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In the KMV universe of listed firms, it is always possible to use the modelled Edf. In the private firm universe, there are no equity prices. Few firms belong to the KMV universe of modelled individual asset returns, among those of banks portfolios. In addition, credit officers prefer to assign ratings using both quantitative criteria and judgmental information to capture all credit risk dimensions, rather than using mechanical models. In this case, a common practice is to map internal ratings to default probabilities. Outside the KMV universe, it becomes necessary to de-link the model from asset values because they are unobservable, and to re-link the option model of default to the preset default probability. The option framework extends, in a simplified version, to the modelling correlations between credit events, migrations and defaults, of different firms when preset default probabilities are assigned rather than modelled.

Chapter 44 explains how to derive correlated default events using the simplified model. The solution is to use a simplified default option framework, given that the assigned default probability embeds all relevant information about the underlying asset value. This simplified model assumes that asset values follow standardized normal distributions (expectation 0 and standard deviation 1). Threshold values corresponding to a given default probability derive from the tables of the standardized normal distribution. If the default probability is 2.5%, the corresponding standardized asset value is —1.96; if the probability of default is 1%, it becomes —2.33. If the asset value falls between two threshold values, we are in the risk class bounded by the upper and lower default probabilities. Considering only one obligor, there is a one-to-one correspondence between the asset value A_X triggering the default of obligor X and the default probability DP_X if we stick to a standardized distribution of asset values (mean 0 and volatility 1) (Figure 38.3).

This correspondence results from: Prob(asset value < A_X) = DP_X. Formally, $(A_X) = DP_X is equivalent to A_X = $^—1(DP_X),where $(A_X) is the cumulative function of a

FIGURE 38.3   Asset value and default probability

standard normal distribution, of mean 0 and volatility 1. The absolute values of A_X deviations and the corresponding default probabilities are given in Table 38.5 and Figure 38.4.

As an example, an obligor X has a 1% default probability. Under the simplified standardized distance to default model, the current distance to default is 2.33. The standardized asset value should move down by 2.33 to trigger default or, equivalently, the default threshold for Xs asset value is A_X = —2.33. The financial interpretation is that the debt value is lower than the asset value by 2.33 units. The initial default probability suffices entirely to define a standardized normal distribution of asset values. The next section uses this model to derive transition probabilities from an initial state defined by a preset default probability.