In this paper we modelled portfolio credit risk as an evolving function of time.
The default probability of any obligor at any time depended on common systemic covariates, class dependent covariate and idiosyncratic random variables - we allowed a fairly general representation of conditional default probabilities that subsumes popular logit, default intensity based representations, as well as threshold based Gaussian and related copula models for defaults. The evolution of systemic covariates was modelled as a VAR(1) process. The evolution of class dependent covariates was modelled as an independent AR process (independent of systemic and other class co- variates). We further assumed that these random variables had a Gaussian distribution.
In this framework we analyzed occurrence of large losses as a function of time. In particular, we characterized the large deviations rate function of large losses. We also observed that this rate function is independent of the representation selected for conditional default probabilities.
This was a short note meant to highlight some of the essential issues. In our ongoing effort we build in more realistic and practically relevant features including:
1. We conduct large deviations analysis
a. when the class and the systemic covariates are dependent with additional relaxations including allowing exposures and recoveries to be random. Fur- ther, as in Duffie et al. (2007), we also model firms exiting due to other reasons besides default, e.g., due to merger and acquisitions. We also allow defaults at any time to explicitly depend upon the level of defaults occurring in previous time periods (see, e.g., Sirignano and Giesecke2015).
b. when the covariates are allowed to have more general fatter-tailed distribu- tions.
c. when the portfolio composition is time varying.
2. Portfolio large loss probabilities tend to be small requiring massive computational effort in estimation when estimation is conducted using naive Monte Carlo. Fast simulation techniques are developed that exploit the large deviations structure of large losses (see, e.g., Juneja and Shahabuddin2006; Asmussen and Glynn2007 for introduction to rare event simulation).
Appendix: Some Proofs
Let||x||2 =n
i=1xi2. Consider the optimization problem
min||x||2 (3.13)
s.t. n
j=1
ai,jxj≥bi i=1, . . . ,m, (3.14) and let x∗ denote the unique optimal solution of this optimization problem. It is easy to see from first order conditions that if(bi :i ≤m,bi >0), is replaced by (αbi :i ≤m),α >0, then the solution changes toαx∗.
Let(Xi :i ≤n)denote i.i.d. Gaussian mean zero variance 1 random variables and letd(n)denote any increasing function ofnsuch thatd(n)→ ∞asn → ∞.
The following lemma is well known and stated without proof (see, e.g., Glasser- man et al.2007).
Lemma 3.3 The following holds:
nlim→∞
1 d(n)logP
⎛
⎝n
j=1
ai,jXj ≥bid(n)+o(d(n)) i=1, . . . ,m
⎞
⎠= −||x∗||2.
Proof of Lemma3.1: Recall that we need to show that
nlim→∞
1
rn2logP(H)= −q∗(t). (3.15) where P(H)denotes the probability of the event that
t j=1
d k=1
ht−j,kEj,k+ t k=1
ηt−kΛ1,k ≥rnα1+rnδ
and t˜
j=1
d k=1
ht˜−j,kEj,k+
˜
t k=1
ηt˜−kΛ1,k ≤rnα1−rnδ for 1≤ ˜t≤t−1.
From Lemma3.3, to evaluate (3.15), it suffices to consider the optimization prob- lem (call itO1),
min t
k=1
d p=1
ek2,p+ t k=1
l2k (3.16)
s. t.
t k=1
d p=1
ht−k,pek,p+ t k=1
η1t−klk≥α1, (3.17)
and
˜
t k=1
d p=1
ht˜−k,pek,p+
˜
t k=1
ηt1˜−klk≤α1. (3.18)
for 1≤ ˜t ≤t−1.
We first argue that inO1, under the optimal solution, the constraints (3.18) hold as strict inequalities.
This is easily seen through a contradiction. Suppose there exists an optimal solu- tion(eˆk,p,lˆk,k≤t,p ≤d)such that fortˆ<t,
ˆ
t k=1
d p=1
htˆ−k,peˆk,p+
ˆ
t k=1
ηt1ˆ−klˆk=α1
and iftˆ>1, then for allt˜<tˆ(3.18) are always strict. We can construct a new feasible solution with objective function at least as small with the property that constraints (3.18) are always strict.
