Proceedings of the Annual Meeting of the American Statistical Association, August 5-9, 2001
STABLE MODELING OF CREDIT RISK Svetlozar Rachev, Eduardo Schwartz, and Irina Khindanova University of California, Santa Barbara, CA 93106-3110; University of California, The Anderson School, 110 Westwood Plaza, Box 951481, Los Angeles, CA 90095; and Colorado School of Mines, Golden, CO 80401-1887 KEY WORDS: Risk Management, Credit Risk, Value at Risk, Stable Distributions, Correlation Estimation, Heavy Tails, Skewed Distributions 1. Introduction One of the important tasks of financial institutions is evaluating the exposure to credit risks. Credit risks refer to potential losses that might occur because of a change in the counterparty’s credit quality such as a rating migration or a default. A commonly used methodology for estimation of risks is the Value at Risk (VaR). In the text below, the credit VaR means the VaR linked to credit risks. The paper examines the use of stable Paretian distributions in modeling credit VaR. The stable distributions are described by four parameters: α - tail index, β - skewness, µ - location, and σ scale. Modeling with such parameters will depict fat tails and skewness of distributions. Empirical analysis reported here confirms that, indeed, stable modeling captures heavy-tailedness and asymmetry of credit returns1, and, therefore, produces more accurate risk estimates. A symmetric stable random variable can be interpreted as a transformation of a normal random variable. Based on this property, a new technique is developed here for estimating correllation. A stable random variable can be decomposed into the “symmetry” and “skewness” parts. Building on this feature, we construct a new method for simulating a distribution of portfolio values. We apply this method for portfolio risk evaluation in two cases of credit instruments’ distributions: symmetric dependent and skewed dependent. The remainder of the paper is organized as follows. In Section 2 we investigate stable modeling of credit returns and discuss risk assessment for individual credit instruments. Sections 3 and 4 present, respectively, evaluation of portfolio risk in two cases of dependent
1
portfolio instruments’: symmetric and skewed. Section 5 states conclusions. 2. Stable Modeling and Risk Assessment for Individual Credit Returns In this section we confirm that, indeed: (i) credit returns exhibit asymmetry and heavy-tails; (ii) stable modeling captures these features of the returns. Also, we assess credit risk of the individual credit returns using stable modeling. The “assets” used in the study are the Merrill Lynch indices of the US government and corporate bonds with maturities from one to 10 years and credit ratings from “BB” to “AAA”, in total, 21 indices . Returns on indices are modeled as stable-distributed: Ri ∼ S α i (σ i , β i , µ i ) , where i=1,…,21. Figure 1 demonstrates stable modeling of the high yield index HOA1.
Figure 1. Stable and Normal Fitting of the HOA1 index.
Basle Committee on Banking Supervision (1999).
In order to assess riskiness of the individual credit series and properties of stable modeling in the credit risk evaluation, the 99% and 95%
Value at Risk (VaR) measurements were computed. The stable and normal VaR estimates for some indices are reported in Table 12. Normal VaR measurements are given for comparison purposes. Figure 2 shows VaR Estimation for the HOA1 index. Results of VaR
Table 1. Empirical, Normal, and Stable Estimates for Bond Indices 99% VaR estimates Index Empirical Normal C1A4 0.262 0.231 C2A4 0.478 0.392 C3A4 0.711 0.545 C4A4 0.862 0.702 H0A1 0.557 0.403 95% VaR estimates C1A4 Empirical Normal C2A4 0.124 0.155 C3A4 0.243 0.268 C4A4 0.361 0.375 H0A1 0.467 0.486 0.258 0.277
VaR
Stable 0.290 0.511 0.741 0.960 0.570 Stable 0.119 0.228 0.343 0.451 0.245
3. Stable Modeling of Portfolio Risk for Symmetric Dependent Credit Returns In this section we suppose that distributions of credit returns are symmetric α-stable and dependent. We interpret a symmetric random variable as a transformation of a normal random variable. Based on this interpretation, we develop a new methodology for correlation estimation. We apply the methodology for portfolio risk assessment. We evaluate portfolio risk by determining portfolio VaR: (i) simulating a distribution of the
Figure 2. VaR Estimation for the HOA1 index
estimations lead to the following conclusions3: the stable modeling produces conservative and accurate 99% VaR estimates, which is preferred by financial institutions and regulators. “Conservative” VaR estimates exceed empirical VaR, which implies that forecasts of losses were greater than observed losses; the stable modeling underestimates the 95% VaR; the normal modeling gives overly optimistic forecasts of losses in the 99% VaR estimation; the normal modeling is acceptable for the 95% VaR estimation. Since credit returns have skewed and heavy-tailed distributions, VaR mesurements provide more adequate indication of risk than symmetric measurements (standard deviation or, in case of stable distributions, scale parameter) do. Our empirical analysis demonstrates advantages of stable modeling in evaluation of riskiness of single credit returns series. The next step is to examine properties of stable modeling in evaluation of portfolio risk.
