Risk with additional structural dependence information Ludger Rüschendorf University of Freiburg Montreal,September 2017 VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 1
1. VaR with marginal information 2.,higher dimensional marginals 3. Variance moment constraints 4. stard 4.1 One-sided dependence information 4.2 (Partial) independence s 4.3 Two-sided improved 5. factor 6. 7. VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 2
1. VaR with marginal information Stochastic Dependence a) dependence modelling X = (X 1,..., X d ), X i R d X i P i marginal dependence : Copula copula Sklar's theorem b) HoedingFréchet f ϑ P ϑ : f ϑ marginals; : given marginals stochastic ordering, extremal dependence for risk functionals Conferences: Probability with given marginals Rome 1990, Seattle 1993, Prague 1996, Barcelona 1998, Montreal 2004, Tartu 2007, Sao Paulo 2010 VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 3
X = (X 1,..., X n ) risk vector marginal information: X i F i high model risk for VaR, TVaR,... maximal upper risk M(s) = sup X i F i { ( n P i=1 )} X i s VaR,higher Variance stard VaR α (S n ) lower bound VaR α (S n ) independent VaR α (S c n ) comonotonic VaR α (S n ) upper bound c Rüschendorf, Uni Freiburg; 4
Theorem (unconstrained ) A := n LTVaR α (X i ) = LTVaR α (Sn c ) i=1 LTVaR α (X i ) := 1 α α 0 VaR α (S n ) TVaR α (S n ) n TVaR α (Sn c ) = TVaR α (X i ) =: B VaR u (X i )du, i=1 S c n = comonotonic sum Bernard, Rü, Vuel (2013), Puccetti, Rü (2012), Wang, Wang (2011) Embrechts, Puccetti (2006), Embrechts, Puccetti, Rü (2012), Puccetti, Rü (2013) VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 5
VaR α (S n ) TVaR α (S c n ), n VaR α (S n ) LTVaR α (S c n ), n Puccetti, Rü (2012), Puccetti, Wang (2013); Wang, Wang (2014); Embrechts, Wang, Wang (2015) note: mixing (= negative dependence) in upper domain allows to increase VaR upper bound c B:=TVaR q(s ) + c VaR q(s ) VaR + c u(s ) VaR,higher Variance stard c A:= LTVaR q(s ) q 1 u c Rüschendorf, Uni Freiburg; 6
Rearrangement = Dependence Theorem (Rü (1979/83)) Let F(F 1,..., F d ) be the set of all joint dfs on R d with marginals F 1,..., F d. Let U be a rom variable with F U = U(0, 1). Then: F(F 1,..., F d ) = {F (f1 (U),...,f d (U)); f i r F 1 i, 1 i d}. { ( n ) } M(s) = sup P L i s ; L i F i i=1 { } n = 1 inf α ; fj α r F 1 [α,1], fj α s j j=1 RA-algorithm, precise determination of VaR VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 7
Dependence Uncertainty VaR,higher Estimates for VaRα(L) VaR α (L) for rom vectors of Pareto(2)-distributed risks. Variance stard VaR range (5), comonotonic VaR(8) (in log-scale on the right) for the sum of d = 8 GPD risks with parameters following Moscadelli (2004), based on RA for N = 1 : 0e05. c Rüschendorf, Uni Freiburg; 8
How to reduce risk using partial dependence information? marginals (reduced ) positive, negative dependence restrictions (improved stard ) information on variance of S n, correlations of X i, X j partially factor with subgroup intuition: positive dependence information allows to increase lower risk (but not upper ) negative dependence information allows to decrease upper risk (but not lower risk ) VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 9
