Conjuntos de confianza para la correlacion de activos en el riesgo de credito de un portafolio.

• Author: Castro, Carlos
• Position: Art
• Pages: 19(40)

Confidence sets for asset correlations in portfolio credit risk

1 Introduction

Asset dependence in portfolio credit risk management is a topic of growing importance for practitioners and academics. Changes in the most common form of dependence (correlation) across assets transfer some of the risk from the mean towards the tail of the loss distribution. Any increase in correlation between the assets fattens the tail of the loss distribution and therefore requires a greater amount of capital set aside to cover unexpected losses. Hence asset correlation is a cornerstone parameter in the estimation of a bank's capital requirements. Tarashev et al. (2007) show that misspecified or incorrectly calibrated correlations can lead to significant inaccuracies in the measures of portfolio credit risk and economic capital.

The Basel accord of 1988 was a first attempt to establish an international standard on a bank's capital requirements. However a significant drawback was the accord's crude approach to determine the risk weights assigned to different positions in a bank's portfolio. For example, a private firm with a top rating would receive a weight a hundred times higher than any type of sovereign debt, regardless of the rating of the former. The second Basel (henceforth Basel II) corrected the imbalance by accounting for the relative credit quality of the issuers. (1) Under Basel II regulators gave more leeway to banks with the hope that they are able to perform a more accurate measure of the risk heterogeneity of their portfolio. Hence bank managers have greater freedom to calibrate the assigned risk weights and derive more accurate loss distributions for their portfolios.

In general, techniques to derive the loss distribution for the portfolio require simulation. Considering the dependence across all individual names would be very cumbersome, hence factor models provide simple ways to map the dependence structure in the portfolio. Under the internal-rating-based (IRB) approach to determine the risk weights, proposed in Basel II, the underlying structure behind default dependence is a one factor model. Basel II suggests a value for the correlation parameter, the unique dependence parameter of this one factor model, between 0.12 and 0.24. The literature proposes various estimates for these values in the ranges (0.01-0.1) Chernih et al. (2006) and (0.05-0.21) Akhavein et al. (2005).

The risk factor model, used extensively in the literature on dynamic modeling of default risk, provides the structure to estimate asset correlations with rating data. This model decomposes credit risk into systematic (macro related) and idiosyncratic (issuer specific) components. In this context (McNeil et al., 2005; McNeil and Wendin, 2007) and Koopman and Lucas (2008) provide estimation methods to fit the dynamics of default. The former explores heterogeneity among industry sectors and also across rating classes whereas the later only explores heterogeneity across ratings. Both articles recognize two difficulties inherent to rating data and, particularly, default: i) rating transitions are scarce events and ii) defaults are extremely rare events. These two elements make statistical inference more difficult. However, rating transition data is a preferred proxy for changes in the creditworthiness of issuers because it is more direct than using other proxies such as equity or spread data. Moreover, equity based correlation is not readily available for some types of issuers. For example, for sovereigns and structured products there is no information available for equity or debt. Therefore, it is not possible to link equity and assets through the option-theoretic framework due to Merton (1974).

The aim of this article is twofold: First, estimate asset correlations within and across different identifiable forms of grouping the issuers. Second, provide a sensibility analysis of these estimates with respect to the model assumptions. We use Moody's rating data for Corporate defaults and Structured products. The Corporate default database contains information on 51,542 rating actions affecting 12,292 corporate and financial institutions during the period 1970 to 2009. The Structured products database contains information on 377,005 rating actions affecting 134,554 structured products during the period 1981 to 2009. The database contains information on group affiliation of the issuers such as type of product, or economic sector and country where a firm carries out its business. (2) According to the group affiliation, firms are organized into 11 sectors, 7 world regions and 6 structured products (Table 1).

The disaggregated approach (first objective), with respect to a world aggregate, contributes to the existing literature on asset correlation since most of the literature has focused on estimating these models on aggregate data (in particular aggregate US default count). With respect to world region affiliation and structured products, the results in this article are a novelty. Furthermore, if accounting for heterogeneity in a bank's portfolio is an important part of Basel II, it is senseless estimating models based on the aggregated data. By moving away from the aggregated data, the few historical observations that are available on rating transitions (especially default) become even more sparse. Therefore the existing methodologies encounter problems due to the sparsity of the data.

The sensibility of the estimates of asset correlation with respect to the model assumptions (second objectives) goes beyond the Basel II benchmark: the one factor model. The elements of the model that are analyzed, with regard to their effect over the parameters of interests, are the following: i) introduction of additional group specific factors (i.e. a two factor model), ii) the nature for the factors (i.e. observed or unobserved), iii) the data generating process of the factors, iv) the functional form of the default probability (i.e. probit or logit), and v) for a given rating system, the implications of migrating from different ratings to default (i.e. correlation asymmetry).

We use a generalized linear mixed model (GLMM hereafter) for estimation. This model considers the observed number of firms that perform some migration (possibly to the default state), out of a total number of firms within a given group (say an economic sector or world region), as a realization of a binomial distribution conditional on the state of some unobserved systematic factor. A one or two factor model (1-F, 2-F, henceforth) allows the decomposition of default risk into the estimated factor(s) and the idiosyncratic component. A set of identifying assumptions on the model allows for the estimation of both the factor(s) and the factor(s) loading(s). The state space model built from this setup has a measurement equation that has the form of a binomial distribution (making the model non-Gaussian and non-linear).

We find that the loading parameters of the factors across the 11 sectors, 7 world regions and 6 structured products are in general statistically significant. We recover the asset correlations from the factor loadings and observe that in some cases, the asset correlations are higher than the Basel II recommended values. For most models there is even a null or very small probability that the asset correlation parameter is within the bounds recommended in the Basel II document. Asset correlation is in particular very sensitive to the assumptions of the statistical model; for instance if the unobserved component is autoregressive, as opposed to i.i.d.. Moreover, the two factor model with AR(1) dynamics for the global factor and with a local systemic factor (called it sectorial, regional, or product) is able to reproduce better the observed number of defaults than the 1-F framework recommended by Basel II.

The results have two direct implications on the measurement of economic capital. First, they show that the one factor model is too restrictive to account in a proper manner for the dependence structure in the data and hence the portfolio. A two factor model provides a hierarchical structure to the banks portfolio while still being parsimonious in terms of the parameters. This model includes a global systemic factor plus a local systemic factor in addition to the idiosyncratic component. The set of local systemic factors account for significant difference across identifiable grouping characteristic within the portfolio such as economic sectors and world regions. Second the estimates of the dependent structure are strongly reliant on modeling assumptions, hence they convey significant model risk. This source of model risk should be taken into account in the process of model validation by the regulators.

The outline of the paper is as follows: Section 2 presents the dynamic default risk model. Section 3 describes the data. Section 4 presents the estimation methods, the results and the implications on the estimation of economic capital. Sections 5 provides methodological solution to the common data scarcity problem found in default models. Section 6 concludes.

2 Dynamic factor model of default risk

The default risk model has its roots in the work by Merton (1974). In the last 10 years this model has been at the center of the literature on portfolio credit risk modeling. The most general version of the Merton model considers the asset value of a firm i = 1,..,N at time t = 1,.., T, [V.sub.i,t] as a latent stochastic variable. Let [V.sub.i,t] follow a standard normal distribution. If [V.sub.i,t] falls below a predetermined threshold [[micro].sub.i,t] (related to the level of debt) then a particular event is triggered. This event refers to a transition between states defined under some rating system. For capital adequacy purposes, the most important event is default. However, since historically this is a rare event, it is also interesting to consider a larger state-space to...