# LO 9.2: Describe the Gaussian copula and explain how to use it to derive the joint

LO 9.2: Describe the Gaussian copula and explain how to use it to derive the joint probability o f default o f two assets.
A Gaussian copula maps the marginal distribution of each variable to the standard normal distribution which, by definition, has a mean of zero and a standard deviation of one. The key property of a copula correlation model is preserving the original marginal distributions while defining a correlation between them. The mapping of each variable to the new distribution is done on percentile-to-percentile basis.
Figure 1 illustrates that the variables of two unknown distributions Xand Khave unique marginal distributions. The observations of the unknown distributions are mapped to the standard normal distribution on a percentile-to-percentile basis to create a Gaussian copula.
Figure 1: Mapping a Gaussian Copula to the Standard Normal Distribution
Distribution of X
Distribution of Y
For example, the 5 th percentile observation for marginal distribution X is mapped to the 5th percentile point on the univariate standard normal distribution. When the 5th percentile is mapped, it will have a value of 1.645. This is repeated for each observation on
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a percentile-to-percentile basis. Likewise, every observation on the marginal distribution of Fis mapped to the corresponding percentile on the univariate standard normal distribution. The new joint distribution is now a multivariate standard normal distribution.
Now a correlation structure can be defined between the two variables X and Y The unique marginal distributions of X and Y are not well-behaved structures, and therefore, it is difficult to define a relationship between the two variables. However, the standard normal distribution is a well-behaved distribution. Therefore, a copula is a way to indirectly define a correlation relationship between two variables when it is not possible to directly define a correlation.
A Gaussian copula, CG, is defined in the following expression for an ^-variate example. The joint standard multivariate normal distribution is denoted as Mn. The inverse of the univariate standard normal distribution is denoted as A^-1. The notation denotes the n x n correlation matrix for the joint standard multivariate normal distribution M n.
C q [Gi(ui),…,G n(un)] Mn ^ HG^uijjj.-.jNn 1(Gn(un));
– l
In finance, the Gaussian copula is a common approach for measuring default risk. The approach can be transformed to define the Gaussian default time copula, CGD, in the following expression:
CGD[Qi. . . . . . Qn(t)] = Mn N f 1(Q 1(t)),…,N~1(Q n(t)); pM 1
1
Marginal distributions of cumulative default probabilities, Q(t), for assets / = 1 to n for fixed time periods t are mapped to the single w-variate standard normal distribution M with a correlation structure of pM. The term N j’^Q ^t)) maps each individual cumulative default probability for asset i for time period t on a percentile-to-percentile basis to the standard normal distribution.
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Example: Applying a Gaussian copula
Suppose a risk manager owns two non-investment grade assets. Figure 2 lists the default probabilities for the next five years for companies B and C that have B and C credit ratings, respectively. How can a Gaussian copula be constructed to estimate the joint default probability, Q, of these two companies in the next year, assuming a one-year Gaussian default correlation of 0.4?
Figure 2: Default Probabilities of Companies B and C
Time, t
B Default Probability
C Default Probability
1 2 3 4 5
0.065 0.081 0.072 0.064 0.059
0.238 0.152 0.113 0.092 0.072
Professor’s Note: Non-investment grade companies have a higher probability o f default in the near term during the company crisis state. I f the company survives past the near term crisis, the probability o f default will go down over time.
In this example, there are only two companies, B and C. Thus, a bivariate standard normal distribution, M2, with a default correlation coefficient of p can be applied. With two companies, only a single correlation coefficient is required, and not a correlation matrix of
CGD [Q b (t).Qc(t)l = M2 N – 1 (Q b (t)),N -1 (Qc(t)) p
-1
Figure 3 illustrates the percentile-to-percentile mapping of cumulative default probabilities for each company to the standard normal distribution.
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Figure 3: Mapping Cumulative Default Probabilities to Standard Normal Distribution
Time, t
B Default Probability
1 2 3 4 5
0.065 0.081 0.072 0.064 0.059
Q a(t)
0.065 0.146 0.218 0.282 0.341
N”;(Q s(t))
C Default Probability
-1.513 -1.053 -0.779 -0.577 -0.409
0.238 0.152 0.113 0.092 0.072
Q cto 0.238 0.390 0.503 0.595 0.667
N -‘(Q c (t))
-0.712 -0.279 0.008 0.241 0.432
Columns 3 and 6 represent the cumulative default probabilities Q^(t) and Qc (t) for companies B and C, respectively. The values in columns 4 and 7 map the respective cumulative default probabilities, Q^(t) and Qc (t), to the standard normal distribution via N _1(Q(t)). The values for the standard normal distribution are determined using the Microsoft Excel function =NORMSINV(Q(t)) or the MATLAB function =NORMINV(Q(t)). This process was illustrated graphically in Figure 1.
The joint probability of both Company B and Company C defaulting within one year is calculated as:
Q (tB < i r n c < l ) = M ( X B < -1 .5 1 3 n X c < -0 .7 1 2 ,p = 0.4) = 3.4% Professors Note: You will not be asked to calculate the percentiles for mapping to the standard normal distribution because it requires the use o f Microsoft Excel or MATLAB. In addition, you will not be asked to calculate the joint probability o f default for a bivariate normal distribution due to its complexity. C o r r e l a t e d D e f a u l t T i m e