This is done as follows: Lets=t− ˆt. Sete¯k+s,p = ˆek,pfor allk≤ ˆtandp≤d.
Similarly, setl¯k+s = ˆlkfor allk≤ ˆt. Set the remaining variables to zero.
Also, since the variables(¯ek,p,l¯kk≤t,p≤d)satisfy constraint (3.18) with vari- ables(¯ek,p,l¯kk≤s,p≤d)set to zero, the objective function can be further improved by allowing these to be positive. This provides the desired contradiction. The specific form ofq∗(t)follows from the straightforward observation in Remark3.4. . Proof of Theorem3.1:
Now,
P(N )≥ P(N |H1τ)P(H1τ).
We argue that P(N|H1τ)converges to 1 asn → ∞. This term equals P
N1(τ)
n ≥aτ, τ−1
t=1 N1(t)
n ≤c1−aτ|H1τ
.
This may be further decomposed as P
τ−1
t=1 N1(t)
n ≤c1−aτ|(∩τ−t=11H˜1,t)
(3.19) times
P
N1(τ) n ≥aτ|
τ−1 t=1 N1(t)
n ≤c1−aτ,H1,τ
. (3.20)
To see that (3.19) converges to 1 asn → ∞, note that it is lower bounded by 1−
τ−1 t=1
P N1(t)
n ≥ε|(∩τ−1t=1H˜1,t)
forε=(c1−aτ)/(τ−1). Consider now, P
N1(1)
n ≥ε| ˜H1,1
This is bounded from above by
2c1nP(Z1,1(n)≥rnδ)εn
where 2c1n is a bound on number of ways at leastεn obligors of Class 1 can be selected fromc1nobligors. Equation3.19now easily follows.
To see (3.19), observe that this is bounded from above by 2c1nP(Zi,τ(n)≤ −rnδ)(c1−aτ)n Since this decays to zero asn→ ∞, (3.19) follows.
In view of Lemma3.1, we then have that
nlim→∞
1
rn2logP(N ∩H1τ)= −q∗(τ),
and thus large deviations lower bound follows. To achieve the upper bound, we need to show that
lim sup
n→∞
1
rn2 logP(N )≤ −q∗(τ). (3.21) Observe that
P(N)≤ P(Hτ+Y1,τ ≥rnα1−rnδ)+P
N1(τ)
n ≥aτ,Hτ+Y1,τ ≤rnα1−rnδ
.
Now, from Lemma3.3and proof of Lemma3.1,
nlim→∞
1
rn2logP(Hτ+Y1,τ ≥rnα1−rnδ)= −q∗(τ).
Now,
P
N1(τ)
n ≥aτ,Hτ+Y1,τ ≤rnα1−rnδ
is bounded from above by
2nP(Zi,τ >rnδ)naτ so that due to Assumption2,
lim sup
n→∞
1 rn2 logP
N1(τ)
n ≥aτ,Hτ+Y1,τ ≤rnα1−rnδ
= −∞,
and (3.21) follows.
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Extreme Eigenvector Analysis of Global Financial Correlation Matrices
Pradeep Bhadola and Nivedita Deo
Abstract The correlation between the 31 global financial indices from American, European and Asia-Pacific region are studied for a period before, during and after the 2008 crash. A spectral study of the moving window correlations gives significant information about the interactions between different financial indices. Eigenvalue spectra for each window is compared with the random matrix results on Wishart matrices. The upper side of the spectra outside the random matrix bound consists of the same number of eigenvalues for all windows where as significant differences can be seen in the lower side of the spectra. Analysis of the eigenvectors indicates that the second largest eigenvector clearly gives the sectors indicating the geographical location of each country i.e. the countries with geographical proximity giving similar contributions to the second largest eigenvector. The eigenvalues on the lower side of spectra outside the random matrix bounds changes before during and after the crisis.
A quantitative way of specifying information based on the eigenvectors is constructed defined as the “eigenvector entropy” which gives the localization of eigenvectors.
Most of the dynamics is captured by the low eigenvectors. The lowest eigenvector shows how the financial ties changes before, during and after the 2008 crisis.