Complexity

Volume 2018, Article ID 6076173, 15 pages

https://doi.org/10.1155/2018/6076173

## Incorporating Contagion in Portfolio Credit Risk Models Using Network Theory

^{1}Computational Science Lab, University of Amsterdam, Science Park 904, 1098XH Amsterdam, Netherlands^{2}Quantitative Analytics, ING Bank, Foppingadreef 7, 1102BD Amsterdam, Netherlands

Correspondence should be addressed to Ioannis Anagnostou; ln.avu@uotsongana.i

Received 20 September 2017; Accepted 29 November 2017; Published 8 January 2018

Academic Editor: Thiago C. Silva

Copyright © 2018 Ioannis Anagnostou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Portfolio credit risk models estimate the range of potential losses due to defaults or deteriorations in credit quality. Most of these models perceive default correlation as fully captured by the dependence on a set of common underlying risk factors. In light of empirical evidence, the ability of such a conditional independence framework to accommodate for the occasional default clustering has been questioned repeatedly. Thus, financial institutions have relied on stressed correlations or alternative copulas with more extreme tail dependence. In this paper, we propose a different remedy—augmenting systematic risk factors with a contagious default mechanism which affects the entire universe of credits. We construct credit stress propagation networks and calibrate contagion parameters for infectious defaults. The resulting framework is implemented on synthetic test portfolios wherein the contagion effect is shown to have a significant impact on the tails of the loss distributions.

#### 1. Introduction

One of the main challenges in measuring the risk of a bank’s portfolio is modelling the dependence between default events. Joint defaults of many issuers over a fixed period of time may lead to extreme losses; therefore, understanding the structure and the impact of default dependence is essential. To address this problem, one has to take into consideration the existence of two distinct sources of default dependence. On the one hand, performance of different issuers depends on certain common underlying factors, such as interest rates or economic growth. These factors drive the evolution of a company’s financial success, which is measured in terms of its rating class or the probability of default. On the other hand, default of an issuer may, too, have a direct impact on the probability of default of a second dependent issuer, a phenomenon known as contagion. Through contagion, economic distress initially affecting only one issuer can spread to a significant part of the portfolio or even the entire system. A good example of such a transmission of pressure is the Russian crisis of 1998-1999 which saw the defaults of corporate and subsovereign issuers heavily clustered following the sovereign default [1].

Most portfolio credit risk models used by financial institutions neglect contagion and rely on the conditional independence assumption according to which, conditional on a set of common underlying factors, defaults occur independently. Examples of this approach include the Asymptotic Single Risk Factor (ASRF) model [2], industry extensions of the model presented by Merton [3] such as the KMV [4, 5] and CreditMetrics [6] models, and the two-factor model proposed recently by Basel Committee on Banking Supervision for the calculation of Default Risk Charge (DRC) to capture the default risk of trading book exposures [7]. A considerable amount of literature has been published on the conditional independence framework in standard portfolio models; see, for example, [8, 9].

Although conditional independence is a statistically and computationally convenient property, its empirical validity has been questioned on a number of occasions, where researchers investigated whether dependence on common factors can sufficiently explain the default clustering which occurs from time to time. Schönbucher and Schubert [10] suggest that the default correlations that can be achieved with this approach are typically too low in comparison with empirical default correlations, although this problem becomes less severe when dealing with large diversified portfolios. Das et al. [11] use data on US corporations from 1979 to 2004 and reject the hypothesis that factor correlations can sufficiently explain the empirically observed default correlations in the presence of contagion. Since a realistic credit risk model is required to put the appropriate weight on scenarios where many joint defaults occur, one may choose to use alternative copulas with tail dependence which have the tendency to generate large losses simultaneously [12]. In that case, however, the probability distribution of large losses is specified a priori by the chosen copula, which seems rather unintuitive [13].