2. Higher dimensional marginals,reduced F E = F(F J ; J E) F(F 1,..., F n ) F J = F XJ, X J = (X j ) j J for J E, J E J = {1,..., n} F E generalized Fréchet class E = {{1},..., {n}} F E = F(F 1,..., F n ) simple marginal class E = {{j, j +1}, 1 j n 1} F E = F(F 1,2, F 2,3,..., F n 1,n ) series system E = {{1, j}, 2 j n} F(F 1,2, F 1,3,..., F 1,n ) starlike system J 2 VaR,higher Variance stard J 1 J 3 J 5 J 4 c Rüschendorf, Uni Freiburg; 10
M E (s) = sup{p(x 1 + + X n s); F X F E } m E (s) = inf{p(x 1 + + X n s); F X F E } marginal problem: F E Φ (Rü (1991)) decomposable case duality theorem M E Φ { } M E (ϕ) : = sup ϕdp; P M E { = inf f J dp J ; } f J π J ϕ, ϕ usc J E J E Rü (1984), Kellerer (1987) VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 11
reduced systems E = {J 1,..., J m } η i := #{J r E; i J r }, 1 i n VaR For X risk vector, F X F E dene: H = F(H 1,..., H m ) Y r := i J r X i η i, H r := F Yr, r = 1,..., m Fréchet class Proposition (reduced ) F E = consistent marginal system, then for s R M E (s) M H (s) m E (s) m H (s),higher Variance stard Embrechts, Puccetti (2010), Puccetti, Rü (2012) c Rüschendorf, Uni Freiburg; 12
Series case F i,i +1 2-dim Pareto Estimates for VaRα(L) for a rom vector of d = 600 Pareto(2)-distributed risks under dierent dependence scenarios: VaR + α (L) ((L1,..., L600) has copula C = M); VaR r α(l), (A): the bivariate marginals F2j 1,2j are independent; VaR r α(l), (B): the bivariate marginals F2j 1,2j have Pareto copula with δ = 1.5; VaRα(L): no dependence assumptions are made. VaR,higher Variance stard VaR VaRα(L) (see (5)) reduced VaR r α(l) (see (24a)) for a rom vector of d = 600 Pareto(2)-distributed risks with xed bivariate marginals F2j 1,2j generated by a Pareto copula with δ = 1.5, comonotone (left) by the independence copula (right). c Rüschendorf, Uni Freiburg; 13
3. Risk with variance higher order moment constraints information: X i F i, 1 i n Var(S n ) s 2 ( ) partial information on dependence alternatively information on Cov(X i, X j ), Bernard, Rü, Vuel (2015) { M = sup{var α (S n ); m = inf{var α (S n ); S n satises ( )} S n satises ( )} VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 14
Theorem α (0, 1), Var(S n ) s 2, then ( ) α a := max µ s 1 α, A m VaR α (S n ) M ( ) α b := min µ + s 1 α, B, µ = ES n Remark VaR convex order worst case dependence has relation to convex order minima in upper lower part {S n VaR α (S n )} resp. {S n < VaR α (S n )} VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 15
maximizing VaR maximizing minimal support over all Y i Fi α is implied by convex order B := TVaRα(S c ) VaR,higher A := LTVaRα(S c ) α 1 VaR convex order u Variance stard c Rüschendorf, Uni Freiburg; 16
Extended Rearrangement Algorithm (ERA) two alternating steps 1. choice of domain, starting from largest α-domain 2. Rearrangement in upper α-part in lower 1 α-part 3. check variance constraint fullled 4. shift of domain iterate b VaR,higher Variance stard a a p Variation of ERA: Self determined split of domains. c Rüschendorf, Uni Freiburg; 17
