LO 9.3: Summarize the process o f finding the default time o f an asset correlated to | Ken Li, FRM

LO 9.3: Summarize the process o f finding the default time o f an asset correlated to all other assets in a portfolio using the Gaussian copula.
When a Gaussian copula is used to derive the default time relationship for more than two assets, a Cholesky decomposition is used to derive a sample Mn () from a multivariate copula Mn() C [0,1]. The default correlations of the sample are determined by the default correlation matrix pM for the -variate standard normal distribution, Mn.
The first step is to equate the sample Mn() to the cumulative individual default probability, Q, for asset i at time t Microsoft Excel or a Newton-Raphson search procedure.
using the following equation. This is accomplished using
M n W = Q i ( T i )
Next, the random samples are repeatedly drawn from the -variate standard normal distribution Mn() to determine the expected default time using the Gaussian copula.
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Random samples are drawn to estimate the default times, because there is no closed form solution for this equation.
Example: Estimating default time
Illustrate how a risk manager estimates the expected default time of asset i using an ^-variate Gaussian copula.
Answer: of 3.5 years. This process is then repeated 100,000 Suppose a risk manager draws a 25% cumulative default probability for asset i from a random 72-variate standard normal distribution, Mn(). The 72-variate standard normal distribution includes a default correlation matrix, pM, that has the default correlations of asset i with all n assets. Figure 4 illustrates how to equate this 25% with the market determined cumulative individual default probability Q^Tj). Suppose the first random sample equates to a default time t of 3.5 years. This process is then repeated 100,000 times to estimate the default time of asset i.
Figure 4: Mapping Default Time for a Random Sample
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K e y C o n c e p t s
LO 9.1
The general equation for correlation copula, C, is defined as:
C [G i(ui),…,Gn(un)] Fn
1(G1(u^)),…,Fn 1(Gn(un));pp
The notation for this copula equation is translated as: G-(uJ are marginal distributions, F is the joint cumulative distribution function, Zq-1 is the inverse function of i 7 , and pF is the correlation matrix structure of the joint cumulative function F .
LO 9.2
The Gaussian default time copula is defined as:
Cg d [QiM- QnW] Mn N f 1(Q i(t)),…,N -1(Q n(t));pM
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 Gaussian copula for the bivariate standard normal distribution, M2, for two assets with a default correlation coefficient of p is defined as:
CGD [ Q b (t). Q c Wl = M2 N – 1 (Q B (t)), N – 1 (Qc (t)) ; p
1
LO 9.3
Random samples are drawn from an -variate standard normal distribution sample, Mn (), to estimate expected default times using the Gaussian copula:
M n W = Q i ( T i )
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C o n c e p t C h e c k e r s
1.
Suppose a risk manager creates a copula function, C, defined by the equation:
C[G1(u1),…,Gn(un)] = Fn Fj 1 ( G j ( u j F n 1 (Gn(un)); pF
-l
Which of the following statements does not accurately describe this copula function? A. G-fu-J are standard normal univariate distributions. B. F is the joint cumulative distribution function. C. Zq-1 is the inverse function of F that is used in the mapping process. D. /?p is the correlation matrix structure of the joint cumulative function F .
Which of the following statements best describes a Gaussian copula? A. A major disadvantage of a Gaussian copula model is the transformation of the
original marginal distributions in order to define the correlation matrix.
B. The mapping of each variable to the new distribution is done by defining a
mathematical relationship between marginal and unknown distributions. C. A Gaussian copula maps the marginal distribution of each variable to the
standard normal distribution.
D. A Gaussian copula is seldom used in financial models because ordinal numbers
are required.
A Gaussian copula is constructed to estimate the joint default probability of two assets within a one-year time period. Which of the following statements regarding this type of copula is incorrect? A. This copula requires that the respective cumulative default probabilities are
mapped to a bivariate standard normal distribution.
B. This copula defines the relationship between the variables using a default
correlation matrix, pM.
C. The term A ^ ^ Q ^ t)) maps each individual cumulative default probability for
asset i for time period t on a percentile-to-percentile basis.
D. This copula is a common approach used in finance to estimate joint default
probabilities.
A risk manager is trying to estimate the default time for asset i based on the default correlation copula of asset i to n assets. Which of the following equations best defines the process that the risk manager should use to generate and map random samples to estimate the default time? A. CGD [QB(t),Q c (t)] = M2 [n -V Q b M J.N-‘CQ c M) ; p
B. C[GI(u1),…,Gn(un)] = Fn F[ 1 ( G ^ u , F n ‘( G J u J ) ; pF
-1
C. CGD[Q i(t),…,Q n(t)]-M n Nj 1(Q 1(t)),…,Nn 1(Q n(t)); pM
i
D. Mn(.) = Q i ( T i )
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Suppose a risk manager owns two non-investment grade assets and has determined their individual default probabilities for the next five years. Which of the following equations best defines how a Gaussian copula is constructed by the risk manager to estimate the joint probability of these two companies defaulting within the next year, assuming a Gaussian default correlation of 0.35? A. CGD [QB(t),Qc (t)] = M2 [n _ 1 (Q b (t)). N 1 (Qc (t)); p
B. CfG^u,)…. G(un)] = F F f1(G,(u1)),…,Fn-I(Gn(un));pF c. CGD [Qi(t),…,Qn(t)] = Mn [Nj (Q1(t)),…,N ‘(Q n(t)); pM D. Mn(.) = Q i(T=)
-1
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C o n c e p t C h e c k e r An s w e r s
1. A
are marginal distributions that do not have well-known distribution properties.
2. C Observations of the unknown marginal distributions are mapped to the standard normal
distribution on a percentile-to-percentile basis to create a Gaussian copula.
3. B Because there are only two companies, only a single correlation coefficient is required and
not a correlation matrix, pM.
4. D The equation Mn() = Q i ( T i ) is used to repeatedly generate random drawings from the
72-variate standard normal distribution to determine the expected default time using the Gaussian copula.
5. A Because there are only two assets, the risk manager should use this equation to define the
bivariate standard normal distribution, Mv with a single default correlation coefficient of p.
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The following is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in:
E m p i r i c a l A p p r o a c h e s t o R i s k M e t r i c s a n d H e d g i n g
T opic 10
E x a m F o c u s
This topic discusses how dollar value of a basis point (DVOl)-style hedges can be improved. Regression-based hedges enhance DV01-style hedges by examining yield changes over time. Principal components analysis (PCA) greatly simplifies bond hedging techniques. For the exam, understand the drawbacks of a standard DVOl-netural hedge, and know how to compute the face value of an offsetting position using DV01 and how to adjust this position using regression-based hedging techniques.
DVO 1-N e u t r a l H e d g e