Dependent Conditional Value-at-Risk for Aggregate Risk Models
DD EPENDENT C ONDITIONAL V ALUE - AT -R ISKFOR A GGREGATE R ISK M ODELS
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Bony Josaphat
Statistics Research DivisionInstitut Teknologi BandungJl. Ganesha No.10, Bandung 40132 [email protected]
Khreshna Syuhada
Statistics Research DivisionInstitut Teknologi BandungJl. Ganesha No.10, Bandung 40132 [email protected]
September 8, 2020 A BSTRACT
Risk measure forecast and model have been developed in order to not only provide better forecastbut also preserve its (empirical) property especially coherent property. Whilst the widely used riskmeasure of Value-at-Risk (VaR) has shown its performance and benefit in many applications, it is infact not a coherent risk measure. Conditional VaR (CoVaR), defined as mean of losses beyond VaR,is one of alternative risk measures that satisfies coherent property. There has been several extensionsof CoVaR such as Modified CoVaR (MCoVaR) and Copula CoVaR (CCoVaR). In this paper, wepropose another risk measure, called Dependent CoVaR (DCoVaR), for a target loss that depends onanother random loss, including model parameter treated as random loss. It is found that our DCoVaRoutperforms than both MCoVaR and CCoVaR. Numerical simulation is carried out to illustrate theproposed DCoVaR. In addition, we do an empirical study of financial returns data to compute theDCoVaR forecast for heteroscedastic process.
Keywords
Archimedean Copula; Farlie-Gumbel-Morgenstern family; GARCH model; Pareto distribution; asset returns
Risk measure forecast has been one of major interests in finance and insurance and developed by academia andpractitioners. The common and widely used risk measure is Value-at-Risk (VaR), see e.g. McNeil et al. (2005), Kabailaand Mainzer (2018), Syuhada et al. (2020); Nieto and Ruiz (2016) provided latest review on VaR and its backtesting.It forecasts maximum tolerated risk at certain level of significance. Basically, VaR is calculated through the quantileof its loss distribution. Whilst the widely used risk measure of VaR has shown its performance and benefit in manyapplications, it is in fact not a coherent risk measure.There have been some efforts done by authors to seek an improvement of VaR, beside describing formulas of VaRand CoVaR as shown in Nadarajah et al. (2016). Their works were derived in two different directions. The first isimprovement of VaR forecast accuracy i.e. the coverage probability of VaR forecast is closer to the target nominalor significant level. The example of this is an improved VaR in which the method was developed by Kabaila andSyuhada (2008, 2010) and Syuhada (2020) whilst estimating confidence region by adjusted empirical likelihood toobtain better coverage was proposed by Yan and Zhang (2016). Furthermore, Kabaila and Mainzer (2018) consideredlinear regression model that consists of approximate VaR and exact VaR in which the former is an unbiased estimatorfor the latter.The second improvement to VaR is seeking alternative risk measure(s) that capture coherent property. The commonlyused coherent risk measure is the Conditional VaR (CoVaR), defined as mean of losses beyond VaR, see e.g. Artzner etal. (1999), McNeil et al. (2005), Jadhav et al. (2009, 2013), Righi and Ceretta (2015), and Brahim et al. (2018). Severalextensions of CoVaR have proposed. Jadhav et al. (2013) has modified CoVaR by introducing fixed boundary, insteadof infinity, for values beyond VaR. They named the risk measure as Modified CoVaR (MCoVaR). Meanwhile, another a r X i v : . [ q -f i n . R M ] S e p ependent Conditional Value-at-Riskfor Aggregate Risk Models A P
REPRINT extension of CoVaR, called Copula CoVaR (CCoVaR), was suggested by Brahim et al. (2018) in which they forecast atarget risk by involving another dependent risk or associate risk. The use of Copula in this dependent case is crucial.The application of this method may be found when we forecast risk premia (as a target risk) that depends on claim size(as an associate risk). Note that Kang et al. (2019) considered such premium and claim size dependence to forecast VaRand CoVaR by involving Copula.Motivated by the work of Jadhav et al. (2013) and Brahim et al. (2018), in this paper, we propose an alternative coherentrisk measure that is not only “considering a fixed upper bound of losses beyond VaR” but also “taking into account adependent risk”. Our proposed risk measure is called Dependent CoVaR (DCoVaR). When we compute an MCoVaRforecast, it will reduce number of losses beyond VaR and thus make this forecast smaller than the corresponding CoVaR.We argue that this forecast must also be accompanied by a dependent risk since this risk scenario occurs in practice, seefor instance Zhang et al. (2018) and Kang et al. (2019).This paper is organized as follows. Section 2 describes our proposed risk measure of DCoVaR in which its formularelies on joint distribution either classical or Copula. Properties of DCoVaR are also stated. The DCoVaR forecast forPareto random loss is explained in Section 3. Such forecast is computed for target risk of Pareto and associate risk ofPareto as well. Farlie-Gumbel-Morgenstern and Archimedean Copulas are employed. The target risk may be extendedto an aggregate risk. Numerical simulation is carried out. Section 4 considers a real application of DCoVaR forecastfor financial returns data (NASDAQ and TWIEX) in which such returns are modeled by heteroscedastic process ofGARCH. Appendix contains all proofs.