Panel A: Approximate sharp obtained by the ERA (m d, M d ) n = 10 n = 100 ϱ = 0 ϱ = 0.15 ϱ = 0.3 ϱ = 0 ϱ = 0.15 ϱ = 0.3 VaR 95% (4.401; 15.72) (4.091; 21.85) (3.863; 26.19) (47.96; 84.72) (42.48; 188.9) (39.61; 243.3) d = 10,000 VaR 99% (5.486; 28.69) (4.591; 43.45) (4.492; 53.22) (48.99; 129.5) (46.61; 366.0) (45.36; 489.5) VaR 99.5% (6.820; 39.48) (5.471; 59.60) (4.850; 73.11) (49.23; 162.8) (47.54; 499.1) (46.68; 671.5) Panel B: Variance-constrained (a d, b d ) n = 10 n = 100 ϱ = 0 ϱ = 0.15 ϱ = 0.3 ϱ = 0 ϱ = 0.15 ϱ = 0.3 VaR 95% (4.398; 16.03) (4.089; 21.92) (3.861; 26.23) (47.96; 84.74) (42.48; 188.9) (39.61; 243.4) d = 10,000 VaR 99% (4.725; 30.20) (4.589; 43.64) (4.490; 53.50) (48.99; 129.6) (46.59; 367.3) (45.33; 491.7) VaR 99.5% (4.800; 40.74) (4.705; 59.80) (4.634; 73.77) (49.23; 162.9) (47.54; 500.0) (46.65; 676.3) VaR 95% (4.372; 16.94) (4.037; 23.30) (3.791; 27.96) (48.01; 87.75) (42.09; 200.3) (38.99; 259.2) d = + VaR 99% (4.725; 32.25) (4.578; 46.77) (4.470; 57.41) (49.13; 136.2) (46.53; 393.1) (45.18; 527.4) VaR 99.5% (4.806; 43.63) (4.702; 64.22) (4.634; 77.72) (49.39; 172.2) (47.56; 536.4) (46.60; 726.9) Panel C: Unconstrained independent of ϱ (A d, B d ) n = 10 n = 100 VaR 95%, (3.646; 30.33) (36.46; 303.3) d = 10,000 VaR 99% (4.447; 57.76) (44.47; 577.6) VaR 99.5% (4.633; 74.11) (46.33; 741.1) VaR 95% (3.647; 30.72) (36.47; 307.2) d = + VaR 99% (4.448; 59.62) (44.48; 596.2) VaR 99.5% (4.635; 77.72) (46.35; 777.2) Bounds on Value-at-Risk of sums of Pareto distributed risks (θ = 3) VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 18
Application to Credit Risk portfolios asset correlations ϱ A default correlations ϱ D, loans X j B(p) example: n = 10,000, p = 0.049 default probability, ϱ D = 0.0157 (McNeil et al. (2005)), s 2 = np(1 p) + n(n 1)p(1 p)ϱ D (A d, B d) (a d, b d) (m d, M d) KMV Beta CreditMetrics VaR0.8 (0%; 24.50%) (3.54%; 10.33%) (3.63%; 10%) 6.84% 6.95% 6.71% VaR0.9 (0%; 49.00%) (4.00%; 13.04%) (4.00%; 13%) 8.51% 8.54% 8.41% VaR0.95 (0%; 98.00%) (4.28%; 16.73%) (4.32%; 16%) 10.10% 10.01% 10.11% VaR0.995 (4.42%; 100.00%) (4.71%; 43.18%) (4.73%; 40%) 15.15% 14.34% 15.87% The table provides VaR VaR computed in dierent (KMV, Beta, CreditMetrics). A d, B d from marginal information a d, b d with variance constraints VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 19
p = 0.25% p = 1% (A, B) (a, b) KMV (A, B) (a, b) KMV ϱ A = 0% (0%; 50%) (0.25%; 0.25%) 0.25% (0.50%; 100%) (1.00%; 1.00%) 1.0% ϱ A = 6% (0%; 50%) (0.23%; 3.27%) 1.2% (0.50%; 100%) (0.95%; 10.98%) 4.0% ϱ A = 12% (0%; 50%) (0.23%; 5.05%) 2.1% (0.50%; 100%) (0.92%; 16.27%) 6.3% ϱ A = 18% (0%; 50%) (0.23%; 6.84%) 2.9% (0.50%; 100%) (0.90%; 21.18%) 8.7% ϱ A = 24% (0%; 50%) (0.21%; 8.76%) 3.8% (0.50%; 100%) (0.87%; 26.09%) 11.1% ϱ A = 30% (0%; 50%) (0.20%; 10.85%) 4.8% (0.50%; 100%) (0.85%; 31.13%) 13.7% Unconstrained constrained upper lower 0.995-VaR for several combinations of default probability correlation VaR in the (one-factor) KMV model signicant model error, ex. ϱ A = 6 %, p = 0.25 %, then 99.5 % VaR 0.2 %3.3 % VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 20
