The purpose of this chapter is to determine a loss distribution for a simple portfolio of two obligors. This shows how correlations alter the loss distribution of portfolios. The measures of risk are the expected loss (EL), the loss volatility (LV), and VaR, which is the loss percentile value L(a) at confidence level a.

After providing the example data, Section 48.2 details the loss distribution with a two-obligor portfolio with independent individual risks. In Section 48.3, the defaults are dependent. The correlation between the individual credit risks of the two obligors is modeled by defining standalone default probability and one conditional probability in this example. In both cases, the loss statistics characteristics are the expected loss, the loss volatility loss percentiles and compared. The comparison follows in Section 48.4.

The next chapters model the dependency effect in the general case, using the techniques of the main credit portfolio models. The example in this chapter is used in the subsequent chapters on risk allocation (Chapters 53 and 54).

TABLE 48.1 Portfolio of two obligors

PORTFOLIO OF TWO OBLIGORS

The next sections develop the case of a simple portfolio with two obligors A and B subject to default risk. Table 48.1 describes the portfolio, with the unconditional (or standalone) default probabilities and the exposures of each. The loss given default is 100% of exposure. The loss distribution includes four points1. The tables provide the intermediate calculations of the corresponding probabilities using the "tree" form used previously for modeling joint probabilities (Chapter 31). Correlation implies that conditional default probabilities differ from the unconditional default probabilities. In order to calculate the probabilities of the four possible loss values, we use the structure of conditional loss probabilities. We consider two cases: the simple case of independency between credit events and the case where there is a positive correlation. The case of independence serves for comparing loss statistics under independence and dependence.

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