财务管理英文论文1
Research on CreditRisk+ Model Based on Severity Variation and Sector Correlation
of Multi-system Risk Factors1
Jiangang Peng * Zhihua Lv
Research Center of Financial Management Research Center of Financial Management
Hunan University Hunan University Changsha, P. R. China 410079 Changsha, P. R. China 410079 [email protected] [email protected] Jing He
Research Center of Financial Management Hunan University Changsha, P. R. China 410079 [email protected]
Lei Tong
Research Center of Financial Management
Hunan University
Changsha, P. R. China 410079 [email protected]
Abstract
On the basis of multi-system risk factors CreditRisk+ model based on sector character, this paper takes account of the variation of loss given default, and proposes the CreditRisk+ model on severity variation and sector correlation of multi-system risk factors. This new model fully takes account of the correlation between the sector risk factors and the variation of the obligor’s loss given default, so the accuracy of calculating unexpected loss of loan portfolio with this model is enhanced. In this paper, we give a practical example to analyze this feasibility of the new model by taking use of saddlepoint approximation algorithm.
Keywords: CreditRisk+ model; loss given default; probability of default; saddlepoint approximation; default correlation
1 Introduction
CreditRisk+ (CR+) model was launched by the Credit Suisse Financial Products (CSFP) in 1997, which is a model of credit risk measurement and portfolio management [1]. Since it has small data requirement and provides the closed form solution, this model has drawn significant interest of practitioners and theorists. However, this model still has some shortcomings.
One of the obvious limitations of the CR+ model is the assumption that the sector risk factors determining the default probabilities are mutually independent. In reality, because sectors are affected by the same macroeconomic variables, there exist some correlations among different sector risk factors. If those factors are assumed to be independent, the risk of loan portfolio will be underestimated. Burgisser et al. (1999) were the first to introduce correlations among the sector risk factors and
1
Support by National Natural Science Foundation of China (NSFC) (No. 70673021)
proposed the single-factor model (SF model) [2]. On the condition that the covariance matrix was given, they computed the relative default variance of the loss distribution with the weighted-average method. Then, on the assumption that all the loans’ default probabilities are only affected by one common system risk factor which follows gamma distribution with mean of 1 and variance as the relative default variance. On this basis, they computed the loss distribution of the loan portfolio. However, the SF model has some obvious shortcomings. For a specific loan portfolio, the VaR of a certain confidence level computed by the SF model is even smaller than that calculated by the original CR+ model. Knowing the shortcomings of the SF model, Giese (2003) proposed the Compound Gamma CreditRisk+ model (CG model) after deeply researching the correlation of the sector risk factors [3]. However, the CG model requires that all covariance between any two sector risk factors are equal, which is not realistic. Based on the CG model, Our team (2008) introduced multi-system risk factors, and denoted the shape parameter of sector risk factor by the product of the linear combination of system risk factors multiplying a parameter that reflects sector character, then we proposed the multi-system risk factors CR+ model based on sector character (MSRF model) [4]. The MSRF model can adopt general covariance matrix of sector risk factors into the framework of the model, thus overcomes the prominent drawback that covariance between different sector risk factors are required to be equal in the CG model and is a qualitative expansion to the original CR+ model. So it is a breakthrough development of the CR+ model.
