Connectedness Measures of Spatial Contagion in the Banking and Insurance Sector Fabrizio Durante1, , Enrico Foscolo2 , Piotr Jaworski3, and Hao Wang4 1
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School of Economics and Management, Free University of Bozen-Bolzano, Bolzano, Italy
[email protected] 2 School of Economics and Management, Free University of Bozen-Bolzano, Bolzano, Italy
[email protected] 3 Institute of Mathematics, University of Warsaw, Warszawa, Poland
[email protected] Department “Methods and Models for Economics, Territory and Finance”, Sapienza University of Rome, Rome, Italy
[email protected]
Abstract. We present some connectedness measures for an economic system that are derived from the spatial contagion measure. These measures are calculated directly from time series data and do not require any parametric assumption. The given definitions are illustrated in an empirical analysis of the behavior of European banking and insurance sector in the recent years. Keywords: Contagion, Copula, Tail dependence.
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Introduction
The recent financial crisis has renewed the interest in the interconnectedness among different financial institutions located in various countries. In particular, “systemic risk” has become a standard concept that relates to the risk imposed by interlinkages and interdependencies in a system or market, where the failure of a single entity or group of entities can cause potential difficulties to other entities, which could potentially bankrupt or bring down the entire system. As stressed by [2], studies about systemic risk can be divided into two major groups. One approach consists of using network analysis and works directly on the structure and the nature of relationships between financial institutions in the market. Another approach investigates the impact of one institution on the market and its contribution to the global system risk [1]. Hence, the latter methodology requires the knowledge of the joint behavior of the financial institutions and is related to previous works about the so-called financial contagion
This work was supported by Free University of Bozen-Bolzano via the project MODEX.
c Springer International Publishing Switzerland 2015 217 P. Grzegorzewski et al. (eds.), Strengthening Links between Data Analysis & Soft Computing, Advances in Intelligent Systems and Computing 315, DOI: 10.1007/978-3-319-10765-3_26
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[9]. Roughly speaking, contagion refers to a significant increase in comovements of prices and quantities across markets, conditional on a crisis occurring in one market or group of markets. Recently, the notion of financial contagion has been reformulated in terms of copulas in [7] (see also [4,6,8]). Specifically, it refers to the change of strength of dependence in the tail and in the center of the joint distribution associated with two financial positions. This concept has been further developed in [5], where a spatial contagion measure has been defined in order to quantify (in a normalized scale) the influence of one market over the others. In this contribution, we review the notion of spatial contagion measure by pointing some of its main features. Then, inspired by [3], we define some simple measures of connectedness that can be used in order to investigate systemic risk in a set of financial institutions. The second part of the contribution is devoted to the empirical investigation of spatial contagion in a set of asset returns related to banking and insurance sectors, which have become increasingly interconnected especially during the last decade.
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The Spatial Contagion Measure
The notion of spatial contagion measure has been introduced in [5]. Basically, it focuses on the discrepancies between tail and central sets of probability distribution function of two financial returns. This approach is based on the geometry of the underlying distribution and, for this reason, it is called spatial contagion. Formally, it is defined in the following way. Let X and Y be two random variables on a suitable probability space representing the returns (or log-returns) of financial markets whose dependence is described by means of a copula C. Consider the following Borel sets of R2 : – the tail set Tα1 ,α2 given by Tα1 ,α2 = [−∞, qX (α1 )] × [−∞, qY (α2 )], where α1 , α2 ∈ [0, 1] and qX and qY are the quantile functions associated with X and Y , respectively. – the central set (or mediocre set ) Mβ1 ,β2 given by Mβ1 ,β2 = [qX (β1 ), qX (1 − β1 )] × [qY (β2 ), qY (1 − β2 )] where β1 , β2 ∈ [0, 1/2]. Intuitively, Tα1 ,α2 represents the “risky scenario” for the pair (X, Y ), since it includes the bivariate observations that are less than a given threshold; while Mβ1 ,β2 represents the so-called “untroubled scenario”, since it is related to all the observations that are in the central region of the joint distribution (being the extreme values excluded).
