familiar distribution is contradicted by empirical data. In order to relax the independence assumption, Moodys proposes to group the obligors into 1 ≤ D ≤ I classes. Within each class, obligors are assumed to be completely dependent, and can then be treated as a single firm with exposure LI/D, while the different classes are deemed to be fully independent. For example, with D = I we recover full independence between obligors, while for D = 1 we have complete dependence. The parameter D is called the diversity score of the portfolio, and the intermediate cases between full independence and complete dependence can now be analyzed for varying values of D.
For a fixed D, the distribution of defaults can be calculated as a binomial distribution.
Therefore, the BET loss distribution for a diversity score D is given by P [X ≤ x] =
This method is not based on any theoretical model, and the parameter D used by Moodys is determined by a complicated recipe. Moreover, its loss resolution is limited to intervals of size I/D, which are often too coarse. In order to have a deeper understanding of default correlation, we must resort to our previous models for single-name default.
Nevertheless, the BET method became something of a market standard to which more detailed models should be compared.
Warning: This method seriously underestimates “tail risk” of large losses!
7.4 Default Correlation in Structural Models
Default correlation can be incorporated quite naturally into structural models by making the asset value dynamics of different firms correlated through time. We illustrate the main ideas with the example of two firms whose assets follow the dynamics
dAit= Ait[(µi− δi)dt + σidWti], Ai0 > 0, i = 1, 2 (7.10) where W1 and W2 are correlated Brownian motions with constant correlation ρ. Then for the classical Merton model, where default of firm i occurs at time Ti if AiT
i < Ki, we obtain that the joint default probability is
p(T1, T2) = P [A1T1 < K1, A2T2 < K2]
7.4.1 CreditMetrics Approach
CreditMetrics, an industry method developed by Moody’s and described in [15], extends the Merton model by assuming that the credit rating of a firm is its basic measure of credit quality, and that changes in credit rating can be attributed to changes in the firm’s asset value. One chooses a fixed time duration ∆t, and a specific rating transition matrix Π := Π∆tsimilar in form to (6.31). Then for each possible initial rating class k = 1, . . . , K one computes a set of marks dk0 = −∞ < dk1 < · · · < dkK+1 := ∞ such that the final rating is determined by the draw of a standard normal variate Xi:
Πkj := P [Y∆ti = j|Y0i = k] = P [dkj < Xi ≤ dkj+1] = Φ(dkj+1) − Φ(dkj) (7.13) Roughly this can be interpreted by relating the asset return over the period to Xi:
log(Ai∆t/Ai0) = (µi− δi− (σi)2/2)∆t + σi√
∆tXi. (7.14)
Figure 7.1: CreditMetrics mapping of the transition matrix to the normal distribution.
In the multifirm setting, one treats the vector of asset returns (X1, X2, . . . , XI) as a correlated multivariate normal N (0, Σ) with N (0, 1) marginals, and a specified correlation matrix Σ (a positive definite matrix with 1 in each diagonal entry). Several alternatives exist for determining Σ. One can assume a strong correlation between Ai and the firm’s market capitalization (stock price times the number of shares), and thus simply take Σ to be exactly the correlation matrix determined by historically observed stock returns. The CreditMetrics people do this somewhat more precisely, by extending the method described in Section 4.4 to back out the asset return correlation matrix Σ from the observations of the multivariate stock returns.
7.4. DEFAULT CORRELATION IN STRUCTURAL MODELS 103 Remark 9. The CreditMetrics method is widely used, but should be viewed with caution for credit risk management. Its weakness is of course its reliance on the multivariate normal distribution: the kinds of default dependence that it produces are of the tame
“thin tailed” type that have proved to be utterly useless in explaining the 2007-2008 credit crunch.
7.4.2 First Passage Models
To illustrate this approach, we consider a two-firm first-passage model with barriers D1 and D2. Then the joint probability for firm 1 to default before time T1 and firm 2 to default before time T2 is given in closed form by
p(T1, T2) = Ψ2 ρT1∧ T2 distribution function with correlation ρ and parameters m1, a, m2, b [38]. A similar result exists for an exponentially growing time-dependent barrier in terms of modified Bessel functions.
No general analytical first passage formulas are known for three or more firms, and one must resort to Monte Carlo simulations to compute.
7.4.3 Factor Models of Correlation
In all of these approaches, we can obtain a full range of Gaussian correlations between default events. The results carry over in principle for a larger number of firms, although the number of parameters gets quickly out of hand. We would need to specify a full I × I correlation matrix in order to obtain the entire dependence structure for I firms. This is still small compared to the 2I correlated default events which would have to be considered if we were not dealing with Gaussian random variables, but it is nevertheless a paralyzing task if we have, say, 100 obligors.
Instead, one simplifies the picture by introducing factor models, also known as Bernoulli mixture models (see [28]). For example, in the CreditMetrics framework, a one-factor model corresponds to assuming that Xi = (σi√
∆t)−1(log(Ai∆t/Ai0)−(µi−δi−(σi)2/2)∆t) ∼ N (0, 1) is such that
Xi = βiZ + ¯βii, β¯i :=p
1 − (βi)2 (7.16)
where Z and iare independent standard normal random variables. This has the interpre-tation that asset values for all firms are driven by one common factor Z plus firm specific (idiosyncratic) factors i. Then the entire dependence structure is reduced to specifying the correlation parameters βi. One can attempt to connect Z to some observed macroe-conomic factor such as an emacroe-conomic activity indicator like GDP: there have been many studies of this type, for example [9] and the references therein.
To further investigate the implications of the one-factor model, let us assume that the credit state of individual firms depends only on their rating, so that βi = β for all firms.
Then the two-firm transition probabilities are computed by P [Y∆t1 = j1, Y∆t2 = j2|Y01 = k1, Y02 = k2] := Π(2)∆t(j1, j2; k1, k2) where Φ2 is the bivariate standard normal distribution function.
The probability of observing up to n defaults in a set of I firms in rating k in time
∆T is where φ is the standard normal density and the probability of one firm’s default conditional on Z = z is
P [τ ≤ ∆t|Z = z; k] := pk(z) = Φdk1− βz β¯
. (7.19)
Note the unconditional default probability for a k rated firm in the time period [0, ∆t] is F (∆t; k) =
Z
R
pk(z)φ(z)dz . (7.20)
Most generally, one can determine the probability that exactly n1, . . . , nK defaults occur from each of the K rating classes, given that there are initially I1, . . . , IK in each
We can obtain more explicit results than (7.18) in the large portfolio approximation, that is, assuming that I → ∞. Then, by LLN, the Law of Large Numbers, pk(z) given by (7.19) represents the conditional average of the fraction of a portfolio of I firms all with rat-ing k that experiences default over the period [0, ∆T ]. Assumrat-ing we normalize the expo-sures Li = 1 and take zero-recovery, the fractional loss per firm is L = I−1PI
i=11{Yi
∆t=0}. This fraction is exactly the portfolio default loss per firm and by LLN, its CDF is then approximated by
7.5. DEFAULT CORRELATION IN REDUCED-FORM MODELS 105