Higher order moment constraints Bernard, Rü, Vuel, Yao (2017), Bernard, Denuit, Vuel (2015) X i F i, 1 i n ESn k c k, k = 2,..., K strengthened upper for VaR α (S n ), modication of RA-algorithm theoretical VaR assessment of a corporate portfolio q = KMV Comon. Unconstrained K = 2 K = 3 K = 4 95% 281.3 393.3 (34.0 ; 2083.3) (111.8 ; 483.1) (111.8 ; 433.0) (111.8 ; 412.8) ϱ = 99% 398.7 2374.1 (56.5 ; 6973.1) (115.0 ; 943.9) (117.4 ; 713.3) (118.2 ; 610.9) 0.05 99.5% 448.5 5088.5 (89.4 ; 10119.9) (116.9 ; 1285.9) (118.9 ; 889.5) (119.8 ; 723.2) 99.9% 573.1 12905.1 (111.8 ; 14784.9) (120.2 ; 2718.1) (121.2 ; 1499.6) (121.8 ; 1075.9) 95% 340.6 393.3 (34.0 ; 2083.3) (97.3 ; 614.8) (100.9 ; 562.8) (100.9 ; 560.6) ϱ = 99% 539.4 2374.1 (56.5 ; 6973.1) (111.8 ; 1245.0) (115.0 ; 941.2) (115.9 ; 834.7) 0.10 99.5% 631.5 5088.5 (89.4 ; 10119.9) (114.9 ; 1709.4) (117.6 ; 1177.8) (118.5 ; 989.5) 99.9% 862.4 12905.1 (111.8 ; 14784.9) (119.2 ; 3692.3) (120.8 ; 1995.9) (121.2 ; 1472.7) 95% 388.4 393.3 (34.0 ; 2083.3) (91.5 ; 735.9) (93.4 ; 697.0) (92.0 ; 727.9) ϱ = 99% 675.8 2374.1 (56.5 ; 6973.1) (111.8 ; 1519.5) (112.4 ; 1174.5) (113.7 ; 1083.9) 0.15 99.5% 816.1 5088.5 (89.4 ; 10119.9) (112.8 ; 2098.0) (115.9 ; 1472.7) (116.9 ; 1287.6) 99.9% 1178.4 12905.1 (111.8 ; 14784.9) (118.4 ; 4531.3) (120.7 ; 2501.8) (120.9 ; 1916.6) We report for various asset correlation levels ϱ condence levels q the VaRs under the KMV framework (second column), the comonotonic VaRs (third column) the VaR in the unconstrained the constrained case (in the last four columns between brackets K reects the number of moments of the portfolio sum that are known). The VaR are obtained using Algorithm 1. VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 21
: impact of variance moment constraints on VaR considerable amount of model risk knowledge of marginals + variance (moments) does not always allow to determine VaR's with condence stard risk methods (based on factor ) like KMV, Beta, Credit Metrics report similarly (why? on what basis?) Variance (moment) restriction is a (global) negative dependence assumption; it implies reduction of upper VaR. VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 22
4. stard How does positive/negative dependence information inuence risk? X positive upper orthant dependence (PUOD) n n if F X (x) = P(X > x) P(X i > x i ) = F i (x i ) i=1 X positive lower orthant dependence (PLOD) n if F X (x) F i (x i ), x i=1 X POD if X PLOD PUOD i=1 VaR,higher Variance stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 23
4.1 One-sided dependence information F = F X, F = F X one-sided dependence information Let G, H be increasing functions, F G, H F + G PLOD F positive dependence restriction (lower tail) G PUOD F positive dependence restriction (upper tail) example: similarly: G(x) = F i (x i ), X is POD F PLOD H, F PUOD H negative dependence restriction Williamson, Downs (1990), Denuit, Genest, Marceau (1999), Denuit, Dhaene, Ribas (2001), Embrechts, Höing, Juri (2003), Rü (2005), Embrechts, Puccetti (2006), Puccetti, Rü (2012) VaR,higher Variance stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 24
