Risk Aggregation. Paul Embrechts. Department of Mathematics, ETH Zurich Senior SFI Professor.

Transcription

1 Risk Aggregation Paul Embrechts Department of Mathematics, ETH Zurich Senior SFI Professor Joint work with P. Arbenz and G. Puccetti 1 / 33

2 The background Query by practitioner (2005): Calculate VaR for the sum of three random variables with given marginals (Pareto, gamma, lognormal) and across a variety of dependence structures (copulas) Research project: Numerical evaluation of (generalized) copula convolutions, leading to (G)AEP. 2 / 33

3 Risk aggregation is relevant for: portfolio analysis understanding diversification & concentration for regulatory capital calculations between risk categories within risk categories within the Basel III, Solvency 2, SST frameworks better understanding of diversification we shall only touch upon some aspects 3 / 33

4 New publication by the Bank for International Settlements 4 / 33

5 A canonical set-up X 1,..., X d one-period risks ψ : R d R aggregation function R a risk measure Task: calculate R(ψ(X 1,..., X d )) Example: ψ(x 1,..., X d ) = X 1 + X X d, R = VaR α, α (0, 1) VaR α (X 1 + X X d ) At best: R L R(ψ(X)) R U depending on the underlying model assumptions! 5 / 33

6 Key issues Conditions: Xi F i, i = 1,..., d known?/unknown?/unknowable? risk versus uncertainty statistical uncertainty model uncertainty Dimensionality: small: d 5, say, versus large: d 100s Extremes matter: in the tails: Extreme Value Theory (EVT) in the interdependence: copulas (may) enter P[X 1 x 1,..., X d x d ] = C (F 1 (x 1 ),..., F d (x d )) 6 / 33

7 Return to canonical example: Issues: VaR α (X 1 + X X d ) Relevance: sense or nonsense? Estimation, calculation additive (=) for comonotonic risks subadditive ( ) for elliptical risks superadditive (>) for very heavy-tailed risks very skewed risks risks with a special interdependence does it matter? measure of frequency (if), not severity (what if) 7 / 33

8 VaR in finance and insurance Concerning VaR-calculations in finance and insurance: the VaR-number is just the final-final issue getting the risk-factor-mapping, clean-p&l are far more important recall: VaR is a statistical estimate often upper (lower) bounds can be found find (best) worst case VaR given some side conditions 8 / 33

9 Example for an upper bound for VaR Theorem (Embrechts-Puccetti) Let (X 1,..., X d ) be continuous with equal margins F i = F, i = 1,..., d. Then for α (0, 1), where VaR α (X X d ) D 1 d (1 α), D d (s) = inf r [0,s/d) s (d 1)r r (1 F (x))dx s/d r 9 / 33

10 This talk (as an example): Numerically calculate, for α close to 1, VaR α (X 1 + X X d ) (1) or equivalently, calculate, typically for s large: P[X 1 + X X d s] (2) numerically in terms of F 1,..., F d and C which are assumed to be known analytically Remark: in order to calculate (1) for a given α, use a root-finding procedure based on (2) 10 / 33

11 Standard solution Monte Carlo: simulate i.i.d. and estimate (X1, i X2, i..., Xd i ), i = 1,..., n P [X 1 + X X d s] 1 n n 1 { X1 i + X2 i + + Xd i s} i=1 (Dis)advantages: A sampling algorithm must be available The convergence rate is relatively slow: O(1/ n) The convergence rate is independent of the dimension d 11 / 33

12 The AEP algorithm: First assumption First assumption: The components of (X 1, X 2,..., X d ) are positive: P[X i > 0] = 1 (or bounded from below) Consequence: Suppose d = 2. Due to X 1 > 0 and X 2 > 0 we get x 2 P [X 1 + X 2 s] = P [(X 1, X 2 ) S] s S = {(x 1, x 2 ) : x 1 > 0, x 2 > 0, x 1 + x 2 s} S 0 s x 1 12 / 33

