Sum of Bernoulli Mixtures: Beyond Conditional Independence

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.

Valuing Retail Credit Tranches with Structural, Double Mixture Models

“…Note that, under the typical conditional independence assumption, the distributional property of heavy-tailedness of the systematic factor, is the single source of dependence between the default indicator random variables. As shown theoretically in Bae and Iscoe (2014), the conditional default correlation under stress converges to zero under the typical conditional independence assumption, regardless of the distributional assumption on the systematic factor. This result naturally motivates the construction of a credit-risk model with additional sources of dependence, especially for stress situations.…”

“…Typical credit‐risk models assume that the X it s are independent, given P t . Bae and Iscoe () proposed a way to incorporate further dependence among the Bernoulli mixtures beyond conditional independence and studied distributional properties of the sum, . Here, in the interest of convenience, we state the converse part of the theorem in Bae and Iscoe ().…”

“…The choice of the second mixing component,, is arbitrary. Bae and Iscoe () considered the following two examples:…”

“…For the double mixture model, based on a family of probability measures, , and the mixing distribution (4), the large‐sample distribution of the default rate is See Bae and Iscoe () for a derivation of (7).…”

“…Particular attention is paid to the valuation of structured credit tranches on the portfolio. The new class of models was introduced in Bae and Iscoe () and termed double mixture models.…”