R P = ∑ wi Ri values; (ii) finding a certain
i =1
n
quantile of the RP distribution, say, the 1% quantile, which corresponds to the 99% VaR confidence level. The simulations must account for dependence among individual credit returns Ri, i=1,…,n. A traditional approach of quantifying dependence is to calculate the covariance matrix. Under the α-stable assumption for distributions of Ri, computation of the covariance matrix is impossible. We suggest a new method for deriving the dependence (association) structure. The method assumes that Ri are symmetric strictly stable: Ri ∼ Sα Ri (σ Ri ,0,0) . A symmetric α-stable (SαS) random variable can be interpreted as a random rescaling transformation of a normal random variable (see Property 1 below). If a collection of SαS variables is obtained by applying a similar transformation to dependent normal variables, the dependence structure among variables will remain. Thus, the dependence among SαS
2
VaR estimates for other indices can be provided by authors upon request. 3 This section computes “in-sample” VaR. Hence, the conclusions discuss in-sample VaR properties.
random variables can be explained by the dependence among underlying normal random variables. Property 14. Assume that: (i) G is a normal random variable with a zero mean: G ∼ S 2 (σ G ,0,0) = N (0,2σ ) ,
2 G
Si is independent of Gi, i=1,…, n. Random rescaling transformations of normal variables Gi into Ri preserve the dependence structure. Hence, the dependence among Ri can be explained by the dependence among Gi, i=1,…, n. Based on this property, we generate dependent normal variables Gi , maintaining the initial dependence5, then, we generate
1 ~ ~ ~ ~ Ri = S i 2 Gi , where S i is a simulated value of
~
(ii)
Y is a symmetric α-stable random variable, α<2: Y ∼ Sα(σY, 0, 0),
Si;
(iii)
α S is a positive -stable random 2
variable:
~ (ii) computing R P =
∑wR
i =1 i
n
~ . i
2 S∼ S σY α 2 σG 2
(iv)
2 πα α cos ,1,0 , 4
S and G are independent.
Then, the symmetric α-stable random variable Y can be represented as a random rescaling transformation of the normal random variable G:
Y = S G.
Simulations of the portfolio return values RP can be divided into two fragments: (i) generating individual returns Ri with the same dependence structure as the Ri’s. We derive the dependence among Ri supposing that Ri ∼ Sα Ri (σ Ri ,0,0) . Based on Property 1, Ri can be expressed as a transformation of a normal random variable:
1 2
~
We evaluate portfolio risk for equally weighted returns on the Merrill Lynch indices of the investment grade corporate bonds: C1A1, C2A1, C3A1, C4A1, C1A2, C2A2, C3A2, C4A2, C1A3,C2A3, C3A3, C4A3, C1A4, C2A4, C3A4, and C4A4. We impose an assumption that returns on these indices are symmetric-α-stable. We compute the 99% and 95% VaR measurements in two procedures: (i) simulation of portfolio returns following the above described procedure; (ii) calculation of the 99% (95%) VaR as the negative of the 1% (5%) quantile. The VaR estimates are reported in Table 2. In order to estimate accuracy of simulations, we calculate the Kolmogorov Distance (KD) and Anderson-Darling (AD) statistics. Table 2. Portfolio VaR for Dependent Credit Returns DeTrunc Portfolio VaR cay ation 99% 95% factor points VaR VaR (%) θ 0.85 10-90 7.508 4.886 5-95 7.777 5.153 No 8.286 5.346 0.94 10-90 7.793 5.147 5-95 8.076 5.248 Symmetric KD AD
Ri = S i Gi ,
where Gi∼ S 2 (σ Gi ,0,0) = N (0,2σ Gi ) ,
2
1 2
(1)
(2)
3.880 3.736 4.859 3.556 4.362
0.086 0.093 0.111 0.081 0.104
1-99
No
5
8.389
8.114
5.434
5.252
5.650
5.212
0.128
0.117
Si∼ S
α Ri 2
2 σ Ri σ 2 Gi
πα cos 4
2 α Ri
,1,0 ,
4
Property 1 is a slightly modified version of Proposition 1.3.1 in Samorodnitsky and Taqqu (1994).
Variables Gi, which enter formulas (1) and (2), are not observable. We suppose the dependence structure of Gaussian variables (Gi)1≤i≤n is “inherited” from the dependence structure of truncated values of stable variables (Ri) 1≤i≤n. Because we believe that the “outliers” are very important for the description of the dependence structure, we take the truncation value for Ri sufficiently large.
