to the chronology of systemic events and classification for financial crises, financial shocks, systemic crises, and systemic events. We detail each specific stage of a systemic crisis and finally propose guidelines for correctly measuring systemic risk. Eventually, we propose a brief empirical example of our framework using multiple systemic risk measures on major European and American financial institutions from 2006 to 2011. Our results show that the Subprime crisis displayed similar stages to the ones we proposed. Moreover, we propose specific timing for regulatory actions for mitigating systemic risk. Our framework does not invalidate most of the famous systemic risk measures proposed up to now but advocates that each of them only captured one dimension and hence, can not correctly describe systemic risk

Keywords: Theory, Systemic Risk, Definitions

JEL Codes: G00, G01, G10

<footnote n="1"><sup>1</sup>E-mail address: mjfmourey@hhs.nl</footnote>
# 1. Introduction

The subprime crisis showed doubtlessly that the complexity of the financial system's structure went above our comprehension. It appeared impossible to predict the numerous positive feedback effects that took place (IMF, 2009; Kapadia et al., 2012; Hautsch et al., 2014); the contagion's dynamics over financial institutions and markets (Jorion and Zhang, 2019; Kritzman et al., 2011; Acharya and Thakor, 2016; Azizpour et al., 2018); the illiquidity issues arising from fire sales and loss in trust (Bryant, 1980; Diamond and Dybvig, 1983; Shleifer and Vishny, 1992; Mishkin, 1996; Greenwood et al., 2015; Bian et al., 2018); or the consequences on the real economy (Reinhart and Rogoff, 2009). The variety of risks involved fueled a new surge in research on systemic risk. An outstanding amount of papers on measures of systemic risk were proposed (Billio et al., 2012; Adrian and Brunnermeier, 2016; Engle, 2016; Acharya, 2009; Cont et al., 2013). Each of the papers focuses on a specific dimension of the crisis and aims to find a reliable indicator of systemic risk. When considering a single measure for a systemic event, the underlying assumption is that the systemic event is unidimensional. The result of measuring systemic risk unidimensionally is noisy<fnref n="2" /> measure which fits precisely to a type of crisis.

We argue that systemic risk is multidimensional and should be assessed using different types of measures. This paper aims to go further by redefining the basics of systemic risk. Numerous academic definitions exist (Aglietta, 1993; De Bandt and Hartmann, 2000; Acharya, 2009) as well as from regulatory instances (IMF; ECB; ...). However, a consensus has yet to be reached. This issue is of first importance as definitions lead to measurements. A theoretical framework seems essential to unify academics on a specific definition. Currently, we note the work of Chen et al. (2013), which presents an axiomatic framework for measuring systemic risk. However, their definition of systemic risk is limited to systemic losses primarily, hence overlooks over aspects of systemic risk.

This paper proposes to tackle a fundamental problem of systemic risk measurement. The current approach to systemic risk measurement is almost entirely based on econometric approaches, while almost none propose a theoretical approach<fnref n="3" />. Thus, this research proposes the Systemic Event Hypothesis (SEH). The SEH is an assumption on how Systemic Event arise. We construct the SEH on a large portion of the existing literature on financial crises, systemic crises, and financial risks in general. Our result is a

<footnote n="2">²Here, noisy means that some increases in the indicators do not correspond to an actual increase in systemic risk.</footnote>

<footnote n="3">³As Zigrand (2014) does for instance.</footnote>
detailed process of the stages a systemic event goes through. We propose times during which regulatory
instances should take actions and offer a draft of a measurement framework for systemic risk. Finally,
we put the SEH to the test. Using various systemic risk measures on a European and a American sample
of financial firms, we show the validity of the SEH for the Subprime crisis.

Before starting, we should also present a brief disclaimer to avoid future confusion. Systemic risk is
by no means exclusive to finance. There exist a multitude of systems that could be physical, biological,
or environmental. Even in economics alone, various systems could be considered. This article only
presents our conceptual view of systemic risk in finance. Hence, we will focus on how systemic risk
affects financial firms.

**2. On the definition of risk**

This section introduces the concept of risk. Firstly, we discuss the concept of risk and the definitions
used so far. In a second time, we propose a definition of risk for our measurement framework. We also
discuss the existing frameworks of risk measurements.

**2.1. *A conceptual discussion***

Knowledge about risk is knowledge about the lack of knowledge. This sentence illustrates the com-
plexity existing around risk and the interest it arose through the years. Intrinsically, the notion is bound
to the unknown and how we dare to approach it. Because each area of research has different unknowns of
interest, there exist as many definitions as domains. This article constitutes the real challenge of defining
risk. It is a matter of finding a definition that will be appropriate for all areas. Let us try to approach why
such a well-known concept is so difficult to corner. An easy place to start is dictionaries' definitions. The
Cambridge Dictionary defines risk as *"the possibility of something bad happening"*. The Stanford Ency-
clopedia of Philosophy gives out a more open description: *"related to a situation in which it is possible
but not certain that an undesirable event will occur"*. Two main concepts emerge from these definitions.
First, risk is related to an event whose realization is uncertain. However, the details of what kind of
uncertainty are not precise yet. Additionally, the conclusion is undesirable. Similarly, the concept of
undesirable is tricky as it depends mainly on social constructs and subjective points of view. The issues
related to the precisions over the notions of undesirability and uncertainty are well exposed by Renn and
Walker (2008). Renn and Walker (2008) add up a third concern about the rule of risk aggregation for the
sake of comparability in the operational sense. Let us first tackle the issue of uncertainty. Pioneer's work of Knight (1921) has defined the sense of uncertainty in the economic theory. The Knightian uncertainty is when an event cannot be fully formalized, and no actual probability (or measure of the likelihood of realization) can be proposed. Consequently, the Knightian risk regards all events that can be defined and with which one can associate a likelihood of realization. This approach to risk directly leads to the use of probabilistic methods for risk assessment and disregards unknown unknowns (Luft and Ingham, 1955). Indeed, probability theory provides a powerful tool for analyzing the unknown. The development that followed the work of Bachelier (1900) and led to the existing option theory constitutes a perfect example. The question posed by the Knightian approach regards our ability to quantify the uncertainty surrounding a future event. What about the cases when it is impossible to assign a numerical value? Also, just because an event is difficult to formalize appropriately, should we overlook it? The gray area in the Knightian approach lies in dealing with unknown unknowns. Insurers, for instance, have for long recognized that insuring all risks would not be possible, and one of the reasons for it is that some risks are not easy to formalize and quantify. Hence, the existence of non-insurable risks, which leads to the second component of risk: undesirability. The question asked by Renn and Walker (2008): "who determines what is undesirable?" is of first importance. The differences in definitions come from the fact that undesirability is conditional to one's area of interest. In finance, a significant part of the literature has based its definition of risk on the probability of losing money. Even if the measures take different forms depending on the precise context (we would look at the variance in portfolio management, when you would look at the probability of death in life insurance, for instance), all are associated with financial losses. An important point to note is the distinction of Luhmann (1990). The author differentiates between danger and risk. The danger is an external threat to the system considered, while risk would emerge from within. The approach of Luhmann is interesting in the sense that he roots his approach on the fact that there is always a system considered. This point is of utmost importance: every event's analysis is implicitly based on a system. The system corresponds to the environment (both physical and conceptual) in which the event occurs. Risk assessment needs to initially consider a system in order to be able to define what variable is of interest and how one can approach the event. The importance of the system on Systemic Risk measurement is especially shown by Benoit (2017).

## 2.2. From discussing to defining

The discussion above on the definition of risk might seem trivial, but it is an essential step to what follows. A concept's definition frames its measurement. As it would seem preposterous to measure
a perfectly and imperfectly defined notion, the way one defines a notion leads one's way to measure it. Up to now, most of the definitions focus on the apparition of an undesirable event. Nevertheless, does an event emerges instantaneously and without causes? We argue in our approach that no event is without cause. Hence an event is the causality of an association of its causes. From here, we propose the following definition of risk:

**Definition 1. risk:** Possibility, that is measurable (or not), that an association of elements of a system leads to a realization of an undesirable event. The measurability of the event being a function of the complexity of the system considered.

At this point, one might question the necessity of providing yet another relatively ordinary definition of risk. We provide an example as an answer. Let us consider the risk of an avalanche occurring in a given place. A previous definition would have stated that we should look at the probability of occurrence of avalanches. As the avalanche is an event that likely happened in the past, one could consider empirically checking the number of occurrences in the past and derive an empirical probability (which is no doubt not a good approach to this kind of problem but let us assume it for the sake of the example). Such a measure would fit the previous definition but not ours. Our definition would advise looking at what causes the avalanche and, hence, looking at the (joint) probability of the cause being reunited simultaneously. Hence, our definition encompasses the coherent measures accepted by the previous definitions while refusing the simplistic measures they also allowed. This also constitutes an example of how a definition frames the measurement of a concept. Additionally to a definition, a proper measure of risk should respond to a given set of rules. This point echoes the third question of Renn and Walker (2008) on the comparison and aggregation of risks. Often, risk managers have to mitigate risk or optimize their decisions given a risk target. Hence, measuring risk so that one can aggregate or mitigate it is vital in risk management. In finance particularly, the initial work of Artzner et al. (1999), followed by Biagini et al. (2019) and Biagini et al. (2020), propose a set of axioms to follow when constructing a risk measure. Chen et al. (2013) offers an axiomatic framework for systemic risk measurement. The axioms proposed by Artzner et al. (1999) are as follows:

**Definition 2. Risk measure:** A measure of risk is a mapping from G to R.

* Axiom T: Translation invariance. For all X belonging to G and all real numbers α, qe have:

$$ \rho(X + \alpha r) = \rho(X) - \alpha. $$
* Axiom S: Subadditivity. For all $X_1$, and $X_2$ belonging to G, we have: $\rho(X_1 + X_2) \le \rho(X_1) + \rho(X_2)$.

* Axiom PH: Positive Homogeneity. For all $\lambda \ge 0$, and all $X$ belonging to G, $\rho(\lambda X) = \lambda \rho(X)$.

* Axiom M: Monotonicity. For all $X$ and $Y$ belonging to G with $X \le Y$, we have $\rho(Y) \le \rho(X)$.

* Axiom R: Relevance. For all $X$ belonging to G with $X \le 0$ and $X \ne 0$, we have $\rho(X) \ge 0$.

