STABLE MODELING OF MARKET AND CREDIT VALUE AT RISK
8. Stable modeling of portfolio risk for skewed dependent credit returns
We quantify portfolio riskRP by generating a distribution of its possible values and deriv- ing a portfolio VaR from the constructed distribution ofRP. In a case of portfolio assets with skewed dependent credit returns, simulations of the RP values should reflect the “cu- mulative” skewness and maintain the dependence (association) among them. In order to do that, we decompose single credit returnsRi 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 skew- ness parts of individual credit returns. Simulations of the portfolio credit returns valuesRP 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 ofRP as a sum of the two generated components. Explanations of our methodology are provided below.
A stable random variableR∼Sα(σ, β,0)can be decomposed (in distribution) into two independent stable random variablesR(1)andR(2):
R=d R(1)+R(2), where
R(1)∼Sα(σ1, β1,0), R(2)∼Sα(σ2, β2,0), σ=
σ1α+σ2α1/α
, (12)
β=β1σ1α+β2σ2α
σ1α+σ2α . (13)
Suppose that: (i)R(1)is a symmetric stable variable:β1=0; (ii)σ1=σ2=σ∗. Then, formulas (12) and (13) can be reduced to the following expressions:
σ=21/ασ∗, (14)
β=1
2β2. (15)
From Equations (14) and (15), we have σ∗=2−1/ασ, β2=2β.
In sum, a stable random variableR∼Sα(σ, β,0)can be decomposed (in distribution) into two independent stable random variables: symmetricR(1)and skewedR(2):
R=d R(1)+R(2), (16)
Ch. 7: Stable Modeling of Market and Credit Value at Risk 297
where
R(1)∼Sα
2−1/ασ,0,0
, (17)
R(2)∼Sα
2−1/ασ,2β,0
. (18)
Using methodology (16)–(18), we can divide individual credit returnsRi ∼SαRi(σRi, βRi,0)into the “dependence” and “skewness” parts. First, we partitionRi into the “sym- metry” and “skewness” fragments:
Ri=d R(1)i +Ri(2), where
R(1)i ∼SαRi
2−1/αRiσRi,0,0
, Ri(2)∼SαRi
2−1/αRiσRi,2βRi,0 ,
partsR(1)i andR(2)i are independent,i=1, . . . , n. Second, we suppose: (i)Ri(1), i=1, . . . , n, are dependent and (ii)R(2)i , i=1, . . . , n, are independent. Consequently, symmet- ric terms R(1)i explain dependence (association) amongRi’s and termsRi(2) account for skewness ofRi’s.
Based on Property 1 (see Section 7),Ri(1)∼SαRi(2−1/αRiσRi,0,0)can be written as a transformation of a normal random variable:
R(1)i =Si1/2Gi, where
Gi∼S2(σGi,0,0)=N 0,2σG2i
, Si∼SαRi/2 2−2/αRiσR2
i
σG2
i
cos π α 4
2/αRi
,1,0
, Si is independent ofGi, i=1, . . . , n.
Random rescaling transformations of normal variablesGi intoR(1)i maintain the depen- dence structure. Therefore, from the dependence amongGi’s we can determine the depen- dence amongR(1)i , or the dependence amongRi.
Adding separately the dependence and skewness terms ofRi’s, we obtain the two com- ponents of the portfolio returnsRP:
RP=R(1)P +RP(2), (19)
whereR(1)P =n
i=1wiR(1)i =n
i=1wiSi1/2Gi is the “dependence” component andR(2)P = n
i=1wiR(2)i is the “skewness” component.
298 S.T. Rachev et al.
We simulate theRP values based on decomposition (19):RP =R(1)P +R(2)P . The simu- lations are executed according to the next algorithm:45
Step 1: Estimate stable parameters ofRi:αRi,βRi,σRi,àRi.
Step 2: “Center” theRi observations:R∗i =Ri−àRi. Further on, we shall assumeàRi= 0 and considerRi∗asRi:Ri∼SαRi(σRi, βRi,0), i=1, . . . , n.
Step 3: Evaluate the covariance matrix of normal random variables(Gi)1in at time T +1,ΣT+1= {cij,T+1|T}, i=1, . . . , n, j=1, . . . , n, using exponential weighting:
ci,T2 +1|T =(1−θ ) K
k=0
θkRi,T2 −k,
cij,T2 +1|T =(1−θ ) K
k=0
θkRi,T−kRj,T−k,
whereT+1|T denotes a forecast for timeT+1 conditional on information up to timeT; θis a decay factor, 0< θ <1;Kis a number of observations’ lags.
Step 4: Generate a value of the multivariate normal random variableG=(G1, G2, . . . , Gn)with the covariance matrixΣT+1.
Step 5: Simulate values of stable random variables
Si ∼SαRi/2 21−2/αRiσR2
i
c2i cos π α 4
2/αRi
,1,0
, i=1, . . . , n.
Step 6: ComputeR(1)i =Si1/2Gi, i=1, . . . , n.
Step 7: GenerateRi(2)∼SαRi(2−1/αRiσRi,2βRi,0), i=1, . . . , n.
Step 8: CalculateRP =n
i=1wiRi(1)+n
i=1wiR(2)i .
Step 9: Repeat Steps 4–8 a large number of times to form anRP-distribution.
Derive a portfolio VaR estimate as the negative of a chosen quantile of theRP-distri- bution.
We implement the suggested procedure (Step 1–Step 9) for the risk assessment of the same portfolio of indices as in Section 7. We suppose that returns on indices are dependent skewed-α-stable. The portfolio VaR estimates are presented in Table 16.
The 99% portfolio VaR estimates fall within the 99% VaR range (3.518, 9.813) of Sec- tion 6. From Table 16, the VaR magnitude generally: (i) increases when the decay factor θincreases from 0.85 to 0.94; (ii) declines whenθchanges from 0.94 to 0.975. Thus, the decay factorθ=0.94 leads to more conservative VaR estimates. The 1%–99% truncation band appears to produce the lowest KD and AD statistics. Based on our observations, we
45This algorithm is an extended version of the algorithm in Section 7.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 299 Table 16
Portfolio VaR for skewed dependent credit returns
Decay Truncation Portfolio VaR Kolmogorov Anderson–
factorθ points (%) distance Darling
99% VaR 95% VaR
0.85 10−90 4.939 2.904 7.22 0.20
5−95 5.380 3.162 5.64 0.18
No 5.449 3.236 5.43 0.17
0.94 10−90 5.101 3.009 6.53 0.19
5−95 5.456 3.248 5.24 0.17
1−99 5.596 3.363 4.70 0.14
No 5.455 3.231 5.13 0.17
0.975 10−90 5.112 3.021 6.54 0.19
5−95 5.416 3.238 5.34 0.17
1−99 5.471 3.307 4.37 0.14
No 5.298 3.238 5.43 0.15
would recommend to employθ=0.94 and the 1%–99% truncation band in VaR deriva- tions under the assumption of skewed dependent credit returns. We computed marginal VaRs for the same combinations of the decay factor and the truncation band as in Ta- ble 16. The marginal VaR estimates were smaller than the corresponding stand-alone VaR measurements, which supports feasibility of suggested procedure for simulating portfolio returns.
We have applied stable modeling to the total risk assessment of credit returns. Below we analyze stable modeling of isolated credit risk.