Suppose that an aggregate loss model S N − k is constructed by a collection of dependent random losses X , X , . . . , X N − k given by S N − k = X + . . . + X N − k , for k = 0 , , . . . , N − . The VaR forecast of S N − k , at asignificant level α , is obtained by the inverse of distribution function of S N − k , i.e. VaR α ( S N − k ) = F − S N − k ( α ) = Q α .In practice, the parameter of the model must be estimated from data. Thus, the coverage probability of this VaR forecastis bounded to O ( n − ) since it takes into account the parameter estimation error. Provided VaR forecast, Q α , the meanof losses beyond VaR to infinity may be calculated, called Conditional VaR (CoVaR). Unlike VaR, the CoVaR forecastpreserves subadditivity (thus satisfies coherent property) that makes diversification reasonable. Furthermore, as statedby Koji and Kijima (2003), any coherent risk measure can be represented as a convex combination of CoVaR.We aim to find a risk measure forecast that calculates the mean of S N − k beyond its VaR up to a fixed value of lossesand the S N − k depends on another dependent or associate random loss. Our proposed risk measure forecast, namelyDependent Conditional VaR (DCoVaR), calculates the mean of S in which Q α ≤ S N − k ≤ Q α and S N − k dependson another random loss Y as follows DCoVaR (cid:16) S N − k (cid:12)(cid:12)(cid:12) Y (cid:17) = E (cid:104) S N − k (cid:12)(cid:12)(cid:12) Q α ≤ S N − k ≤ Q α , Q δ ( Y ) ≤ Y ≤ Q δ ( Y ) (cid:105) , (1)where α = α + (1 − α ) a +1 and δ = δ + (1 − δ ) d +1 for a specified a and d . Note that such random loss Y may be(i) a single component of S N − k , (ii) another aggregate risk model S N − l , or (iii) a parameter model. Note also that inmany applications, the distribution of S N − k and Y may be either non-normal or not specified so that we need a Copula.In what follows, we state our proposed DCoVaR in the two propositions below.P ROPOSITION S N − k and Y be two random losses with a joint probability function f S ,Y . Let α, δ ∈ (0 , . The DependentConditional VaR (DCoVaR) of S N − k given values beyond its VaR up to a fixed value of losses and a random loss Y isgiven by DCoVaR ( δ,d )( α,a ) (cid:16) S N − k (cid:12)(cid:12)(cid:12) Y (cid:17) = Q δ (cid:82) Q δ Q α (cid:82) Q α s f S N − k ,Y ( s, y ) ds dy Q δ (cid:82) Q δ Q α (cid:82) Q α f S N − k ,Y ( s, y ) ds dy , (2)where Q α = Q α ( S N − k ) , Q δ = Q δ ( Y ) , α = α + (1 − α ) a +1 and δ = δ + (1 − δ ) d +1 .In practice, joint probability function is difficult to find unless a bivariate normal distribution is assumed. For the caseof joint exponential distribution, we may refer to Kang et al. (2019) for Sarmanov’s bivariate exponential distribution.In most cases, two or more dependent risks rely on Copula in order to have explicit formula of its joint distribution.2ependent Conditional Value-at-Riskfor Aggregate Risk Models A P
REPRINT P ROPOSITION S N − k and Y be two random losses with a joint distribution function represented by a Copula C . The DependentConditional VaR (DCoVaR) of S N − k given values beyond its VaR up to a fixed value of losses and a random loss Y isgiven by DCoVaR ( δ,d )( α,a ) ( S N − k | Y ; C ) = Q α (cid:82) Q α Q δ (cid:82) Q δ s c ( F S N − k ( s ) , F Y ( y )) f S N − k ( s ) f Y ( y ) dy dsC ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) , (3)where F S N − k denote distribution function of S N − k , α = α + (1 − α ) a +1 and δ = δ + (1 − δ ) d +1 .R EMARK . According to the method of Brahim et al. (2018), the DCoVaR formula is represented by