This research gives a proper framework for future constructions of risk measures. Even though the framework is not perfect (Dhaene et al., 2008), it serves as a great set of guidelines. The choice that such guidelines are purely mathematical or plainly explained is in fine moot. The mathematical formulation reflects a more precise degree of operationality. Our approach tends to the same goal as Chen et al. (2013) as we aim for a set of rules of construction for (systemic) risk measures. More than revocating all the existing literature on risk measures, it builds upon it. The necessity of early risk measures is that they allow a better understanding of the phenomenon considered. Moreover, as the measures get better, our understanding of the nature of the risk grows better.

### 3. Anatomy of a systemic event

This section presents our most important insights on what a systemic event is. We start by defining the concepts of financial instability, financial fragility, financial shock, and crisis. In a second time, we detail the complete process of a systemic event. We provide our working definition of systemic risk and conclude with the regulatory proposals following our view of a systemic event.

#### 3.1. *Financial instability, fragility, crisis*

Let us apply the systemic approach (Walliser, 1977) to present our conception of the financial system. The financial system consists of agents (nodes) that interact (links). It has rules for how agents may interact at any time $t$ (synchronic properties) and a framework on how these rules may change over time (diachronic properties). A *state* of the financial system is defined as the set of characteristics that the system displays at time $t$. In this sense, the choice of the system considered is central. One could consider the international financial system or observe national financial systems. The importance of the choice of the system for systemic risk measurement is highlighted in the research of Benoit (2017). A
simple model allows for an over-simplistic<fnref n="4" /> two states’ conception: stable and unstable. In this framework, any phenomena can be described by a succession of interactions (or simultaneous interactions) between agents, taking into account the synchronic properties at the time. For instance, the subprime crisis was first a consequence of the housing bubble in the U.S. and significant common exposures of banks to default risk through CDS. Every phenomenon may impact the diachronic properties of the system, hence the future structure of the system. We define the structure of the system as the set of the types and amount of existing agents, the types and amount of links/interactions between them, and the synchronic and diachronic properties. One assumption is the atomicity of financial institutions. In such a way, small changes in the number of financial institutions from one period to another will not drastically affect the whole system’s structure. We hence define a structural change as a significant variation in any of these components. We will keep this framework in mind for the rest of the paper and, more importantly, for describing the stages of a systemic event.

Both academics and regulators well know systemic risk. It is usually defined as a “know-when-you-see-it” concept, which is a diplomatic way to say it is not well defined. This inaccuracy stems from the imprecise definitions of subsequent notions. For instance, we often approach systemic risk as related to the instability of a system (De Bandt et al., 2013). However, financial stability is not well defined. The financial system is doomed to crash now and then, hence the difficulty of defining financial stability. It exists two distinct views on financial instability. On the one hand, academics consider that we define negatively what financial stability is. In other words, we define financial stability by what is not: financial instability. De Bandt et al. (2013) presents the general characteristics of financial instability. On the other hand, researchers that tried to define financial stability by what it is: a complex concept. Schinasi (2004)<fnref n="5" /> proposes the following definition:

**Definition 3.** *Financial stability: A financial system is in a range of stability whenever it is capable of facilitating (rather than impeding) the performance of an economy, and of dissipating financial imbalances that arise endogenously or as a result of significant adverse and unanticipated events.*
*(Schinasi, 2004)*

In the view of Schinasi (2004), financial stability is a continuum. It implies that financial sectors

<footnote n="4">⁴Conceptually, states are probably not clearly defined into crisis and stable. The concept of multiple states is probably a continuum of states. Hence the difficulty of settling on beginning and ending dates of crises.</footnote>

<footnote n="5">⁵To see alternative definitions of financial (in)stability, see the survey at the end of Schinasi (2004)</footnote>
are well-functioning<fnref n="6" />. More importantly, even though the financial system can be unstable (have an unstable state), stability is considered a property. That is to say, stability is an asynchronous property of the financial system, which limits the development of large imbalances that threaten the longevity of the stable state. Conceptually, one could consider that the financial system allows for two distinct states: stable and unstable. The diachronic properties of the system allow it to pass from one to another with probabilities that may vary with time. The whole purpose of systemic risk measurement is to reach (or approximate correctly) these probabilities. We have left vague a few key concepts, such as financial fragility, financial crisis, and systemic crisis. The concepts are closely intertwined but by no means equal. First, we know that financial fragility is:

**Definition 4.** **Financial fragility:** Situations in which small shocks have a significant impact on the financial system. (Allen and Gale, 2007)

This definition is linked to the resilience of financial institutions. For instance, an institution can be individually resilient. The most important ones are thanks to the current prudential oversight and still can be put into distress due to common sectoral exposures. The fact is that the network (system) must be resilient as well (Allen et al., 2010; Acemoglu et al., 2015). Knowing that the financial system is composed of various agents, linked in numerous ways, and has implicit diachronic and synchronic properties; we can now define a financial crisis as:

**Definition 5.** **Financial crisis:** Event introducing the transition from stable to the unstable state of the financial system. Its risk is accordingly, the possibility that an association of elements of the financial system leads to a state of financial instability.

One could argue that the real goal is finding the transition matrix for the financial system, that is to say, the probabilities of going from one state to another or staying in the same state. In such a case, a straight forward-approach could lead to a Hidden Markov Model (HMM), allowing for two states. Such a method indeed gives us the transition matrix. However, it seems difficult to expect such a model to differentiate between “usual” financial and systemic crises. Finally, it does not tell us anything about the causes of the crisis. Hence, it does not help the regulatory response needed during the crisis and

<footnote n="6">⁶The well-function of financial markets implies the optimization of resources allocation; mobilization of savings; facilitation of wealth accumulation, development, and growth; and smooth function of the payment system (Schinasi, 2004).</footnote>
afterward. The complexity of regulators’ task lies in the fact that financial crises come in all sorts and flavors. Allen et al. (2009) provide an insightful survey on the work on financial crises through the past decades. Troubles stem from the nature of the cause, the consequences being always identically undesirable (Eichengreen and Portes, 1987; Mishkin, 1996; Reinhart and Rogoff, 2009). Thus, the challenge with systemic risk measurement is to extract generic characteristics from historical systemic crises. The first step consists of agreeing on the systemically of a crisis: which crises are systemic and which are not. An important point to note here is that we should also identify which crises could have been systemic and were managed in time. Without including such crises, we would produce an “inverse survivorship bias” by taking into account just the cases we could not deal with at the time and missing all the situations that we managed appropriately. Here lies a central piece of insight. A crisis *becomes* systemic. Another difficulty of systemic risk measurement stems precisely from this fact. It is complex to determine whether a crisis will remain benign or spread out and worsen. Generally, we admit that a financial crisis has three possible resolutions (LaBrosse et al., 2009; Laeven and Valencia, 2010). First, the positive feedback effects create a downward spiral of losses and financial market(s) crumbles. Second, prudential authorities and central banks intervene before it is too late and resolve the crisis. Lastly, the financial markets themselves absorbed the crisis. Even though this case means that the financial system was resilient and it would be inappropriate to say that the crisis was systemic. Hence, any study on systemic crises should prioritize the first two resolutions. In this article, we believe, and thus assume, that a systemic crisis consists of a specific succession of steps.

## 3.2. Chronology

Having stated the above, let us take a closer look at the anatomy of a systemic crisis. The underlying rationale of the following lines is the assumption that systemic crises consist of identifiable steps that remain, regardless of the precise specifics of each crisis. Through academic literature and empirical observations, we have identified the following steps:

### 1. Displacement, opportunity of profit & latent risk.

As Abreu and Brunnermeier (2003) observed for bubbles, each agent will realize the situation they are in at different times. Hence the beginning of a crisis or a major structural change is not publicly known. Thus, searching for a precise date for a beginning of a crisis is moot. We base the beginning of our analysis on the Minsky anatomy of a financial crisis (Kindleberger, 1978; Minsky, 1991). His original
insight was based on the fact that a financial crisis starts with a ‘displacement’. In his view, the displace-
ment consists of the apparition of a new opportunity for profit in at least one important economic sector.
The “displacement” could be anything, but his view focuses on financial innovations. We argue that this
‘displacement’ is a structural change (as defined earlier). Note that our conception includes Minsky’s.
This new opportunity for profit changes the very structure of the financial systems. In other words, it
might alter the way financial agents interact or the way the financial system evolves. Aglietta (1996) also
agreed that the roots of a crisis were in the financial system’s structure. This first step, often overlooked
by the existing literature, is essential to determine the form of the coming crisis. Hence, each type of
structural change might lead to a correspondingly different type of systemic crisis. Structural events are
various and have been documented in the past: Collective change in investor’s behavior (Guesnerie and
Azariadis, 1982), Financial innovation (Minsky, 1991), New institution/agent, New regulation (Allen
and Gale, 2007), etc. Most of such events could appear harmless, but the fact is that they modify the
existing structure of the financial system hence, modifying its dynamics. For instance, in Minsky’s anal-
ysis, if the displacement is large and pervasive enough, there will be an improvement in anticipation of
profits in at least one important sector of the economy. This will accentuate potential losses, as we will
see in the next section.

The existence of the structural change brings a new type of risk. Here, the rationale is actually taken
from the Modern Portfolio Theory (MPT). If the anticipation of returns increases, the risk should also
increase (there is no free lunch in finance). Let us consider a financial firm as an example. We already
know the various risks it is exposed to: market, liquidity, credit, operational, etc. We will call such risks
“usual risks” henceforward. For usual risks, there is a doubt on *whether* the risky event will occur. Such
risks are reasonably known, and a broad documentation exists on how to manage them. Contrary to
the risk born from the displacement, where the uncertainty lies in *when* the risky event will occur. We
will call such risk “latent risk” from now onward. One might wonder why we are going through such a
distinction. In our view, once a displacement takes place, the financial system is doomed to go through a
crisis<sup>7</sup>. The structural change creates a new profit opportunity that will lead to the creation of latent risk.
We cannot hedge perfectly an unknown risk. Hence, we will try to manage this risk without perfectly
knowing how to measure it, inevitably leading to the second step of a systemic event.

