# On the correlation of systemic dimensions

Mathis Mourey

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# On the correlation of systemic dimensions

Mathis MOUREY¹

¹University Grenoble-Alpes, Grenoble INP, CERAG, 38000 Grenoble, France
mathis.mourey@univ-grenoble-alpes.fr

March 30, 2021

## Abstract

We propose an analysis of correlation between three systemic risk measures: ΔCoVaR, Granger causalities and ILLIQ. Based on previous research, we interpret the measures above as proxies for systemic dimensions; respectively, Losses, Connectedness and Illiquidity. Our study shows that systemic dimensions are uncorrelated in stable periods and lose their orthogonality in crisis periods. This result confirms that systemic risk is a composition of different type of risk that occur concomitantly. Moreover, we show that, by applying a specific lag for each systemic dimensions, we improve the identification of systemic crisis periods and reduce the noise of the measure. This constitutes an evidence of a peculiar order in the occurrence of systemic dimensions; hinting that a specific chronology of systemic events might exist.

**Keywords:** Systemic Risk, PCA, CoVaR, ILLIQ, Connectedness, Subprime crisis.

**JEL Classification:** G01, G21, G22, G23, G28.
# 1 Introduction

“A financial crisis is a moment when all goes badly”. This sentence illustrates the most common view of a financial crisis. Numerous academics have tried to define and study these abnormal events, but never agreed on a specific definition. Hence, the sentence above is one that comes the closest to the truth. It is possible that a financial crisis is not defined by the amount lost, or the number of firms’ failure but simply by the fact that “all goes badly”. When looking at the history of financial crisis, it provides a relevant, although imprecise, description. During a systemic crisis, which, most would agree, is worse than a financial crisis, it is even more accurate. A systemic crisis combines different risks occurring simultaneously. This article builds up on this insight and poses the question of the existing correlations between risk measures. Although there is an extensive literature on correlation risk<fnref n="1" />, there is very few studies on how different risks are correlated. We postulate that, since systemic risk is a combination of different risks, a good proxy for systemic risk measurement is the correlation between different risk indicators.

The first challenge of the article is to find and show that there exists different² systemic risk measures, and, thus defining how two measures can be different. The following challenge is straightforward: in the case where there exists more than two “different” measures of systemic risk, how can we find a measure of correlation for more than two variables. We based our initial thinking on a previous work of ours Madies et al. 2020, which implies that systemic risk is based on three major dimensions (systemic dimensions): Losses, Connectedness and Liquidity. The measure taken to proxy this systemic dimensions, come from the work of Adrian & Brunnermeier (2016) for CoVaR, Billio et al. (2012) for connectedness measures, and Amihud et al. (2012) for illiquidity measure. This paper shows that the systemic dimensions are orthogonal in stable times and become correlated in crisis periods. The tests are performed on sample spawning from June 2000 to June 2020; and are based on 75 American firms (25 banks, 25 insurers, 25 investment firms). Robustness is assured by an additional analysis on the European market (from 2005 to 2018, for 100 firms). We use Principal Component Analysis (PCA) as a correlation measure for our systemic dimensions. We applied the PCA over a rolling forward window, and find that the first component’s inertia shows clearly the period of systemic crises in our sample. By showing that the correlation of different (systemic) risk measures is enough to find systemic crises in our sample, we argue that the very nature of systemic risk lies in these correlations.

The implications of this research are twofold. First, our study provides an innovative point of view on what systemic risk really is, by taking the correlations of systemic risk measures in order to identify systemic crises periods. Moreover, in our view, systemic risk is a succession of different risks. Its measurement remains a challenge due to the temporal aspect as well as the variety of risks involved. By looking at the appropriately lagged risk measure and using PCA for finding a principal component, we

<footnote n="1">¹That is the fact that financial returns of different firms are highly correlated, hence providing evidence of herding behavior and/or common exposure on a specific risk.

²Different in the sense that they all measure Systemic Risk, but not from the same point of view. This will be explicated afterwards.</footnote>
provide a draft of an answer for this challenge. Second, it advocates for a new prudential regulation, which would be based on categories of Systemically Important Financial Institutions (SIFIs) instead of buckets. As shown in Madies et al. (2020), each type of financial firm is more prone to a specific type of risk, which occurs at a specific time. Prudential regulation should consequently aim for mitigating specific risks of specific firms, instead of compartmentalizing firms in levels of capital buffer.

The structure of this article is as follows. The literature review covers the necessary knowledge required for this article, starting from the initial systemic risk theories, followed by the empirical measures and concluded by the notion of risks dependency. We present the data and methods in the following section. The section results and discussion concludes with our findings and their implications.

# 2 Literature Review

In this section, we intend to provide a sufficient theoretical background for any reader to be able to follow the relevancy of this article in current literature. We first introduce the genesis of systemic risk research. In a second time, we present the most famous measurements proposed by academics over the last two decades. Eventually, we conclude the section by measures of dependency and correlations in finance. The last section is not directly related to the assessment of systemic risk, but will be more than relevant for the sake of this article.

## 2.1 Theory of systemic risk

The study of systemic risk directly stems from banking crises. One can easily argue that the globalization of finance was due to banking activity. Through the years, banking evolved and allowed businesses to invest, grow and spread. This fact is observable by the extent of banking regulation that took place over the years (Banking Act or Glass-Steagall Act, Dodd-Frank Act, Basel agreements, ...). The initial endeavors were to discover the natural properties of such phenomena. Among the most known papers, we note the work of Bryant (1980), Diamond & Dybvig (1986), and Kiyotaki & Moore (1997) on banking crises<fnref n="3" />. Such papers highlighted the actual risk associated with banking activity and what were our solutions to mitigate it. Kiyotaki & Moore (1997) particularly focused on the importance of credit cycles in a shock's transmission mechanism. They make an explicit link between banking activity and financial crises. Also, the risk of bank run was one of the major issues tackled in the early researches, and the creation of deposit insurance directly stemmed from it. Although, the total efficiency of deposit insurance as a solution to bank runs remains arguable (Aglietta, 1993).