DCoVaR ( δ,d )( α,a ) ( S N − k | Y ; C ) = α (cid:82) α δ (cid:82) δ F − S N − k ( u ) c ( u, v ) dv duC ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) , (4)where F − S N − k denote quantile function of S N − k , u = F S N − k ( s ) , v = F Y ( y ) , α = α + (1 − α ) a +1 and δ = δ + (1 − δ ) d +1 . This formula, however, may not be obtained when no closed form expression of the quantile functionis given.The following properties apply to our proposed DCoVaR. The first property is to argue that the DCoVaR satisfiescoherent property of risk measure in particular the subadditivity i.e. the DCoVaR of aggregate risk is no more thanaggregate of DCoVaR of individual risk. Meanwhile, the second property is to show that the DCoVaR outperforms thanMCoVaR and CCoVaR.P ROPERTY
1. The Dependent Conditional VaR (DCoVaR) is a coherent risk measure.P
ROPERTY
2. The Dependent Conditional VaR (DCoVaR) has larger risk than or equal to MCoVaR and lower risk thanor equal to CCoVaR.
Suppose that X i , component for aggregate risk S N − k , is a Pareto random loss with parameter (1 , β i ) . We consider adependent random loss Y that follows a Pareto distribution with parameter (1 , β a ) . The distribution functions of X i and Y are, respectively, F X i ( x ) = 1 − ( β i / ( x + β i )) , for x i ≥ , and F Y ( y ) = 1 − ( β a / ( y + β a )) , for y ≥ . Theirinverses are easy to find and thus their VaR’s are straightforward i.e. VaR α ( X i ) = Q α = β i (cid:2) (1 − α ) − − (cid:3) . In whatfollows, we provide some examples.E XAMPLE -1: DC O V A R FORECAST OF A P ARETO RISK WITH A P ARETO MARGINAL . The risk measure of DCoVaRforecast for S = X i , given Y , may be found by using Proposition 2 since we apply a Copula for their distributionfunction. Specifically, we employ the Farlie-Gumbel-Morgenstern (FGM): C FGM θ ( u, v ) = u v + θ u v (1 − u ) (1 − v ) ,where u, v ∈ [0 , , θ ∈ [ − , . Suppose that the joint distribution of S and Y , defined by an FGM copula, is F S ,Y ( s, x ) = C θ SY ( F S ( s ) , F Y ( y )) , where θ ∈ [ − , . Then, the DCoVaR of S at levels of α and δ, < α, δ < , is given by DCoVaR ( δ,d )( α,a ) ( S | Y ; C ) = β ( δ − δ ) C ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) × (cid:110)(cid:104) α − α − ln(1 − (1 − α ) a ) (cid:105)(cid:104) θ SY (1 − δ − δ ) (cid:105) − θ SY (1 − δ − δ ) (cid:104) α ( α − − α ( α − −
12 ln (cid:0) − (1 − α ) α (cid:1)(cid:105)(cid:111) (5)where the Copulas are C ( α , δ ) = α δ + θ i α δ (1 − α ) (1 − δ ) , C ( α, δ ) = αδ + θ i αδ (1 − α ) (1 − δ ) , C ( α , δ ) = α δ + θ i α δ (1 − α ) (1 − δ ) , and C ( α, δ ) = αδ + θ i αδ (1 − α ) (1 − δ ) .3ependent Conditional Value-at-Riskfor Aggregate Risk Models A P
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Meanwhile, when applying the method of Brahim et al., we find the DCoVaR as follows
DCoVaR ( δ,d )( α,a ) ( S | Y ; C ) = β ( δ − δ ) C ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) × (cid:110)(cid:104) ln(1 − (1 − α ) a ) (cid:105)(cid:104) θ SY (1 − δ − δ ) − (cid:105) − (cid:104) θ SY (1 + α − α )(1 − δ − δ ) − (cid:105) ( α − α ) (cid:111) (6)E XAMPLE -2. DCoVaR of Pareto risk in Example-1 may be carried out by using a Clayton Copula (which is anArchimedean Copula): C C θ ( u, v ) = (cid:0) u − θ + v − θ − (cid:1) − /θ . The resulting DCoVaR forecast, however, is not in a closedform expression. DCoVaR ( δ,d )( α,a ) ( S | Y ; C ) = β i C ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) × (cid:34) α (cid:90) α (cid:0) u − θ + δ − θ − (cid:1) − θθ − (cid:0) u − θ + δ − θ − (cid:1) − θθ (1 − u ) u θ +1 du − (cid:110)(cid:0) α − θ + δ − θ − (cid:1) − θθ − (cid:0) α − θ + δ − θ − (cid:1) − θθ (cid:111) + (cid:110)(cid:0) α − θ + δ − θ − (cid:1) − θθ − (cid:0) α − θ + δ − θ − (cid:1) − θθ (cid:111)(cid:35) . (7)E XAMPLE -3. DCoVaR for multivariate risk