<footnote n="7"><sup>7</sup>Whether it is a financial shock, financial crisis, or a systemic crisis. The distinction will be made later on.</footnote>
## 2. A rise in financial fragility

Financial firms do not go bankrupt without a strong event forcing them to do so. As put by Borio et al. (2014), ‘Financial crises are not like meteorite strikes from outer space. They resemble volcanic eruptions or earthquakes: they reflect the sudden and violent release of pressure gradually built up over time. [...] The build-up of such financial imbalances gives rise to endogenous boom-bust processes, or “financial cycles”' (Borio et al. (2014), p.3). Similarly, in our approach, the triggering event of a systemic crisis only occurs due to a build-up of financial fragility. Financial fragility is inherently connected to the structure of the financial system (Cont et al., 2013; Acemoglu et al., 2015; Caccioli et al., 2018). Hence, when a new profit opportunity appears and its latent risk, it serves as a foundation for financial fragility growth. The inability to assess latent risk properly makes engaging in the profit opportunity riskier than it looks. Depending on the nature of the initial displacement, the opportunity for profit can go to multiple markets<fnref n="8" />. This fact is highlighted by the international contagion effects occurring during and after financial crises (Guo et al., 2011; Aloui et al., 2011; Lee, 2012; Cipollini and Muzzioli, 2015). Our approach to financial fragility differs slightly from the one of Allen and Gale (2007), not in the definition but its origin. Allen and Gale (2007) describe how financial fragility arises from the interactions between banks and financial institutions, as well as the central role of liquidity. In our approach, financial fragility stems from the growing common exposure to a given risk in the financial system. Note that both conceptions fit the definition given earlier (see Def. 4). We root our analysis, of this stage, on what Brunnermeier and Oehmke (2013) view as the run-up phase and link to the literature on financial bubble’s creation. In our view, the opportunity for profit creates herding behavior on the market (Cai, 2020). Acharya and Yorulmazer (2008) shows that this herding behavior can be a strategy against the negative effects of information contagion. Numerous agents will react similarly and try to profit from the structural change. This translates operationally to a rise in cross-correlations of returns of financial firms. Figure 1 shows such an increase in the cross-correlation (or commonality) in returns before 2008.

In essence, the rise in financial fragility does not appear straightforwardly. That is to say, from an exterior point of view, it only seems as if financial firms are making a profit and the market appears bullish. However, the more financial firms engage in the opportunity of profit, the more significant the impact will be when latent losses occur. Consequently, in our view, the crises where multiple markets were affected are the ones for which the opportunity for profit was exploited hence creating common

<footnote n="8">⁸There is a need for studying pre-crisis building upon common exposure over multiple markets).</footnote>
<caption>Figure 1: **Commonality in stock returns.**</caption>

The graph displays the explained variance of the first components for rolling-forward Principal Components Analysis (PCA) on the stock returns of 53 major U.S. financial institutions from 2001 to 2020. In green, the explained variance of Principal Components (PC) 1 to 3. In orange, the explained variance of PC 1 to 2, and in blue the explained variance of the first PC only.

<figure><img src="image_3.png" /></figure>

exposure over multiple markets (Guo et al., 2011; Aloui et al., 2011; Lee, 2012; Cipollini and Muzzioli, 2015). Also, we know that derivatives markets can work as a channel of contagion (Aglietta, 1993, 1996; Weithers, 2007; Schwarcz, 2008). Hence, correlated hedging on the latent risk could build up financial fragility via derivative markets. In a sense, this stage can be thought of as the creation of common exposure to latent risk across the financial system (sometimes across markets) and the increase of individual exposure to macroeconomic risk (Weithers, 2007). The combination of these effects translates into the transition from Hedging to Speculative units; and Speculative to Ponzi units, presented by Minsky (1991). As he described so accurately, the increase in financial fragility leads to an extreme reaction to a “not-so-unusual” event. Measuring the exposure to macroeconomic shocks and how a specific firm will react to what types of shocks are arduous tasks. Most of the existing measures focus on assessing the economic uncertainty and how it will impact countries/firms (Oxelheim and Wihlborg, 1995; Bali et al., 2014; Briguglio, 2016).

### 3. Revealing event & realization of latent losses

Any hidden exposure will be uncovered at some point in time. Our framework considers that the high financial fragility is related to common exposure to a particular event. Our insight comes from the work of Hellwig (1995, 1998), which studied how aggregating hidden exposures in the banking system can lead to extreme shocks in conjunction with macroeconomic shocks. The general build-up of financial fragility, by definition, in the financial system inevitably leads to an extreme reaction to an otherwise normal event. We call such an event the revealing or triggering event. The nature of the triggering event
might be the most difficult to predict, mainly because it relies on the nature of the structural change. Additionally, the revealing event can be endogenous or exogenous to the financial system. One could argue that financial firms have generally developed a higher sensitivity to macro events. Thus, leading to extreme reactions when the revealing event occurs. The primary effect of the macroeconomic event will be to make latent losses occur. It is, in fact, the occurrence of common idiosyncratic shocks for firms that engaged in the profit opportunity. The existing literature on how to assess systemic loss exposure is substantially large. The subprime crisis brought a global academic interest in finding a proper systemic risk measure. Among the most famous, we find the work of Billio et al. (2012), Adrian and Brunnermeier (2016), Brownlees and Engle (2017), Acharya et al. (2017). We argue that the measures mentioned above only capture a single dimension of systemic risk. Also, because of the difficulty in assessing the extent of latent risk, we argue that the measures will also tend to underestimate the actual depth of losses. Danielsson et al. (2016) showed that our risk models also contain an inherent error, called model risk. In a similar fashion, we could argue that the model's error observed empirically stem from the lack of knowledge of latent risk. Due to the gap between estimated losses and losses that occurred, some financial firms will find themselves in a temporary period of distress, leading to the realization of other usual risks (as defined earlier). The combination of losses puts the firm in an even more precarious state, and if enough firms experience the same, it pushes the financial system into a financial crisis. We have defined earlier the concept of a financial crisis. However, it seems to be the time to present a precision. A financial crisis pushes the financial system from a stable state into an unstable one. More accurately, it represents the transition. Given our definition, we cannot call this stage a financial crisis yet. We argue that the financial system only steps into an unstable state when there is a global disturbance in financial activity. Consequently, when some idiosyncratic shocks do not disrupt global financial activity, such events are referred to as financial shocks.

**Definition 6.** **Financial shock:** *event causing one or more idiosyncratic losses for one or more financial firms. If the losses are experienced by firms of a given sector/type, we will refer to the shock as systematic. The event may develop into a financial crisis, given that the idiosyncratic shocks lead to a global occurrence of contagion and/or amplification of the losses.*

Such a definition put forward the temporal aspect present in a systemic event. We observe first the triggering event (macroeconomic shock, for instance), which turns into a financial shock, given that multiple financial firms suffer losses. Finally, when observing a contagion and/or amplification of losses, the financial shock may turn into a financial crisis.
## 4. Contagion and amplification

The financial world is one of (positive) feedback relationships. Any given system is constituted of simple rules and, yet, can display complex behavior (Walliser, 1977). The existence of feedback effects creates such complexity. Such effects highlight why studying systemic risk as a dynamic phenomenon is tricky. The stage of contagion and amplification in a systemic event is one of feedback effects as well. For every loss incurred, the situation in the financial market worsens, making some financial institutions' failure more likely. For every marginal failure, the losses incurred increase, making the financial system's situation worse and worse. We often take the example of the banking system in order to illustrate the power of feedback effects in a crisis (Freixas et al., 2000; Degryse and Nguyen, 2007; Azizpour et al., 2018). As every bank is linked with the others via the interbank market, every failure in the banking system translates to a joint increase in default risk. The more defaults occur, the more banks will be in distress and/or fail. This will translates into additional default risk realization, etc. The propensity with which the banking market is exposed to feedback effects (contagion) explains the amount of empirical work on the matter (Allen and Gale, 2000; Acharya, 2009; Huang et al., 2009; Giesecke and Kim, 2011; Billio et al., 2012; Hautsch et al., 2014).

In our chronology, the contagion/amplification stage marks the transition from a financial shock to a financial crisis. Due to the extent of the previous idiosyncratic financial shock(s), financial firms will try to recapitalize by selling liquid assets and offloading their positions related to the initial profit opportunity. As we know, correlated selling behavior that is not accompanied by a corresponding rise in demand will translate into a crash in the asset value. Hence, financial firms will worsen their situation and market liquidity by trying to recapitalize and sell away their risk exposure. In some cases, a large common offloading of positions will, although propagate the shock to other markets (Rajan, 2005; Roy and Roy, 2017). Moreover, positive feedback effects between contagion and illiquidity can easily speed up a financial system crash. Boyson et al. (2010) show that the probability of contagion of Hedge Funds is linked to liquidity shocks. Numerous studies put forward the importance of liquidity in a systemic crisis (Aglietta, 1993; De Bandt and Hartmann, 2000; Amihud, 2002; Allen and Gale, 2007; Brunnermeier and Pedersen, 2009; Adrian and Boyarchenko, 2018). Shleifer and Vishny (1992), Stein (2012) and Greenwood et al. (2015) also describe the mechanisms of a fire sales event. It seems important to point out that other amplification mechanisms (besides liquidity events) exist, such as market freezes,
payment system breakdowns, or coordination failures. More generally, one can find a comprehensive
review of all documented contagion and amplification mechanisms in Benoit et al. (2019).

At a given point, the amount in losses incurred previously by financial institutions summed up
enough for a few to fail. The first failures are determinants of whether other institutions will be ‘con-
taminated’<fnref n="9" /> or not. Cont et al. (2013) show that the most systemic nodes are not necessarily the most
interconnected but the ones for which their creditors are (eigencentrality). The general disruption in
the financial market has already started causing negative externalities in the real economy. The more
financial institutions fail, the more the payment system is disrupted, which causes problems for day to
day operations of commercial firms. The more the financing mechanisms are disrupted, which interrupts
firms’ development and cancels investments’ opportunities. The more financial markets are disrupted,
which poses essential issues for the price discovery process and immediate liquidity. We can quote the
crisis of Cyprus in 2013, where the banking system situation forced the freezing of deposit withdrawals,
for instance.