Theorem (improved stard ) X risk vector, marginals X i F i, G, F G F +, then a) If G PLOD F X, then ( d P b) If G PUOD F X, ( then d P c) If G is POD, then ( d i=1 i=1 i=1 ) X i s G(s); ) X i < s 1 G(s); ( d F i )(s) P i=1 ) X i s, ( d ) P X i < s 1 ( d F i )(s). i=1 i=1 U(s) := { x R n ; n i=1 x i = s }, G(s) := infx U(s) G(x) G -inmal convolution, G(s) := supx U(s) G(x) G -supremal convolution VaR,higher Variance stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 25
Fréchet : higher dimensional marginals various types of Bonferroni parameter uncertainty,robust neighbourhood `known domains' VaR,higher F (x) = Γ(x), x S Variance ( or or ) d = 2 Rachev, Rü (1994), Nelsen, Quesada-Molina, Rodríguez-Lallena, Úbeda-Flores (2001, 2004), Tankov (2011) d 2 Puccetti, Rü, Manko (2016), Lux, Papapantoleon (2016) digital options on default times for result: improved VaR- for options stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 26
Model for lower Bignozzi, Puccetti, Rü (2014) X = (X 1,..., X d ) risk vector, F i = F Xi {1,..., d} = k k-subgroups I j j=1 Y = (Y 1,..., Y d ) satises: i.e. F Y (x) = k j=1 min G j (x i ) i I j Y has k independent, homogenous subgroups components within subgroups comonotonic Assumption: ( ) Y X, positive dependence restriction where is uo or lo, typically: F i = G j for i I j If k = d F j = G j then ( ) to PUOD resp. PLOD of X k = 1 F i = G j X comonotonic VaR,higher Variance stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 27
Example: Pareto portfolio lower, homogeneous portfolio, d Pareto(2) risks, k subgroups, d/k variables in each subgroup VaR,higher Variance lower, inhomogeneous portfolio, d/2 Exp(2) risks d/2 Exp(4) risks stard One sided (Partial) Two-sided essential improvement of lower for k = 1, 2, 4; POD alone does not improve lower c Rüschendorf, Uni Freiburg; 28
4.2 (Partial) independence s Puccetti, Rü, Small, Vuel (2014) Assumption I) I 1... a) independent subgroups I 1,..., I k I 2 I k b) any dependence within subgroups k n i n i S = X i,j, Y i = independent S c,k = Theorem i=1 j=1 k n i F 1 i, j (U i) i=1 j=1 j=1 X i,j Under independence assumption I) a I := LTVaR α (S c,k ) VaR I α VaR I α b I := TVaR α (S c,k ). c Rüschendorf, Uni Freiburg; 29 VaR,higher Variance stard One sided (Partial) Two-sided
Gamma distributed groups: d = 8 k = 1 k = 2 k = 4 VaR + α VaR α b I e α b I e α b I e α α = 0.990 33.37 38.26 38.27 29.15 23.8% 23.29 39.1% α = 0.995 36.82 41.63 41.63 31.15 25.2% 24.52 41.1% α = 0.999 44.59 49.27 49.27 35.63 27.7% 27.21 44.8% d = 8, 4 Gamma(2,1/2), 4 Gamma(4,1/2), eα = 1 bi a I. VaRα VaR α Pareto distributed groups: (a I ; b I ) k = 1 k = 2 k = 5 k = 10 k = 25 k = 50 α = 0.95 (18.23; 153.72) (20.21; 116.32) (22.03; 81.54) (22.95; 63.93) (23.76; 48.57) (24.15; 41.09) α = 0.99 (22.24; 297.84) (23.14; 208.2) (23.92; 132.28) (24.28; 95.97) (24.59; 65.87) (24.73; 51.98) α = 0.995 (23.17; 388.91) (23.8; 269.08) (24.31; 163.37) (24.55; 115.34) (24.74; 76.06) (24.83; 58.25) (VaR α ; VaR α) α = 0.95 (18.24; 153.3) α = 0.99 (22.26; 297.64) α = 0.995 (23.2; 388) Monte Carlo simulation of marginal independence, Pareto case with d = 50, θi = θ = 3 ci = 1 for i = 1,..., k. VaR,higher Variance stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 30