One of the first models to consider contagion in credit portfolios was developed by Davis and Lo [14]. They suggest a way of modelling default dependence through infection in a static framework. The main idea is that any defaulting issuer may infect any other issuer in the portfolio. Giesecke and Weber [15] propose a reduced-form model for contagion phenomena, assuming that they are due to the local interaction of companies in a business partner network. The authors provide an explicit Gaussian approximation of the distribution of portfolio losses and find that, typically, contagion processes have a second-order effect on portfolio losses. Lando and Nielsen [16] use a dynamic model in continuous time based on the notion of mutually exciting point processes. Apart from reduced-form models for contagion, which aim to capture the influence of infectious defaults to the default intensities of other issuers, structural models were developed as well. Jarrow and Yu [17] generalize existing models to include issuer-specific counterparty risks and illustrate their effect on the pricing of defaultable bonds and credit derivatives. Egloff et al. [18] use network-like connections between issuers that allow for a variety of infections between firms. However, their structural approach requires a detailed microeconomic knowledge of debt structure, making the application of this model in practice more difficult than that of Davis and Lo’s simple model. In general, since the interdependencies between borrowers and lenders are complicated, structural analysis has mostly been applied to a small number of individual risks only.

Network theory can provide us with tools and insights that enable us to make sense of the complex interconnected nature of financial systems. Hence, following the 2008 crisis, network-based models have been frequently used to measure systemic risk in finance. Among the first papers to study contagion using network models was [19], where Allen and Gale show that a fully connected and homogeneous financial network results in an increased system stability. Contagion effects using network models have also been investigated in a number of related articles; see, for example, [20–24]. The issue of too-central-to-fail was shown to be possibly more important than too-big-to-fail by Battiston et al. in [25], where DebtRank, a metric for the systemic impact of financial institutions, was introduced. DebtRank was further extended in a series of articles; see, for example, [26–28]. The need for development of complexity-based tools in order to complement existing financial modelling approaches was emphasized by Battiston et al. [29], who called for a more integrated approach among academics from multiple disciplines, regulators, and practitioners.

Despite substantial literature on portfolio credit risk models and contagion in finance, specifying models, which take into account both common factors and contagion while distinguishing between the two effects clearly, still proves challenging. Moreover, most of the studies on contagion using network models focus on systemic risk and the resilience of the financial system to shocks. The qualitative nature of this line of research can hardly provide quantitative risk metrics that can be applied to models for measuring the risk of individual portfolios. The aforementioned drawback is perceived as an opportunity for expanding the current body of research by contributing a model that would account for common factors and contagion in networks alike. Given the wide use of factor models for calculating regulatory and economic capital, as well as for rating and analyzing structured credit products, an extended model that can also accommodate for infectious default events seems crucial.

Our paper takes up this challenge by introducing a portfolio credit risk model that can account for two channels of default dependence: common underlying factors and financial distress propagated from sovereigns to corporates and subsovereigns. We augment systematic factors with a contagion mechanism affecting the entire universe of credits, where the default probabilities of issuers in the portfolio are immediately affected by the default of the country where they are registered and operating. Our model allows for extreme scenarios with realistic numbers of joint defaults, while ensuring that the portfolio risk characteristics and the average loss remain unchanged. To estimate the contagion effect, we construct a network using credit default swaps (CDS) time series. We then use CountryRank, a network-based metric, introduced in [30] to quantify the impact of a sovereign default event on the credit quality of corporate issuers in the portfolio. In order to investigate the impact of our model on credit losses, we use synthetic test portfolios for which we generate loss distributions and study the effect of contagion on the associated risk measures. Finally, we analyze the sensitivity of the contagion impact to rating levels and CountryRank. Our analysis shows that credit losses increase significantly in the presence of contagion. Our contributions in this paper are thus threefold: First, we introduce a portfolio credit risk model which incorporates both common factors and contagion. Second, we use a credit stress propagation network constructed from real data to quantify the impact of deterioration of credit quality of the sovereigns on corporates. Third, we present the impact of accounting for contagion which can be useful for banks and regulators to quantify credit, model, or concentration risk in their portfolios.

The rest of the paper is organized as follows. Section 2 provides an overview of the general modelling framework. Section 3 presents the portfolio model with default contagion and illustrates the network model for the estimation of contagion effects. In Section 4 we present empirical analysis of two synthetic portfolios. Finally, in Section 5, we summarize our findings and draw conclusions.

#### 2. Merton-Type Models for Portfolio Credit Risk

Most financial institutions use models that are based on some form of the conditional independence assumption, according to which issuers depend on a set of common underlying factors. Factor models based on the Merton model are particularly popular for portfolio credit risk. Our model extends the multifactor Merton model to allow for credit contagion. In this section, we present the basic portfolio modelling setup, outline the model of Merton, and explain how it can be specified as a factor model. A more detailed presentation of the multivariate Merton model is provided by [9].

