3.2 Asset Value Model
3.2.1 Model dependence structure
The general assumption of the model is that the driver of credit events is the asset value of a company. This means one should be able to model the dependence structure between asset values of the counterparties. As in Cred- itMetrics, at this point we assume that the dependence structure between asset values of two firms can be approximated by the dependence structure between the stock prices of those firms. This fact offers a very natural solu- tion to the problem: if we are successful in modeling dependence structure between the stock prices and all relevant market risk factors (interest rates, exchange rates, etc.), then we accomplish simultaneously two goals:
Figure 3.1: The schema summarizes the calculation process by showing the data required for each step and highlighting the key components of the model.
• We construct dependency between credit risk events of our obligors
• We model dependency between market risk factors and the credit risk drivers
If one uses as a measure of dependence the correlation between risk factors (as in CreditMetrics), the above task is trivial — all one needs is to estimate the correlation matrix for the stock prices and the relevant market risk factors.
The correlation is a widespread concept in modern finance and insurance and stands for a measure of dependence between random variates. However, as we noted in Section 1.5.6, this term is very often incorrectly used to mean
any notion of dependence. Actually correlation is one particular measure of dependence among many. Of course, in the world of multivariate normal distribution and, more generally in the world of spherical and elliptical dis- tributions, it is the accepted measure, see Chapter 1 for more information regarding spherical and elliptical distributions. Yet empirical research shows that real data seldom seems to have been generated from a distribution be- longing to this class.
There are at least three major drawbacks of the correlation method. Let us consider the case of two real-valued random variables X and Y.
1. The variances of X and Y must be finite or the correlation is not defined. This assumption causes problems when working with heavy- tailed data. For instance the variances of the components of a bivariate t(n) distributed random vector forn≤2 are infinite, hence the corre- lation between them is not defined.
2. Independence of two random variables implies correlation equal to zero, the opposite, generally speaking, is not correct — zero correlation does not imply independence. A simple example is the following: Let X ∼ N(0,1) and Y = X2. Since the third moment of the standard normal distribution is zero, the correlation between X and Y is zero despite the fact that Y is a function of X which means that they are dependent. Only in the case of multivariate normal distribution zero correlation and independence are interchangeable notions. This statement is not valid if only the marginal distributions are normal and the joint distribution is non-normal. The example on Figure 3.2 illustrates this fact
3. The correlation is not invariant under non-linear strictly increasing transformationsT: R→Rwhich is a serious disadvantage. In general
corr(T(X), T(Y))6= corr(X, Y).
A more prevalent approach is to model dependency using copulas, see Sec- tion 1.5.6 for more details on copulas. The use of copulas offers the following advantages:
Figure 3.2: 5000 simulated data from two bivariate distributions with nor- mal marginals and identical correlation of 0.9 but different dependence struc- tures.
• The nature of dependency that can be modeled is more general. In comparison, only linear dependence can be explained by the correla- tion.
• Dependence of extreme events can be modeled.
• Copulas are indifferent to continuously increasing transformations (not only linear as it is true for correlations): If (X1, . . . , Xn)t has copula
C and T1, . . . , Tn are increasing continuous functions, then the vector
(T1(X1), . . . , Tn(Xn))talso has copula C.
This is extremely important in Asset-Value models for credit risk, be- cause this property postulates that the asset values of two firms shall have exactly the same copula as the stock prices of these two companies. The latter is true if we consider the stock price of a company as a call option on its assets and if the option pricing function giving the stock price is con- tinuously increasing with respect to the asset values. Cognity Credit Risk Module supplies both models for describing dependence structure:
• The simplified approach using correlations like a measure for depen- dency
• The copula approach. Our model accounts for the joint extreme move- ments at the same time not incurring significant computational bur- den. This is done by different treatment of the extremes and the body of the distribution trying to avoid in this way the curse of dimension- ality.
As a conclusion to this part of the discussion it is worth saying that in case there is no information about stock prices for a given obligor we employ the idea of segmentation described in CreditMetrics. The essence of this approach is that the user determines the percentage of the allocation of obligor volatility among the volatilities of certain market indexes and explains the dependence between obligors by the dependence of the market indexes that drive obligor volatility.