13 The AEP algorithm: Second assumption Second assumption: The joint distribution function (df) H(x 1,..., x d ) = P [X 1 x 1, X 2 x 2,..., X d x d ] is known analytically or can be numerically evaluated Example: H is given by a copula model: H(x 1,..., x d ) = C(F 1 (x 1 ),..., F d (x d )) 13 / 33

14 The probability mass of a rectangle is easy to calculate For d = 2 x 2 d c Q = (a, b] (c, d] 0 a b x 1 Then P[(X 1, X 2 ) Q] = H(b, d) H(a, d) H(b, c) + H(a, c) Idea behind the AEP algorithm: approximate the triangle S by rectangles! 14 / 33

15 First approximation (d = 2) Recall: S = {(x 1, x 2 ) R 2 : x 1 > 0, x 2 > 0, x 1 + x 2 s} Set: Q = (0, 2/3s] (0, 2/3s] (later: why 2/3) Use Q as a first approximation of S x 2 s P [(X 1, X 2 ) S] P [(X 1, X 2 ) Q] }{{} easy to calculate Q = (0, 2/3s] (0, 2/3s] 0 s x 1 15 / 33

16 Error of the first approximation x 2 s 0 s x 1 [ ] P [S] = P + P [ ] P [ ] + P [ ] The error of the first approximation P [(X 1, X 2 ) Q] can again be expressed in terms of triangles! Idea: again approximate those triangles by squares! 16 / 33

17 Approximate new triangles by squares + = + + = + + = + + With these geometric approximations of S, define a sequence P n of approximations of P[X 1 + X 2 s] = P[(X 1, X 2 ) S]: [ ] P 1 = P [ ] P + P [ ] P [ ] + P [ ] 2 = P. 17 / 33

18 Set representation of P 1, P 2 and P 3 P 1 P 2 P 3 Triangles are iteratively approximated by squares and the left over triangles are then passed on to the next iteration 18 / 33

19 AEP algorithm for d = 3 In higher dimensions, the AEP can also be used. For instance, for d = 3, the set representation of P 1, P 2 and P 3 is Analogous decomposition possible in any dimension d N, but resulting simplexes are overlapping for d 4! 19 / 33

20 Choice of the sidelengths of the approximating hypercubes How to choose the sidelengths of the approximating hypercubes? Answer: For an optimal rate of convergence, take a hypercube with sidelength h = 2 (sidelength of the triangle) d + 1 Hence the choice of Q = (0, 2/3s] (0, 2/3s] before for d = 2 20 / 33

21 Convergence Theorem Let d 5 and suppose (X 1,..., X d ) has a density in a neighbourhood of {x R d : x i = s}, then lim n P n = P [X X d s] Remark: reason for convergence problems in high dimensions: simplex decomposition is overlapping for d 4 21 / 33

22 Richardson extrapolation Define the extrapolated estimator P n of P [X X d s] by P n = P n + a (P n P n 1 ), where a = 2 d (d + 1) d /d! 1. The additional term cancels the dominant error term of P n Theorem Let d 8 and suppose (X 1,..., X d ) has a twice continuously differentiable density in a neighbourhood of {x R d : x i = s}, then lim n P n = P [X X d s] Remark: for d > 8, higher order extrapolation may be useful for proving convergence 22 / 33

23 Convergence rates Theorem Let d 5 and suppose (X 1,..., X d ) has a density in a neighbourhood of {x R d : x i = s}, then P n P [X X d s] = O ( (A d ) n) Let d 8 and suppose (X 1,..., X d ) has a twice continuously differentiable density in a neighbourhood of {x R d : x i = s}, then P n P [X X d s] = O ( (A d )n) d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = 8 A d A d / 33

24 Convergence rates, cont. The calculation of P n and P n requires N(n) = O ( (B d ) n) evaluations of the joint distribution function d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 d = 8 B d Both convergence rate and numerical complexity of P n and Pn are exponential. Combining both, we get ( ) P n P [X X d s] = O N(n) γ d ( ) Pn P [X X d s] = O N(n) γ d where γ d and γ d determine the rate of convergence. 24 / 33

