Abstract

We consider the distribution of the sum of Bernoulli mixtures under a general dependence structure. The level of dependence is measured in terms of a limiting conditional correlation between two of the Bernoulli random variables. The conditioning event is that the mixing random variable is larger than a threshold and the limit is with respect to the threshold tending to one. The large-sample distribution of the empirical frequency and its use in approximating the risk measures, value at risk and conditional tail expectation, are presented for a new class of models which we call double mixtures. Several illustrative examples with a Beta mixing distribution, are given. As well, some data from the area of credit risk are fit with the models, and comparisons are made between the new models and also the classical Beta-binomial model.

1. Introduction

Since the U.S. subprime mortgage crisis which led the world economy into the global recession in the late 2000s, there has been serious criticism about the use of inappropriate dependent default models for portfolio credit risk (see Donnelly and Embrechts [1]). Specifically, the Gaussian copula model (Basel II Accord [2] and Li [3]) had become the industry standard model and has been widely used for both the pricing of portfolio credit derivatives such as collateralized debt obligations (CDOs) or mortgage backed securities and the credit risk management. Its inability to incorporate extreme tail dependence, however, resulted in a serious lack of default clustering, especially under stressful economic situations.

In the literature, several alternative approaches were proposed to cope with the issue of an insufficient level of dependency among defaults. As a simple extension of the Gaussian copula model, Andersen and Sidenius [4] and Burtschell et al. [5] considered stochastic asset correlation models. A more popular way of incorporating further dependence beyond the Gaussian copula model is the use of heavy tail copulas such as Archimedean copulas. Amongst many others, we mention Burtschell et al. [6] and Schönbucher and Schubert [7] for applications of copula models under the structural and the intensity-based credit risk model framework, respectively.

It is important to note that factor copula models under the typical conditional independence assumption have a close link with Bernoulli mixture models. Specifically, by de Finettis theorem, there exists a common mixing random variable on which the default indicator random variables, in an exchangeable credit portfolio, are dependent. The mixing random variable is essentially the random probability of default which can be represented as a function of the common systematic factor in a factor copula model. Under the typical conditional independence assumption, however, the level of dependence between the default indicator random variables is controlled solely by the distributional property of the systematic factor, which can impose limitations for some applications where extreme levels of dependence are required. See Bluhm et al. [8], Cousin and Laurent [9], Frey and McNeil [10, 11], KMV [12], McNeil et al. [13], McNeil and Wendin [14], Moraux [15], and RiskMetrics Group [16] for more references on credit risk applications of the Bernoulli mixture models under the conditional independence modeling framework.

Formally, a Bernoulli mixture model is defined as follows. Let be a sequence of identically distributed indicator random variables. We assume that each , , follows the Bernoulli distribution with the common default probability . The probability is randomly drawn from a distribution with cumulative distribution function (cdf) , . At this point, the dependency structure for is still arbitrary.

In credit applications, the statistical properties of the sum of the Bernoulli random variables are the main interest, and the following assumption plays an important role in alleviating computational difficulties: are conditionally independent given . In this case, given , the sum follows the binomial distribution with mean and variance . Then, the unconditional probability mass function (pmf) is given as follows: where the superscript “” on indicates the assumed conditional independence. The most commonly used mixing distribution in modeling the credit risk of a homogeneous portfolio such as a mortgage or credit card portfolio is the Beta distribution. This model is referred to as the Beta-binomial (-I) model.

The present paper is concerned with the level of dependence incorporated in Bernoulli mixture models, especially under stress situations. Together with the asset correlation, the default correlation is used as the standard measure of dependence in portfolio credit risk models. In particular, we consider the behavior of the correlation between two arbitrary terms, as the random default probability is conditioned to be larger than , tending to 1. Specifically, we define the conditional default correlation between two terms, and , given , , as follows (Bae and Iscoe [17] used an analogous definition to study the asset correlations under stress for various single-factor credit risk models—some dynamic—while Kalkbrener and Packham [18] also studied asset correlations under stress, for static, normal variance mixture models. In both studies, in place of the conditioning on , conditioning is done on an auxiliary risk factor which, for example, is related to a systematic factor): where

Then, the limit of the conditional correlation as tends to 1 (referred to as the limiting correlation) is defined as provided the limit exists.