**5. Consequences on the real economy**

Inevitably, the succession of events described above leads to a severe disruption of financial markets.
Based on the insights of Aglietta (1993), we denote three major types of disruptions: **1. Payments System**. Numerous counterparty default transfer risk onto central counterparties. Thanks to Acemoglu et al. (2015), we know that star networks<fnref n="10" /> are prone to failure if the central node fails. Hence, transferring counterparty risk from financial institutions to central counterparties does not reduce systemic risk. It merely changes its nature. **2. Credit system**. Correlated failures in the banking system, possibly combined with the freezing of the interbank market, lead to credit-providing disruption. Banks are more focused on trying to recapitalize and prove the reliability of their collaterals. Hence, they tend to provide credit less efficiently. Such episodes of decrease in aggregated credit and increase in borrowing costs are denominated as *Credit Crunch*. **3. Stock Markets**. Chaos in the financial system also leads to higher informational asymmetry. Such fact stems from higher market illiquidity, as developed above, which leads ‘true’ or fundamental values of stock to be mistaken with noise (Mishkin, 1992) or with irrational expectations (Kindleberger, 1978; Blanchard and Watson, 1982; Minsky, 1991). Irregularities in finan-

<footnote n="9">⁹Here, contaminated means whether the firm *i* will fail due to the losses incurred by the failure of firm *j*.</footnote>

<footnote n="10">¹⁰Networks where all connections go towards a single node. Central counterparties naturally create such network (Jorion and Zhang, 2019).</footnote>
cial markets are the logical outcomes of previous stages of the systemic event. Hoggarth et al. (2002) compute the cost of a banking crisis on the loss of GDP experienced by countries. Reinhart and Rogoff (2009) develop on the nature of a crisis’ aftermath. Their research shows that housing and equity prices stumble down after severe financial crises that can arguably be compared to systemic crises. Economic output and employment drop severely. Moreover, public debt tends to increase drastically. Importantly, they argue that the increase in government debt is more due to the decrease in tax revenue and general consumption than to the costs of bailing out financial institutions. Atkinson et al. (2013) reach the same conclusion as Reinhart and Rogoff (2009), and include the additional fiscal costs of saving banks from bankruptcy, as well as the indirect costs of ‘national trauma’. Indeed, crises lead to waves of job loss in the economy and tend to increase the duration of unemployment. Goodhart (2010) identifies five costs of a failure of a Systemically Important Financial Institution (SIFI). The direct (accounting and juridical) costs of bankruptcies; the potential breakdown of financial markets and the costs associated with the loss of resource allocation; loss of information leading to temporary (frictional) unemployment; losses incurred by all creditors of firms that crumbled down; loss of funding for commercial firms. All of these categories of negative externalities can translate straightforwardly into an actual shock for the real economy, ending in a shock on national outputs, international trade, and countries’ development.

### 3.3. A definition for systemic risk

We have now defined precisely all the underlying concepts of systemic risk. Furthermore, we have proposed a detailed chronology of systemic events based on empirical observations and confirmed by the existing literature. Knowing all of the above, we propose the following definition for systemic risk.

**Definition 7.** Systemic Risk: Risk associated with the occurrence of a systemic event.

The definition above is by no means revolutionary and might appear as disappointing to most. It links itself quite closely with the one given by Zigrand (2014)<fnref n="11" />. The actual contribution of this paper does not lie in the definition but its implications. We have defined what a systemic event is and all its subsequent notions. Hence, the risk of occurrence of a systemic event roots in the possible consecutive occurrence of all stages mentioned above. The resulting challenge is determining at which stage one is in any specific time and obtaining the transitions’ probabilities. It seems important to point out

<footnote n="11">¹¹For the definition, see in Appendix A.</footnote>
that, given our definition of risk and the complexity of systemic events, it is, by no means, obvious that
systemic risk can be properly measured. Like most of the concepts involving human behavior, there
is always a chance that, when understood by most of the system's participants, the system's dynamics
would change. This relates directly to Lucas' critique (Lucas, 1976) on the implications of changes in
human behavior on econometric models. Systemic risk also links directly to the risk inherent in a given
system. Looking at the extent of the financial system's evolution and the speed with which it occurred
gives a unique definition of systemic risk<fnref n="12" /> seems foolish. The definition should be related to the sys-
tem's structure, which makes the definition generic and remains relevant as the system changes over
time.

In order to sum up graphically the chronology of systemic events, we propose the following graph
(see Figure 2). The figure summarizes all the concept previously mentioned and clarify our view on
what a systemic event should look like. Even though straight lines naively represent the systemic risk, it
does not imply that the increase in systemic risk must be linear.

<caption>Figure 2: Process of a systemic event.</caption>

Schematic view of the process of a systemic event. The process is composed of 4 events and 4 periods. In chronological order: Structural change (event), Increase in financial fragility (period), Revealing event (event), Latent losses occurrence (period), First major failure (event), Contagion and amplification (period), Negative externalities (period), Financial system breakdown (event).

<figure><img src="image_4.png" /></figure>

<footnote n="12">¹²Here, the unique definition means a definition that is not generalized. That is to say, a definition based on specific values and/or events.</footnote>
### 3.4. Regulatory opportunities

This section explores the opportunities that oversight authorities can exploit from our framework. In our view, the opportunities lie in the timing of a systemic event. Primarily, it is essential to know at which stage the economy currently lies. We classify prudential authorities under three groups: Structural, Monetary, and Market regulators. We summarize their role and prudential-related instruments and discuss their place in the chronology of a systemic event.

#### 3.4.1. Structural Regulation

The structural regulators are the institutions that will impose restraints on financial institutions in order to limit their systemic potential. Which, we find in Europe the European Systemic Risk Board (ESRB), the Basel Committee on Banking Supervision (BCBS), the Financial Stability Board (FSB), the European Banking Authority (EBA), and many others abroad. The adjective structural comes from the fact that the directives created by such institutions will change the ways financial institutions can interact and/or function. It changes the ‘rules’ of the system, thus its structure. The set of macroprudential instruments in the European Monetary Union is already quite restrictive and covers most of the risks considered above. It prevents the excessive fragility of the financial system using counter-cyclical buffers, requirements on Loan-to-Value and Loan-to-Income, and a leverage ratio. Additionally, the liquidity coverage ratio (LCR), the net stable funding ratio (NSFR), and the Loan-to-Deposit ratio prevent financial institutions from suffering a liquidity crisis. It also includes restrictions on the build-up of large common exposure and additional capital requirements for SIFIs. The extent of the current regulation results from decades of improvement of the prudential oversight framework. Banks have been regulated the earliest with Basel I (1988), followed by Basel II (2004) and Basel III (2010) agreements which came to complete the previous agreements. Insurers have also been identified as systemic and have been accordingly regulated (Solvency II in 2009). The structural regulation has had no choice but to fill the gaps exposed during financial crises left by the earliest reforms. It is inevitable for the system to go through structural changes. However, regulators should expect the risks created due to the change and react accordingly. For instance, the regulatory arbitrage shown by Adrian (2014) consists of a structural failure that creates a possibility for a common reaction of large credit institutions to turn to shadow banking. This constitutes a significant rise in financial instability due to the opacity of this practice<fnref n="13" />.

<footnote n="13">¹³Lysandrou and Nesvetailova (2015); Prates and Farhi (2015) provide elements of understanding for the impact of the shadow banking system on the financial system (in)stability in 2007-09.</footnote>
Given the above, we identify two critical times for the structural regulators to be involved. The first is well known. Structural oversight has the role of implementing regulatory tools to increase individual firms’ resilience. In other words, it focuses on how to prevent the triggering event from having a signif- icantly large impact on the financial system, limiting the chances of the shock turning into a financial crisis. This particular aspect is well accounted for. Numerous ratios and buffers are engineered to make the financial system more resilient. The second aspect, however, is currently overlooked. Structural regulators are responsible for forecasting what particular risk might be created from the initial structural displacement. The forecasting of potential impacts of a structural change is an arduous task. However, methodologies exist to assess risks that happen in the past, or that may occur given a particular scenario (e.g., stress testing). This phase would imply two steps. First, one must identify what the structural dis- placement is and what might be the risk associated. Secondly, one would construct a relevant scenario to test financial institutions’ sensitivity to that risk. The use and construction of such tests are already recommended by the Dodd-Franck Act (2010) and the European Banking Authority (EBA). The use of stress tests for regulatory purposes is described in the work of Weber (2014).

3.4.2. *Monetary intervention*

Monetary policy is also a central regulatory tool during a systemic crisis. Numerous studies show the link between monetary policy and systemic risk. Laséen et al. (2017) shows that a surprise tightening of monetary policy does not necessarily reduce systemic risk. Furthermore, an excessive attempt to stabilize output through monetary policy will tend to decrease risk premia while increasing leverage, hence increasing the vulnerability of the financial system<fnref n="14" />. Colletaz et al. (2018) explains that the impact of monetary policy on systemic risk is essentially appearing in the long run. Additionally, their work shows that excessively loose monetary policy can lead to a build-up of systemic risk. The fact that loose monetary policy may lead to ‘systemic risk taking’ is also highlighted by Kabundi and De Simone (2020).

In our framework, monetary policy has two major tasks. The first comes in during the building up of financial fragility. As Borio and Lowe (2002), and other suggest, excessive leverage tend to precede periods of financial stress and/or financial crises (See Figure 3). As the primary goal of Central Banks is to ensure the stability of prices and control inflation (as is the case for the European Central Bank),

<footnote n="14">¹⁴This part relates of the volatility paradox exposed in Brunnermeier and Sannikov (2014).</footnote>
preventing an excessive increase in leverage falls under their responsibilities. The literature on how monetary policy can control credit granting in the economy is substantial, and the methods are now well known. Central banks may increase the primary refinancing operations rate to increase credit funding costs, limiting credit supply indirectly. Alternative solutions also exist, such as varying the mandatory reserve ratio or performing open market operations.

<figure><img src="image_5.png" /><figcaption>Figure 3: Loans issued by commercial banks in the U.S. Commercial and industrial loans of U.S. Commercials banks from 1996 to 2020. The loans are in billion USD. Shaded areas are the dotcom crisis, subprime crisis, and covid crisis (ongoing). The dates of the crises are the ones provided by the FRED. Source: Federal Reserve Bank of St Louis.</figcaption></figure>

The second task is to deal with short-term liquidity issues (Rochet and Tirole, 1996). Central banks are known to have the capacity/role in acting as a Lender of Last Resort (LOLR) given an essential<fnref n="15" /> Institutions might fail. The function of LOLR, especially during the subprime crisis, has already reached a significant place in the existing literature (Bordo, 1990; Zalewski, 2011; Herr et al., 2016). The role of LOLR is of major importance. Indeed, there might be times when the measures taken by structural regulators will fall short to the extent of losses suffered by financial institutions. In such a case, it is vital for the survival of the financial system to save TBTF institutions to prevent a potential cascade of failure.