The assumption that loss given default (LGD) keeps constant is another obvious shortcomings of the CR+ model. In reality, LGD is usually changing. In addition, when CR+ model is used to measure credit risk of loan portfolio in a long horizon, the volatility of the LGD affected by the change of macroeconomic, the value of collaterals and other factors must be taken into account. Without considering the change of the collateral value of the loan portfolio, using a constant LGD, the result will be inaccurate calculation of unexpected loss. On the basis of the SF model, Burgisser et al. (2001) revised the constant LGD limitation of the CR+ model [5]. They assumed that the volatility of the probability of default (PD) and the volatility of LGD are independent, and treat with the volatility of LGD in this model with the similar method used to deal with the volatility of PD in SF model. On this basis, they computed the variance of the loan portfolio loss distribution. Then they computed the probability generating function (pgf) of the loss distribution, and used panjer algorithm to obtain the unexpected loss on certain confidence level. However, because of the shortcomings of the SF model and the panjer algorithm, the outcome of the model is inaccurate. On the basis of original CR+ model, Cai Feng-jing et al. (2004) assumed that LGD followed Beta distribution, took the volatility of LGD into the CR+ model, and obtained the outcome by the saddlepoint approximation algorithm [6]. However, as we have pointed out, the original CR+ model doesn’t consider the correlation of the sector risk factors, so the model of Cai Feng-jing still has shortcomings. In this paper, on the basis of considering the correlation of the sector risk factors, we adopt the volatility of LGD into the CR+ model.
On the basis of the MSRF model and the method of measuring unexpected losses of loan portfolio with CR+ model for Chinese commercial banks which our team proposed [7], we take the volatility of LGD into consideration like Cai Feng-jing et al. (2004) and propose the CR+ model based on severity
variation and sector correlation of multi-system risk factors (SVSCM model). The SVSCM model fully takes account of the correlation between the sector risk factors and the variation of the obligor’s LGD, develop the CR+ model and improve the accuracy of the unexpected loss calculation of loan portfolio.
2 Proposing the CR+ model based on severity variation and sector
correlation of multi-system risk factors
2.1 Ca lcula t ion of the proba b ility generating functio n of default lo ss In the original CR+ model, for obligor A, affected by the sector risk factorsγk , k can be expressed as follows:
K
=1, 2, , K , the PD
p A (γ) =p A ∑g k A γk (1)
k =1
Where
∑g
k =1
K
A k
=1, p A is the unconditioned PD of obligor A.
The exposure of obligor A is v A , LGD is εA , then the pgf of default risk of obligor A can be expressed:
G A (z γ, εA ) =1−p A (γ) +p A (γ) z v A εA . (2)
2.1.1 Trea tment for the vola tility of LGD
In the original CR+ model, the obligor’s LGD is constant. In order to overcome this shortcoming, in this paper, we follow the assumption of CreditMetrics model that the LGD follows Beta distribution [8]. So (2) can be written as follows:
G A (z γ)=∫(1−p A (γ)+p A (γ)z v A x )⋅f a A , b A (x )dx
1
=1−p A (γ) +p A (γ) ⋅∫z v A x ⋅
1
Γ(a A +b A ) a A −1
⋅x ⋅(1−x ) b A −1dx . (3)
Γ(a A ) ⋅Γ(b A )
Where f a A , b A (x ) is the probability density function of εA . Since z
v A x
=e
v A x ln z
(v A x ln z ) n
, (4) =∑n ! n =0
∞
(v A x ln z ) n Γ(a A +b A )
⋅⋅x a A −1⋅(1−x ) b A −1dx So G A (z γ) =1−p A (γ) +p A (γ) ⋅∫∑0n ! Γ(a A ) ⋅Γ(b A ) n =0
1∞
a A +r (v A ln z ) n
=1−p A (γ) +p A (γ)[1+∑(∏) . (5)
n ! n =1r =0a A +b A +r
∞
n −1
a A +r (v A ln z ) n
⋅, (6) Let h A (z ) =1+∑(∏a +b +r n ! n =1r =0A A
∞
n −1
So G A
(z γ)=1−p (γ)+p (γ)h (z )=1+p
A
A
A
p A (γ)(h A (z )−1)
p A (γ)(h A (z ) −1)
(γ)(h (z ) −1) ≈e . (7) A A
So the pgf of the loan portfolio can be expressed:
G (z γ)=∏e
A
=e
∑
A
p A (γ)(h A (z )−1)
=e
∑∑
p A
A
k =1
K
A
g k γk (h A (z )−1)
=e k =1
∑γk P k (z )
K
. (8)
Where P k (z ) =
∑p
A
A
g k A (h A (z ) −1) .
