The impact of the dependence structure in risk management: a focus on credit-risk

ABSTRACT

The aim of the paper is to discuss the important role of the dependence structure in risk management. Therefore, we focus on credit-risk and propose an innovative model to value the credit risk of a portfolio. This new approach (HYC for short) is based on a hierarchical hybrid copula and involves a clusterization of the portfolio in several risk's classes. The HYC model is classified as hybrid because the computation of the loss cdf depends on the class's cardinality: for large groups one is justified to apply a limiting approach, while for small ones one applies a procedure preserving the granularity of the group itself. In order to appreciate the impact of the dependence structure in credit-risk evaluation, a VaR analysis based on the HYC loss function is here compared to the CreditMetrics approach in an in-sample exercise and to the empirical VaR in an out-of sample exercise aimed to test the forecasting effectiveness of the model. This comparison allows us to appreciate over/under-valuation of the capital detained from the financial institution. Moreover, the impact of an enlargement of the dependence structure is discussed with respect to the systemic/contagious effects in the context of a portfolio optimisation with constraint on a sub-portfolio's risk.

KEYWORDS:

• Risk management
• copula functions
• risk analysis
• linear programming

MSC[2010]::

• 62H15
• 62P05
• 62H99

JEL CLASSIFICATIONS:

• C63
• C55
• C31
• C33
• G11
• D89

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Enrico Bernardi  http://orcid.org/0000-0003-3923-1407

Silvia Romagnoli  http://orcid.org/0000-0002-4945-0673

Notes

1 In Bernardi, Falangi, and Romagnoli (2015b) we focused on a world-wide sovereign debt portfolio whose dependence structure is hierarchical too; this feature allows us to represent the portfolio as a set of classes which are modelled as a not-exchangeable hierarchical copula. This way we empirically detailed the distance between the HYC loss and the limiting one as sum of two components, i.e. the concentration (depending on the homogeneous feature of every class) and the granularity errors (depending on the granularity approximation).

2 We need to fix a bound to allow for the classification of small group. Coherently with (Schönbucher 2004), we set a bound of 20.

3 The acronym stands for Self Organising Map which refers to a particular type of artificial neural network advanced by Kohonen (1982).

4 All data are drawn from Thomson Reuters Datastream.

5 In the Bootstrap procedure we used the IRS (Interest Rate Swap) checked on the 26th of June 2014 for different maturities as the rates in the discounting functions.

6 We observe that, being the bootstrapped method a reverse-engineering approach of market prices, the recovered probabilities are risk-neutral because compatible with the martingale pricing measure and hence with the assumption of absence of arbitrage opportunities and rational pricing. In a risk-management perspective and in line with the aim of the paper, we have chosen for these probabilities that allow us to deal with forward-looking strategies taking into account for the market sentiment. Nevertheless in case of a discussion based on regulatory requirements involving the explicit consideration of credit-risk premium, as in the context of state aid assessment, the real-world estimates should be recovered. Among the existing literature, we recall (Hull 2012; Hull, Predescu, and White 2005) where the existence of risk premium is explained, mainly due to liquidity risk, traders' expectations and systemic/contagious effects, providing empirical estimations of a coverage ratio, i.e. the ratio between the risk-neutral and the real-world default probabilities on average greater than one. An interesting feature of the coverage ratio, deeply documented for the Eurozone in Heynderickx et al. (2016), is its decreasingness of the credit quality along with the convergence toward one for highly distress bond.

7 Sometimes the dimension is said to be equivalent in the meaning of the Diversity Score approach (see Cifuentes and O'Connor 1996 and Cifuentes et al. 1999).

8 This variable, linked to a second level dependence structure, corresponds to the variable $L^j$ yet introduced into the large classes.

9 See Cherubini and Romagnoli (2009) and Bernardi and Romagnoli (2011) for details on the computing algorithm for copula volume.

10 c.c.d. stands for combinatoric distributions which are coherent with the cardinalities of risk's classes.