forecast may be expressed for the case of N identical dependent Paretorandom risks: X i , · · · , X N . Their joint probability function is given by f ( x , · · · , x n ; γ, N ) = Γ( γ + N )Γ( γ ) β N (cid:16) β (cid:80) Ni =1 x i (cid:17) γ + N . Let S N = X + · · · + X N and Y be another Pareto random risk with parameter (1 , β a ) . Suppose that the joint distributionof S N and Y is defined by a bivariate FGM Copula F S N ,Y ( s, y ) = C θ SY ( F S N ( s ) , F Y ( y )) , where θ ∈ [ − , . Then,for N even, the DCoVaR of S N at levels α and δ, < α, δ < , is given byDCoVaR ( δ,d )( α,a ) (cid:16) S N | Y ; C (cid:17) = N γ ( δ − δ ) C ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) × (cid:40)(cid:16) δ − δ (cid:17)(cid:16) θ SY (1 − δ − δ ) (cid:17) × (cid:34) ln 1 − α /N − α /N + N (cid:16) α /N − α /N (cid:17) − · · · + NN − (cid:20)(cid:16) − α /N (cid:17) N − − (cid:16) − α /N (cid:17) N − (cid:21) − N (cid:20)(cid:16) − α /N (cid:17) N − (cid:16) − α /N (cid:17) N (cid:21) (cid:35) + 2 θ SY (1 − δ − δ ) × (cid:34) ln 1 − α /N − α /N + N (cid:16) α /N − α /N (cid:17) − · · · + 2 N N − (cid:20)(cid:16) − α /N (cid:17) N − − (cid:16) − α /N (cid:17) N − (cid:21) − N (cid:20)(cid:16) − α /N (cid:17) N − (cid:16) − α /N (cid:17) N (cid:21) (cid:35)(cid:41) (8)whilst for N odd, the DCoVaR of S N at levels α and δ, < α, δ < , is given by4ependent Conditional Value-at-Riskfor Aggregate Risk Models A P
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DCoVaR ( δ,d )( α,a ) (cid:16) S N | Y ; C (cid:17) = N γ ( δ − δ ) C ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) × (cid:40)(cid:16) δ − δ (cid:17)(cid:16) θ SY (1 − δ − δ ) (cid:17) × (cid:34) ln 1 − α /N − α /N + N (cid:16) α /N − α /N (cid:17) − · · ·− NN − (cid:20)(cid:16) − α /N (cid:17) N − − (cid:16) − α /N (cid:17) N − (cid:21) + 1 N (cid:20)(cid:16) − α /N (cid:17) N − (cid:16) − α /N (cid:17) N (cid:21) (cid:35) + 2 θ SY (cid:16) − δ − δ (cid:17) × (cid:34) ln 1 − α /N − α /N + N (cid:16) α /N − α /N (cid:17) − · · · + 2 N N − (cid:20)(cid:16) − α /N (cid:17) N − − (cid:16) − α /N (cid:17) N − (cid:21) − N (cid:20)(cid:16) − α /N (cid:17) N − (cid:16) − α /N (cid:17) N (cid:21) (cid:35)(cid:41) (9)DC O V A R FORECAST FOR P ARETO RANDOM LOSS : A
SIMULATION RESULT
We carry out a simulation study for calculating DCoVaR forecast. The parameters of Pareto distribution of X and Y are, respectively, β = 1 . and β a = 1 . . Suppose also the model parameter Λ is gamma distributed with shape andscale parameters τ = ω = 1 . The significance level for α (and δ ) is set above 0.9 whilst we set a = d = 0 . . Figure1-3 show the DCoVaR forecast for the above parameters set up. As for comparison, we also plot the MCoVaR forecast.For each figure, we have an associate or dependent random loss Y which is a Pareto random loss, an aggregate Y = S of Pareto losses, and a parameter model Y = Λ of gamma distributed. It is shown from the figures that the DCoVaRforecast tends to increase as δ increases whilst the MCoVaR forecast remains the same. As for the CCoVaR forecast, itis larger than the DCoVaR forecast (not shown in the figures).Figure 1: DCoVaR forecast of S with various of Y and using Clayton Copula; Y is Pareto distributed, Y = S isPareto distributed, Y = Λ is Gamma distributed; such forecasts are in comparison to MCoVaR forecast.Note that, as for the Copula choices, we have used Archimedean Copulas. The Clayton Copula (Figure 1) functionis given by C Cθ ( u, v ) = (cid:16) u − θ + v − θ − (cid:17) − /θ , θ ∈ [ − , ∞ ) . Meanwhile, for other Copulas of Gumbel (Figure 2)5ependent Conditional Value-at-Riskfor Aggregate Risk Models A P