It is intuitive to expect that the correlations in (2) increase monotonically as the threshold approaches 1. Unfortunately, this is not the case for the Beta-binomial model. In fact, the conditional correlation always converges to zero regardless of the mixing distribution, as long as conditional independence is postulated (Corollary 2). This type of model failure can introduce a significant bias in the measurement and management of risk in stress situations.

The objectives of the present paper are threefold: (i) to derive the relationship between the limiting correlation and the probability structure; (ii) to construct general Bernoulli mixture models with nontrivial limiting correlations; (iii) to demonstrate implications of the constructed models, for tail risk measures such as value at risk and conditional tail expectation.

The theoretical results for these objectives are presented in Section 2. We first examine the limiting behavior of correlations for Bernoulli mixture models in Section 2.1. Then, Section 2.2 introduces a general model framework—the class of double mixture models—which allows further positive dependence between entities, beyond conditional independence (Theorem 4). Section 2.3 is devoted to the third objective, and it considers the large-sample distribution of the sum of Bernoulli mixtures under the general model framework and its application to the approximation of risk measures is discussed. As a specific case, Section 3 considers a Beta Bernoulli mixture, and Section 4 provides the results for an example with specific parameter values. Section 5 demonstrates the fitting results of the double mixture models to a dataset from the area of credit risk. Finally, Section 6 concludes this paper with a summary. Technical details of some proofs are given in the appendices.

2. Main Results

2.1. Limiting Correlation for Bernoulli Mixtures

We denote by the regular conditional probability, given , that all terms , of a fixed sequence of Bernoulli random variables, , take on the value 1. (For this definition, we allow .) Formally, where . Note that, under an assumption of conditional independence, .

The following hypothesis on the properties of in a deleted neighbourhood of will be part of the assumptions in Theorem 1 and its Corollary 2 below.

Hypothesis H. is absolutely continuous in a neighbourhood of , with probability density function (pdf) thereon, such that (i);(ii) is continuous in a deleted neighbourhood of ;(iii) converges to a finite limit as tends to 1. The significance of (i) is clear: it is required for the conditioning on to be well defined. Parts (ii) and (iii) are technical assumptions which play a role in the proof of Theorem 1. Part (iii) of Hypothesis is satisfied in many cases of practical interest, where has a simple asymptotic behaviour near . For example, (iii) is satisfied if for some positive (finite) constant and some (finite) constants and (the relation “” denotes asymptotic equivalence, in the sense that the ratio tends to 1, as )

Here is the verification that (6) implies that (iii) holds, in the case that . (The case is similar but easier, so the details will be omitted.) Applying (6) in the first step and L’Hospital’s rule in the second step, as tends to 1, we have so (iii) is satisfied with the value of the limit being . It will be clear that the proof of Theorem 1 is easily generalized to allow the limit in (iii) to be infinite. However, it is not apparent, given the previous example, whether such a situation can actually occur. For this reason, we have taken the limit to be finite in the formulation of (iii).

Based on the conditional joint probability , we obtain the following result for the limiting correlation.

Theorem 1. Let and be two Bernoulli mixture random variables with correlation, , as in (2). Suppose that Hypothesis holds. One further assumes that , is differentiable for in a deleted neighbourhood of 1, and exists. Then, the limiting correlation in (4) exists and satisfies

Proof. See the appendices.

In the context of a sequence of Bernoulli mixture random variables, the notation, “,” suppresses the dependence on and ; more precisely, it should be written as . In order to have a result which does not actually depend on or , an extra hypothesis on the Bernoulli mixtures must be imposed. We will return to this point in Section 2.2.