It is important to note that research on systemic risk also started with the study of financial crises. The pioneer work of Kindleberger (1978), Minsky (1991) and Mishkin (1992) have laid a solid foundation for our understanding of the underlying process of financial crises. Kindleberger (1978) and Minsky

<footnote n="3">³For a general overview on bank regulation see Dewatripont & Tirole (1994).</footnote>
(1991) share a common analysis on how financial crises build up. In their conception, a financial crisis is inherent to capitalism and develop regularly according to a given pattern: A change, called “displacement”, occurs and provoke a shift in agent’s behavior. Afterwards, firms that were working regularly (hedge units) develop a need to borrow in order to pay interest of their loans (speculative units). And firms that were already risky (speculative units) become forced to borrow continuously in order to survive (Ponzi units). This creation, and increase, in financial fragility leads to a point, where most of the financial system is composed of Ponzi units. A “not-so-unusual”<fnref n="4" /> event happens, pushing the first firms into bankruptcy. In Minsky (1991) and Kindleberger (1978)’s view, the crisis is created from irrational behavior<fnref n="5" />. On another hand, Mishkin believe that financial crises root from information asymmetry. Mishkin (1992) presents the different causes of a financial crisis and a chronology of how they entangle together. Information asymmetry creates adverse selection and moral hazard issues that worsen the crisis.

Due to the crises which spawned at the beginning of our millennium (Dot-Com bubble, Subprime crisis, European debt crisis, ...), we have observed a rise in systemic risk interest. Leading theoretical work such as, and among others, Acharya (2009), Freixas & Rochet (2010), Brunnermeier & Oehmke (2013) and Chen et al. (2013); have demonstrated the general interest in providing a theoretical framework for systemic risk. As said by Brunnermeier & Oehmke (2013), in systemic assessment, theory should lead the operational measurement. The traditional view of treating the banking sector as the only systemic institution slowly became less and less relevant. Recent theories include also insurers and hedge funds into their scope. The logic behind this adaption of scope comes from the fact that the mechanisms linking all financial agents together constitute the roots of systemic risk.

## 2.2 Measures of systemic risk

The subprime crisis created a relentless stream of research on how to measure systemic risk. The impact of the crisis was so unexpectedly large, that it forced both academics and regulators to tackle the issue of measurement. We will present in this section only the most famous measures and recommend the curious reader to consult the comprehensive work of Bisias et al. (2012) and Benoit et al. (2017) on systemic risk measures; and Smaga (2014) on systemic risk's definitions/concept. Existing measures separate in in two main categories. The ones trying to give a global index of systemic risk; and the ones proposing a firm-level contribution to systemic risk. As already presented in Borio (2003) and Bisias et al. (2012), both types of measure are complementary.

From one hand, measures providing a global index, are essential for macroprudential supervision. Among these, the most famous are the following: The Composite Indicator of Systemic Stress (CISS), created by Hollo et al. (2012), focuses on how much of the financial system is at risk at any time point.

<footnote n="4">⁴In Minsky’s view, the first failures of a crisis start because of a given macroeconomic event. It is called “not-so-unusual” because, such event, would not have incur any failure in regular times. It is because of the large degree of financial fragility that the failures occur.</footnote>

<footnote n="5">⁵See Blanchard & Watson (1982) for a study on how bubbles can develop under rational expectations</footnote>
The indicator is created from various financial stress indicators. It implies that systemic risk is higher when many markets are at risk, taking into account various types of risk. In the same vein, Kritzman et al. (2011) propose a systemic indicator called the Absorption Ratio (AR) based on a PCA on financial returns. Hu et al. (2010) propose a measure of noise based on bond price. The rationale of the measure is that, in stable times, the Treasury market should be almost noiseless. Although, in crisis periods, the amount of capital in the bond's market is reduced, resulting in more noise. Their measure gives an indicator of liquidity crises and liquidity risk over time. All these measures allow for an analysis of how general systemic risk evolves over time. They also permit regulators to evaluate the performance of their actions.

On another hand, we also need individual risk measures in order to identify which agent is causing the most risk in the financial market. This type of studies is, from far, the largest part of the existing literature on systemic risk. Their objectives is to display the risks, which can cause an institution to fail. We find first the measures linked to financial losses: the ΔCoVaR of Adrian & Brunnermeier (2016), Systemic Expected Shortfall (SES) of Acharya et al. (2017) and Co-risk created by Chan Lau et al. (2009). Such articles try to display the size of losses associated with an unusually risky period. Another stream of systemic risk measures focuses on the connectedness of financial institutions. Freixas et al. (2000) represent the interbank system as a network and show the effect of one (or more) bank insolvency over the whole network<fnref n="6" />. Giesecke & Kim (2011) propose a model timing banks' default. The model is made to display systemic linkages in an economy. Billio et al. (2012) show, through Granger causality regressions and PCA, the systemic importance of connectedness. The PCA allows showing the general level of correlation among financial returns, while Granger causalities identifies important linkages between financial firms. The importance of such studies is as vital as the previous ones [to clarify]. We need to know how institutions can fail, individually or in clusters, as much as keeping an idea of where the general level of systemic risk rests.

## 2.3 Dependency measures

Dependency is a trickier concept that one might initially think. The first step of researching dependences lies in how we define it. A fair introduction to the concept of dependency is provided by Lehmann (1966). He describes three distinct types of dependency:

1. *Quadrant Dependency*: $P(X \le x, Y \le y) \ge P(X \le x)P(Y \le y)$. In such case, there is dependency if the joint probability of an event is greater the product of the marginal probabilities.

2. *Regression Dependency*: $P(Y \le y|X \le x) \ge P(Y \le y)$. Such dependency exists when, the probability of an event is greater when taking into account that the other event already occurred.

<footnote n="6">⁶For an update on the network models of systemic risk, see Caccioli et al. (2018)</footnote>
3. *Likelihood Ratio Dependence:* $f(x, y')f(x', y) \le f(x, y)f(x', y')$, for $x' < x$ and $y' < y$, with $f(x, y)$ being the joint density of X and Y. The dependence here lies in the fact that probability of observing large (small) values of the couple $(x, y)$ is greater than the one of observing disparate (one large associated with one small) values in the couple $(x, y)$.

We see that the appropriate approach will differ greatly depending on the chosen case. In this study, we present the correlations between our systemic dimensions. Hence, our approach would tend to fit the first definition of dependence above. Although, results in the chronological order of occurrence of systemic dimensions would advocate for using dependency measures associated with the second definition. We leave this gap for future research. Unfortunately, the current literature regarding correlation in financial risks is almost inexistent. However, there exists a burgeoning one on correlation risk, that is to say the risk associated with the correlation between financial returns. Among most famous works, we find:

* Copulas: A copula is a concept coming from probability theory and, whose development is based on Sklar's theorem. Simply put, it corresponds to a joint cumulative distribution function of a random vector with uniform marginals. It is used to study the dependence between random variables. As an introduction, Embrechts (2009) provides solid and thorough explanations, as a must-read section that allows one to easily get a comprehensive view on copulas<fnref n="7" />. In finance, Malevergne & Sornette (2003) tests whether the existing dependences between pairs of financial assets can be modeled convincingly by the Gaussian (and Student) Copula. Finally, Kole et al. (2007) show the importance of a copula's choice for modeling dependences.

