Sharp bounds on the VaR for sums of dependent risks
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1 Paul Embrechts Sharp bounds on the VaR for sums of dependent risks joint work with Giovanni Puccetti (university of Firenze, Italy) and Ludger Rüschendorf (university of Freiburg, Germany)
2 Mathematical Problem Assumptions: L 1,...,L d one period risks with statistically estimated marginals. L L d total loss exposure. VaR (L L d ) amount of capital to be reserved. (if, then ) VaR (L L d ) = s P(L L d s) apple 1 Task: for a fixed (high) level of probability, calculate:!!! n o VaR = sup VaR (L L d ):L j F j, 1 apple j apple d n o VaR = inf VaR (L L d ):L j F j, 1 apple j apple d
3 Motivation (QRM) L 1 F 1, L 2 F 2,..., L d F d marginal distributions d dependence model = VaR (L L d )
4 Motivation (QRM) L 1 F 1, L 2 F 2,..., L d F d marginal distributions d dependence model = VaR (L L d ) VaR VaR + = P d j=1 VaR (L j ) VaR
5 Known results F j = F, 1 apple j apple d - In the homogeneous case, the bound has been recently given for d > 2 in [PR11] and [WW11] under different assumptions. VaR - In the homogeneous case, VaR is very easy to calculate in arbitrary dimensions. VaR - In the inhomogeneous case, the computation of poses serious problems. And the computation of is not possible. VaR
6 Homogeneous marginals, d 3 In the case that L 1,...,L d are identically distributed, we have M(s) = sup n P(L L d s); L j F, 1 apple j apple d o Duality theorem (reduced) ( Z ) M(s) = inf d g df; g 2 A(s), where A(s) = {g : R! [0, 1] such that g(x 1 ) + + g(x d ) 1{x x d s}}
7 Dual bounds Embrechts and Puccetti (2006) introduce the following class of piecewise-linear functions for a < s/d 1 g a 0 a b = s (d 1)a M(s) apple D(s) = inf a<s/d d Z g a df! = d inf a<s/d R b a F(x) dx s da. D(s) The dual bound is better than the standard bound produced by choosing piecewise-constant dual functions.
8 Referee report on Embrechts and Puccetti (2006)
9 Timeline to the result
10 Timeline to the result 1981 Makarov gives the optimal coupling for the sum of two risks answering a question by Kolmogorov
11 Timeline to the result Rüschendorf gives independently the same optimal coupling and the dual solution Makarov gives the optimal coupling for the sum of two risks answering a question by Kolmogorov
12 Timeline to the result Rüschendorf gives independently the same optimal coupling and the dual solution Makarov gives the optimal coupling for the sum of two risks answering a question by Kolmogorov 2006 Embrechts and Puccetti introduce dual bounds in the homogeneous case
13 Timeline to the result Rüschendorf gives independently the same optimal coupling and the dual solution Wang and Wang gives optimal couplings for the sum of arbitrary risks in some specific examples 2011 Makarov gives the optimal coupling for the sum of two risks answering a question by Kolmogorov Embrechts and Puccetti introduce dual bounds in the homogeneous case
14 Timeline to the result Rüschendorf gives independently the same optimal coupling and the dual solution Wang and Wang gives optimal couplings for the sum of arbitrary risks in some specific examples Makarov gives the optimal coupling for the sum of two risks answering a question by Kolmogorov Embrechts and Puccetti introduce dual bounds in the homogeneous case sharpness of dual bounds is stated for a general class of distributions
15 The Rearrangement Algorithm (RA) a new numerical approximation procedure
16 Game X =
17 Game X = Rearrange the second column to obtain sum with minimal variance
18 Game X = Rearrange the second column to obtain sum with minimal variance
19 Game (more difficult) X = Rearrange the entries within each column in order to obtain sum with minimal variance
20 Game (more difficult) X = Rearrange the entries within each column in order to obtain sum with minimal variance
21 Game (more difficult) X = Rearrange the entries within each column in order to obtain sum with minimal variance
22 Game (more difficult) X = Rearrange the entries within each column in order to obtain sum with minimal variance
23 Game (more difficult) X = Rearrange the entries within each column in order to obtain sum with minimal variance
24 Game (more difficult) X = Rearrange the entries within each column in order to obtain sum with minimal variance
25
26
27
28
29 Summary: For a given matrix, rearrange the entries in the columns until you find an ordered matrix, i.e. a matrix in which each column is oppositely ordered to the sum of the others. We call a matrix optimal if the minimal component of the vector of the sum is maximized and the maximal component of that vector is minimized (min-max problem).