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Figure 2: DCoVaR forecast of S with various of Y and using Gumbel Copula; Y is Pareto distributed, Y = S isPareto distributed, Y = Λ is Gamma distributed; such forecasts are in comparison to MCoVaR forecastFigure 3: DCoVaR forecast of S with various of Y and using Frank Copula; Y is Pareto distributed, Y = S is Paretodistributed, Y = Λ is Gamma distributed; such forecasts are in comparison to MCoVaR forecast6ependent Conditional Value-at-Riskfor Aggregate Risk Models A P
REPRINT and Frank (Figure 3) the functions are C Gθ ( u, v ) = exp (cid:110) − (cid:104) ( − ln u ) θ + ( − ln u ) θ (cid:105) /θ (cid:111) , θ ∈ [0 , ∞ ) and C Fθ ( u, v ) = − θ ln (cid:16) − (1 − e − θu )(1 − e − θv )1 − e − θ (cid:17) , θ ∈ ( −∞ , ∪ (0 , ∞ ) , respectively.The significance level for α (and δ ) is set above 0.9. Unlike calculating CoVaR forecast, the DCoVaR forecastcomputation requires two significance levels. In particular, the joint significance level is given by P (cid:16) Q α ≤ S ≤ Q α , Q δ ≤ Y ≤ Q δ (cid:17) = C θ ( α , δ ) − C θ ( α, δ ) − C θ ( α , δ ) + C θ ( α, δ ) . For the case of a = d = 0 , the joint significance level is − α − δ + C θ ( α, δ ) . We use joint significance level tomeasure the number of violations of the DCoVaR forecast. We generate data of 3000 observations for each X , Y , and Λ . The DCoVaR forecast is computed by using Proposition 2.Assessment of accuracy for the DCoVaR forecast is carried out by first observing joint significance level. For example,in Table 1 (first row, first column), 2.79% joint significance level is lower than 10%. This means that the DCoVaRforecast is quite accurate. Then, by calculating the number of violations of the DCoVaR ( δ, . α, . , it is obtained 1.83%(number of violations is 55, total observations 3000; 55/3000=0.0183). Basically, the numbers of violations are thenumber of sample observations located out of the critical value i.e. more than or equal to DCoVaR ( δ, . α, . forecast.These computations are shown in Table 1-3, for Clayton, Gumbel, and Frank Copulas, respectively. In summary, usingClayton Copula provides more accurate forecast due to lower joint significance level and number of violations.Table 1: Joint significance level and number of violations of the DCoVaR ( δ, . α, . forecast of S associated with Y withClayton Copula ( θ = 7 . , a = d = 0 . α = 0 . α = 0 . sig. level (%) no. violations sig. level (%) no. violations(%) (%) δ ( δ, . α, . forecast of S associated with Y withGumbel Copula ( θ = 6 . , a = d = 0 . α = 0 . α = 0 . sig. level (%) no. violations sig. level (%) no. violations(%) (%) δ We carry out a numerical analysis of returns data and model it with stochastic volatility processes. In particular, weemploy the Generalized Autoregressive Conditional Heteroscedastic (GARCH) model of order one. Consider tworeturns processes, { X t } and { X t } . Suppose that each process follows a GARCH(1,1) model defined as X it = ε t (cid:112) h t , h t = κ + κ X t − + η h t − , i = 1 , A P
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Table 3: Joint significance level and number of violations of the DCoVaR ( δ, . α, . forecast of S associated with Y withFrank Copula ( θ = 25) , a = d = 0 . α = 0 . α = 0 . sig. level (%) no. violations sig. level (%) no. violations(%) (%) δ κ > , κ ≥ , η ≥ , and κ + η < . Let S t = X t and Y t = X t . The DCoVaR forecast of S t with anassociate risk Y t is given byDCoVaR ( δ,d )( α,a ) ( S t | Y t ; C ) = E ( S t | Q αt < S t < Q α t , Q δt < Y t < Q δ t )= Q α t (cid:82) Q αt Q δ t (cid:82) Q δt s t c ( F ( s t ) , F ( y t ) |G t − ) f ( s t |G t − ) f ( y t |G t − ) dy t ds t P ( Q αt < S t < Q α t , Q δt < Y t < Q δ t ) (10)where f S t ( ·|G t − ) is the conditional probability function of the target risk S t on G t − . The denominator of (10) is givenby P (cid:16) Q αt < S t < Q α t , Q δt < Y t < Q δ t (cid:17) = (cid:90) Q α t Q αt (cid:90) Q δ t Q δt c ( F ( s t ) , F ( y t ) |G t − ) f ( s t |G t − ) f ( y t |G t − ) dy t ds t , and Q αt as well as Q δt satisfy P ( S t ≤ Q αt |G t − ) = Q αt (cid:90) −∞ f ( s t |G t − ) ds t , P ( Y t ≤ Q δt |G t − ) = Q δt (cid:90) −∞ f ( y t |G t − ) dy t . E MPIRICAL RESULTS X it = − ln (cid:18) P it P i,t − (cid:19) , where P it is the price of an i -th asset at time t , i = 1 , . Figure 4 shows such daily returns. In addition, we may observethat one of the stylized facts of returns, known as volatility clustering, occurs in both NASDAQ and TWIEX returns.Huang et al. (2009) argued that the GARCH- t (1,1) model were appropriate for the returns of NASDAQ and TWIEX.Accordingly, we presents the maximum likelihood estimates for such model parameter as in Table 4.Table 4: Parameters estimates of GARCH- t (1,1) model for NASDAQ and TWIEX. (cid:98) κ (cid:98) κ (cid:98) η (cid:98) ν NASDAQ 0.0064 0.0266 0.9678 6.4188TWIEX 0.0368 0.0643 0.9082 6.9057In order to calculate the DCoVaR forecast, Figure 5, we do in-sample forecast in which we have used 1000 firstobservations whilst the out-of-sample is to evaluate forecasting performance. As in Table 4 above, Student’s t distribution is assumed for innovation. Meanwhile, Archimedean Copula are used for the joint distribution function.8ependent Conditional Value-at-Riskfor Aggregate Risk Models A P
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Figure 4: Daily returns of NASDAQ and TWIEX. A stylized fact of volatility clustering may be observed for bothreturns. Figure 5: DCoVaR forecast of the NASDAQ returns, given the TWIEX returns.9ependent Conditional Value-at-Riskfor Aggregate Risk Models
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Table 5: Joint significance level (%) and number of violations (%) of the DCoVaR ( δ, α, forecast by using Clayton andGumbel Copulas.Copula ParameterClayton α = δ = 0 . α = δ = 0 . α = 0 . , δ = 0 . α = 0 . , δ = 0 . ( (cid:98) θ = 0 . ) C θ (0 . , . C θ (0 . , . C θ (0 . , . C θ (0 . , . Joint sig. level 3.49 5.72 4.41 4.41No. violations 0.16 0.65 0.16 0.65Copula ParameterGumbel α = δ = 0 . α = δ = 0 . α = 0 . , δ = 0 . α = 0 . , δ = 0 . ( (cid:98) θ = 1 . ) C θ (0 . , . C θ (0 . , . C θ (0 . , . C θ (0 . , . Joint sig. level 1.95 3.90 2.74 2.74No. violations 0.16 0.49 0.16 0.49In particular, we employ Clayton and Gumbel Copulas. The parameter θ for each Copula is estimated by maximumlikelihood method. We obtain (cid:98) θ C = 0 . and (cid:98) θ G = 1 . , respectively.The number of violations of the DCoVaR ( δ, α, forecast for both Clayton and Gumbel Copulas are presented in Table 5.It is the number of sample observations located out of the critical value i.e. less than or equal the DCoVaR forecast. It isshown from the table that the DCoVaR forecast with Gumbel Copula has lower joint significance level in comparison tothe DCoVaR forecast with Clayton Copula. As for the number of violations, it conforms the use of Gumbel Copula. Inshort, it suggests that Gumbel Copula is more appropriate Copula for describing the joint distribution of NASDAQ andTWIEX returns. The use of GARCH model for marginal of asset returns may be replaced by its extensions such as ARMA-GARCH andGJR-GARCH models. In addition, any innovations may also be applied to such volatility models. Syuhada (2020) hascarried out VaR forecast and compared such observable stochastic volatility process (GARCH) class of models) withthe latent one i.e the Stochastic Volatility Autoregressive (SVAR) model.