* GARCH models: Generalized Auto-Regressive Conditional Heroskedastic (GARCH) models are one of the most used tools to model dynamic variance and correlations. An initial method was the BEKK model<fnref n="8" /> based on the work of Engle & Kroner (1995). Bollerslev (1990) constructed a model for modeling conditional correlation, called Constant Conditional Correlation (CCC). Engle (2002) built up on the work of Bollerslev (1990) in order to make a model which can model a Dynamic Conditional Correlation (DCC), that is to say, model time-varying conditional correlation.

* PCA: Additionally, the papers cited above that applied PCA directly to systemic risk (Billio et al. 2012; Kritzman et al. 2011) have been using it for measuring commonality among financial assets. Avellaneda & Serur (2020) used a Hierarchical PCA to expose clusters of assets based on their principal components. Fenn et al. (2011) show the time varying behavior of correlations and principal components for financial returns. Shapira et al. (2009) study the correlations among stocks taking into account the market's index.

Over the techniques mentioned above, we have decided to apply PCA onto our systemic dimensions.

<footnote n="7">⁷For a detailed description of the most important area of application, see the work of Genest et al. (2009)</footnote>

<footnote n="8">⁸based on the names of the authors: Baba, Engle, Kraft and Kroner</footnote>
This choice stems from the fact that PCA allows for a clear and simple graphical representation of correlation, as well as for its cheap computer power needs.

# 3 Data & Methodology

In this section, we present the data used for constructing the systemic risk measures. In a second time, we detail the mathematical formulation of each measures. Finally, we present the methodology of the PCA used as a correlation indicator among our systemic dimensions.

## 3.1 Summary Statistics

We computed the systemic dimensions over 75 U.S. firms. The sample goes from June 2000 to June 2020, and is separated in 25 banks, 25 insurers/reinsurers and 25 investment firms. The firms were first selected on Osiris database, by looking at their market capitalization. We have extracted daily prices, daily volumes and market capitalization from Eikon database. In addition, we extracted daily prices, daily volumes and market capitalization for the underlying stock of the S&P100, which will be used for constructing a proxy for market liquidity. The summary statistics are provided in the Appendix C Table 1. Additionally, the financial system, as used in the ΔCoVaR of Adrian & Brunnermeier (2016), is simply an index constructed from the 75 financial firms in our sample, which are market capitalization weighted. Formally:

$$R_{system}(t) = \frac{1}{MV_{tot}} \sum_{i=1}^{N} R_{i}(t) MV_{i}$$

## 3.2 Systemic Risk measures

### 3.2.1 CoVaR

The Conditional Value-at-Risk (CoVaR) results from the work of Adrian & Brunnermeier (2016). Simply put, it represents the q% worst loss possible of a firm i, given that a firm j is already at its p% worst loss. Originally, the authors use quantile regressions, but we decided on using the bivariate Gaussian model in order to have daily observations for our δCoVaRs. By definition, CoVaR is a VaR of an institution j conditional to some event C(X_i) on firm i's returns:

$$CoVaR^{j|i}(q) \equiv \Pr(X_j \le CoVaR^{j|i}(q) | C(X_i)) = q \quad (1)$$

The formulation of the CoVaR is useful when it comes to changing the condition terms, which generates multiple values and multiple interpretations of the measurement. In this paper, we propose that the event C(X_i) consists in the fact that the i-th firm's returns are at their VaR, then:

$$CoVaR^{j|i}(q, p) \equiv \Pr(X_j \le CoVaR^{j|i}(q, p) | X_i = VaR_i(p)) = q \quad (2)$$

In order to construct the ΔCoVaR, we need to subtract the 50% level CoVaR to the q% level
CoVaR. This specification describes differences in CoVaRs between when the conditioning event is that firm i is either distressed and when firm i is “stable” or at its usual level. Formally, it gives:

$$ \Delta \text{CoVaR}^{j|i}(q, p) = \text{CoVaR}^{j|i}(q, p) - \text{CoVaR}^{j|i}(q, 50\%) \quad (3) $$

As presented in the appendices of Adrian & Brunnermeier (2016), we use the Gaussian model by assuming that the returns of firms i and j follow normal bivariate distribution. The mathematical development is made available in the appendix A. We then obtain the following expression for CoVaR:

$$ \text{CoVaR}^{j|i}(q, p, t) = N^{-1}(q) \sqrt{1 - \rho_t^2} \sigma_{j,t} + N^{-1}(p) \sigma_{j,t} \rho_t \quad (4) $$

With $\rho_t$ denoting the correlation between firms i and j. Like Adrian & Brunnermeier (2016), we focus our study of the CoVaR not on the relationships between two firms but rather on the analysis of one firm and the system. As for the VaR computation, we decided to implement time-varying correlations in our CoVaR. One obstacle, when trying to compute the CoVaR, as described above, is estimating the dynamic conditional correlation (DCC). In our computation, we follow the work of Engle (2002). We propose $R(t)$ as:

$$ R(t) = \begin{pmatrix} R_i(t) \\ R_j(t) \end{pmatrix} \quad (5) $$

With $R_i(t)$ and $R_j(t)$ being the returns of firms i and j. We assume that $R(t)$ follows a bivariate normal distribution with mean equal to zero and with the conditional covariance matrix $H(t)$.

$$ R(t) | \mathbb{F}(t) \sim N_{\mathbb{R}^2}(0, H(t)) \quad (6) $$

Where:

$$ H(t) = D(t) \Lambda(t) D(t) \quad (7) $$

With $D(t)$ being a diagonal vector of conditional standard deviations $\sqrt{h_i(t)}$ and $\Lambda(t)$ being the conditional correlation matrix. Generally, the $h_i(t)$ term is the volatility extracted from a univariate GARCH model; in our case it comes from the EGARCH performed for the computation of VaRs. We specify that:

$$ \varepsilon(t) = D(t)^{-1} R(t) \quad (8) $$

As all variances need to have positive values, $\Lambda(t)$ must be positive definite. A matrix is positive definite if it has nonnegative eigenvalues. In order to respect that constraint, Engle gives the following proxy,<fnref n="9" /> which is developed to be a matrix version of a GARCH model:

$$ Q(t) = S(1 - \alpha - \beta) + \alpha(\varepsilon(t - 1)\varepsilon(t - 1)') + \beta Q(t - 1) \quad (9) $$

