Journal of Econoics and Manageent, 016, Vol. 1, No., 157-176 Calculating Value-at-Risk Using the Granularity Adjustent Method in the Portfolio Credit Risk Model with Rando Loss Given Default Yi-Ping Chang Departent of Financial Engineering and Actuarial Matheatics, Soochow University, Taiwan Jing-Xiu Lin Departent of Financial Engineering and Actuarial Matheatics, Soochow University, Taiwan Chih-Tun Yu Departent of Statistics, National Chengchi University, Taiwan According to the Basel Coittee on Banking Supervision (BCBS), the internal ratings-based approach of Basel II and Basel III allows a bank to calculate the Valueat-Risk (VaR) for portfolio credit risk by using its own credit risk odel. In this paper we use the Granularity Adjustent (GA) ethod proposed by Martin and Wilde (00) to calculate VaR in the portfolio credit risk odel with rando loss given default. Moreover, we utilize a Monte Carlo siulation to study the ipact of concentration risk on VaR. Keywords: granularity adjustent ethod, loss given default, portfolio credit risk odel, Value-at-Risk JEL classification: C15, G3 Received January 8, 016, accepted January 8, 016. Correspondence to: Departent of Financial Engineering and Actuarial Matheatics, Soochow University, No.56, Section 1, Kueiyang Street, Chungcheng District, Taipei City 100, Taiwan. E-ail: ypchang@scu.edu.tw.
158 Journal of Econoics and Manageent 1 Introduction In loan portfolios of a bank the ain risk is the occurrence of defaults. A default in a loan portfolio eans a borrower fails to eet its contractual obligation to repay a debt with the agreed ters. Loan portfolio defaults lead to huge losses for a bank, which is called portfolio credit risk. Under the Basel II and Basel III Accords, banks are allowed to establish their internal portfolio credit risk odel to estiate their credit risk factor. The purpose of estiating this credit risk factor is to calculate regulatory capital for credit risk. This is called the internal ratings-based (IRB) approach of Basel II and Basel III. Under IRB of Basel II and Basel III, banks use the Value-at- Risk (VaR) to easure their portfolio credit risk and capital cushion. Thus, estiating VaR is an iportant issue. Many articles have proposed several ethods to calculate VaR. As a general rule, the Exposure at Default (EAD), Probability of Default (PD), and Loss Given Default (LGD) of each asset are input values to calculate VaR, where LGD is defined as the ratio between the actual loss and aount of loan in a default event. The earliest credit risk studies usually assue LGD to be fixed/constant (Vasicek, 1987, 1991, 00; Eer and Tasche, 005) or rando but independent of the default rate (Pykhtin and Dev, 00; Gordy, 003, 004). However, any epirical studies point out that there is a strong correlation between LGD and the default rate. Thus, LGD should be rando and correlated with the default rate (Frye, 000; Andersen and Sidenius 004; Altan et al. 005; Bruche and González-Aguado, 010; Van Dae, 011; Farinelli and Shkolnikov, 01). For siplicity, the uncorrelated rando LGD odel represents that LGD is rando and uncorrelated with the default rate, and the correlated rando LGD odel represents that LGD is rando and correlated with the default rate. Vasicek (1987, 1991, 00) use the law of large nubers ethod to calculate VaR under the asyptotic single risk factor (ASRF) assuption with constant LGD. The ASRF approach assues that the portfolio is infinitely fine grained and only one sys-