2.1.2 Tr e a t men t fo r t h e c o rr e l a ti on of the s ector risk fa cto rs
In the original CR+ model, the sector risk factors γ1, γ2, , γK are mutually independent, and follow gamma distribution with mean 1. Their shape parameters and scale parameters are αk and
βk ,
k =1, 2, , K .
In order to introduce general correlation for the sector risk factors, this paper adopts the method
proposed by our team. We select some system risk factors Y 1, Y 2, , Y N from macroeconomic variables,
and assume those system risk factors follow gamma distribution with mean 1 and varianceδi ,
2
i =1, 2, , N . Sector risk factors are affected by system risk factor, so shape parameters can be
expressed:
αk =(b k 1Y 1+b k 2Y 2+ +b kN Y N ) αk
Where k is constant and
. (9)
∑b
i =1
N
ki
=1, k =1, 2, , K .
∞
∞
So, by (8), G (z
Y 1, Y 2, , Y N ) =∫ ∫e
∑γk P k (z )
k =1
K
∏g α
k =1
K
k
, βk
(γk ) d γ1 d γK
=∏
k =1
K
1
. (10) k
(1−βk P k (z ))
Where g αk , βk
(γk )is the probability density function of γk .
. (11)
K
Since the expectations of the sector risk factors is 1, so:
αk =
b k 1Y 1+b k 2Y 2+ +b kN Y N
βk
N
−
So G (z Y 1, Y 2, , Y N ) =e Where A i (z ) =−
∑Y i ∑βki k ln(1−βk P k (z ))
i =1
k =1
b
=e i =1
∑A i (z ) ⋅Y i
N
. (12)
∑β
k =1
K
b ki
k
ln(1−βk P k (z )) .
N
So G (z ) =
∫
+∞
∫e
+∞
∑Y i A i (z )
i =1
∏g αβ(Y ) dY dY
i =1
i , i
N
i 1N
∑αi ⋅ln(1−βi A i (z )) 1
. (13) =∏=e i =1
i
β(1()) −A z i =1i i
N
−
N
Since we have assumed that Y i followed Gamma distribution with mean 1 and varianceso αi
δi 2
,
=
1
δi 2
, βi
N
=δi 2.
1
−
So G (z ) =e
∑δi 2ln(1−δi 2A i (z ))
i =1
, (14)
Where A i (z ) =−
∑β
k =1
K
b ki
k
ln(1−βk P k (z )) , P k (z ) =∑p A g k A (h A (z ) −1) ,
A
n −1
a A +r (v A ln z ) n
h A (z ) =1+∑(∏⋅.
a +b +r n ! n =1r =0A A
∞
Above, on the assumption that the volatility of PD and LGD is independent, we obtain the pgf of the loan portfolio loss distribution. By analyzing (14), we find that the only difference between the pgf of the loan portfolio loss distribution of the SVSCM model and that of the MSRF model is the form ofh A (z ). In SVSCM model, whenb A =0, h A (z )=z
v A
, then the pgf of the SVSCM model degenerates into the pgf of
the MSRF model. So the SVSCM model is the general form of the MSRF model. In addition, seen from the deduction process, the SVSCM model can flexibly handle the volatility of LGD, for if the LGD of obligor doesn’t follow Beta distribution, but follows other distributions, we only need to modify the form ofh A (z ), the rest of the pgf remains the same.
2.2 Application of the SVSCM mo del
Like the MSRF model, in the SVSCM model, the variance and the covariance of the sector risk factors have the following relationships:
N
⎧22⎪Var [γk ]=βk +∑b ki δi ⎪i =1
, (15) ⎨N
⎪Cov (γ, γ) =b b δ2, k ≠l
∑k l ki li i
⎪i =1⎩
In practical application, according to the inner character of sector risk factors and their covariance matrix, we need to estimate the scale parameterβk , which is the variance of the sector risk factors if they are not affected by the system risk factors. Then we can calculate the corresponding parameters according to the covariance matrix of the sector risk factors with (15).