Where $\alpha$ and $\beta$ are nonnegative parameters and, so the process is mean-reverting, $\alpha + \beta < 1$. S is

<footnote n="9">⁹The development is available either in the original article of Engle (2002) or in the documentation for rmgarch models Ghalanos (2019)</footnote>
the unconditional correlation matrix of $R(t)$, obtained simply by computing:

$$S = \frac{\sigma_{R_i, R_j}}{\sigma_{R_i} \sigma_{R_j}} \qquad (10)$$

Where $\sigma_{R_i, R_j}$ is the covariance of $R_i$ and $R_j$. We find our final DCC by computing:

$$\Lambda(t) = \mathrm{diag}\{Q(t)\}^{-1} Q(t) \mathrm{diag}\{Q(t)\}^{-1} \qquad (11)$$

As all the CoVaRs that we compute are either on the $i^{th}$ firm conditionally to the system; or the opposite, we only need to compute the time-varying correlations of each firm against the system. By allowing our measure to be time-varying, we can refine our analysis and study the behavior of the $\Delta$CoVaR values through our whole sample.

### 3.2.2 Interconnectedness

We base our methodology on that of Billio et al. (2012) and develop Granger causality tests (Granger, 1969) to study the interconnectedness of firms in our sample. We perform linear Granger regressions on a rolling window of 200 days in order to get the evolution of the Granger adjacency matrix over the sample. Granger causality regressions are presented as in the original methodology of Granger (1969). The measure consists in performing two regressions, on two random variables and their lagged values. In other words, for the first regression, we regress the returns of the $i^{th}$ firm against the lagged value of the $j^{th}$ firm's returns and its own lagged value. The second regression is identical but inverts $i$ and $j$ in the regression. Formally it gives:

$$R_i(t+1) = \alpha_i R_i(t) + \beta_{ij} R_j(t) + e_i(t+1) \qquad (12)$$

$$R_j(t+1) = \alpha_j R_j(t) + \beta_{ji} R_i(t) + e_j(t+1) \qquad (12)$$

Where $R_i(t)$ is the return for the $i^{th}$ firm and $R_j(t)$ the return for the $j^{th}$ firm, $e_i(t+1)$ and $e_j(t+1)$ are two uncorrelated white noises. The lagged value prevents the $\beta$ to be biased by autocorrelation effects in the timeseries. We accept that the returns of firm $i$ have caused firm $j$'s returns if $\beta_{ji} \neq 0$; conversely, if $\beta_{ij} \neq 0$ then it is firm $j$ that causes the returns of firm $i$. We denote the fact that firm $i$'s returns cause firm $j$'s returns by the following:

$$(i \to j) \quad \text{if} \quad b_{ji} \neq 0 \qquad (13)$$

$$(j \to i) \quad \text{if} \quad b_{ij} \neq 0 \qquad (13)$$

All causality, as defined above, is computed to be significant at the 95% level of confidence. As stated by Billio et al. (2012), if both coefficients are significantly different from 0, there is a feedback relationship between the two asset returns. In a similar fashion of Billio et al. (2012), we develop connectedness indicators from causality regressions: IN, OUT and DGC.

* The 'IN' measure counts the number of firms whose returns caused the $j^{th}$ firm's returns, in the
Granger sense.

$$ \text{IN} : (S \to j) = \frac{1}{N-1} \sum_{i \neq j} (i \to j) $$

* In contrast to the 'IN' measure, 'OUT' gives the number of Granger connections the $j^{th}$ firm caused.

$$ \text{OUT} : (j \to S) = \frac{1}{N-1} \sum_{i \neq j} (j \to i) $$

* The DGC or Degree of Granger Causality, is an index on how much causality relationships exist in a given system. Operationally, we sum up the IN and OUT measure and divide by the maximal possible amount of connections (74 given our sample). The resulting index gives an idea of how much the system is connected.

### 3.2.3 Market illiquidity

To gain an idea of how sectors' returns interact with market illiquidity, we follow the methodology of Amihud et al. (2012) in developing AILLIQ. We use AILLIQ as an indicator of market illiquidity. By regressing this proxy against financial returns of the firms in our sample, we assess their exposure to market illiquidity. To avoid endogeneity, we use the S&P 100, which contains only a few financial firms. The measure can be interpreted as the daily volume adjusted price variation or, alternatively, the response of the price against one dollar of trading volume. Although ILLIQ can also be considered a price impact measure, it remains an attractive illiquidity proxy due to its simplicity and the availability of its components. We begin by computing the ratio between the absolute value of daily returns on one firm's returns and the daily euro trading volume for the same firm. When computing the mean of this ratio (i.e., by dividing by the number of trading days), we get the ILLIQ measure for the firm:

$$ \text{ILLIQ}_i(y) = 1/D_i(y) \sum_{i=1}^{D_i(y)} |R_i(y, d)|/\text{VOLE}_i(y, d) \quad (14) $$

With $R_i(y, d)$ being the return for firm $i$ on day $d$ and in year $y$. Similarly, $\text{VOLE}_i(y, d)$ represents the euro volume for firm $i$ on day $d$ and in year $y$. Also, $D_i(y)$ is the number of trading days for year $y$. Evidently, the ratio follows the definition of illiquidity as defined by Kyle (1985), which is to say the percentage of price change per unit's daily trading volume, here, in euro. Given our methodology, we need an indicator of market illiquidity rather than one of the illiquidity of each stock of the index. We therefore use the AILLIQ measure, as it is essentially the mean of all the $\text{ILLIQ}_i$. Formally:

$$ \text{AILLIQ}(y) = 1/N_i(y) \sum_{i=1}^{N_i(y)} \text{ILLIQ}_i(y) \quad (15) $$

We construct a rolling-forward linear regression between the firm's returns and the AILLIQ indicator, taking a 200-day window. By doing so, we obtain a value of $\beta_i(t)$ for each firm $i$ and each day $t$.
Formally:

$$ AILLIQ(t) = \alpha_i(t) + \beta_i(t)R_i(t) + e_i(t) \quad (16) $$

By posing $\alpha = 1, 2, 3$ and $M_\alpha$ as the length of the sample in sector $\alpha$, we construct the sector's average beta:

$$ \beta_{\alpha}(t) = 1/M_{\alpha} \sum_{i=1}^{M_{\alpha}} \beta_{i}(t) \quad (17) $$