(e α, e α ) k = 1 k = 2 k = 5 k = 10 k = 25 k = 50 α = 0.95 ( 0.05; 0.27) (10.8; 24.12) (20.78; 46.81) (25.82; 58.3) (30.26; 68.32) (32.4; 73.2) α = 0.99 ( 0.09; 0.07) (3.95; 30.05) (7.46; 55.56) (9.07; 67.76) (10.47; 77.87) (11.1; 82.54) α = 0.995 ( 0.13; 0.23) (2.59; 30.65) (4.78; 57.89) (5.82; 70.27) (6.64; 80.4) (7.03; 84.99) Monte Carlo simulation of marginal independence, Pareto case with d = 50, θi = θ = 3 ci = 1 for i = 1,..., k, e α = VaRα bi. VaRα Partial independent subs: {1,..., n} = k I j, (X Ij ) independent for j H {1,..., k} j=1 Partial independent subs. VaR,higher Variance stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 31
Theorem (partial independent subs) For α (0, 1) the following VaR hold: ( a p = a p (α, H) := LTVaR(Si c ) + LTVaR VaR(S d ) =: b p (α, H) = b p. Si c i H i {1,...,k}\H i {1,...,k}\H is an independent sum, TVaR(Si c) = n i j=1 are simple to calculate. ( TVaR(Si c ) + TVaR i H TVaR(X ij ) LTVaR(Si c) = n i j=1 S c i i H ) S c i ) LTVaR(X ij ) VaR,higher Variance stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 32
Remark α = 0.95 α = 0.995 α = 0.995 Fi Gamma(κ (1) i, 1) Fi N(µi, 1) Fi N(0, 1) (a I ; b I ) (27.58; 76.02) (149.67; 214.67) ( 0.33; 64.66) H = {2, 3, 4, 5} (26.83; 90.4) (149.57; 236.76) ( 0.44; 86.76) H = {3, 4, 5} (25.85; 108.7) (149.47; 257.93) ( 0.55; 107.93) H = {4, 5} (24.8; 128.81) (149.36; 277.66) ( 0.64; 127.66) H = {5} (23.75; 148.66) (149.28; 294.6) ( 0.73; 144.60) (VaR α ; VaRα ) (23.76; 148.63) (149.29; 294.59) ( 0.71; 144.59) Partial independence with variation of independent sub, d = 50, k = 5, µi = i. a) partial independent graph s Partial independent graph s. VaR,higher Variance stard One sided (Partial) Two-sided Partial independent reduction. c Rüschendorf, Uni Freiburg; 33
b) combination with variance VaR,higher Variance constrained versus independence + variance constrained a V, a p,v resp. b V, b p,v. s V d = 10 d = 100 α = 0.95 22.39 2239.26 α = 0.99 7.17 717.49 α = 0.995 4.20 420.27 Approximations of critical value s V by Monte Carlo simulation with 102 repetitions of 10 5 simulations. (a p,v ; b p,v ) (a V ; b V ) d = 100, k = 10 s 2 = 20 s 2 = 50 s 2 = 100 s 2 = 200 s 2 = 500 α = 0.95 ( 1.03; 19.49) ( 1.62; 30.82) ( 2.29; 43.59) ( 3.24; 61.64) ( 3.43; 65.23) α = 0.99 ( 0.45; 44.5) ( 0.71; 70.36) ( 0.85; 84.28) ( 0.85; 84.28) ( 0.86; 84.28) α = 0.995 ( 0.32; 63.09) ( 0.46; 91.45) ( 0.46; 91.45) ( 0.45; 91.45) ( 0.46; 91.45) α = 0.95 ( 1.03; 19.49) ( 1.62; 30.82) ( 2.29; 43.59) ( 3.24; 61.64) ( 5.13; 97.47) α = 0.99 ( 0.45; 44.5) ( 0.71; 70.36) ( 1.01; 99.5) ( 1.42; 140.71) ( 2.25; 222.49) α = 0.995 ( 0.32; 63.09) ( 0.5; 99.75) ( 0.71; 141.07) ( 1; 199.5) ( 1.45; 289.2) Variance stard One sided (Partial) Two-sided Approximation of (a p,v, b p,v ) by Monte Carlo simulation with 10 2 iterations of 10 5 simulations. c Rüschendorf, Uni Freiburg; 34