11 We justifies the BBB-rating by an analysis done on the rating assigned to the names at the evaluation date. More precisely for every group/subgroup we identified the names which are “the nearest” to the corresponding centroid based on the clustering features, i.e. Group A: Compass (SubGroup A1: Compass; SubGroup A2: Peugeot), Group B: Roche Holding (SubGroup B1: Roche Holding; SubGroup B2: Air France), Group C: Unicredit (SubGroup C1: Air France; SubGroup C2: Carrefour) and Group D: Continental (SubGroup D1: Rallye; SubGroup D2: Continental). Their mean rating from the balance sheets 2010–2014 leaded to a BBB-grade (they were all BBB-rated, flowing in the range {BBB-, BBB+}).

12 The one-year and five-year transition matrixes were derived from CreditMetrics paper issued by J.P. Morgan. Ratings are provided by the Agency Standard & Poor's issued in 2014.

13 The credit thresholds for our BBB credit rating-portfolio identified the following regions in term of R, the annual portfolio's return: $AAA:R\ge 4\mathrm{\%}$, $AA:4\mathrm{\%}<R\le 2.2\mathrm{\%}$, $A:2.2\mathrm{\%}<R\le 1.6\mathrm{\%}$, $BBB:1.6\mathrm{\%}<R\le 1.16\mathrm{\%}$, $BB:1.16\mathrm{\%}<R\le -1.62\mathrm{\%}$, $B:-1.62\mathrm{\%}<R\le -3.75\mathrm{\%}$, $CCC:-3.75\mathrm{\%}<R\le -4.10\mathrm{\%}$ and $D:R<-4.1\mathrm{\%}$. Here $Z_{CCC}=-4.1\mathrm{\%}$ is the threshold point to trigger default, i.e. it is “distance-to-default”.

14 Considering a period overlapping the calibration period of the dependencies, i.e. 2007–2014, the mean coupon rate of bonds issued by the most representative names of every group, i.e. the “nearest” to the centroids based on the clustering features, is 5%.

15 The data regarding spot rates have been provided from IRS Agency. The IRS is a bureau of the Department of the Treasury and one of the world's most efficient tax administrators. The website is the following: “http://www.irs.gov/Retirement-Plans/Monthly-Yield-Curve-Tables”.

16 For our rolling procedure a longer maturity is meaningful.

17 As observed in Bernardi, Falangi, and Romagnoli (2015b), if the HYC model is implemented with a localised approach and with a Gaussian copula (and Gaussian marginals), the total dependence is represented by linear correlations within and between groups; in this case HYC aggregation approach converges to the simple aggregation methodology QIS 5 whose purpose is to aggregate the solvency capital requirements as potential solvency rules by the European Insurance and Occupational Pension Authority (see CEIOPS 2010).

18 This important feature is missing if the minimised risk measure is not convex as for example the VaR.

Additional information

Notes on contributors

Enrico Bernardi

Enrico Bernardi is Full Professor of Mathematics and Probability at the University of Bologna. His scientific research concerns the Mathematical modelling through nonlinear ordinary differential equation, Cauchy problem for Hyperbolic PDEs, Stability for stochastic Hamiltonian systems, Big data and complexity reduction. He has published extensively in international journals.

Silvia Romagnoli

Silvia Romagnoli is Associate Professor of Mathematical Methods for Economics and Actuarial/Financial Sciences and Director of the Master in Quantitative Finance at the University of Bologna, Italy. Her scientific research is mainly addressed to the applications of stochastic models to finance and insurance, with special emphasis on multidimensional problems. Thanks to her original contributions to copulas' theory, she is frequently invited as speaker in international conferences. She has published extensively in international top reviews and she is co-authos of Dynamic Copula Methods in Finance, John Wiley & Sons, Ltd., 2012.

Matteo Doti

Matteo Doti is Assistant Professor of Applied Mathematics at the University of Bologna. He studied at the University of Bologna and later at the Grenoble Business School. Currently He is working on several research projects in Bologna and Grenoble, involving mathematical modelling for risk/asset management.