In the work of Amihud et al. (2012), such regressions were made with inverse dependent and independent variables. Nevertheless, the rationale remains identical: the measure assesses the relationship between market illiquidity and each sector's returns. Amihud et al. (2012) did observe that the regressions give a positive $\beta$ if illiquidity is expected. The measure is therefore able to show us when the illiquidity is not expected, hence, theoretically, when the risk associated with illiquidity is high. We consider the $\beta_\alpha(t)$ as a measure of systemic risk for the following reasons. In a general sense, a market with high illiquidity is prone to greater variations in stock prices. Since excess asset returns will increase the variance, using ILLIQ's structure, it will increase the illiquidity measurement. It is then straightforward to link high volatility of an asset's returns with its high illiquidity, *ceteris paribus*. In the context of systemic risk management, we have to recognize that high market illiquidity has a substantial potential in amplifying a fire sale type of event. The measure allows for an additional specification that only a negative $\beta$ will indicate significant systemic risk, mostly because unpredicted illiquidity is more dangerous than expected illiquidity.

## 3.3 Principal Component Analysis

Principal Component Analysis (PCA) is a statistical technique allowing to decompose the covariance matrix ($\Sigma$) of the m x n matrix ($X$), into a diagonal matrix of eigenvalues ($\Lambda$). Such as:

$$ H^T \Sigma H = \Lambda \equiv \text{diag}(\lambda_1, ..., \lambda_m) \quad (18) $$

With $H = (h_1, ..., h_m)$ being an orthogonal matrix of size m x m, and $h_i$ is an eigenvector corresponding to $\lambda_i$. Principal components are then computed as:

$$ U = H^T X = (U_1, ..., U_m) \quad (19) $$

In more details, the first component consists in $U_1 = h_1X$, with variance $\lambda_1$. In a similar fashion, the second $U_2 = h_2X$, with variance $\lambda_2$. Necessarily, all components must be orthogonal, hence $Cov(U_i, U_j) = 0$ for all $i \neq j$. In our case, the PCA serves as an alternative measure for correlation. This article aims to study the correlation structure of systemic risk measures through time. By applying PCA of our systemic risk measures on a rolling forward window, we expose the changes in their common uncorrelated factors. Additionally, we use the quartimax rotation on our PCA. Generally, the unrotated PCA tries to maximize the variances of each factors and forces orthogonality on the principal compo-
nents. Such variances of principal components should be strictly decreasing. A rotation can simplify the output in making it more understandable. Specifically, the quartimax method is an orthogonal rotation, which will result in a little amount of explaining factors. It does so by making large loading on a specific factor larger and conversely on small loadings, resulting in less large principal components.

We will look at the evolution of the proportion of explained variance of the three first principal components. The rationale behind this methodology is to capture a common increase in the correlation of the three systemic dimensions during a crisis. Such fact should be described by the increase of the first principal component, which will be describing systemic risk and/or the correlation behind systemic risk measures. An intuitive thought would be: “it would appear normal that all risk measures increase in time of crisis, since a crisis represents times when all goes badly”. Following this intuition, systemic risk measures should be more correlated in time of crisis.

# 4 Results & Discussion

First, we introduce results for the whole U.S. financial system. That is to say, we study the existing correlations between the three systemic dimensions (Losses, Connectedness and Liquidity). We add a brief analysis of an identical analysis, that is sector-specific, in order to observe potential differences in systemic risk inside the financial sector. Eventually, we propose an improvement of systemic crises’ period identification, based on lagging appropriately systemic dimensions before performing PCA. We see that the lagging procedure allows for an improvement of the identification, and provide a discussion on the reason why, below.

## 4.1 Othorgonality of systemic dimensions

We postulate the following. Measures of systemic risk typically represent a noisy signal for the actual risk. Since they focus on different types of risk, the noise among each of them is uncorrelated. However, as they are systemic risk measure, the signal is correlated. Logically, the stable periods, where systemic risk is at the lowest, are caracterised by a predominance of noise. Since noise is uncorrelated, the measure are uncorrelated accordingly. However, during crisis period, the proportion of noise to risk decreases. The signal for actual risk increases and, hence, so are the measures. The general rationale starts with systemic dimensions: Losses, Connectedness, Liquidity. Systemic dimensions constitute the main risks involved in a systemic crisis. The assumption made in this study is that the orthogonality of the systemic dimensions is assured in stable period, and breaks in crisis times. Thus, this research aims to validate this assumption by, first, proving that systemic dimensions are orthogonal in stable periods, and that is possible to identify crisis periods by looking at the time-varying correlation of systemic dimensions. Intuitively, the PCA is used as a proxy for the correlation amongst the three dimensions. We expect the first component to increase in crisis periods and it would become the driving factor of all risks.
<figure><img src="image_2.png" /><figcaption>Figure 1: Systemic Dimensions</figcaption></figure>

As Fig. 1 tells, all three systemic dimensions show significant spikes during crisis periods. We can identify residual stress from the dot-com bubble (2000-2002), the subprime crisis (2007-2009), the European debt crisis (2011-2012) and, more recently, the coronavirus situation (2020). A valid remark would be about the reason to use different measures if they all show clearly systemic risk. The answer is twofold. It is first a question of meaning. We probably could find various indicators that show spikes during crisis period. Does it mean that they represent systemic risk? The relationships we tend to positulate too quickly can seldom be spurious. A second reason is about noise. Even though each of the measures above serve as a decent indicator of systemic risk, one could argue that they show noise; an increase in these indicators could mean something else than an increase in systemic risk. An underlying reason of why we chose to determine three systemic dimensions comes from that last point. We argue that an increase in systemic risk could only be “significant” if there is a common increase in all three dimensions. Thus, by studying the time-varying correlation between the dimensions, we can get an accurate view of systemic risk.

<figure><img src="image_3.png" /><figcaption>Figure 2: Orthogonality of Systemic Dimensions</figcaption></figure>

The orthogonality of the systemic dimensions shown above constitutes a simple heuristic to un-
derstand the nature of systemic risk. In times of financial stability<fnref n="10" /> all systemic dimensions are relatively close to being perfectly orthogonal, that is to say, uncorrelated. While, in crisis periods, they appear to be less orthogonal. The result is clearer for the connectedness and losses measures. The conclusion is straight forward, all dimensions are measuring different risks in stable times, which can increase/decrease ephemerally (i.e. noise). However, during crises, all risks occur simultaneously, which is directly translated statistically with the inertia of the first component. As a robustness test, we performed an identical procedure on the European market (25 banks, 25 brokers, 25 insurers, 25 investment firms) and obtained similar results (see Appendix). A weak point of our argumentation comes from the arbitrary choice for the stable and crises periods. In order to prove the robustness of the method, as well as its usefulness, we computed the PCA on a rolling forward window. The time-varying inertia of the first component is shown below.