Examples (application to insurance portfolio) d = 11, k = 4 Market Asset Credit Insur. Busin. Non life Reput. Life Gaussian marginals Insurance risk portfolio.??? Reinsurance Operational Catastrophic LogN LogN Pareto VaR,higher Variance b I VaR + α VaR α b I /VaR α 1 α = 99% 147.34 148.46 149.66 168.37 209.59 29.2% b I VaR + α VaR α VaR α (L + t ) α = 99.5% 173.37 175.18 176.96 202.89 249.55 29.8% b I VaR + α VaR α VaR α (L + 6 ) α = 99.9% 250.41 256.04 262.47 304.63 367.70 30.4% upper b I, VaR + α = comonotonic VaR VaRα for 11-dimensional insurance portfolio stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 35
4.3 Two-sided improved improved : positive dependence: G F X or F X G ; or negative dependence problem: needs strong positive dependence d small two-sided : Q C Q, Q, Q quasi-copulas result: example: two-sided improved based on multiset-inclusion exclusion principle 1 B1 B 2 B 3 = 1 B1 + 1 B2 + 1 B3 1 B1 B 2 1 B2 B 3 1 B1 B 3 + 1 B1 B 2 B 3 needs upper lower! parsimonious representation reduction scheme Lux, Rü (2017) VaR,higher Variance stard One sided (Partial) Two-sided c Rüschendorf, Uni Freiburg; 36
Examples 1. C (u) δ C C (u) + δ, C Gaussian equi-correlated ϱ = 0.1 ϱ = 0.4 ϱ = 0.8 i. stard scheme impr. i. stard scheme impr. i. stard scheme impr. α (low : up) (low : up) % (low : up) (low : up) % (low : up) (low : up) % 0.95 3.4 : 45.0 8.2 : 24.8 60 3.6 : 41.2 7.2 : 28.1 44 7.8 : 31.4 9.2 : 26.2 28 0.99 9.0 : 106.2 15.9 : 56.7 58 9.0 : 105.3 14.9 : 80.8 32 17.4 : 84.9 18.6 : 82.2 6 0.995 13.3 : 153.0 19.0 : 90.0 49 13.3 : 153.0 18.0 : 153.0 3 23.4 : 126.0 22.8 : 125.0 0 stard on VaR of X1+...+X5 VaR estimates via reduction schemes for δ = 0.0005. 2. C Σ C C Σ, Gaussian-copula ϱ = 0.1, ϱ = 0.2 ϱ = 0.3, ϱ = 0.5 i. stard scheme impr. i. stard scheme impr. α (low : up) (low : up) % (low : up) (low : up) % 0.95 3 : 32 8 : 26 38 1 : 30 7 : 29 24 0.99 9 : 74 20 : 52 51 2 : 74 18 : 63 37 0.995 13 : 104 26 : 70 52 3 : 104 25 : 86 40 stard on VaR of X1 +... + X4 VaR estimates computed via reduction schemes using C Σ C Σ. 3., C θ 1 C m C θ 2 for subgroups copulas by Frank-copulas m = 8 m = 4 m = 2 i. stard scheme impr. i. stard scheme impr. i. stard scheme impr. α (low : up) (low : up) % (low : up) (low : up) % (low : up) (low : up) % 0.95 42 : 113 59 : 86 62 22 : 150 39 : 112 43 12 : 193 28 : 150 33 0.99 82 : 210 108 : 147 70 42 : 264 67 : 175 51 21 : 329 42 : 218 43 0.995 105 : 266 135 : 180 72 53 : 329 83 : 206 55 43 : 403 51 : 252 44 stard VaR estimates via reduction schemes for X1 +... + X16 given distributions of subgroups. c Rüschendorf, Uni Freiburg; 37 VaR,higher Variance stard One sided (Partial) Two-sided