### 3.4.3. Market authorities

Market authorities as the Securities and Exchanges Commission (SEC) in the U.S. or the European Securities and Markets Authority (ESMA) also have a role to play in a systemic event. As we have presented above (see Figure 2), a systemic event will have to go through a financial shock. Such shock will be observable in financial markets, and we know that financial market disruptions directly impact financial firms. Thus, the first task of market authorities is to mitigate excessive volatility and sharp de-

<footnote n="15">¹⁵Essential here means essential to the survival of the financial system. Such institutions are known to be Too Big to Fail.</footnote>
creases in liquidity, which, in essence, consists in ensuring market stability. We already discussed above that a financial crash might significantly affect the values of collateral and erase market liquidity in case of fire sales. Moreover, there exists evidence of volatility contagion through markets (Hamao et al., 1990; Xiong and Han, 2015), and that such volatility spillovers have systemic implications (Yang et al., 2020). Market authorities have various tools to enforce market stability, and research has shown their effectiveness. Subrahmanyam (1994) show that circuit breakers reduce volatility in the particular asset price but will, however, tend to reduce its liquidity along and move volatility to a ‘satellite’<fnref n="16" /> market. Buss et al. (2014) compare other regulatory tools used to reduce/mitigate market volatility. Moreover, the Dodd-Franck Act (2010) also put in charge the SEC to monitor rating agencies. Since the European sovereign debt crisis, regulators noticed the systemic importance of rating agencies and assigned their regulatory oversight. In the European Union, directive No 462/2013 of the European Parliament forbids financial institutions to rely on credit ratings mechanically.

### 3.4.4. Regulatory actions

In order to summarize, we provide below (see Figure 4) a schematic representation of points of action for regulatory institutions in a systemic event. Chronologically, structural regulators' task is first to analyze any potential structural change and assess its systemic importance while assuring the financial system's resilience through prudential measures on financial institutions. Monetary authorities should then try to limit the build of large imbalances and the creation of excessive leverage. In case of the triggering event, market authorities should enforce market stability and prevent volatility spillovers across markets. Eventually, should TBTF institutions find themselves in harm's way, monetary authorities will act as a safety net for the financial system's survival, assuming their role of LOLR.

It seems clear that only coordination of all regulatory institutions will be able to cover the entire systemic event process. Kabundi and De Simone (2020) advocate for an improvement in the coordination of monetary and macroprudential policies. In the same vein, Schwarcz (2019) campaigns for the need for global coordination of regulators. The major gap in current systemic oversight is the lack of coordination between these institutions. However, communicating insights and coordinating policies may be the way to tame systemic events.

<footnote n="16">¹⁶Satellite is used here as a meaning that a very similar asset is traded in this market. Hence, while investors will stop trading the asset on the first market due to the trading halt, they will switch to the ‘satellite’ market to continue trading.</footnote>
<caption>Figure 4: Points of action for regulatory oversight</caption>

Schematic view of where each regulatory institutions should intervene in a systemic event. (1): Risk assessment of the structural change and scenario analysis of the impact of the structural change. (2) Requirements and ratios on financial institutions (Basel I to III, Solvency). (3) Limitation of the excessive build-up of leverage via monetary policy. (4) Lender of Last Resort for financial institutions in distress. (5) Prevention of a market crash (circuit breakers, liquidity monitoring/provision, or others)

<figure><img src="image_6.png" /></figure>

# 4. Systemic Risk measurement

This section will be twofold. We will first present the coherence propositions for proper systemic risk measurement based on the description of a systemic event above. In a second time, we propose a measure as an initial example of what systemic risk measurement could be. Given that there exist two types of systemic risk measures: individual and global, we precise that our framework regards primarily global measures.

## 4.1. Coherence of systemic risk measures

Similarly to Artzner et al. (1999), and Chen et al. (2013), we start the construction of our measure by presenting first the conditions to which it should respond. Based on the empirical evidence on systemic crises, we believe that the following characteristics are intrinsic to systemic risk and hence should be included somehow in its measure.

### 1. Temporality

This first aspect may appear trivial; it is, however, essential. Any measure of systemic risk must be dynamic because the concept itself is. The main complexity in measuring systemic risk lies in this fact. It changes with time. More importantly, the absolute value of the measure is generally not that important. The variations of the measure matter. Measures based on accounting data, for
instance, will tend to be late in describing systemic risk. Any useful measure should be able to
detect changes in systemic risk levels at a reasonable frequency.

## 2. Multi-dimensionality

A systemic event is composed of different risks. Often, existing literature has proposed measures based on the risk of a single step in our framework. We argue that all these studies were partially right. We refer to these step-specific risks as *systemic dimensions*. Each stage presented above can be linked to a specific systemic dimension. Each dimension is, in stable times, uncorrelated (Mourey, 2020). Additional complexity comes from this characteristic: each dimension can have a different unit. Constructing a measurement becomes then a matter of associating measures having different units.

## 3. Chronological order

We know that a systemic event is composed of successive stages. The order of the stages is naturally essential. It implies two main facts for measurements.

### (a) Nullity of systemic dimensions

As in Artzner et al. (1999), if the loading of a given risk measure is zero, then the risk is null accordingly. In our case, the logic is somewhat similar. Since all stages happen in a given order, a systemic crisis will not appear if any stage does not occur. The direct implication is that if the probability associated with a systemic dimension is null, then the systemic risk is null.

### (b) Convexity

The sequentiality of a systemic event implies that systemic risk must increase with each realized step. Logically, there is a higher chance of being in a systemic crisis if there is contagion in financial markets than if we are just in the build-up of financial fragility. Hence, systemic risk increases both the probability of each step; and the occurrence of each additional step.

We can then say that systemic risk is a convex function of its systemic dimensions.

## 4. Feedback effects

One of the complexities of systemic risk measurement lies in the existing feedback effects. The correlation between systemic dimensions in crisis times is not linear. Some types of risk will feed others to grow higher. For instance, there are known feedback effects between illiquidity and contagion. A firm that is already highly illiquid is prone to fail. Furthermore, its bankruptcy will increase the illiquidity of firms owning their debts, increasing the probability of bankruptcy of these firms. It seems essential that a measure of systemic risk must include the dynamics stem-
ming from feedback effects.

## 4.2. Data & Methods

In this section, we propose to empirically validate the chronology presented above using well-known Systemic Risk measures. We propose to analyze the Subprime crisis as it is a well-known systemic crisis with widely accessible data. Our first sample consists of the daily returns of 53 of the largest financial firms<fnref n="17" /> in the U.S. from 01-01-2006 to 01-01-2011, over which we compute measures designed to capture the effect of each stage of a systemic event. In order to show the robustness of the SEH, we apply the same methodology to a second sample. The latter consists of the daily returns of 50 of the largest financial firms in Europe. Our data starts the 01-01-2005 and ends the 30-12-2011. The summary statistics of both samples are presented in the Appendix B (see Table B.1 and Table B.2). The results for the European sample are presented in Appendix C, Figure C.6

Similarly to Kritzman et al. (2011) and Billio et al. (2012), we use a rolling-forward PCA and compute the explained variance of the first principal component in order to proxy the level of commonality in stock returns. This indicator focuses on the rise in financial fragility. For the second stage of a systemic event, we use two widely accepted measures of systemic losses, the ΔCoVaR (Adrian and Brunnermeier, 2016) and the CAViaR (Engle and Manganelli, 2004; White et al., 2015). For the contagion and amplification of losses, we use, respectively, the Average Chi of Balla et al. (2014) and illiquidity measures, ILLIQ (Amihud, 2002) and the Roll Implicit Spread, or RIS, of Roll (1984); Hasbrouck (2009). Finally, we show the negative externalities of the crisis on the real economy, with the U.S. unemployment rate and U.S. Gross Domestic Product. We detail the computation of the measures below.

### 4.2.1. Commonality in Asset Returns

The proxy for the commonality in stock returns is straightforward. Billio et al. (2012) uses a rolling PCA on weekly returns to show an increase in the explained variance of the first component before the subprime crisis, and Kritzman et al. (2011) use PCA to derive a turbulence index based on the Mahanabolis distance. In this paper, we perform PCA on daily returns of U.S. financial firms. Initially, we start by decomposing the covariance matrix of our stock returns.

$$H^T \Sigma H = \Lambda. \qquad (1)$$

<footnote n="17">¹⁷Including banks, insurance companies, and investment firms.</footnote>
Where, Σ is the variance-covariance matrix, H are orthogonal matrices or size mxm of eigenvector, such that H = (h₁, h₂, ..., hₘ), with h₁ being the first eigenvector. Λ is the diagonal matrix of eigenvalues, such that, diag(Λ) = (λ₁, λ₂, ..., λₘ). We extract principal components (U) by multiplying H to our initial matrix of returns (R),

$$U = H^T R. \qquad (2)$$

From this, we compute the explained variance of the principal component as:

$$\pi_i = \frac{\lambda_i}{\sum_{j=1}^{m} \lambda_j}. \qquad (3)$$

Where πᵢ is the explained variance of the i'h principal component.

