MODELING DEFAULTS IN A UNIFORM PORTFOLIO:THE LIMIT DISTRIBUTION

The limit distribution refers to the loss distribution of a granular portfolio widely diversified, with a uniform correlation resulting from systematic risk due to a single common factor. Each obligor is characterized by a standard normal asset value and a default threshold.

The Uniform Granular Portfolio

The default probability is identical for all obligors and all pairs of asset values have the same uniform correlation. This default probability is equal to the expected loss, if loss given defaults are also identical and equal to 1. The weights of each loan are identical. This is a limit case, but a proxy for a widely diversified granular portfolio of which exposures and correlations are similar. Loans are assumed to have same maturity M. This simplifies the modeling of the loss distribution, since we stay in default mode, ignoring migrations if horizon extends beyond maturity.

Such restrictive assumptions can be relaxed by using Monte Carlo simulation. For example, single factor models allow generating correlated asset values without making any restrictive assumptions on the default point, or default probability, and size of exposure. The Cholesky decomposition can impose a variance-covariance structure on obligor's asset values. In the next chapter on modeling loss distribution for credit portfolio (Chapter 50), the equal exposure and equal default probability assumptions are relaxed.

The case of uniform correlation is of interest for "homogeneous portfolios" with same default probability and uniform asset correlation in the same credit class, such as Basel 2 pools of loans. The portfolio is made of TV credits, with subscript i, with all asset values A. following bivariate normal distributions correlated with a single factor Z, with a uniform correlation p:

X. is the specific risk of the obligor "i," equal to the standardized normal asset value variable. For A. and Z to follow a bivariate distribution, Z and all X. have to be independent standard normal variables.

Each individual loss is /.. The individual exposures / are equal to a common value /, equal to the exposure multiplied by the percentage loss under default. Default is a Bernoulli variable 1 with value 1 when an obligor defaults and 0 under no default, or an indicator function. Using the indicator function of default allows expressing the random individual loss as 1./. The random portfolio loss Lp is the summation of losses of individual obligors, or Lp = t Nll.

The individual probability of default is equal to d. Each individual loss 11 has an expected value for a single loan equal to default probability times the loss:

The portfolio loss Lp has an expectation equal to the sum of the expectations of individual loans: E(L7) = ¿(1./) = Ndl. The portfolio loss, in percentage of total portfolio exposure, is /p. It is the ratio of total portfolio loss divided by the total of individual exposures X ■ x N/ — N1. The expected value of the portfolio loss percentage is:

If the default probability depends on the state of the economy, all conditional default probabilities are equal to the same a, conditional on the state of the economy, and named a(Z).

The portfolio loss is equal to conditional default probability on the state of the economy. It is random because asset values depend on Z, and any value of asset value lower than the default point matching the unconditional default probability triggers default. As shown above, conditional default probability varies inversely with the state of the economy.

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