30 Sudoku 9 9 X = 9 9 9
31 Sudoku X = Hint!
32 Sudoku X = Solution!
33 Permutation matrices X2 X3 X1 ordered optimal
34 BEST/WORST VAR?
35 BEST/WORST VAR? dependence=rearrangement
36 VaR 0.99 (L 1 ) Fix and assume that each values all having the same 2 (0, 1) F 1 takes only N probability (1 )/N. j [, 1] 0 3X 1 P B@ j=1 L j min(rowsums(x)) CA 1 VaR 1 (L 1 )
37 VaR 0.99 (L 1 ) Fix and assume that each values all having the same 2 (0, 1) F 1 takes only N probability (1 )/N. j [, 1] 0 3X 1 P B@ j=1 L j min(rowsums(x)) CA 1 VaR (L L d ) min(rowsums(x)) VaR 1 (L 1 )
38 VaR 0.99 (L 1 ) Fix and assume that each values all having the same 2 (0, 1) F 1 takes only N probability (1 )/N. j [, 1] 0 3X 1 P B@ j=1 L j min(rowsums(x)) CA 1 VaR (L L d ) min(rowsums(x)) VaR = max X2P(X) min(rowsums( X)) (idea of the proof) VaR 1 (L 1 )
39
40 Pareto(2) marginals and = 0.99
41 Pareto(2) marginals and = 0.99
42 Pareto(2) marginals and = 0.99 VaR = ( )
43 Pareto(2) marginals and = 0.99 VaR = ( ) N = 10 5 ) VaR = 45.99
44 Pareto(2) marginals and = 0.99 OPTIMAL COUPLING! VaR = ( ) N = 10 5 ) VaR = 45.99
45 Optimal coupling yields a dependence in which: - - either the (three) rvs are very close to each other and sum up to something very close to the minimal sum (-> complete mixability ) or one of the components is large and the other (two) are small (-> mutual exclusivity )
46 Complete mixability Definition A distribution F is called d-completely mixable if there exist d random variables identically distributed as F such that Examples - Gaussian, Cauchy, t - Uniform X 1,...,X d P(X X d = constant) = 1 - Binomial(n,r/s) is s-completely mixable, n,r,s integers - Multivariate extensions can be given [Rüschendorf and Uckelmann (2002)]
47 Complete mixability Sufficient conditions for complete mixability (Wang and Wang (2011), Puccetti, Wang and Wang (2012)) - F is continuous with a symmetric and unimodal density. [Rüschendorf and Uckelmann (2002)] - F is continuous with a monotone density on a bounded support and satisfies a moderate mean condition. [Wang and Wang (2011)] - F is continuous with a concave density on a bounded support. [Puccetti, Wang and Wang (2012)]
48 X2 X3 X1 ordered X With N=100000, we obtain the first two decimals of VaR in 0.2 sec.
49 Rearrangement algorithm 1) Approximate the (1 ) upper part of the support of each marginal from above and below: F j F j F j F j and create two matrices X and Y with N columns and d rows. 2) Iteratively rearrange the column of each matrix until the matrices X* and Y* with each column oppositely ordered to the sum of the other columns. 3) min(rowsums(x )) apple VaR min(rowsums(y apple )). 4) Run the algorithm with N large enough.
50
51 Application: superadditivity ratio Define the superadditivity ratio as: (d) = VaR (L + ) VaR + (L + ) and investigate its properties as a function of the dimension d, the level and the parameters of the underlying model. Investigate the limit, given it exists, = lim d!+1 (d) exists. For large dim
52 Examples quantile quantile quantile quantile d θ Figure 5: Left: plot of the function (d) versus the dimensionality d of the portfolio for a risk vector of Pareto( )-distributed risks, for two di erent quantile levels and = 2. Right: Plot of the limit constant versus the tail parameter of the Pareto distribution.
53 Conclusions The rearrangement algorithm calculates numerically sharp bounds for the VaR of a sum of dependent random variables. - it is accurate, fast and computationally less demanding wrt to the methods in the literature. - can be used with inhomogeneous marginals, in high dimensions. - computes also the best-possible Value-at-Risk. - can be used with any marginal df and any quantile level. - can be used also to compute bounds on the distribution function of different operators such as, min, max.
54 Further work Find optimal couplings for the best VaR Interpret these couplings wrt realistic scenarios Add statistical uncertainty Compute VaR sharp bounds with some additional dependence information Compare and contrast with other approaches: Robust Optimization...
55 References Makarov, G.D.(1981):Estimates for the distribution function of the sum of two random variables with given marginal distributions.theory Probab. Appl. 26, Embrechts, P. and G. Puccetti (2006b). Bounds for functions of dependent risks. Finance Stoch. 10(3), Embrechts,P, Puccetti, G. and L. Rüschendorf (2012). Model uncertainty and VaR aggregation, preprint. Puccetti, G. and L. Rüschendorf (2012). Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. App. Math. 236 (7), Puccetti, G. and L. Rüschendorf (2012). Sharp bounds for sums of dependent risks, preprint. Puccetti, G., Wang, B., and R. Wang (2012). Advances in complete mixability. Forthcoming in J. Appl. Probab. Rüschendorf, L. and L. Uckelmann (2002). Variance minimization and random variables with constant sum. In Distributions with given marginals and statistical modelling, pp Dordrecht: Kluwer Acad. Publ. Rüschendorf, L. (1982). Random variables with maximum sums. Adv. in Appl. Probab. 14(3), Wang, B. and R. Wang (2011). The complete mixability and convex minimization problems with monotone marginal densities. J. Multivariate Anal., 102,
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