25 Convergence rates, cont. The following table shows γ d and γ d d = 2 d = 3 d = 4 d = 5 d = 6 d = 7 γ d γd Convergence rate of Monte Carlo: O ( N 0.5), where N is the number of simulations. BUT: a (complex?) sampling algorithm must be available. Convergence rate of Quasi Monte Carlo O ( N 1 (log N) d). BUT: the algorithm must be tailored for each application. AEP does not need any tailoring or simulation. Only requirement: able to evaluate the joint distribution function of (X 1,..., X d ). 25 / 33

26 Numerical example d = 2, 3, 4 X i are Pareto(i) distributed (P[X i x] = 1 (1 + x) i ) Clayton copula with θ = 2 (pairwise Kendall s tau = 0.5) s = 100 plot shows logarithm absolute errors: difference between estimate (extrapolated AEP & MC) and reference value x-axis, execution time on log scale 1e 1 AEP 1e 1 Monte Carlo absolute error 1e 4 1e 7 1e 10 1e 13 d=2 d=3 d=4 absolute error 1e 4 1e 7 1e 10 1e 13 d=2 d=3 d=4 0.1msec 10msec 1sec execution time 0.1msec 10msec 1sec execution time 26 / 33

27 Numerical example: Conclusion In two and three dimensions, AEP is much faster than Monte Carlo For d 4, Monte Carlo beats AEP Memory requirements to calculate P n with AEP grow exponentially in n and in the dimension d, hence only low dimensions are numerically feasible AEP in general: INPUT: marginal dfs F i copula C threshold s OUTPUT: sequence P n of estimates of P[X X d s] SOFTWARE: available in C++ 27 / 33

28 Open problem Recall: using Richardson extrapolation, P n = P n + a(p n P n 1 ) for some a R converges faster and in higher dimensions than P n Further work: Extend Richardson extrapolation to cancel higher order error terms! Possibly through estimators of the following form? P n = P n + b 1 (P n P n 1 ) + b 2 (P n 1 P n 2 ) P n = P n + c 1 (P n P n 1 ) + c 2 (P n 1 P n 2 ) + c 3 (P n 2 P n 3 ). 28 / 33

29 The GAEP algorithm GAEP (Generalized AEP) concerns more general aggregation functionals, i.e. the estimation of P[ψ(X 1,..., X d ) s], where ψ : R d R is a continuous function that is strictly increasing in each coordinate. This probability can be represented as the mass of some generalized triangle : x 2 {x R 2 + : ψ(x) = s} 0 x 1 29 / 33

30 GAEP generalized triangle decomposition Analogous to the AEP algorithm, we can decompose a generalized triangle into a rectangle and further generalized triangles: x 2 {x R 2 + : ψ(x) = s} x 2 0 x 1 0 x 1 30 / 33

31 GAEP: short summary Issue: how to choose the sidelengths of the approximating hypercubes (rectangles)? Paper proposes different possibilities Performance: Similar to AEP, very good for d = 2, 3, acceptable for d = 4 and not competitive for d 5 Open problems: A proof for an optimal choice of the hypercube sidelengths Extension of the extrapolation technique as used for AEP 31 / 33

32 References P. Arbenz, P. Embrechts, G. Puccetti: The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables. Bernoulli 17(2), 2011, P. Arbenz, P. Embrechts, G. Puccetti: The GAEP algorithm for the fast computation of the distribution of a function of dependent random variables. (Forthcoming in Stochastics, 2011) P. Embrechts, G. Puccetti: Risk Aggregation. In: Copula Theory and its Applications, P. Jaworski, F. Durante, W. Haerdle, and T. Rychlik (Eds.). Lecture Notes in Statistics - Proceedings 198, Springer Berlin/Heidelberg, pp Software (C++ code) to be obtained through the authors 32 / 33

33 Thank you 33 / 33