<figure><img src="image_4.png" /><figcaption>Figure 3: Inertia of the three first components

*Explained variance of the three first components of the PCA. The PCA is realized on a 300 days rolling-forward window and with a quartimax rotation. 1st PC in black, 2nd PC in red and 3rd PC in green.*</figcaption></figure>

The variations of inertia of the first component allows to identify clearly the crises periods, with, as presented before, the residual stress for the dot-com bubble, the subprime crisis, the European debt crisis and the current coronavirus crisis (spike at the end of the graph). We can see, that the realization of a systemic crisis actually happens when the increase in the explained variance of the first component occurs simultaneously with a decrease of the inertia of the other two components. In other words, a systemic crisis happens when all of the systemic dimensions stop to be orthogonal.

## 4.2 Lagged systemic dimensions

The measure is still far from optimal though. We argued that noise was one of the reasons why we needed a new measure. Yet, the inertia clearly shows transitory spike between 2012 and 2020. There was not any

<footnote n="10">¹⁰Here, financial stability is intended as outside of financial instability.</footnote>
notable financial crises in the American market over these period, and one could argue that the spike are
actually noise, or, equivalently, a concomitant increase in systemic dimensions that was not evidence of
a crisis. This particular issue comes from the temporality of systemic risk. We assumed above that all
systemic dimensions were highly correlated *simultaneously* during crisis times. It turns out not to be
precisely true. From a historical point of view, we know that systemic crises follow steps. For instance,
contagion over financial institutions is seldom the initial shock. There is an order in which things get
progressively worst. From this assumption, it would seem reasonable to further assume that systemic
dimensions evolve at different time, in a given order. Theoretically, the measurement would get better if
we could find the appropriate lag. Practically, we lag our variables according to the supposed order, and
the following results.

<figure><img src="image_5.png" /><figcaption>Figure 4: Comparaison of inertia of first components</figcaption></figure>

Based on preliminary research of ours (Mourey, 2020), we propose the following process of a sys-
temic event: first, the build of common exposure should happen early in the timeline, followed by losses
and, finally, by the amplification via market illiquidity<fnref n="11" />. Since Granger causalities are performed on
a rolling window, the measure is naturally delayed forward. We have tried iteratively different lags in
order to find an appropriate fit for the illiquidity measure. As Fig. 4 displays, we can get rid of the
transitory spikes, by lagging our systemic dimensions.

Such results put forward evidence of an order of occurrence in systemic dimensions. As the measure
conforms to both theoretical idea of a systemic event and empirical periods of systemic crisis, we argue
that it constitutes an initial proof that a systemic event is composed of various risks (major ones being
denoted as systemic dimensions) which occur in a specific order. The results are also visible on the
European market between 2005 and 2018. The orthogonality of systemic dimensions is also showed on

<footnote n="11">¹¹The amplification is usually accompanied by contagion. One could pose that Granger causalities constitute a proxy for contagion. We argue that Granger causalities display a potential for contagion due to a common exposure. Hence, it is a better proxy of buildup of financial fragility.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 5: Orthogonality of Lagged Systemic Dimensions</figcaption></figure>

The loss of orthogonality during the crisis is clearer for lagged risk measures. In our view, the implications of this result are substantial. Up to now, most of the existing literature on systemic risk measurement, was implicitly assuming that only one dimension mattered<fnref n="12" />. Our result shows first, that different measures of systemic risk are orthogonal in stable times. This means that they measure drastically different concepts, and only measure (together) systemic risk in crisis periods. Hence, taking into account only one, blocks the analysis from capturing the full picture of systemic risk. Secondly, there exists a chronological order in systemic crises. This second fact confirms the importance of the first. Indeed, each dimension should be considered as they all participate in a systemic event. Our results show that, for both the U.S. and the European financial system, an increase in interconnectedness arises first. The rise in connectedness can be interpreted in two ways: First, it represents the potential for a wide spread shock, by saying that temporal cross-correlations proxy common exposures. Second, it represents the potential for contagion. In a second time occurs the spike in financial losses, captured by $\Delta CoVaR$. And finally, occurs the increase of exposure to market illiquidity. This sequence of events allows for an initial idea of how a systemic event develops. We let future research study more precisely whether this sequence is empirically robust and whether it can be applied to forecasting. Consequently, our results advocate for taking into account the multiplicity of systemic dimensions and their specific entanglement.

The regulatory implications are quite significant as well. Even though the SIFI assessment (Bcbs, 2013) uses various types of indicators, the constraints posed on SIFIs are standardized. The results of this study combined by those of a previous study of ours Madies et al. (2020) preach that: Firstly,

<footnote n="12">¹²Each paper basically chose a given dimension among the following: Losses, Liquidity, Connectedness, Contagion. And, assumes that this measurement proxies systemic risk realistically.</footnote>
systemic risk depends on various risks. Secondly, each type of financial firms has a specific profile of systemic risk. Hence, prudential regulation should identify a specific systemic risk profile<fnref n="13" /> per SIFI, which details where the SIFI locates in the systemic event unfolding. The changes are substantial in a way that the regulation should have to become more flexible in order to adapt to the profile of the SIFI. The main questions are the following: What firms participates in the increase of common exposure and how restrict it? How to prevent financial losses to occur and how to prevent a contagion/amplification of these losses? Such questions have already been asked separately. Our results advise such questions to be considered conjointly.

## Sectorial systemic dimensions correlation.

When applying the PCA on the systemic dimensions only computed for a specific type of financial institution, we can see how much this particular sector was at risk. In a general sense, the method applied in this article to identify systemic crisis, is actually one for assessing the risk of a system considered. When taking into account of financial firms, the results will regard the risk of the financial system. Accordingly, we can study the risk of a given type of institution.