Calculating VaR Using the GA in the Portfolio Credit Risk Model 159 teatic risk factor could affect the default risk of all assets in the portfolio. However, the ASRF fraework cannot capture the concentration risk that affects the accuracy of the estiate for VaR. For ore details about the ASRF approach, please refer to Gordy (003, 004) and Overbeck and Wanger (003). Wilde (001) and Martin and Wilde (00) propose a Granularity Adjustent (GA) ethod to calculate VaR by using the Taylor expansion of the quantile and the results of Gouriéroux et al. (000). The ipact of concentration risk on VaR can be approxiated analytically through the GA ethod. Pykhtin and Dev (00), Eer and Tasche (005), and Bellalah et al. (015) use the GA ethod to calculate VaR in the portfolio credit risk odel with rando and uncorrelated LGD and constant LGD, respectively. Gordy (003, 004) also eploys the GA ethod to calculate VaR in the CreditRisk + odel with rando and uncorrelated LGD. Gürtler et al. (010) ake an extension of the GA ethod to obtain the closed for of VaR in the ulti-factor odel with constant LGD. Note that these studies do not consider the rando and correlated LGD for calculating VaR. Lin (010) utilize the GA ethod to calculate VaR in the portfolio credit risk odel with rando and correlated LGD. The ain contribution of this paper is that we show that the GA ethod can successfully calculate VaR in portfolio credit risk with stochastic and correlated LGD. We also use a Monte Carlo siulation to study how the concentration risk affects VaR. Note that, in general, there are two types of concentration risk. One is referred to as single-nae concentration, i.e., the portfolio with a large EAD on highly rated obligors. The other one is referred to as sectoral concentration, i.e., EAD to obligors of the portfolio in the sae sectors. For ore studies on the concentration risk of portfolio, please refer to Lütkebohert (009). In this study we focus on how the single-nae concentration affects VaR. Eer and Tasche (005) also ake a siilar study with constant LGD. This paper is organized as follows. Section introduces the portfolio credit risk
160 Journal of Econoics and Manageent odel and description of VaR. Section 3 describes the approxiate closed for of VaR with rando and correlated LGD. Section 4 shows the Monte Carlo siulation results. Section 5 suarizes the conclusions. Finally, Appendix presents all proofs of the leas. Portfolio Credit Risk Model and VaR Consider a portfolio with assets. We define the standardized asset value X i of the ith asset in the portfolio as: X i ρ i Z + 1 ρ i U i, i 1,,, (1) where 0 ρ i 1, and Z,U 1,,U be i.i.d. N (0,1). By a siple coputation, X i also follow N(0,1), and ρ i ρ j denotes the correlation between X i and X j (i j). Let Y i represent the default indicator function of the ith asset: Y i I(X i c i ), () where c i is the default threshold and I( ) is an indicator function. Asset i is assued to be in default when its standardized asset value X i falls below the threshold value c i. We define the probability of default of asset i as PD i : PD i P(Y i 1) P(X i < c i ) Φ(c i ), where Φ( ) is the cuulative distribution function of N (0,1), and thus: c i Φ 1 (PD i ), where Φ 1 ( ) is the inverse function of Φ( ).
Calculating VaR Using the GA in the Portfolio Credit Risk Model 161 In this paper LGD is rando and correlated, as proposed by Andersen and Sidenius (004), i.e., LGD i of the ith asset is: LGD i 1 Φ(u i + σ i η i ), (3) where < u i <, σ i > 0. The rando variables η i are assued to be: η i λ i Z + 1 λ i ε i, (4) where 0 λ i 1, and Z,ε 1,,ε be i.i.d. N (0,1). Note that η i also follows N (0,1). When λ i 0, i 1,,, LGD is rando and uncorrelated with the default rate and called the uncorrelated stochastic LGD odel. When λ i > 0, i 1,,, LGD is rando and correlated with the default rate and called the correlated rando LGD odel. Lea 1. Under the correlated rando LGD odel, we have: E(LGD i ) Φ i, Var(LGD i ) Φ u i u i σ, i ; Φ i, Cov(LGD i,lgd j ) Φ u i u j σ i σ j λi λ j, ; 1 + σ j ( )(1 j ) Φ i Φ j 1 + σ j, Cov(LGD i,y i ) Φ u i 1 + σ i,φ 1 (PD i ); σ i λi ρ 1 + σ i