### 4.2.2. Delta-Conditional Value-at-Risk

The work of Adrian and Brunnermeier (2016) is now famous in the literature on systemic risk measurement. We compute the ΔCoVaR using quantile regressions<fnref n="18" />. The quantile regression allows the use of state variables, which lead the CoVaR to be more linked to macroeconomic variables. Initially, we define the CoVaR as,

$$Pr(R_i \le CoVaR^i | m(q, p) | R_m = VaR_m(p)) = q. \qquad (4)$$

Where Rᵢ, Rₘ are the vector of returns of firm j, and the market. VaRₘ(p) is the Value-at-Risk (VaR) of the market index at the p level of confidence. And CoVaR^i|m(q, p) is the VaR of firm j at the q level of confidence, conditional to the market being at its VaR at the p level of confidence. ΔCoVaR is then defined as,

$$\Delta CoVaR^i|m(q, p) = CoVaR^i|m(q, p) - CoVaR^i|m(q, 50\%). \qquad (5)$$

### 4.2.3. CAViaR

Based on the work of Engle and Manganelli (2004), the Conditional Autoregressive Value-at-Risk by regression quantiles (CAViaR) is an alternative to the ΔCoVaR, for measuring the exposure to systemic losses. A multivariate application of the initial CAViaR was proposed by White et al. (2015), using a bivariate Vector AutoRegressive (VAR) model. Formally,

$$\begin{aligned} Q_1(q,t) &= R(t-1)\beta_1 + b_{11}Q_1(q,t-1) + b_{12}Q_2(q,t-1), \\ Q_2(q,t) &= R(t-1)\beta_2 + b_{22}Q_2(q,t-1) + b_{21}Q_1(q,t-1). \end{aligned} \qquad (6)$$

<footnote n="18">¹⁸There exists other ways to compute it, as the bivariate Gaussian model.</footnote>
$Q_i(q)$ is the quantile function for the returns of firm i, such that,

$$Q_i(q,t) = \inf\{R_i(t) \in \mathbb{R}: q \le F(R_i(t))\}. \quad (7)$$

With $F(R_i(t))$ being the distribution function of the returns of firm i, and q is the probability level desired. $R(t-1)$ is a two columns-matrix of lagged returns for firms i and j. Correspondingly, $\beta_1, \beta_2$ are the vectors of parameters associated with the lagged returns. In the paper of White et al. (2015), the CAViaR is directly derived from the Equation 6. Contrary to other systemic risk measures, the loss is considered as the lower quantile of the stock returns (which is the definition of VaR) and then, is estimated in a bivariate framework to incorporate bilateral dependencies in the quantiles in the firm's returns.

### 4.2.4. Asymptotic Tail Dependence

Balla et al. (2014) develop a measure based on the asymptotic tail dependence in the stock returns of large U.S. depositories. The article proposes two closely-related measures, founded on the Multivariate Extreme Value Theory, the Asymptotic Dependence Rate and the Average Chi. Our study uses the Average Chi on the lower tail of stock returns in order to assess system-wide common losses, thus providing a proxy for contagion effects in our sample. Initially, the $\chi_{ij}$ corresponds to the probability of the returns of firm i to fall below a given quantile, conditionally to the firm j experiencing itself such low returns. Mathematically,

$$\chi_{ij}(q) = \Pr(R_i < Q_i(q) | R_j < Q_j(q)), \quad (8)$$

$$\chi_{ij} = \lim_{q \to 0} \chi_{ij}(q). \quad (9)$$

Where $R_i, R_j$ are vector of returns for firm i and j. q is the target probability level. Finally, we define the Average Chi as,

$$\text{Avg}\chi = \frac{\sum_{i=1}^{N} \sum_{j=1}^{N} \chi_{ij}}{N(N-1)}. \quad (10)$$

With N being the number of firms in our sample. In essence, if $\chi = 0$, then the returns of firm i and j are said to be asymptotically independent. The larger $\chi_{ij}$ gets, the more probable it will be for firm i to experience extreme returns, given that firm j is already experiencing extreme returns.
### 4.2.5. ILLIQ

The measure of Amihud (2002) (ILLIQ) is one of the most widely used proxy for illiquidity. The reason for that lies in the simplicity of the measure and the fact that it relies on market-based data. ILLIQ essentially is a ratio between the absolute asset returns ($R_i$) at time $t$ and the volume in dollar ($V$) at time $t$, such that,

$$ILLIQ(t) = \frac{|R_i(t)|}{V(t)}. \qquad (11)$$

Following the recommendation from the Volatility-Lab of New York University (NYU), we apply a Multiplicative Error Model (MEM) on ILLIQ in order to extract its conditional mean ($\mu$). The ILLIQ presented in Figure 5 is actually its conditional mean. We apply an Asymmetric MEM that give higher importance of illiquidity when returns are negative. Formally,

$$ILLIQ(t) = \mu(t)\varepsilon(t), \qquad (12)$$

$$\mu(t) = \omega + (\alpha + \mathbb{I}(t-1)\gamma)ILLIQ(t-1) + \beta\mu(t-1). \qquad (13)$$

Where $\varepsilon$ is Gaussian white noise, $\mu$ the conditional mean of ILLIQ, and $\mathbb{I}$ is the indicator function, such that,

$$\mathbb{I}(t-1) = \begin{cases} 0, & \text{if } R_i(t-1) \ge 0, \\ 1, & \text{if } R_i(t-1) < 0. \end{cases} \qquad (14)$$

### 4.2.6. Roll Implicit Spread

The RIS is a measure founded by Roll (1984). Roll (1984) propose a model that relies on two major assumptions. First, the asset considered is traded in an informationally efficient market. Second, the returns' probability distribution must be stationary. The second assumption may be nuanced for long time horizons but is strict for short time intervals. The measure is then straightforward,

$$RIS = 2 * \sqrt{-cov}. \qquad (15)$$

Where $cov$ is the first-order serial covariance of the changes of price of the firm. In other words,

$$cov = Cov(\Delta P(t), \Delta P(t-1)). \qquad (16)$$

Where $P(t)$ is the stock price at time $t$, and

$$\Delta P(t) = P(t) - P(t-1). \qquad (17)$$

In our case, the higher value of RIS will imply higher illiquidity. This interpretation comes from the
fact that the *cov* is inversely related to the effective bid-ask spread (Roll, 1984).

### 4.3. *Results*

We compute the measures presented above on our sample of 53 of the largest U.S. financial firms over the 2006-2011 period. Our results are summarized in Figure 5. Our framework postulates the existence of four main periods in a systemic event: Financial Fragility increase, the realization of Latent Losses, Contagion & amplification of losses, and the occurrence of negative externalities on the real economy. The choice of the measures was driven by the type of risk they assessed. The commonality of asset returns is approximated by the explained variance of Kritzman et al. (2011); ΔCoVaR (Adrian and Brunnermeier, 2016) and CAViaR (Engle and Manganelli, 2004) measure the extent of systemic losses; the average $\chi$ (Balla et al., 2014) assess the contagion potential; ILLIQ (Amihud, 2002) and RIS (Roll, 1984) proxy the average illiquidity of financial firms; and the negative externalities are expressed in terms of unemployment rate and loss in real GDP.

The commonality in asset returns increases significantly (from 40% to 60%) at the beginning of 2007, while the other measures stay stable. From a logical point of view, the increase in financial fragility has to occur in advance. It comes from the fact that the exposure to the latent risk must grow enough to have a significant impact. The growth corresponds to the growth in real GDP and house prices in the U.S.. The exponentially growing development of the U.S. housing market, fueled by the appari- tion of new financial instruments (e.g. CDOs), has led most major banks and insurance companies to create a major common exposure to a potential housing market decline/crash.

When the housing market started to decline in 2007, and banks realized the extent of their current exposure, some got exposed to unexpectedly high insolvency and bankruptcy. The failure of Lehman Brothers gave the gentle push that threw the financial system into a state of instability. The financial shock<fnref n="19" /> starts at the end of 2008, right after the Bear Stearns avoided bankruptcy and got acquired by JP Morgan Chase, and shortly after the fall of Lehman Brothers.

However, the potential for contagion or illiquidity has not peaked yet. The amplification of losses starts in the U.S., where multiple firms experience insolvency due to their toxic portfolio of CDOs, such as the American Insurance Group (AIG), which benefited from a massive bail-out from the Federal Re-

<footnote n="19">¹⁹From the Definition 5, a financial shock marks the transition from a stable to an unstable state.</footnote>
<caption>Figure 5: Chronology of systemic dimensions during the subprime crisis</caption>

The chart displays systemic risk measures (dimensions) from 2006 to 2011. We proposed to divide the graph given the periods given above (see Figure 2). Blue corresponds to the rise in financial fragility, yellow relates to the financial shock and realization of latent losses, red corresponds to the rise of contagion and amplification effects. Finally, the hatched area corresponds to the period of negative externalities. Commonality corresponds to the variance explained by the first factor of a PCA on asset' returns (Kritzman et al., 2011; Billio et al., 2012), CAViaR to the measure of White et al. (2015), CoVaR is the ΔCoVaR of Adrian and Brunnermeier (2016), ACHI is the Average Chi coefficient (lower tail) of Balla et al. (2014), RIS is the liquidity measures from Roll (1984), ILLIQ is the illiquidity measure from Amihud (2002). Finally, the last graph displays the unemployment rate and the real GDP of the U.S.A.

<figure><img src="image_7.png" /></figure>

serve. In October 2008, the crisis spread to Europe, and the international contagion started, and cause the interbank market to freeze at the end of the month. Empirically, we observe that the increase in lower tail asymptotic dependencies happens in delay compared to the losses. Similarly, both illiquidity measures only summit at the beginning of the year 2009. Shortly after, the financial crisis turns into a major economic downturn, manifested by a high unemployment rate and a high decrease in real GDP. Furthermore, we observe a common decrease in systemic dimensions occurring through the middle/end of the Contagion & Amplification phase (except for the Average $\chi^2$). Macroeconomic variables, however, tend to be more sticky and take more time to adjust back to their previous respective levels.
## 5. Conclusion

The chronology of systemic events should have been the primary goal of researchers interested in measuring systemic risk. The definitions of systemic risk being central in its measurement context, knowing the stages and nature of risks involved seems capital. The current literature lacks theoretical approaches to systemic risk. This fact is highlighted by the evident variety of measures proposed so far (Allen et al., 2012; Billio et al., 2012; Balla et al., 2014; Adrian and Brunnermeier, 2016; Acharya et al., 2017; Brownlees and Engle, 2017). Academics have taken the perspective that systemic risk can be boiled down to a single indicator. It is, however, an arduous task to find one estimator/factor capable of taking into account the multiplicity of risks and mechanisms taking action in a systemic event. The numerous feedback loops, latent exposures, and other phenomena are already challenging to model by themselves.

We start from the fact that studies focusing on one specific aspect of systemic risk are not in the wrong. Such work overlooks other dimensions of the problem. The Systemic Events Hypothesis (SEH) proposes an alternative view. There exists an actual succession of steps in systemic events, and each step can be assessed using an appropriate measure. We propose a complete chronology of a systemic event that can be expressed in the following steps. First, a structural displacement gives rise to a profit opportunity. Such opportunity is linked to a latent risk, difficult to assess. Many companies identify the profit opportunity and engage in a common, yet not coordinated, investment, leading inevitably to the building up of a large common exposure. An increasing state of financial fragility arises from this correlated exposure and leads to the realization of the (latent) risk of the profit opportunity. The losses incurred simultaneously by financial firms push some to go bankrupt and make the financial system fall into a crisis state. Individual and common losses increase due to contagion, amplification, and feedback effects, eventually leading the financial system to face a potential breakdown. Regardless of whether the crisis is resolved using government/authorities' intervention, the systemic event has caused such a disruption in the financial sphere that negatively impacts the real economy.