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Definition 1. Let L ⊆ (0, 0.5). The (spatial) contagion measure from X to Y is defined by the formula γ(X → Y ) =
1 λ({α ∈ L | ρ(Tα,1 ) − ρ(Mα,0 ) > 0}), λ(L)
(1)
where λ is the Lebesgue measure, ρ(Tα,1 ) (respectively, ρ(Mα,0 )) denotes the Spearman’s correlation of the conditional distribution function of [(X, Y ) | (X, Y ) ∈ Tα,1 ] (respectively, [(X, Y ) | (X, Y ) ∈ Mα,0 ] ). The introduced measure depends hence on the Spearman’s rank correlation and avoids to restrict to the use of linear correlation. Moreover, the estimation of the contagion measure can be done mainly in a non-parametric way, as illustrated in [5]. Roughly speaking, the contagion measure counts how many times the correlation in the tail of the joint distribution is larger than the correlation in the central region for some predefined set of possible levels α ∈ L, where L is usually chosen by the decision maker according to her/his risky attitude. Notice that the mapping α → Δα = ρ(Tα,1 ) − ρ(Mα,0 ) for every α ∈ L = [a, b] depends only on the copula of (X, Y ) (since it is based on rank correlation). Moreover, it can be positive, negative or changing in sign depending on the involved dependence. For instance: – If (X, Y ) has copula equal to the ordinal sum of comonotonicity and independence copula with respect to the partition ([0, a], [a, 1]), then Δα ≥ 0 for every α ∈ L. – If (X, Y ) has copula equal to the ordinal sum of independence and comonotonicity copula with respect to the partition ([0, a], [a, 1]), then Δα ≤ 0 for every α ∈ L. – If (X, Y ) has Gaussian copula with correlation ρ > 0, then Δα changes sign in (0, 0.5) (see [8] for more details). Starting with this definition, we consider now some derived measures of connectedness, inspired by the motivations presented in [3]. Let us consider historical time series from different asset returns X1 , . . . , Xd that are operating in the same sector and/or geographic region. Let J be a subset of indices in {1, 2, . . . , d}. Let μ(Xi → Xj ) be the contagion measure from asset i to asset j. Then we can define the following measure of connectedness that may help in the identification of contagion effects from one asset to a group of assets or between groups of assets. Definition 2. Let J be a subset of indices in {1, 2, . . . , d}, let i ∈ {1, . . . , d}\J . We define contagion from Xi to {Xj : j ∈ J } as μ(Xi → {Xj : j ∈ J }) =
1 μ(Xi → Xj ). |J | j∈J
Obviously, μ(Xi → {Xj : j ∈ J }) = 0 when μ(Xi → Xj ) = 0 for each choice of the indices j’s in J . Analogously we can define the following measure.
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Definition 3. Let I, J be disjoint subset of {1, . . . , d}. We define contagion from {Xi : i ∈ I} over {Xj : j ∈ J } as μ({Xi : i ∈ I} → {Xj : j ∈ J }) =
1 μ(Xi → {Xj : j ∈ J }). |I| i∈I
Both these measures of connectedness are obtained by aggregating spatial contagion measures at individual levels. They will be used below to provide some insights in understanding systemic risk.
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Empirical Analysis
We consider the daily log-returns of the European banks and insurance companies characterizing the STOXX Europe 600 Index. The financial list counts 84 institutions and covers 16 countries1 and 7 currencies2 . The companies are divided into two sectors, Bank and Insurance, containing 47 and 37 assets, respectively. Moreover, each sector is divided into three groups according to the market capitalization. As result, we cluster the 84 assets by their capital size and sector into the following groups: 25 large banks, 8 medium banks, 14 small banks, 15 large insurance companies, 8 medium insurance companies, and 14 small insurance companies. Following [3], the emphasis on market returns is motivated by the desire to incorporate the most current information in our measures. On the other hand, the clustering procedure by market capitalization is designed for taking into account possible different trading liquidity and financial instability within groups of large-, medium- and small-sized companies. The dataset refers to time interval January 3rd, 2005-December 31st, 2012. In order to compare the time-variation of the connectedness measures between the sectors and within the sectors, we fix two four-years time windows: 2005-2008 and 2009-2012; shortly, the “before the crisis” and the “after the crisis” period, respectively. As mentioned in Section 2, the definitions of connectedness from one single financial institution to a sector of institutions and from one sector to another
Table 1. Contagion measure between different financial sectors in both periods using Definition 3. Letter B stands for the bank sector, while letter I for the insurance sector. µ(B → B) µ(B → I) µ(I → B) µ(I → I) 2005-2008 2009-2013
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0.43 0.23
0.37 0.24
0.40 0.20
0.32 0.23
Shortly, AT, BE, CH, CZ, DE, DK, ES, FI, FR, GB, IE, IT, NL, NO, PT, and SE. British Pound, Czech Koruna, Danish Krone, Euro, Norwegian Krone, Swedish Krona, and Swiss Franc.