16 Journal of Econoics and Manageent Corr(LGD i,lgd j ) Φ u i PD i, Cov(LGD i,lgd j ) Var(LGD i )Var(LGD j ), Corr(LGD i,y i ) Cov(LGD i,y i ) Var(LGD i )Var(Y i ), where Var(Y i ) PD i PD i. Consider a portfolio with assets. We denote EAD and LGD of the ith asset as EAD i and LGD i, respectively. We discuss the VaR specifications as follows. VaR is the value in which a loss on a portfolio will not exceed a given value over the given tie horizon with a level of probability. More precisely, we define the loss rate of a portfolio as: L w i LGD i Y i, (5) where w i EAD i EAD i denotes the weight of EAD of the ith asset. When L is a rando variable and the confidence level is: α (0 < α < 1), the α quantile of L is: q α (L ) inf{l 0 : P(L l) α}. Under the IRB approach of Basel II and Basel III, α is taken as 99.9%. We then denote the α-quantile of L as VaR: VaR q 99.9% (L ). 3 Calculating Portfolio Credit VaR Using the GA Method
Calculating VaR Using the GA in the Portfolio Credit Risk Model 163 Martin and Wilde (00) use the Taylor expansion and results fro Gouriéroux et al. (000) to obtain the portfolio credit VaR with the GA ethod in ore general odel specifications. For convenience, we let: g(z) E(L Z z) be the conditional expectation of L given Z z. Condition 1. Given the realization of Z z, g(z) is a continuous, differentiable, and decreasing function. Condition. Given the realization of Z z, Var(L Z z) is a continuous and differentiable function. Bluh et al. (003) eploy the law of large nubers ethod to obtain the following results under the ASRF assuption, i.e.: α P(L q α (L )) P(E(L Z) q α (L )). If condition 1 holds, then: q α (L ) E(L Z z α ) g(z α ), where z α is the (1 α) quantile of the standard noral distribution. For siplicity, using g(z α ) to estiate VaR is called the ASRF ethod. According to the results of Martin and Wilde (00), under the correlated rando LGD odel, the approxiate α quantile of L is: where q α (L ) g(z α ) + GA, (6) { GA 1 g (z) z Var(L Z z)
164 Journal of Econoics and Manageent [ g ] } (z) Var(L Z z) g (z) + z. (7) zzα Lea. Under the correlated rando LGD odel, g(z) is given by: where g(z) ψ i (z) w i Φ(ψ i (z))φ(ς i (z)), u i σ i λi z, (8) (1 λ i) ς i (z) Φ 1 (PD i ) ρ i z 1 ρi. (9) Through straightforward calculus, we can obtain g (z) and g (z) as the following. Lea 3. Under the correlated rando LGD odel, g (z) and g (z) are given by: g (z) g (z) w i ψ i φ(ψ i (z))φ(ς i (z)) + w i ψ i ψ i (z)φ(ψ i (z))φ(ς i (z)) + w i ς i ς i (z)φ(ψ i (z))φ(ς i (z)), w i ς i Φ(ψ i (z))φ(ς i (z)), w i ψ i ς i φ(ψ i (z))φ(ς i (z)) where φ( ) is the probability density function of the standard noral distribution: ψ i z ψ σ i λi i(z), (1 λ i) ς i z ς ρi i(z). 1 ρ i According to Lea 3, we observe that g (z) < 0. Thus, condition 1 holds.
Calculating VaR Using the GA in the Portfolio Credit Risk Model 165 Lea 4. Under the correlated rando LGD odel, Var(L Z z) and z Var(L Z z) are given by: and Var(L Z z) z Var(L Z z) + w i Φ (ψ i (z),ψ i (z);ρ i )Φ(ς i (z)) w i Φ (ψ i (z))φ (ς i (z)), w i ψ i φ(ψ i (z))φ ψ i (z) Φ(ς i (z)) (1 i) w i ς iφ (ψ i (z),ψ i (z);ρ i )φ(ς i (z)) w i [ ψ i φ(ψ i (z))φ(ς i (z)) + ς i Φ(ψ i (z))φ(ς i (z))] Φ(ψ i (z))φ(ς i (z)), where Φ (, ;ρi ) is the standard bivariate noral cuulative distribution with correlation ρ i σ i (1 λ i) 1 + σ i (1 λ i). (10) Lastly, we obtain the approxiate closed for of q α (L ) by using (6), (7), and Leas 3 and 4. For siplicity, using the approxiate closed for of q α (L ) to estiate VaR is called the GA ethod. 4 Siulation Results The Monte Carlo studies present how the concentration risk of a portfolio affects VaR. For ore studies on the concentration risk of a portfolio, please refer to Lütkebohert