An interesting example is the case that both terms take on identical values (comonotonicity) (here, comonotonicity refers to the case that all the components (identically distributed) of a Bernoulli random vector coincide in value with probability one, conditional on ); that is, . Under Hypothesis , Theorem 1 implies that the limiting correlation for this case is 1. The following corollary, another important example, states that the limiting correlation between two arbitrary terms from an identically distributed Bernoulli mixture sequence converges to zero, under the assumption of conditional independence.

Corollary 2. Suppose that two Bernoulli mixture random variables and are conditionally independent given the common default probability . Then, under Hypothesis , the limiting correlation in (4) exists and satisfies

Proof. Under the conditional independence assumption, . The desired result follows immediately from Theorem 1:

Remark 3. Results similar to those of Theorem 1 and Corollary 2 can also be given for the other extreme, conditional on with tending to 0. One simply replaces, in Hypothesis and Theorem 1, the phrase “neighbourhood of ” with “neighbourhood of ” and (iii) with “ converges to a finite limit as tends to 0;” then, in Theorem 1 and Corollary 2, one replaces “” with “.” (Note that is automatically 0 because .) However, in our applications, it is the case “” which is important, so we will not dwell further on the case “.”

2.2. Double Mixture Models

We now turn to the construction of a more general probability structure beyond conditional independence—a structure which reveals the fundamental role of the quantities, . We require a stronger assumption on the sequence than their being just conditionally identically distributed, which is a statement about the 1-dimensional marginals. We require that be conditionally exchangeable, meaning that, under each Pr, all permutations of have a common joint distribution which may depend on . (Note that a sequence of conditionally iid random variables is conditionally exchangeable, as is a comonotonic sequence.) In particular, conditional exchangeability is sufficient to guarantee that , , and hence the limiting correlation in Theorem 1, are independent of and .

We assume, for the remainder of the paper, that our sequence of Bernoulli mixture random variables is conditionally exchangeable. The quantities , , then do not depend on the choice of subset of variables. The probability that any of the Bernoulli random variables take on the value 1 is given in the following theorem which shows that the pmf of is essentially determined by the quantities , , which in turn can be specified quite generally as input to the model.

Theorem 4. Let be a sequence of conditionally exchangeable Bernoulli mixture random variables. Then, for , where and .
Conversely, let be a family of probability measures on the interval , such that and is a Borel measurable function of , for each Borel set, . If is defined to be , then , as defined by the right-hand sides of (11) and (12), is a valid pmf. Moreover, suppose that follows a Bernoulli mixtures distribution with conditional distribution for all which have precisely components equal to 1 and the remaining components equal to 0. Then, is conditionally exchangeable; for all ; hence, has conditional pmf given by (11) and (12). Furthermore, therefore, for all which have precisely components equal to 1 and the remaining components equal to 0, and hence

Proof. See the appendices.

Remark 5. For a general characterization of multivariate Bernoulli random variables, see Sharakhmetov and Ibragimov [19]. With the exchangeability assumption, (12) can be deduced from Theorem 3 of the aforementioned paper.

Remark 6. Under the assumption of conditional independence, reduces to the pmf of a binomial distribution (i.e., ).

Remark 7. The converse part of the theorem is connected with the easy part of the classical Hausdorff moment problem and its application to the proof of de Finetti’s theorem for sequences of exchangeable random variables (see, e.g., Chapter VII in Feller [20]).

Definition 8. Let , be as in Theorem 4, and let , for . A Bernoulli mixture model with conditional distribution as specified in (14) (or equivalently, (12)) and (13) is called a Bernoulli double mixture model. It is determined by the following distributions: the family of measures , and the mixing distribution, , and it can be formally denoted as a - Bernoulli double mixture model. Similarly, model (16), for the sum, , of the - Bernoulli double mixture random variables, can be formally referred to as a - binomial double mixture model.
In concrete examples, we will employ some meaningful, worded description or an acronym, rather than through the formal notation of Definition 8. In addition, for the remainder of the paper, all considered models will be Bernoulli (or binomial) double mixture models; therefore, we may omit those words from the names of models, simply retaining the description of the building blocks, and .

The following simple example illustrates the converse of the theorem.