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sector are based on the spatial contagion measure proposed in [5], which requires some ad-hoc algorithm to be computed. In this work, we calculate our measures directly on the data (without any preliminary ARMA-GARCH filter); for the computation, we refer to the Algorithm 4.2 and the Algorithm 4.3 as described in [5]. Firstly, for every single financial institution Xi , we calculate μ(Xi → S) in the two considered periods, where S is formed by all institutions beloning to a specific sector (i.e., bank or insurance). In Figure 1, each line corresponds to the (smoothed) empirical density of the histograms related to all the measurements of type μ(Xi → S) in a specific time period, where Xi is varying in a specific sector, while S equals bank, insurance, or both sectors. As can be seen, regardless of the choice of a different set S, the distribution of the spatial contagion measure moves towards smaller values during the second period. In order to highlight such a finding, we compute the contagion from one sector to another one in both periods; see Table 1. We note that, in the latter period, the overall contagion risk between financial institution sectors seems to be reduced. Nevertheless, for banks this change is more clear, since the two sets of three density curves seem to be unimodal; see the upper panel in Figure 1. For insurance companies, however, the evidence is different because after crisis the density curves seems to be bimodal, implying that for a large subset of these corporations contagion risk remains high; see the lower panel in Figure 1. In order to give a graphical representation of the evolving relations, we provide a network diagram to show the linkages among different financial institutions by plotting a line connection when the contagion measure from institution Xi to institution Xj is larger that 0.5; see Figure 2. The charts highlight the fact that the contagion measures within and between sectors decreased after the crisis since the number of extreme edges in the previous period is much larger than in later period. As can be noticed in Figure 2, when we look at the contagion within sectors before the crisis period, the large-sized banks are heavily connected, but they are less affected by medium- and small-sized banks. When we focus on the insurance sector, the situation is just the opposite. Here, large-sized insurance companies are more easily affected by medium- and small-sized companies, especially the small-sized ones. If we consider the contagion measure between sectors, contagion effects from banks to insurance companies are less likely than those from insurance companies to banks. In the second period, however, all these effects seems to considerably reduce. A flight-to-quality evidence towards different markets and investments (e.g., bond market and cash equivalent) appears a possible explanation.
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Fig. 1. The density curve of the contagion measure from a single financial institution, namely bank (upper chart) and insurance company (lower chart), to different sets: the bank sector (solid ), the insurance sector (dashed ), and both sectors (dotted ). Black and gray colors refer to the before and after the crisis period, respectively.
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(g) µ(I → B), 2005-2008 (h) µ(I → B), 2009-2012 Fig. 2. Extreme (> 0.5) asymmetric contagion measures within the sectors: Panels a and b refer to the bank sector B, while c and d the insurance sector I. Extreme (> 0.5) asymmetric contagion measures between the sectors: Panels e and f show the effects from bank to insurance sector, while g and f from insurance to bank sector. First column charts concern 2005-2008, while second column charts 2009-2012. The red, yellow, and green vertices stand for the large-, medium-, and small-sized banks, respectively; the cyan, blue, and purple vertices stand for the large-, medium-, and small-sized insurance companies, respectively.
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Conclusions
We proposed the spatial contagion measure due to [5] for analyzing the connectedness from a single institution to one sector, and within and between the bank and insurance sectors, before and after the 2008 subprime crisis. We found the contagion measures from single insurance companies to bank sector seem to be larger (or at least equal) than those from banks to the insurance sector. Moreover, while large-sized banks are more connected and affected by each other, insurance companies are more affected by medium- and smallsized companies. Finally, after the crisis, the contagion risk between financial institutions seems to reduce.
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