166 Journal of Econoics and Manageent (009). According to the rule of Basel II and Basel III (BCBS, 011), the value of ρ i is set to be: [ ] [ 1 exp( 50 PDi ) ρ i 0.1 + 0.4 1 1 exp( 50 PD ] i). 1 exp( 50) 1 exp( 50) This paper takes the Monte Carlo siulation (MCS) ethod as a benchark ethod to calculate VaR. We give the steps of the MCS ethod as follows. Step 1: Siulate +1 rando saples Z,U 1,,U fro N (0,1) and obtain X 1,,X fro (1). Step : Take X 1,,X into () to obtain the default indicator functions Y 1,,Y. Step 3: Siulate rando saples ε 1,,ε fro N (0,1) and obtain η 1,, η fro (4). One can then obtain LGD 1,,LGD fro (3). Step 4: Obtain L 1,,L fro (5). Step 5: Repeat Steps 1-4 10 6 ties. Step 6: Take the 10 6 α-largest of siulated L as the estiate for q α (L ). In the siulation studies, we focus on how the single-nae concentration affects VaR. Eer and Tasche (005) also conduct a siilar study with constant LGD. Thus, the choice of paraeter settings in this paper is the sae as that in the siulation of Eer and Tasche (005). The set-up for nuber of assets, PD, and EAD are given as follows. 1. 1000, α 0.999.. PD 1 0.00, PD PD 0.05, i.e., the first asset has a saller PD than other assets. 3. Given the weight of EAD of first asset w 1, assue: w w 1 w 1 1,
Calculating VaR Using the GA in the Portfolio Credit Risk Model 167 i.e., the weight of EAD of assets is the sae except for the first asset. The range of w 1 is [0,0.]. When w 1 1/, w w 1/, it eans that there is no concentration risk in portfolio. When w 1 > 1/ and w 1 becoes large, the concentration risk becoes large. 4. E(LGD i ) 0.6, i 1,,. 5. SD(LGD i ) Var(LGD i ) 0,0.,0.4, i 1,,. 6. Corr(LGD i,lgd j ) 0,0.3,0.6, i j, i, j 1,,. When Corr(LGD i, LGD j ) 0, i j, LGD is rando and uncorrelated. When Corr(LGD i, LGD j ) > 0, i j, LGD is rando and correlated. Note that a constant LGD is assued in the siulation studies of Eer and Tasche (005), naely - LGD i 1, i 1,,. Figure 1 shows VaRs estiated by MCS, GA, and ASRF ethods in the constant LGD odel. Figure illustrates VaRs estiated by MCS, GA, and ASRF ethods in the correlated rando LGD odel with SD(LGD i ) 0. and SD(LGD i ) 0.4. Note that when SD(LGD i ) 0, LGD is constant. When SD(LGD i ) > 0, LGD is rando. Figure 1: Estiated Portfolio VaRs by Using MCS, GA, and ASRF Methods under the Constant LGD Model
168 Journal of Econoics and Manageent Figure : Estiated Portfolio VaRs by Using MCS, GA, and ASRF Methods under the Correlated Rando LGD Model Several conclusions can be observed in Figures 1-. 1. When the weight of EAD of first asset w 1 is increasing fro 1/ to 1, the trend of VaR will first be saller and then larger in both the constant and correlated rando LGD odels. In fact, the sae conclusion is also obtained in Eer and Tasche (005).
Calculating VaR Using the GA in the Portfolio Credit Risk Model 169. The perforance of the estiated VaR using the GA ethod is better than the ASRF ethod in both the constant and correlated rando LGD odels. However, as w > 0.1, the estiated VaRs have a larger error by using the GA and the ASRF ethods. As w 1 becoes large, the error increases. In other words, when the weight of EAD of the first asset is larger than 10%, the estiated VaR using the GA ethod has a larger error and significantly underestiates the risk. In fact, the sae conclusion is also obtained in Eer and Tasche (005). 3. The estiated portfolio VaR under the correlated rando LGD odel is larger than the estiated VaR under the constant LGD odel. This eans that, as the true LGD is rando, the estiated VaR by using the constant LGD odel underestiates the risk. 4. The siulation results also show the econoic/anageent eaning of portfolio allocations. Fund anagers and investors can thus carefully allocate their portfolio to decrease the concentration of assets. Due to the concentration of assets, the portfolio will incur a high concentration risk and VaR. 5 Conclusions The ajor work of this paper is to obtain the approxiate closed for of VaR by using the GA ethod proposed by Martin and Wilde (00) under the correlated rando LGD odel. The results iprove the analysis presented in Eer and Tasche (005). We observe that the VaR perforance using the GA ethod is better than the perforance using the ASRF ethod fro our siulation results. However, we note that when the weight of EAD of one asset is large, the estiated VaR using the GA ethod has a larger error and significantly underestiates the risk.