<figure><img src="image_7.png" /><figcaption>Figure 6: Sector-specific inertia</figcaption></figure>

Figure 6 emphasizes the evolution of inertia for the first components of the PCA for types of financial firms. The measure clearly shows spikes during the crises period in our sample, although remains more noisy than the global one (Figure 3). This fact appears as logical. When considering a larger system, its inherent risk is determined by when most of its components are at risk<fnref n="14" />. For instance, times where only banks are at risk but insurers are not, will not be a high systemic risk period globally. The noisy behavior of the measure then logically grows stronger as the system considered goes smaller. An important insight from this approach lies in identifying the period where a specific type of institution is actually risky. We can see for instance that all types of institutions were at risk during the subprime crisis, hinting the systemic aspect of the crisis. Let us then confirm that there exists a chronology in systemic dimensions. We apply the (identical) lags on the systemic dimensions and compare the accuracy of the method below (see Figure 7).

<footnote n="13">¹³We intend by systemic risk profile, a detailed rapport of what risks are more relevant to this firm (or type of firm) in particular.</footnote>

<footnote n="14">¹⁴This is the underlying assumption for our approach.</footnote>
<figure><img src="image_8.png" /><figcaption>Figure 7: Comparaison sector-specific inertia</figcaption></figure>

Without doubts, the lagging procedure allows for a reduction of the noise for all types of firms (even though slightly less for banks). Moreover, it shows more clearly the systemically risky periods in our samples. We argue that the results provide a solid initial evidence of a peculiar order of occurrence in systemic dimensions.

# 5 Conclusion

Systemic risk measurement has been one of the most challenging field of research since the subprime crisis. The difficulties are numerous. As crises come in all sorts and flavors, it is complex to choose the variables that matter the most. Even though, a model fits perfectly existing crises, it is not obvious that such model will perform well on the next. As finance is mostly driven by human's behavior and investor's expectations, few, if none, of our available variables have stable distributions over time. Our study decided not to choose what indicator matters the most, by choosing to look at the correlation between the indicators.

We provide a correlation analysis on different systemic risk measures computed on the U.S. financial system over the 2000-2020 period. Our sample for the U.S. financial system consists of the largest 25 banks, 25 insurers and 25 investment firms. Because each measure corresponds to a specific type of risk and, as our results show, are usually orthogonal, we denote them as systemic dimensions. “A moment when all goes bad” is a common thought on the nature of financial crises. Building up on this insight, we study whether it can be applied to systemic risk. In such case, a systemic crisis would be when all its subsequent risks occur, hence when systemic dimensions lose orthogonality. We propose a rolling-forward PCA analysis on the systemic dimensions over the 2000-2020 period. We find that systemic dimensions are orthogonal outside of crises period and lose their orthogonality to become correlated during crisis periods. Moreover, we see that realized systemic crises are identified as moments when the explained variance of first principal component increases due to the fall of the two others. This constitutes a solid evidence that systemic risk occurs when multiple risks happen concomitantly,
i.e. when all systemic dimensions become correlated. The results are similar for the European market<fnref n="15" />. Additionally, we show that, by lagging some systemic dimensions in time, we allow for a clearer identification of systemic crisis periods. The result constitutes an evidence of a peculiar order in a systemic event. From our results, an increase in connectedness happens first. It is followed by a spike in financial losses. Finally, the exposure to market illiquidity rockets. We are the first paper, in our knowledge, to provide empirical evidence of a peculiar order of risk occurrence in a systemic event. This results opens research opportunities in systemic risk measurement and predictions; as well as in prudential regulation.

As time passes by, the financial world keeps getting more complex than ever. There is no certitude as if a proper systemic risk model will ever exists. Mostly, due to the Lucas' critique (Lucas, 1976). Indeed, if a perfect model to predict systemic risk existed, it is probable that financial agents will act in such a way that the model would become obsolete. However, models that are flexible, in the sense that they do not rely on precise indicators, have a chance to guide prudential regulation to financial resilience.

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# A Mathematical Developments

## A.1 GAUSSIAN MODEL FOR THE CoVaR

We assume that two random variables $X_i$ and $X_j$ follow a bivariate Gaussian distribution. We note:

$$ (X_i(t), X_j(t)) \sim \Phi \left( 0, \begin{pmatrix} \sigma_i^2(t) & \rho(t)\sigma_i(t)\sigma_j(t) \\ \rho(t)\sigma_j(t)\sigma_i(t) & \sigma_j^2(t) \end{pmatrix} \right) \quad (20) $$

Since both firms i and j follow a bivariate normal distribution, we can write the conditional distribution for firm i as:

$$ X_j(t) | X_i(t) \sim \Phi \left( \frac{X_i(t)\sigma_j(t)\rho(t)}{\sigma_i(t)}, (1 - \rho^2(t))\sigma_j^2(t) \right) \quad (21) $$

We define the standardized value of our random variable $X_j$ as $Z_j$:

$$ Z_j(t) = \frac{X_j(t) - X_i(t)\sigma_j(t)\rho(t)/\sigma_i(t)}{\sqrt{1 - \rho^2(t)\sigma_j^2(t)}} \quad (22) $$

And, as such:

$$ Z_j(t) \sim \Phi(0, 1) \quad (23) $$

We then come back to the definition of the Conditional Value-at-Risk:

$$ \Pr(X_j \le \text{CoVaR}^j | i(q, p) | X_i = \text{VaR}_i(p)) = q \quad (24) $$

Which we can rearrange by standardizing $X_j$ into $Z_j$:

$$ \Pr \left( Z_j(t) < \frac{\text{CoVaR}^j |i - X_i(t)\sigma_j(t)\rho(t)/\sigma_i(t)}{\sqrt{1 - \rho^2(t)\sigma_j^2(t)}} | X_i(t) = \text{VaR}_i(p, t) \right) = q \quad (25) $$

And, finally, we can develop this into its final form:

$$ \text{CoVaR}^j |i(q, p, t) = \Phi^{-1}(q)\sigma_j(t)\sqrt{1 - \rho^2(t)} + \Phi^{-1}(p)\rho(t)\sigma_j(t) \quad (26) $$
B Figures

<figure><img src="image_9.png" /><figcaption>Figure 8: Inertia of the three first components for the European sample</figcaption></figure>

*Explained variance of the three first components of the PCA. The PCA is realized on a 300 days rolling-forward window and with a quartimax rotation. 1st PC in black, 2nd PC in red and 3rd PC in green.*

<figure><img src="image_10.png" /><figcaption>Figure 9: Comparison of inertia of first components for the European sample</figcaption></figure>

*Explained variance of the first component of the PCAs. The black dashed line corresponds to the inertia of the first component for non-lagged systemic dimensions. The red line corresponds to the inertia of the first components for lagged systemic dimensions. The PCA is realized on a 300 days rolling-forward window and with a quartimax rotation.*
C Tables