##### 2.1. Basic Setup and Notations

This subsection introduces the basic notation and terminology that will be used throughout this paper. In addition, we define the main risk characteristics for portfolio credit risk.

The uncertainty of whether an issuer will fail to meet its financial obligations or not is measured by its* probability of default*. For comparison reasons, this is usually specified with respect to a fixed time interval, most commonly one year. The probability of default then describes the probability of a default occurring in the particular time interval. The* exposure at default* is a measure of the extent to which one is exposed to an issuer in the event of, and at the time of, that issuer’s default. The default of an issuer does not necessarily imply that the creditor receives nothing from the issuer. The percentage of loss incurred over the overall exposure in the event of default is given by the* loss given default*. Typical values lie between 45% and 80%.

Consider a portfolio of issuers, indexed by , and a fixed time horizon of year. Denote by the exposure at default of issuer and by its probability of default. Let be the loss given default of issuer . Denote by the default indicator, in the time period . All issuers are assumed to be in a nondefault state at time . The default indicator is then a random variable defined bywhich clearly satisfies . The overall portfolio loss is defined as the random variable

With credit risk in mind, it is useful to distinguish potential losses in* expected losses*, which are relatively predictable and thus can easily be managed, and* unexpected losses*, which are more complicated to measure. Risk managers are more concerned with unexpected losses and focus on risk measures relating to the tail of the distribution of .

##### 2.2. The Model of Merton

Credit risk models are typically distinguished in structural and reduced-form models, according to their methodology. Structural models try to explain the mechanism by which default takes place, using variables such as asset and debt values. The model presented by Merton in [3] serves as the foundation for all these models. Consider an issuer whose asset value follows a stochastic process . The issuer finances itself with equity and debt. No dividends are paid and no new debt can be issued. In Merton’s model the issuer’s debt consists of a single zero-coupon bond with face value and maturity . The values at time of equity and debt are denoted by and and the issuer’s asset value is simply the sum of these; that is,Default occurs if the issuer misses a payment to its debtholders, which can happen only at the bond’s maturity . At time , there are only two possible scenarios: (i): the value of the issuer’s assets is higher than its debt. In this scenario the debtholders receive , the shareholders receive the remainder , and there is no default.(ii): the value of the issuer’s assets is less than its debt. Hence, the issuer cannot meet its financial obligations and defaults. In that case, shareholders hand over control to the bondholders, who liquidate the assets and receive the liquidation value in lieu of the debt. Shareholders pay nothing and receive nothing; therefore we obtain , .

For these simple observations, we obtain the below relations:Equation (4) implies that the issuer’s equity at maturity can be determined as the price of a European call option on the asset value with strike price and maturity , while (5) implies that the value of debt at is the sum of a default-free bond that guarantees payment of plus a short European put option on the issuer’s assets with strike price .

It is assumed that under the physical probability measure the process follows a geometric Brownian motion of the formwhere is the mean rate of return on the assets, is the asset volatility, and is a Wiener process. The unique solution at time of the stochastic differential equation (6) with initial value is given bywhich implies thatHence, the real-world probability of default at time , measured at time , is given byA core assumption of Merton’s model is that asset returns are lognormally distributed, as can be seen in (8). It is widely acknowledged, however, that empirical distributions of asset returns tend to have heavier tails; thus, (9) may not be an accurate description of empirically observed default rates.

##### 2.3. The Multivariate Merton Model

The model presented in Section 2.2 is concerned with the default of a single issuer. In order to estimate credit risk at a portfolio level, a multivariate version of the model is necessary. A multivariate geometric Brownian motion with drift vector , vector of volatilities , and correlation matrix , is assumed for the dynamics of the multivariate asset value process with , so that for all where the multivariate random vector with is satisfying . Default takes place if , where is the debt of company . It is clear that the default probability in the model remains unchanged under simultaneous strictly increasing transformations of and . Thus, one may defineand then default equivalently occurs if and only if . Notice that is the standardized asset value log-return . It can be easily shown that the transformed variables satisfy and their copula is the Gaussian copula. Thus, the probability of default for issuer is satisfying , where denotes the cumulative distribution function of the standard normal distribution. A graphical representation of Merton’s model is shown in Figure 1. In most practical implementations of the model, portfolio losses are modelled by directly considering an -dimensional random vector with containing the standardized asset returns and a deterministic vector containing the critical thresholds with for given default probabilities , . The default probabilities are usually estimated by historical default experience using external ratings by agencies or model-based approaches.