170 Journal of Econoics and Manageent Appendix This section contains all proofs of leas. We first present the three leas herein. Lea A1 (Andersen and Sidenius (004)). Given the constants a and b: ( ) b Φ(ax + b)φ(x)dx Φ. 1 + a Lea A (Andersen and Sidenius (004)). Given the constants a 1,a,b 1, and b : Φ(a 1 x + b 1 )Φ(a x + b )φ(x)dx Φ b 1 b, ; a 1 a. 1 + a 1 1 + a (1 + a 1 )(1 + a ) Lea A3. Given the constants a 1,a,b 1, and b : Φ(a 1 x + b 1 )φ(a x + b )φ(x)dx 1 1 + a φ 1 + a Φ 1 + a b 1 a 1 a b (1 + a )(1 + a 1 + a ). Proof. By Lea A1: Φ(a 1 x + b 1 )φ(a x + b )φ(x)dx φ Φ(a 1 x + b 1 )φ 1 + a 1 + a x + a b dx 1 + a 1 1 + a φ 1 + a
Calculating VaR Using the GA in the Portfolio Credit Risk Model 171 Φ 1 y + b 1 + a b 1 a 1 a b φ(y)dy 1 + a 1 + a 1 φ Φ 1 + a b 1 a 1 a b. 1 + a 1 + a (1 + a )(1 + a 1 + a ) Proof of Lea 1. By Lea A1: E(LGD i ) [1 Φ(u i + σ i η i )]φ(η i )dη i Φ( u i σ i η i )φ(η i )dη i Φ i. (11) By Lea A1, Lea A, and (11): Var(LGD i ) E[ Φ ( u i σ i η i ) ] E (LGD i ) By Lea A and (11): Cov(LGD i,lgd j ) Φ ( u i σ i η i )φ(η i )dη i E (LGD i ) Φ u i u i σ, i ; Φ u i. E[E(LGD i LGD j Z)] E(LGD i ) E(LGD j ) [ ( E Φ ( u i σ i λi Z + )) 1 λ i ε i φ(ε i )dε i ( ] Φ ( u j σ j λ j Z + 1 λ j ε j ))φ(ε j )dε j
17 Journal of Econoics and Manageent E(LGD i ) E(LGD j ) Φ u i σ i λi z Φ u j σ j λ j z φ(z)dz (1 λ i) 1 + σ j (1 λ j) E(LGD i ) E(LGD j ) Φ u i u j σ i σ j λi λ j, ; 1 + σ j ( )(1 j ) Φ i Φ j 1 + σ j. By Lea A and (11): Cov(LGD i,y i ) E[E(LGD i Y i Z)] E(LGD i ) E(Y i ) ( Φ ( u i σ i λi Z + )) 1 λ i ε i φ(ε i ) ( Φ 1 (PD i ) ρ i Z Φ )dε i dz E(LGD i ) E(Y i ) 1 ρi Φ u ( i σ i λi z Φ Φ 1 (PD i ) ρ i Z )φ(z)dz (1 λ 1 ρi i) E(LGD i ) E(Y i ) Φ u i,φ 1 (PD i ); σ i λi ρ Φ u i PD i. Proof of Lea. Given Z z, the conditional default probability of the ith asset is: P(Y i 1 Z z) P(X i < Φ 1 (PD i ) Z z)
Calculating VaR Using the GA in the Portfolio Credit Risk Model 173 P ( Φ(ς i (z)), where ς i (z) is defined in (9). By Lea 3: g(z) E(L Z z) E ( U i < Φ 1 (PD i ) ) ρ i Z Z z 1 ρi w i LGD i Y i Z z ) w i E[Φ( u i σ i η i ) Z z] E(Y i Z z) w i [ P(Y i 1 Z z) ( Φ ( u i σ i λi z + ] 1 λ i ε i ))φ(ε i )dε i w i Φ(ψ i (z))φ(ς i (z)), where ψ i (z) and ς i (z) are defined in (8) and (9), respectively. Proof of Lea 4. By Lea A: Var(L Z z) Var ( w i LGD i Y i Z z ) w i w i E(LGD i Yi Z z) [ Φ(ς i (z)) [ w i w i E (LGD i Y i Z z) Φ ( ( u i σ i λi z + )) ] 1 λ i ε i φ(ε i )dε i Φ ( u i σ i ( λi z + 1 λ i ε i ))φ(ε i )dε i ]
174 Journal of Econoics and Manageent Φ (ς i (z)) w i Φ (ψ i (z),ψ i (z);ρ )Φ(ς i (z)) w i Φ (ψ i (z))φ (ς i (z)), where ψ i (z), ς i (z), and ρ are defined in (8), (9), and (10), respectively. Moreover: z Φ (ψ i (z),ψ i (z);ρ ) Φ ( ( u i σ i λi z + 1 λ i ε i ))φ(ε i )dε i z ) σ i λi Φ ( u i σ i λi z σ i 1 λi ε i ) φ ( u i σ i λi z σ i 1 λi ε i φ(ε i )dε i ψ i φ(ψ i (z))φ ψ i (z). (1 i) By a straightforward coputation, Lea 4 can be proved.
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