Acknowledgment
We are grateful to anonymous referees for their comments that improve the paper. We also thank Prof Ken Seng Tan(University of Waterloo) for a thoughtful discussion. 10ependent Conditional Value-at-Riskfor Aggregate Risk Models
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A P
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Appendix P ROOF FOR P ROPOSITION
1. For simplicity, let Q p = Q p ( S N − k ) , Q p = Q p ( S N − k ) , Q δ = Q δ ( Y ) , and Q δ = Q δ ( Y ) . Then DCoVaR ( δ,d )( α,a ) (cid:16) S N − k (cid:12)(cid:12)(cid:12) Y (cid:17) = 1 P ( Q α ≤ S N − k ≤ Q α , Q δ ≤ S N − k ≤ Q δ ) × E [ S N − k ( Q α ≤ S N − k ≤ Q α ,Q δ ≤ S N − k ≤ Q δ ) ]= Q δ (cid:82) Q δ Q α (cid:82) Q α s f S N − k ,Y ( s, y ) ds dy Q δ (cid:82) Q δ Q α (cid:82) Q α f S N − k ,Y ( s, y ) ds dy P ROOF FOR P ROPOSITION
2. We assume first that s ≤ Q p ( S N − k ) . We obtainP ( S N − k ≤ s | Q p ≤ S N − k ≤ Q p , Q δ ≤ Y ≤ Q δ ) = P ( Q p ≤ S N − k ≤ s, Q δ ≤ Y ≤ Q δ ) P ( Q p ≤ S N − k ≤ Q p , Q δ ≤ Y ≤ Q δ ) , where the denominator may be written asP ( Q p ≤ S N − k ≤ Q p , Q δ ≤ Y ≤ Q δ ) = C ( p , δ ) − C ( p, δ ) − C ( p , δ ) + C ( p, δ ) . Thus, P ( S N − k ≤ s | Q p ≤ S N − k ≤ Q p , Q δ ≤ Y ≤ Q δ )= 1 C ( p , δ ) − C ( p, δ ) − C ( p , δ ) + C ( p, δ ) × Q δ (cid:90) Q δ s (cid:90) Q p ∂ C ( F S N − k ( s ) , F Y ( y )) ∂s ∂y ds dy. For fixed level p = α and a , the DCoVaR of S N − k is given byDCoVaR ( δ,d )( α,a ) ( S N − k | Y ; C )= 1 C ( p , δ ) − C ( p, δ ) − C ( p , δ ) + C ( p, δ ) × Q δ (cid:90) Q δ Q α (cid:90) Q α s ∂ C ( F S N − k ( s ) , F Y ( y )) ∂s ∂y ds dy. A P
REPRINT
We suppose that the densities of F S N − k and F Y are f S N − k and f Y , respectively. Thus,DCoVaR ( δ,d )( α,a ) ( S N − k | Y ; C )= 1 C ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) × Q δ (cid:90) Q δ Q α (cid:90) Q α s c ( F S N − k ( s ) , F Y ( y )) f S N − k ( s ) f Y ( y ) ds dy. P ROOF FOR P ROPERTY
1. To prove coherent property, we follow the proof of the subbaditivity of the CoVaR, given inAcerbi and Tasche (2002) and that of the MCoVaR, given in Jadhav et al. (2013). For simplicity, let S N − k = S. Let F S ( s ) be the distribution function of a continuous random variable S and define the α -quantile of S as Q α = F − S ( α ) for a specified probability α ∈ (0 , and δ -quantile of Y as Q δ = F − Y ( δ ) for some probability δ ∈ (0 , . We maywrite the DCoVaR asDCoVaR ( δ,d )( α,a ) ( S | Y ; C )= 1 C ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) × E (cid:104) S { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } (cid:105) . Let S = S + S . Then (1 − δ ) d +1 + C ( α, δ ) − C ( α, δ ) × (cid:110) DCoVaR ( δ,d )( α,a ) ( S | Y ; C ) + DCoVaR ( δ,d )( α,a ) ( S | Y ; C ) − DCoVaR ( δ,d )( α,a ) ( S | Y ; C ) (cid:111) = E (cid:104) S (cid:16) { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } − { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } (cid:17) + S (cid:16) { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } − { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } (cid:17)(cid:105) ≥ Q α E (cid:104) { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } − { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } (cid:105) + Q α E (cid:104) { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } − { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } (cid:105) = Q α (cid:110) C ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) − C ( α , δ ) + C ( α, δ ) + C ( α , δ ) − C ( α, δ ) (cid:111) + Q α (cid:110) C ( α , δ ) − C ( α, δ ) − C ( α , δ ) + C ( α, δ ) − C ( α , δ ) + C ( α, δ ) + C ( α , δ ) − C ( α, δ ) (cid:111) = 0 . In the above inequality, we have used(*) if
S < Q α , then { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } − { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } ≥ (**) if Q α ≤ S ≤ Q α , then { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } − { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } ≤ This proves that the DCoVaR follows the subadditivity and hence is a coherent risk measure.P
ROOF FOR P ROPERTY
2. Note that the statement in Property 2 is mathematically equivalent to these both inequalities.MCoVaR ( α,a ) ( S ) ≤ DCoVaR ( δ, α,a ) ( S | Y ; C ) , DCoVaR ( δ, α,a ) ( S | Y ; C ) ≤ CCoVaR δα ( S | Y ; C ) . Note that 13ependent Conditional Value-at-Riskfor Aggregate Risk Models
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1. We may write the MCoVaR asMCoVaR ( α,a ) ( S ) = 1(1 − α ) a +1 E [ S { Q α ≤ S ≤ Q α } ] . Thus, (1 − α ) a +1 (cid:104) MCoVaR ( α,a ) ( S ) − DCoVaR ( δ, α,a ) ( S | Y ; C ) (cid:105) = E (cid:110) S (cid:104) { Q α ≤ S ≤ Q α } − (1 − α ) a +1 α − α − δ + C ( α, δ ) { Q α ≤ S ≤ Q α ,Q δ ≤ Y } (cid:105)(cid:111) ≤ Q α [(1 − α ) a +1 − (1 − α ) a +1 ]= 0 . In the above inequality, we have used(a) if Q α ≤ S < Q α , then { Q α ≤ S ≤ Q α } − (1 − α ) a +1 { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } α − α − δ + C ( α, δ ) ≤ (b) if S ≥ Q α , then { Q α ≤ S ≤ Q α } − (1 − α ) a +1 { Q α ≤ S ≤ Q α ,Q δ ≤ Y ≤ Q δ } α − α − δ + C ( α, δ ) ≥ . This proves that MCoVaR has a lower-value than the DCoVaR.2. We may write the Copula CoVaR asCCoVaR δα ( S | Y ; C ) = 11 − α − δ + C ( α, δ ) E [ S { S ≥ Q α ,Y ≥ Q δ } ] . Thus, (1 − α − δ + C ( α, δ )) (cid:104) CCoVaR δα ( S | Y ; C ) − DCoVaR ( δ, α,a ) ( S | Y ; C ) (cid:105) = (1 − α − δ + C ( α, δ )) × (cid:34) E ( S { S ≥ Q α ,Y ≥ Q δ } )1 − α − δ + C ( α, δ ) − E ( S { Q α ≤ S ≤ Q α ,Q δ ≤ Y } ) α − α − δ + C ( α, δ ) (cid:35) ≥ Q α [(1 − α − δ + C ( α, δ )) − (1 − α − δ + C ( α, δ ))]= 0 . In the above inequality, we have used(a) if
S > Q α , then { S ≥ Q α ,Y ≥ Q δ } − (1 − α − δ + C ( α, δ )) { Q α ≤ S ≤ Q α ,Q δ ≤ Y } α − α − δ + C ( α, δ ) ≥ (b) if Q α ≤ S ≤ Q α , then { S ≥ Q α ,Y ≥ Q δ } − (1 − α − δ + C ( α, δ )) { Q α ≤ S ≤ Q α ,Q δ ≤ Y } α − α − δ + C ( α, δ ) ≤ ..