0.975
10-90 5-95 1-99 No
8.028 8.166 8.469 8.516
5.036 5.318 5.493 5.470
3.452 9.085 5.805 7.274
0.077 0.234 0.130 0.167
At each truncation band, increasing the decay factor leads to higher values of the 99% VaR. Thus, as the decay factor grows, the 99% VaR generally rises. At each value of the decay factor, in general, reduction of truncated observations produced higher VaR numbers. We explain the latter observation by positive correlation in tails (concurrent extreme events). Consideration of a larger number of tail observations results in higher VaR. The KD and AD statistics, in general, decline with smaller decay factors. We examine how selection of the decay factor and the truncation method affects estimation of marginal risks. The marginal risk is a risk added by an asset to the portfolio risk. It is computed as the difference between the portfolio risk with an analyzed asset and the portfolio risk without the asset. The decay factor of 0.85 does not produce cases “Marginal VaR > Stand-alone VaR” and “Within one maturity band, higher ratings contribute more risk”. In sum, the decay factor = 0.85 results in the lower KD and AD statistics and does not lead to irregular cases; the no-truncation method better accounts for correlation in tails. Hence, we would recommend the choice of the decay factor=0.85 and the no-truncation method. Marginal VaR estimates are consistent with the expectation that, for a given credit rating, bonds with longer maturities contribute more risk. Having marginal VaR numbers, we can identify concentration risks. We find that the C4A3 bond index makes the highest addition to the portfolio 99% VaR: the C4A3 marginal VaR of 0.920 exceeds all other marginal VaR. Marginal risks for all bond indices are smaller than stand-alone risks, which indicates that, indeed, diversification reduces risk. We notice that the C4A1 and C3A4 bond indices have highest diversification effects. We studied stable modeling of portfolio risk under the assumption of the symmetric dependent instruments. In the next section we consider portfolio risk evaluation in the most general case – skewed dependent instruments.
distribution of its possible values and deriving a portfolio VaR from the constructed distribution of RP. In a case of portfolio assets with skewed dependent credit returns, simulations of the RP values should reflect the “cumulative” skewness and maintain the dependence (association) among them. In order to do that, we decompose single credit returns Ri into two independent parts: the first part accounts for dependence and the second – for skewness. Then, we obtain the portfolio dependence and skewness components separately aggregating the dependence and skewness parts of individual credit returns. Simulations of the portfolio credit returns values RP can be divided into three portions: (i) generation of the portfolio dependence component maintaining the dependence structure among individual credit returns, (ii) generation of the portfolio skewness component, and (iii) computation of RP as a sum of the two generated components. Explanations of our methodology are provided below. A stable random variable R∼Sα(σ, β, 0) can be decomposed (in distribution) into two independent stable random variables: symmetric R(1) and skewed R(2):
R = R (1) + R ( 2) ,
where
d
(3)
−1 R (1) ∼ Sα 2 α σ ,0,0 , −1 R ( 2 ) ∼ S α 2 α σ , 2 β ,0 .
(4)
(5)
Using methodology (4)-(5), we can divide individual credit returns Ri∼ Sα Ri (σ Ri , β Ri ,0) into the “dependence” and “skewness” parts. First, we partition Ri into the “symmetry” and “skewness” fragments:
Ri = Ri(1) + Ri( 2 ) ,
where
d
4. Stable Modeling of Portfolio Risk for Skewed Dependent Credit Returns We quantify portfolio risk RP by generating a
R
(1) i ∼
−α1 Sα Ri 2 Ri σ Ri ,0,0 ,
Ri( 2 ) ∼ Sα Ri 2
parts Ri
(1)
1 − α Ri
σ Ri ,2 β Ri ,0 ,
(2)
and Ri
are independent, i=1,…,n.
(1)
Second, we suppose: (i) Ri , i=1,…,n, are dependent and (ii) Ri , i=1,…,n, are independent. Consequently, symmetric terms
(2)
We implement the suggested procedure for the risk assessment of the same portfolio of indices as in Section 3. We suppose that returns on indices are dependent skewed-α-stable. The portfolio VaR estimates are presented in Table 3. We also compute marginal VaRs. The marginal VaR estimates were smaller than the corresponding stand-alone VaR measurements, which supports feasibility of suggested procedure for simulating portfolio returns. Table 3. Portfolio VaR for Skewed Credit Returns Decay Trunca Portfolio VaR factor tion 99% 95% points, VaR VaR θ (%) 0.85 10-90 4.939 2.904 5-95 5.380 3.162 No 5.449 3.236 0.94 10-90 5.101 3.009 5-95 5.456 3.248 1-99 5.596 3.363 No 5.455 3.231 0.975 10-90 5.112 3.021 5-95 5.416 3.238 1-99 5.471 3.307 No 5.298 3.238 12. Conclusions This work proposes the application of stable distributions in credit VaR estimation. Our empirical analysis verifies that stable modeling well captures skewness and heavy-tails of credit returns. The superior fit allows to derive accurate risk estimates. Based on the properties of stable distributions, we design new methods for the correlation estimation and simulating portfolio values. We employ the methods in evaluation of portfolio and marginal VaR for two cases of the credit returns: symmetric dependent and skewed dependent. The stable Paretian model provides superior fit in modeling credit VaR. However, additional research is needed. Future work is this direction will be construction of models that capture the features of financial empirical data such as heavy tails, time-varying volatility, and short and long range dependence6. In order to describe thick tails, one can employ the conditional heteroskedastic models based on the
6
Dependent KD AD
R
(1) i
explain dependence (association) among
(2)
Ri’s and terms Ri Ri’s.