For the LGD, we have assumed that it follows Beta distribution, denoteεA ∼B (a A , b A ). Beta distribution has the following properties:
a A ⎧
=E [ε]A ⎪a A +b A ⎪
, (16) ⎨
a A b A
⎪Var [ε]=
A
⎪(a A +b A ) 2(a A +b A +1) ⎩
In practical application, owing to the restriction of reality, it is difficult to estimate the mean and variance of LGD for every obligor. But we can group the obligors in the loan portfolio according to the maturity, collateral and seniority, then estimate the average LGD and variance for every group, and use them as the approximation value of the mean and variance of the LGD of the obligor in the group. Based on that, the value of a A and b A can be calculated with (16).
3 Example Analysis
In order to testify the validity of the SVSCM model, the paper then provides the following example analysis. 3.1 Data selection
We assume that the loan portfolio consists of four different sectors, and the sector risk factors areγ1, γ2, γ3 and γ4, which follow gamma distribution with mean 1. We assume that there are 100 loans in every sector and number them from 1 to 100. We assume the unconditioned PD and the exposure as follows:
⎧1%,i =4k −3⎪2%,i =4k −2⎪+
, v ij =i , i ∈N , i.e. i is positive integer. p ij =⎨
⎪3%,i =4k −1⎪⎩4%,i =4k
p ij and v ij is the unconditioned PD and the exposure of the i-th loan in the sector j , respectively.
We assume the covariance matrix of the four sectors is:
⎡0.050.01450.01410.066⎤⎢0.01450.060.01470.0663⎥⎢⎥, ⎢0.01410.01470.070.0666⎥⎢⎥
0.6⎦⎣0.0660.06630.0666
We assume again that the scale parameters of the four sector risk factors are 0.0351, 0.0454, 0.0547 and
0.0811, respectively. According to the covariance matrix and the given scale parameter values, with (15), the shape parameters of the sector risk factors can be expressed as follows:
⎧α1=0.8Y 1+0.1Y 2+0.1Y 3⎪α=0.7Y +0.2Y +0.1Y ⎪2123
, ⎨
α0.6Y 0.3Y 0.1Y =++123⎪3⎪⎩α4=0.1Y 1+0.1Y 2+0.8Y 3
And the variance of Y 1, Y 2, Y 3 is 0.01, 0.04 and 0.81, respectively.
For the sake of convenience, we assume that the loan is only affected by the sector risk factor which the loan belong to, and the 400 loans have the same LGD. 3.2 The selectio n of algorithm
In the original CR+ model, the loss distribution was computed by the Panjer algorithm. However, Gordy (2002) showed that this algorithm was fragile [9]. Under empirically conditions, numerical error could accumulate in the execution of the recurrence rule and produce wildly inaccurate results for VaR. To circumvent these problems, Gordy and Giese (2003) [3] proposed the saddlepoint approximation algorithm and the nested calculation approach, respectively. The nested calculation approach is fit when the pgf of the loss distribution can be expressed as the exponential and logarithmic polynomials. According to the character of the pgf of the SVSCM model, this paper computes the loss distribution by the saddlepoint approximation algorithm. For using the saddlepoint approximation algorithm, we introduce the cumulant generating function (cgf). The cgf and the pgf have the following relationship:
ψy (z ) =log(G y (exp(z ))) (17)
Where: ψy (z )is the cgf of random variable y and G y (z ) is the pgf of random variabley . The saddlepoint approximation algorithm can be expressed as follows:
ˆ denote the unique root of the Let Y be a random variable with distribution G (y )and cgfψ(z ), and let z
equation y =ψ' (z ). The Lugannani-Rice formula for tail of G is:
11
1−G (y ) ≈1−φ(w ) +ϕ(w )(−, (18)
u w
, u =Where w =3.3 Th e r e s u lt an d an a ly sis
, and where φ and ϕ denote the cdf and pdf of the standard In this paper, we program with MatLab 7.0 and use the saddlepoint approximation algorithm to calculate the
loan portfolio loss distribution when the shape parameter is a =1, b =1, a =2, b =2, a =3, b =3,
a =5, b =5 and the LGD is constant at 0.5, respectively. The results are shown in table 1.