<table><caption>Table 1: Summary Statistics</caption><thead><tr><th></th><th>Firms</th><th>n</th><th>Mean</th><th>SD</th><th>Min</th><th>Max</th><th>Skew</th><th>Kurtosis</th></tr></thead><tbody><tr><td rowspan="3">June 2000 - June 2002</td><td>Banks</td><td>522.00</td><td>-0.04</td><td>0.22</td><td>-0.05</td><td>0.05</td><td>0.02</td><td>1.51</td></tr><tr><td>Insurers</td><td>522.00</td><td>0.06</td><td>0.15</td><td>-0.06</td><td>0.04</td><td>-0.45</td><td>4.55</td></tr><tr><td>Invests.F.</td><td>522.00</td><td>0.11</td><td>0.18</td><td>-0.05</td><td>0.05</td><td>0.09</td><td>1.72</td></tr><tr><td rowspan="3">June 2002 - June 2004</td><td>Banks</td><td>523.00</td><td>0.04</td><td>0.21</td><td>-0.05</td><td>0.05</td><td>0.19</td><td>2.18</td></tr><tr><td>Insurers</td><td>523.00</td><td>0.06</td><td>0.20</td><td>-0.04</td><td>0.06</td><td>0.32</td><td>2.87</td></tr><tr><td>Invests.F.</td><td>523.00</td><td>0.08</td><td>0.17</td><td>-0.03</td><td>0.04</td><td>0.22</td><td>1.32</td></tr><tr><td rowspan="3">June 2004 - June 2006</td><td>Banks</td><td>522.00</td><td>0.11</td><td>0.10</td><td>-0.02</td><td>0.02</td><td>0.22</td><td>0.24</td></tr><tr><td>Insurers</td><td>522.00</td><td>0.11</td><td>0.10</td><td>-0.03</td><td>0.03</td><td>0.16</td><td>1.71</td></tr><tr><td>Invests.F.</td><td>522.00</td><td>0.14</td><td>0.11</td><td>-0.02</td><td>0.03</td><td>0.03</td><td>0.31</td></tr><tr><td rowspan="3">June 2006 - June 2008</td><td>Banks</td><td>523.00</td><td>-0.10</td><td>0.23</td><td>-0.05</td><td>0.07</td><td>0.61</td><td>4.65</td></tr><tr><td>Insurers</td><td>523.00</td><td>0.01</td><td>0.15</td><td>-0.04</td><td>0.04</td><td>-0.10</td><td>2.18</td></tr><tr><td>Invests.F.</td><td>523.00</td><td>-0.07</td><td>0.25</td><td>-0.05</td><td>0.07</td><td>0.06</td><td>2.45</td></tr><tr><td rowspan="3">June 2008 - June 2010</td><td>Banks</td><td>523.00</td><td>-0.15</td><td>0.63</td><td>-0.20</td><td>0.16</td><td>-0.09</td><td>3.92</td></tr><tr><td>Insurers</td><td>523.00</td><td>-0.16</td><td>0.50</td><td>-0.13</td><td>0.16</td><td>-0.09</td><td>3.40</td></tr><tr><td>Invests.F.</td><td>523.00</td><td>-0.11</td><td>0.56</td><td>-0.15</td><td>0.13</td><td>-0.38</td><td>2.52</td></tr><tr><td rowspan="3">June 2010 - June 2012</td><td>Banks</td><td>522.00</td><td>-0.06</td><td>0.29</td><td>-0.10</td><td>0.07</td><td>-0.31</td><td>3.30</td></tr><tr><td>Insurers</td><td>522.00</td><td>0.24</td><td>0.00</td><td>-0.08</td><td>0.07</td><td>-0.20</td><td>4.24</td></tr><tr><td>Invests.F.</td><td>522.00</td><td>-0.07</td><td>0.31</td><td>-0.13</td><td>0.07</td><td>-0.55</td><td>5.21</td></tr><tr><td rowspan="3">June 2012 - June 2014</td><td>Banks</td><td>523.00</td><td>0.26</td><td>0.16</td><td>-0.04</td><td>0.03</td><td>-0.20</td><td>1.39</td></tr><tr><td>Insurers</td><td>523.00</td><td>0.22</td><td>0.13</td><td>-0.03</td><td>0.03</td><td>-0.35</td><td>1.06</td></tr><tr><td>Invests.F.</td><td>523.00</td><td>0.30</td><td>0.18</td><td>-0.05</td><td>0.03</td><td>-0.54</td><td>1.54</td></tr><tr><td rowspan="3">June 2014 - June 2016</td><td>Banks</td><td>522.00</td><td>-0.03</td><td>0.19</td><td>-0.05</td><td>0.05</td><td>-0.20</td><td>1.28</td></tr><tr><td>Insurers</td><td>522.00</td><td>0.05</td><td>0.14</td><td>-0.04</td><td>0.03</td><td>-0.22</td><td>1.37</td></tr><tr><td>Invests.F.</td><td>522.00</td><td>-0.16</td><td>0.20</td><td>-0.05</td><td>0.04</td><td>-0.35</td><td>1.57</td></tr><tr><td rowspan="3">June 2016 - June 2018</td><td>Banks</td><td>523.00</td><td>0.19</td><td>0.18</td><td>-0.08</td><td>0.04</td><td>-1.28</td><td>8.56</td></tr><tr><td>Insurers</td><td>523.00</td><td>0.10</td><td>0.13</td><td>-0.06</td><td>0.02</td><td>-1.28</td><td>7.69</td></tr><tr><td>Invests.F.</td><td>523.00</td><td>0.10</td><td>0.16</td><td>-0.06</td><td>0.04</td><td>-0.88</td><td>4.45</td></tr><tr><td rowspan="3">June 2018 - June 2020</td><td>Banks</td><td>523.00</td><td>-0.14</td><td>0.35</td><td>-0.16</td><td>0.13</td><td>-0.72</td><td>15.01</td></tr><tr><td>Insurers</td><td>523.00</td><td>-0.06</td><td>0.29</td><td>-0.15</td><td>0.11</td><td>-1.60</td><td>20.50</td></tr><tr><td>Invests.F.</td><td>523.00</td><td>-0.12</td><td>0.35</td><td>-0.16</td><td>0.10</td><td>-1.22</td><td>13.08</td></tr></tbody></table>

Summary statistics for daily returns of the 75 firms of our sample: June 2000 to June 2020. Of the 75 firms, 25 are banks, 25 insurers, and 25 investment firms. We include the annualized mean, annualized standard deviation, minimum, maximum, skewness, and kurtosis of the annualized daily returns.