account for skewness of
Based on Property 1 (see Section 3),
Ri(1) ∼ Sα Ri 2
(1) i 1 2 i
1 − α Ri
σ Ri ,0,0 can be written as a
transformation of a normal random variable:
R
= S Gi ,
2
where Gi∼ S 2 (σ Gi ,0,0) = N (0,2σ Gi ) ,
7.22 5.64 5.43 6.53 5.24 4.70 5.13 6.54 5.34 4.37 5.43
0.20 0.18 0.17 0.19 0.17 0.14 0.17 0.19 0.17 0.14 0.15
−α2 2 2 Ri σ Ri Si∼ S α Ri 2 σ Gi 2
2 πα α Ri cos ,1,0 , 4
Si is independent of Gi, i=1,…, n. Random rescaling transformations of normal variables Gi into Ri maintain the dependence structure. Therefore, from the dependence among Gi’s we can determine the dependence among
(1)
Ri(1) , or the dependence among Ri.
Adding separately the dependence and skewness terms of Ri’s, we obtain the two components of the portfolio returns RP:
( ( RP = RP1) + RP2 ) ,
( where R P1) =
∑ wi Ri(1) = ∑ wi S i2 Gi is the
i =1 i =1
n
n
1
“dependence” component and
( R P2) = ∑ wi Ri( 2) is the “skewness” i =1 n
component.
For some preliminary results see Liu and Brorsen (1995), Mittnik, Rachev, and Paolella (1998), Mittnik, Paolella, and Rachev (1997, 1998a, and 1998b), Panorska, Mittnik, and Rachev (1995).
stable hypothesis7. ARMA-stable-GARCH models can incorporate both heavy tails and time-varying volatility8. The fractional-stable GARCH model can capture all observed phenomena in financial data: heavy tails, timevarying volatility, and short- and long-range dependence. An analysis of VaR estimation with ARMA-α-stable, ARMA-stable-GARCH, and fractional-stable GARCH models will be provided in a subsequent paper. References 1. Basle Committee on Banking Supervision, 1999, “Credit Risk Modelling: Current Practices and Applications”. Liu, S.-M. and B.W. Brorsen, 1995, "Maximum Likelihood Estimation of a GARCH-stable Model", Journal of Applied Econometrics, 10, 273-285. Mittnik S., M.S. Paolella, and S.T. Rachev, 1998a, "Unconditional and Conditional Distributional Models for the Nikkei's Index", Asia-Pacific Financial Markets, 5, 99-128. Mittnik S., M.S. Paolella, and S.T. Rachev, 1998b, "The Prediction of DownSide Risk with GARCH-stable Models", Technical Report, Institute of Statistics and Econometrics, University of Kiel, Germany. Mittnik S., M.S. Paolella, and S.T. Rachev, 1998c, "A Tail Estimator for the Index of the Stable Paretian Distribution", Communications in Statistics - Theory and Methods, 27, 1239-1262. Mittnik S., M.S. Paolella, and S.T. Rachev, 1997, "Modeling the Persistence of Conditional Volatilities with GARCHstable Processes", Manuscript, Institute of Statistics and Econometrics, University of Kiel, Germany. Mittnik, S., S. T. Rachev, and M. S. Paolella, 1998, “Stable Paretian Modeling
in Finance: Some Empirical and Theoretical Aspects”, in R. Adler et al, eds, "A Practical Guide to Heavy Tails: Statistical Techniques and Applications", Boston: Birkhluser, 79-110. 8. Panorska, A.K., S. Mittnik, and S.T. Rachev, 1995, "Stable GARCH Models for Financial Time Series", Applied Mathematics Letters, 815, 33-37. Samorodnitsky, G. and M.S. Taqqu, 1994, “Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance”, New York: Chapman & Hall.
9.
2.
3.
4.
5.
6.
7.
7 8
These models are named as ARMA-α-stable models. For discussion of stable-GARCH models see Panorska, Mittnik and Rachev (1995) and Mittnik, Paolella, and Rachev (1997).