We choose and compute well-known systemic risk measures that correspond to a specific stage in our systemic event chronology. The measures are applied to a sample of 53 of the largest financial firms in the U.S. from 2006 to 2011. Empirical evidence in our sample makes a strong case for the SEH to be valid. We find that commonality in asset returns increases prior to the crisis indicates a potential common driving factor. Losses occur in the breakout of the crisis, while contagion and illiquidity proxies peak, in
delay, at the end of the crisis. The consequences of the subprime crisis are undeniable and are shown in
our results in terms of loss of real GDP and an increase in the U.S. unemployment rate. We discuss the
optimal timing for different types of regulatory actions and specify what type of actions should be used
in a systemic event. Additionally, we provide definitions and related elements for concepts underlying
systemic risk.

While this study constitutes a preliminary look into SEH, which seems to fit the last systemic crisis
in date, we call for more research on detailing the transitions from one stage to another, validation of the
SEH using other measures/sample, or even the construction of a model capable to estimate the state of a
crisis we are currently in.
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# Appendix A. Systemic Risk definitions

**Definition 1.** **Mishkin (1992):** 'The likelihood of a sudden, usually unexpected, event that disrupts information in financial markets, making them unable to effectively channel funds to those parties with the most productive investment opportunities.'

<figure><img src="image_8.png" /></figure>

**Definition 2.** **Aglietta and Moutot (1993):** 'the possibility for an economy to experience a state where, the agents' responses given the risk they face, far from leading to a better allocation of individual risks, leads to increase general insecurity.'<fnref n="20" />

<figure><img src="image_9.png" /></figure>

**Definition 3.** **Rochet and Tirole (1996):** 'refers to the propagation of an agent's economic distress to other agents linked to that agents through financial transactions.'

<figure><img src="image_10.png" /></figure>

**Definition 4.** **De Bandt and Hartmann (2000):** 'a systemic event that affects a considerable number of financial institutions or markets in a strong sense, thereby severely impairing the general well-functioning of the financial system. [...] At the heart of the concept is the notion of contagion, a particularly strong propagation of failures from one institution, market or system to another'

<figure><img src="image_11.png" /></figure>

**Definition 5.** **Group of Ten (2001):** 'the risk that an event will trigger a loss of economic value or confidence in, and attendant increases in uncertainty about, a substantial portion of the financial system that is serious enough to quite probably have significant adverse effects on the real economy.'

<figure><img src="image_12.png" /></figure>

**Definition 6.** **Kaufmann and Scott (2003):** 'refers to the risk or probability of breakdowns in an entire system, as opposed to breakdowns in individual parts or components, and is evidence by co-movements (correlation) among most or all the parts.'

<figure><img src="image_13.png" /></figure>

**Definition 7.** **Kupiec and Nickerson (2004):** 'the potential for a modest economic shock to induce substantial volatility in asset prices, significant reductions in corporate liquidity, potential bankruptcies and efficiency losses'

<figure><img src="image_14.png" /></figure>

**Definition 8.** **Hendricks et al. (2007):** 'the risk of a phase transition from one equilibrium to another, much less optimal equilibrium, characterized by multiple self-reinforcing feedback mechanisms making it difficult to reverse.'

<figure><img src="image_15.png" /></figure>

<footnote n="20">²⁰Traduction made by the author.</footnote>
**Definition 9.** Schwarcz (2008): ‘the risk that (i) an economic shock such as market or institutional failure triggers (through a panic or otherwise) either (X) the failure of a chain of markets or institutions or (Y) a chain of significant losses to financial institutions, (ii) resulting in increases in the cost of capital or decreases in its availability, often evidenced by substantial financial market price volatility’

**Definition 10.** Acharya (2009): ‘Joint failure risk arising from the correlation of returns on asset-side of bank balance-sheets.’

**Definition 11.** IMF (2009): ‘the risk of disruption of financial services that is (i) caused by an impairment of all or parts of the financial system and (ii) has the potential to have serious negative consequences on the real economy.’

**Definition 12.** Billio et al. (2012): ‘Any set of circumstances that threatens the stability of, or public confidence in, the financial system’

**Definition 13.** Patro et al. (2013): ‘the likelihood of experiencing a systemic failure, a broad-based breakdown of the financial system that is triggered by a strong systemic event (e.g. financial institution failure), which severely and negatively impacts the financial markets and the economy in general’

**Definition 14.** Chen et al. (2013): ‘refers to the risk of collapse of an entire complex system, as a result of actions taken by the individual component entities or agents that comprise the system.’

**Definition 15.** Zigrand (2014): ‘the risk of an event - labeled a systemic event - occurring in a given system that at least temporarily, to an altered and damaged transitional “system” whose proper functioning is impeded. In the extreme, the structure of the system itself is damaged and the system can no longer function’

**Definition 16.** Silva et al. (2017): ‘the risk that several market participants are simultaneously affected by severe losses, which then spread to the entire economy through financial interconnections among economic agents’.

**Definition 17.** Montagna et al. (2021): ‘the probability to have a large number of banks going into distress simultaneously or, more formally, the probability that the number of banks’ default or other credit events in a certain period is higher than a given threshold’
Appendix B. Tables

<caption>Table B.1: Summary Statistics - U.S. Sample.</caption>

Summary statistics for each type of financial firms' daily returns, from 01-01-2006 to 30-12-2010. The sample consists in 53 American financial firms. The mean, standard deviation, max, min and median are expressed annually.

<table><caption>Table B.2: Summary Statistics - European Sample.</caption><thead><tr><th colspan="10">Full Sample</th></tr><tr><th>Sector</th><th>count</th><th>mean</th><th>sd</th><th>max</th><th>min</th><th>median</th><th>skewness</th><th>kurtosis</th><th>auto-corr</th></tr></thead><tbody><tr><td>Bank</td><td>24</td><td>-0.082</td><td>0.636</td><td>0.626</td><td>-0.893</td><td>-0.0</td><td>-0.212</td><td>19.486</td><td>-0.070</td></tr><tr><td>Insurance</td><td>22</td><td>-0.026</td><td>0.495</td><td>0.705</td><td>-0.936</td><td>0.0</td><td>-0.372</td><td>18.521</td><td>-0.114</td></tr><tr><td>Asset Management</td><td>7</td><td>0.075</td><td>0.480</td><td>0.269</td><td>-0.261</td><td>0.0</td><td>0.052</td><td>6.984</td><td>-0.120</td></tr></tbody><thead><tr><th colspan="10">During Subprimes</th></tr><tr><th>Sector</th><th>count</th><th>mean</th><th>sd</th><th>max</th><th>min</th><th>median</th><th>skewness</th><th>kurtosis</th><th>auto-corr</th></tr></thead><tbody><tr><td>Bank</td><td>24</td><td>-0.561</td><td>0.885</td><td>0.626</td><td>-0.893</td><td>-0.003</td><td>-0.327</td><td>10.163</td><td>-0.053</td></tr><tr><td>Insurance</td><td>22</td><td>-0.450</td><td>0.696</td><td>0.705</td><td>-0.936</td><td>-0.002</td><td>-0.473</td><td>11.029</td><td>-0.108</td></tr><tr><td>Asset Management</td><td>7</td><td>-0.312</td><td>0.674</td><td>0.269</td><td>-0.261</td><td>-0.001</td><td>0.043</td><td>3.395</td><td>-0.127</td></tr></tbody><thead><tr><th colspan="10">Out of Subprimes</th></tr><tr><th>Sector</th><th>count</th><th>mean</th><th>sd</th><th>max</th><th>min</th><th>median</th><th>skewness</th><th>kurtosis</th><th>auto-corr</th></tr></thead><tbody><tr><td>Bank</td><td>24</td><td>0.417</td><td>0.446</td><td>0.463</td><td>-0.288</td><td>0.000</td><td>0.937</td><td>15.273</td><td>-0.126</td></tr><tr><td>Insurance</td><td>22</td><td>0.409</td><td>0.341</td><td>0.507</td><td>-0.481</td><td>0.001</td><td>0.571</td><td>11.680</td><td>-0.141</td></tr><tr><td>Asset Management</td><td>7</td><td>0.377</td><td>0.334</td><td>0.167</td><td>-0.145</td><td>0.001</td><td>0.444</td><td>4.866</td><td>-0.118</td></tr></tbody></table>

Summary statistics for each type of financial firms' daily returns, from 01-01-2005 to 30-12-2011. The sample consists in 50 European financial firms. The mean, standard deviation, max, min and median are expressed annually.

<table><thead><tr><th colspan="10">Full Sample</th></tr><tr><th>Sector</th><th>count</th><th>mean</th><th>sd</th><th>max</th><th>min</th><th>median</th><th>skewness</th><th>kurtosis</th><th>auto-corr</th></tr></thead><tbody><tr><td>Bank</td><td>25</td><td>-0.068</td><td>0.482</td><td>0.550</td><td>-1.215</td><td>-0.0</td><td>-0.514</td><td>23.092</td><td>0.033</td></tr><tr><td>Insurance</td><td>14</td><td>0.029</td><td>0.366</td><td>0.233</td><td>-0.406</td><td>0.0</td><td>-0.183</td><td>12.151</td><td>0.007</td></tr><tr><td>Asset Management</td><td>11</td><td>0.021</td><td>0.394</td><td>0.328</td><td>-0.341</td><td>0.0</td><td>0.060</td><td>9.791</td><td>-0.000</td></tr></tbody><thead><tr><th colspan="10">During Subprimes</th></tr><tr><th>Sector</th><th>count</th><th>mean</th><th>sd</th><th>max</th><th>min</th><th>median</th><th>skewness</th><th>kurtosis</th><th>auto-corr</th></tr></thead><tbody><tr><td>Bank</td><td>25</td><td>-0.577</td><td>0.697</td><td>0.550</td><td>-1.215</td><td>-0.003</td><td>-0.422</td><td>12.391</td><td>0.032</td></tr><tr><td>Insurance</td><td>14</td><td>-0.415</td><td>0.539</td><td>0.233</td><td>-0.406</td><td>-0.002</td><td>-0.187</td><td>6.853</td><td>-0.012</td></tr><tr><td>Asset Management</td><td>11</td><td>-0.476</td><td>0.583</td><td>0.280</td><td>-0.341</td><td>-0.002</td><td>0.065</td><td>4.914</td><td>-0.021</td></tr></tbody><thead><tr><th colspan="10">Out of Subprimes</th></tr><tr><th>Sector</th><th>count</th><th>mean</th><th>sd</th><th>max</th><th>min</th><th>median</th><th>skewness</th><th>kurtosis</th><th>auto-corr</th></tr></thead><tbody><tr><td>Bank</td><td>25</td><td>0.229</td><td>0.381</td><td>0.220</td><td>-0.287</td><td>0.0</td><td>0.251</td><td>6.661</td><td>0.023</td></tr><tr><td>Insurance</td><td>14</td><td>0.253</td><td>0.285</td><td>0.198</td><td>-0.157</td><td>0.0</td><td>0.276</td><td>5.982</td><td>0.011</td></tr><tr><td>Asset Management</td><td>11</td><td>0.291</td><td>0.306</td><td>0.328</td><td>-0.192</td><td>0.0</td><td>0.388</td><td>7.570</td><td>0.010</td></tr></tbody></table>
<caption>Table B.3: Summary Statistics for the State Variables - U.S. sample</caption>