7. Risk with marginal information can be calculated, typically (too) wide Various reductions by including additional information Higher dimensional marginals (reduced ) Variance constraints, moment constraints good reduction, when constraints are small enough partial independence (combined with variance information) strong reduction of dependence uncertainty leads in examples to realistic VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 63
(cont.) reduction by partially specified risk factor strong reduction of dependence uncertainty applicable in variety of mixture partial dependence information (improved stard ) one-sided, improved Hoeffding Fréchet needs strong enough dependence constraints, d small two-sided dependence promising tool also in higher dimension d subgroup flexible tool, good reduction VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 64
P. Embrechts, G. Puccetti, L. Rüschendorf: Model uncertainty VaR aggregation. J. Banking Finance 37(8), 27502765, 2013 P. Embrechts, B. Wang, R. Wang: Aggregation-robustness model uncertainty of regulatory risk measures. Finance Stoch. 19(4), 763790, 2015 G. Puccetti, L. Rüschendorf: Asymptotic equivalence of conservative VaR- ES-based capital charges. J. Risk 16(3), 322, 2014 G. Puccetti, B. Wang, R. Wang: Complete mixability asymptotic equivalence of worst-possible VaR ES estimates. Insurance Math. Econom. 53(3), 821828, 2013 R. B. Nelsen, B. Quesada-Molina, J. A. Rodríguez-Lallena, M. Úbeda-Flores: Bounds on bivariate distribution functions with given margins measures of association. Commun. Stat., Theory Methods 30(6), 11551162, 2001 R. B. Nelsen, B. Quesada-Molina, J. A. Rodríguez-Lallena, M. Úbeda-Flores: Best-possible on sets of bivriate distribution functions. J. Multivariate Anal. 90(2), 348358, 2004 VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 65
(cont.) C. Bernard, L. Rüschendorf, S. Vuel: Value-at-Risk with variance constraints. Journal of Risk Insurance 2015; doi:10.1111/jori.12108 G. Puccetti, L. Rüschendorf: Bounds for joint portfolios of dependent risks. Statistics & Risk Modeling 29(2), 107132, 2012 V. Bignozzi, G. Puccetti, L. Rüschendorf: Reducing model risk via positive dependence assumptions. Insurance Math. Econ. 61(1), 1726, 2015 C. Bernard, L. Rüschendorf, S. Vuel, R. Wang: Risk for factor. To appear in: Finance Stoch. 2016 G. Puccetti, D. Small, L. Rüschendorf, S. Vuel: Reduction of Value-at-Risk by independence variance information. Scinavian Actuarial Journal 2017 (3), 245-266; T. Lux, A. Papapantoleon: FréchetHoeding for d-copulas applications in model-free nance. Preprint, 2016. arxiv/1602.08894 VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 66
(cont.) C. Bernard, L. Rüschendorf, S. Vuel, J. Yao: How Robust is the Value-at-Risk of Credit Risk Portfolios? The European Journal of Finance 23(6), 507534, 2017; doi:10.1080/1351847x.2015.1104370 G. Puccetti, L. Rüschendorf, D. Manko: VaR for joint portfolios with dependence constraints. Dependence Modeling 4, 368381, 2016 T. Lux, L. Rüschendorf: VaR with two sided dependence information. Preprint, 2017 L. Rüschendorf, J. Witting: VaR in with partial dependence information on subgroups. Dependence Modeling 5, 5974, 2017 VaR,higher Variance stard c Rüschendorf, Uni Freiburg; 67