Insurance: Mathematics and Economics 61 (2015) 155–169 Contents lists available at ScienceDirect Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime A hierarchical copula-based world-wide valuation of sovereign risk Enrico Bernardi, Federico Falangi, Silvia Romagnoli * University of Bologna, Department of Statistics, via Belle Arti 41, 40126 Bologna, Italy article info Article history: Received May 2014 Received in revised form December 2014 Accepted 13 January 2015 Available online 20 January 2015 Keywords: Sovereign risk Hierarchical copula function Loss distribution Clustering methods Enlargement of the dependence structure abstract We propose a new model for the aggregation of risks that is very flexible and useful in high dimensional problems. We propose a copula-based model that is both hierarchical and hybrid (HYC for short), because: (i) the dependence structure is modeled as a hierarchical copula, (ii) it unifies the idea of the clusterized homogeneous copula-based approach (CHC for short) and its limiting version (LHC for short) proposed in Bernardi and Romagnoli (2012, 2013). Based on this, we compute the loss function of a world-wide sovereign debt portfolio which accounts for a systemic dependence of all countries, in line with a global valuation of financial risks. Our approach enables us to take into account the non-exchangeable behavior of a sovereign debts’ portfolio clustered into several classes with homogeneous risk and to recover a possible risks’ hierarchy. A comparison between the HYC loss surface and those computed through a pure limiting approach, which is commonly used in high dimensional problems, is presented and the impact of the concentration and the granularity errors is appreciated. Finally the impact of an enlargement of the dependence structure is discussed, in the contest of a geographical area sub-portfolios analysis now relevant to determine the risk contributions of subgroups under the presence of a wider dependence structure. This argument is presented in relation to the evaluation of the insurance premium and the collateral related to the designed project of an euro-insurance-bond. © 2015 Elsevier B.V. All rights reserved. 1. Introduction A comprehensive evaluation of the risks that governments could default on their debts became crucial during the recent mar- ket global economic decline that began in December 2007. The recession, beginning in the United States in December 2007 as a consequence of the bursting of the US housing bubble, became in- ternational in September 2008. Moreover the failure of three of the major US investment banks, increased the instability of the global financial system characterized by an exceptionally high level of both private and public debt in the US and in many other coun- tries. This excessive debt played an important role in causing bank crises, that progressed to sovereign debt problems, and appeared first in the euro-area. Many European countries embarked on aus- terity programs which contributed to improve their budget deficits from 2010 to 2011 relative to GDP. Despite this effort, the eurozone unemployment reached record levels during the 2012, indicating that GDP growth was not sufficient to support the variation in the debt-to-GDP ratio. On the other hand financial crisis did not affect emerging and developing countries, like the Africa area, that are * Corresponding author. E-mail address: silvia.romagnoli@unibo.it (S. Romagnoli). not strictly integrated in the world market. The global answer to the crisis focused on the lack of transparency about banks’ risk ex- posures that disable the markets from correctly pricing risks. Euro- pean regulators introduced Basel III whose novelty did not address the problem of faulty risk-weighing, leading indeed to excessive lending to risky governments (see Nouy, 2012 for a detailed dis- cussion about the regulations required from capital institutions to hold adequate regulatory capital associated with sovereign expo- sures). The complex interactions between the economic and financial variables remind us how important it is to improve the modeling of financial crises and sovereign risk. Gray et al. (2007) proposed a model based on the theory and practice of modern contingent claim analysis, where the sectors of a national economy and the sovereign balance sheet are considered as dependent portfolios of assets, liabilities and guarantees. A systemic version of this model appeared in Gray and Jobst (2011) where a multivariate approach represents the interactions and the feedback between the financial sector and the sovereign. We propose a new model (HYC for short), that is essentially copula-based. Multivariate copulas (see Nelsen, 2006 and Joe, 1997), allowing for a very flexible dependence structures and fa- cilitating statistical parameters’ estimations, are commonly used in quantitative finance (see Cherubini et al., 2004 for standard http://dx.doi.org/10.1016/j.insmatheco.2015.01.003 0167-6687/© 2015 Elsevier B.V. All rights reserved.