Summary statistics for the state variables from 01-01-2006 to 30-12-2010. We have separated our summary statistics for the Subprimes period and stable periods. The state variables are used in the computation of the ΔCoVaR. We use the TEDRATE (TED spread), DeltaTB3 (log returns of 3-months T-Bill), DJ_EXCESS (Dow Jones Industrial excess returns compared to S&P500), the FFR (the effective federal fund rate) and the Volatility index (VIXCLS). The mean, standard deviation, min, max and median of the log returns series are annualized.

<table><thead><tr><th colspan="10">Full Sample</th></tr><tr><th>State Variable</th><th>mean</th><th>sd</th><th>max</th><th>min</th><th>median</th><th>skewness</th><th>kurtosis</th><th>auto-corr</th></tr></thead><tbody><tr><td>VIXCLS</td><td>23.415</td><td>11.923</td><td>80.860</td><td>9.890</td><td>21.43</td><td>1.776</td><td>3.767</td><td>0.982</td></tr><tr><td>TEDRATE</td><td>0.741</td><td>0.664</td><td>4.580</td><td>0.090</td><td>0.49</td><td>2.029</td><td>5.696</td><td>0.977</td></tr><tr><td>DeltaTB3</td><td>-0.157</td><td>1.349</td><td>0.885</td><td>-1.447</td><td>0.00</td><td>-1.691</td><td>97.377</td><td>-0.095</td></tr><tr><td>DJ_EXCESS</td><td>0.008</td><td>0.026</td><td>0.013</td><td>-0.010</td><td>0.00</td><td>0.183</td><td>6.628</td><td>-0.142</td></tr><tr><td>FFR</td><td>2.451</td><td>2.234</td><td>5.410</td><td>0.050</td><td>2.01</td><td>0.163</td><td>-1.781</td><td>0.998</td></tr><tr><th colspan="10">During Subprimes</th></tr><tr><td>VIXCLS</td><td>30.583</td><td>14.670</td><td>80.860</td><td>14.210</td><td>24.270</td><td>1.372</td><td>0.917</td><td>0.975</td></tr><tr><td>TEDRATE</td><td>1.428</td><td>0.691</td><td>4.580</td><td>0.200</td><td>1.280</td><td>1.761</td><td>4.287</td><td>0.938</td></tr><tr><td>DeltaTB3</td><td>-0.288</td><td>2.083</td><td>0.885</td><td>-1.447</td><td>-0.002</td><td>-1.390</td><td>47.721</td><td>-0.104</td></tr><tr><td>DJ_EXCESS</td><td>0.020</td><td>0.035</td><td>0.013</td><td>-0.010</td><td>0.000</td><td>0.021</td><td>4.796</td><td>-0.190</td></tr><tr><td>FFR</td><td>2.644</td><td>1.757</td><td>5.410</td><td>0.080</td><td>2.185</td><td>0.071</td><td>-1.308</td><td>0.989</td></tr><tr><th colspan="10">Out of Subprimes</th></tr><tr><td>VIXCLS</td><td>19.664</td><td>7.918</td><td>45.890</td><td>9.890</td><td>18.045</td><td>0.915</td><td>0.372</td><td>0.973</td></tr><tr><td>TEDRATE</td><td>0.381</td><td>0.211</td><td>1.130</td><td>0.090</td><td>0.370</td><td>1.072</td><td>1.282</td><td>0.994</td></tr><tr><td>DeltaTB3</td><td>-0.079</td><td>0.707</td><td>0.398</td><td>-0.301</td><td>0.000</td><td>0.907</td><td>21.809</td><td>-0.055</td></tr><tr><td>DJ_EXCESS</td><td>0.003</td><td>0.021</td><td>0.008</td><td>-0.005</td><td>-0.000</td><td>0.438</td><td>2.773</td><td>-0.068</td></tr><tr><td>FFR</td><td>2.347</td><td>2.442</td><td>5.410</td><td>0.050</td><td>0.200</td><td>0.239</td><td>-1.920</td><td>0.997</td></tr></tbody></table>
<caption>Table B.4: Summary Statistics for the State Variables - European sample</caption>

Summary statistics for the state variables from 01-01-2005 to 30-12-2011. We have separated our summary statistics for the Subprimes period and stable periods. The state variables are used in the computation of the ΔCoVaR. We use MSCI_EU (log returns of the MSCI Europe), EUROSpread (spread between EURIBOR 3-months and French T-Bill 3-months), EONIA, Delta3M (log returns on French T-Bill 3-months), and the Volatility index (VIXCLS). The mean, standard deviation, min, max and median of the log returns series are annualized.

<table><thead><tr><th>State Variable</th><th>mean</th><th>sd</th><th>max</th><th>min</th><th>median</th><th>skewness</th><th>kurtosis</th><th>auto-corr</th></tr></thead><tbody><tr><td>MSCI_EU</td><td>0.009</td><td>0.007</td><td>0.047</td><td>-0.044</td><td>0.000</td><td>-0.080</td><td>6.381</td><td>-0.010</td></tr><tr><td>VIXCLS</td><td>22.006</td><td>11.198</td><td>80.860</td><td>9.890</td><td>19.400</td><td>1.838</td><td>4.273</td><td>0.982</td></tr><tr><td>EUROSpread</td><td>0.005</td><td>0.004</td><td>0.031</td><td>-0.003</td><td>0.004</td><td>2.102</td><td>7.375</td><td>0.989</td></tr><tr><td>EONIA</td><td>0.021</td><td>0.014</td><td>0.046</td><td>0.003</td><td>0.021</td><td>0.159</td><td>-1.466</td><td>0.997</td></tr><tr><td>Delta3M</td><td>-0.078</td><td>0.043</td><td>1.243</td><td>-0.519</td><td>0.000</td><td>12.114</td><td>402.229</td><td>-0.087</td></tr><tr><td colspan="9" style="text-align: center; font-weight: bold;">During Subprimes</td></tr><tr><td>MSCI_EU</td><td>-0.196</td><td>0.010</td><td>0.047</td><td>-0.044</td><td>-0.001</td><td>0.123</td><td>4.656</td><td>-0.033</td></tr><tr><td>VIXCLS</td><td>30.583</td><td>14.670</td><td>80.860</td><td>14.210</td><td>24.270</td><td>1.372</td><td>0.917</td><td>0.975</td></tr><tr><td>EUROSpread</td><td>0.009</td><td>0.005</td><td>0.031</td><td>0.001</td><td>0.008</td><td>1.860</td><td>4.914</td><td>0.986</td></tr><tr><td>EONIA</td><td>0.037</td><td>0.009</td><td>0.046</td><td>0.012</td><td>0.040</td><td>-1.849</td><td>2.175</td><td>0.983</td></tr><tr><td>Delta3M</td><td>-0.335</td><td>0.012</td><td>0.059</td><td>-0.139</td><td>-0.000</td><td>-4.013</td><td>44.082</td><td>0.068</td></tr><tr><td colspan="9" style="text-align: center; font-weight: bold;">Out of Subprimes</td></tr><tr><td>MSCI_EU</td><td>0.087</td><td>0.006</td><td>0.036</td><td>-0.028</td><td>0.000</td><td>-0.109</td><td>3.257</td><td>0.000</td></tr><tr><td>VIXCLS</td><td>19.218</td><td>8.026</td><td>48.000</td><td>9.890</td><td>17.005</td><td>1.090</td><td>0.535</td><td>0.974</td></tr><tr><td>EUROSpread</td><td>0.004</td><td>0.003</td><td>0.015</td><td>-0.003</td><td>0.003</td><td>1.466</td><td>2.386</td><td>0.975</td></tr><tr><td>EONIA</td><td>0.016</td><td>0.011</td><td>0.041</td><td>0.003</td><td>0.011</td><td>0.491</td><td>-1.098</td><td>0.992</td></tr><tr><td>Delta3M</td><td>0.026</td><td>0.049</td><td>1.243</td><td>-0.519</td><td>0.000</td><td>10.820</td><td>313.997</td><td>-0.091</td></tr></tbody></table>
# Appendix C. Figures

<caption>Figure C.6: Chronology of Systemic Dimensions during the subprime crisis - European Sample</caption>

The chart displays Systemic Risk measures (dimensions) from 2005 to 2011, for our sample of 50 European financial firms. We proposed to divide the graph according to the periods given above (see Figure 2). Blue corresponds to the rise in financial fragility, yellow relates to the financial shock and realization of latent losses, red corresponds to the rise of contagion and amplification effects. Finally, the hatched area corresponds to the period of negative externalities. Commonality corresponds to the variance explained by the first factor of a PCA on asset' returns (Kritzman et al., 2011; Billio et al., 2012), CAViaR to the measure of White et al. (2015), CoVaR is the ΔCoVaR of Adrian and Brunnermeier (2016), ACHI is the Average Chi coefficient (lower tail) of Balla et al. (2014), RIS is the liquidity measures from Roll (1984), ILLIQ is the illiquidity measure from Amihud (2002). Finally, the last graphs displays the average unemployment rate and the average real GDP for European Union members (Source: Federal Reserve Bank of St Louis).

<figure><img src="image_16.png" /></figure>