Table 1: The VaR Value of the Certain Confidence Levels
99% 99.5% 99.9%
a =1, b =1 a =2, b =2 a =3, b =3 a =5, b =5
The LGD equal to 0.5
614 669 789 601 653 772 595 646 764 589 640 754 579
628
738
When a =1, b =
1, the relationship of the confidence level and corresponding VaR is shown in figure 1.
Figure 1: when a =1, b =1, the relationship of the Corresponding VaR and the confidence level
On the condition that the mean of LGD keeps constant, it can be seen from table 1 that the VaR at the same
confidence level decreased with the decrease of the variance of LGD. That is because that when other conditions remain unchanged, the mean of the LGD keeps unchanged, the variance diminishes, the
fluctuations of the loan portfolio default loss become smaller, and the corresponding risk is certainly getting smaller. In addition, from table 1, we can obviously find that compared with other cases, when the LGD is a constant, VaR has minimum value corresponding to each confidence level. Therefore, do not consider the volatility of the LGD will underestimate the risk level of the loan portfolio.
In table 1, when a =1, b =1, if the confidence level increases from 99% to 99.5%, the default loss grows from 614 to 669, with an increase of 55; When the confidence level increases from 99.5% to 99.9%, the default
loss increases from 669 to 789, with an increase of 120. That is to say, the closer to the tail, the default loss is necessary to increase greater when the confidence level increases. From figure 1, it can be found that the loss distribution is obvious fat-tail. Moreover, we can see that as the default loss increases, the confidence level is more and more close to 1, which also proves that our result is precise.
4 Conclusion
On the basis of the MSRF model, we group the loan portfolio into different sectors according to the maturity, collateral nature and the seniority, assume every obligor’s LGD follows Beta distribution, and propose the SVSCM model. Moreover, we obtain the loss distribution by the saddlepoint approximation algorithm. The numerical experiment shows that our model can fully reflect the volatility of LGD.
The SVSCM model which our paper proposed can flexibly handle the volatility of LGD. With the development of research on LGD, if other distributions can be used to simulate the distribution of the obligor’s LGD better, the SVSCM model our paper proposed is still applicable, only with corresponding adjustment of the h A (z ) of the pdf of the loan portfolio default loss.
References
[1] Credit Suisse Financial Products (1997), CreditRisk+: A Credit Risk Management Framework, Credit Suisse
Financial Products. [2] P. Burgisser, A. Kurth , Armin Wagner et al. (1999) Integrating Correlations, Journal of Risk, vol. 12: 37-44. [3] G. Giese, (2003) Enhancing CreditRisk+, Risk, vol. 16: 73-77.
[4] J. G. Peng, Z. H. Lv, (2008) Multi-system risk factors CreditRisk+ model based on sector character, Chinese
Sciencepaper Online, http://www.paper.edu.cn/paper.php?serial_number=200809-524. [5] B. Peter, K. Alexandre, W. Armin. (2001) Incorporating Severity Variations into Credit Risk, Journal of Risk,
vol. 3: 5-31. [6] F. J. Cai, Y. D. Yang, Y. Li, (2004) Saddlepoint approximation of CreditRisk+ basing on severity variation,
Chinese Journal of Management Science. [7] J. G. Peng, L. H. Zhang, B. Liu, H. B. Tu, (2008) A research on the application of CreditRisk+ model in
chinese commercial banks, Journal of Financial Research. [8] Morgan J P, (1997) CreditMetrics -Technical Document. http://www.jpmorgan.com.
[9] M. B.Gordy, (2002) Saddlepoint approximation of CreditRisk+, Journal of Banking & Finance, vol. 26:
1335-1353.
Author Brief Introduction:
Jiangang Peng (1955- ), PhD. in economics, Professor of Finance, Ph.D. supervisor, Vice Dean of Research Institute of Hunan University, Director of Research Center of Financial Management of Hunan University.