Example 9 (ICM). We consider a weighted average of independence and comonotonicity. Specifically, for a specified weight , the random -vector is a two-point mixture of a conditionally comonotonic -vector of Bernoulli mixtures and a conditionally independent -vector of Bernoulli mixtures (with common mixing distribution, ), with respective weights and . For such a model, the (conditional) joint probability, , , is expressed as the weighted average of the joint probability of the comonotonic case and that of independence case: In practice, the weight parameter must be estimated from data or be chosen exogenously.
This model (with unspecified) will be denoted by the acronym ICM standing for independent comonotonic mixture (with everything being implicitly conditional). It corresponds to the following choice for , which is a double mixture of point masses: Note that follows from the usual understanding that .
Then, the limiting correlation is (cf. (4)) and the unconditional pmf of the sum, , is with given in (1). The mean and variance of the sum can be easily obtained as follows: Note that both the mean and variance are the weighted averages of those for the independent and comonotone cases. (In general, the th moment is the weighted average of the th moment of the independent and comonotone cases.)

2.3. Large-Sample Distribution of Empirical Frequency

The proportion of observed defaults to the total number of entities in a credit portfolio is of interest in practice. For example, the historical default rate for a homogenous credit portfolio is used to estimate the probability of default for a generic counterparty in the portfolio.

For a fixed , let denote the empirical frequency, which can be considered as the percentage gross loss of a portfolio of loans in equal dollar amounts.

The probability distribution of is

For the general binomial double mixture model, based on a family of probability measures, , and mixing distribution, , we have the following result:

Here is the proof of (22). The case is trivial. For , first note that by (14) where, for a nonnegative number , represents the greatest integer less than or equal to . The integrand is the cdf, at , of where ; so, by the LLN, as , , a.s. Therefore, for , where denotes the indicator function of a set, . (The second term, with the factor “1/2,” comes from an application of the CLT to , when .) For , Integrating these two limiting results with respect to yields the first two cases of (22). The interchange of limit and integrals is justified by the bounded convergence theorem.

Example 10 (ICM reprise). For the -weighted average of conditionally comonotonic and conditionally independent Bernoulli mixtures described in Example 9, we can easily obtain the following limiting distribution (this is the weighted average of the obvious limit for the conditionally comonotonic component and the convergence result in Vasicek [21] for the conditionally independent component of the mixture): Then, the mean and variance of the limiting distribution are In general, the th moment of the limiting distribution is
The limiting result (27) can be applied to the area of financial credit risk management. The tail risk measures of a large-sample portfolio credit loss distribution are of particular interest for financial risk managers. The two well-known risk measures, value at risk (VaR) and conditional tail expectation (CTE), can be approximated by using the limiting distribution, (27). Formally, for a loss random variable with the cumulative distribution function , VaR and CTE at confidence level are defined as follows:

Note that this definition reduces to

if is a continuous random variable (see Hardy [22] or Acerbi and Tasche [23]).

Then, by the scaling property of these risk measures and the limiting result (27), for a large , the VaR of the sum of Bernoulli mixtures at level can be approximated as follows:

where .

Similarly, the CTE at level can be approximated as

3. Beta Mixing Distribution

We now specialize the results of the preceding section to the case where the mixing distribution, , is the Beta distribution with parameters and . Specifically, the density is given by where and are the Gamma and Beta functions, respectively.

For a Beta Bernoulli mixture, several computationally convenient expressions are available. We first introduce a notation which is convenient for the cumulative distribution function: where is the incomplete Beta function. Equation (1), the pmf of the sum under the conditional independence assumption, can be evaluated as The th moments for and in this case are

The following lists several results for the ICM model with the Beta mixing distribution (-ICM model).

Example 11 (-ICM). The pmf of the binomial double mixture is given from (20) as The limiting distribution of the empirical frequency, (27), is Since the inverse cumulative Beta distribution function does not admit a closed-form expression, a numerical method is required to approximate the VaR of a large-sample credit portfolio. Given the VaR at level , the CTE can be written in a concise form. Specifically, where , and where .