Scenario Aggregation

Transcription

1 Scenario Aggregation for Solvency Regulation Mathieu Cambou 1 (joint with Damir Filipović) Ecole Polytechnique Fédérale de Lausanne 1 SCOR ASTIN Colloquium The Hague, 22 May 2013

2 Outline Scenario Aggregation in the Swiss Solvency Test Minimum φ-divergence Approach Minimum φ-divergence Scenario Aggregation Robustness of Capital Requirement Solving the Optimization Problem Case Studies Case Study 1 Case Study 2 2/48

3 Outline Scenario Aggregation in the Swiss Solvency Test Minimum φ-divergence Approach Minimum φ-divergence Scenario Aggregation Robustness of Capital Requirement Solving the Optimization Problem Case Studies Case Study 1 Case Study 2 Scenario Aggregation in the Swiss Solvency Test 3/48

4 Internal Models Probability space (Ω, F, P) Annual loss L: random variable assigning a loss L(ω) to any possible state of the world ω Ω (Ω, F) is universal P and L are insurer specific (internal model) Assumption: Regulator trusts the mapping L : Ω R (no ambiguity) Fact: Regulator wants to challenge P, or distribution F L (x) = P[L x] Scenario Aggregation in the Swiss Solvency Test 4/48

5 Scenarios A scenario is an event S F: a narrative description of a possible state of the world For factor models, this notion includes point scenarios S = {x R n x j = c j for some j}, e.g. 1-year loss of EUR 100 mio in Eurowind, 1-year drop of 20% for SMI, quadrants S = {x R n x j c j for some j}, e.g. 1-year change in EURCHF 20%, 1-year change European Credit Spreads AAA 50%. Scenario Aggregation in the Swiss Solvency Test 5/48

6 Scenarios from a Regulatory point of view Mit den SST-Szenarien sollen die Mängel aus verteilungsbasierten Modellen korrigiert werden. So können analytische Modelle extreme Ereignisse oft nur unzureichend abbilden, sowohl in Bezug auf die Heavy-Tailedness der Randverteilungen als auch in Bezug auf die so genannte Tail-Dependency. FINMA, Wegleitung für die Erarbeitung des SST-Berichtes 2013, 2012 Scenario Aggregation in the Swiss Solvency Test 6/48

7 SST Scenario Aggregation Given by FINMA Scenarios S 1,..., S d along with auxiliary probabilities π i > 0 In general, π i P[S i ] It is natural to set S 0 = Ω and π 0 = 1 d i=1 π i Source: FOPI, SST Technical Documents, 2006 Scenario Aggregation in the Swiss Solvency Test 7/48

8 SST Scenario Aggregation Ansatz Scenario S i causes an extra-ordinary loss l i = E[L S i ] E[L] to be determined by actuary, with l 0 = 0 Loss distribution conditional on scenario S i is F L (x l i ) Aggregation Replace F L (x) by aggregated loss distribution F aggr L (x) = d π i F L (x l i ) i=0 Scenario Aggregation in the Swiss Solvency Test 8/48

9 SST Scenario Aggregation Scenario Aggregation in the Swiss Solvency Test 9/48

10 SST Scenario Aggregation Scenario Aggregation in the Swiss Solvency Test 10/48

11 SST Scenario Aggregation Scenario Aggregation in the Swiss Solvency Test 11/48

12 SST Method: Discussion Fact: F aggr L (x) is the cdf of L + Z for an independent extra-ordinary loss random variable Z with P[Z = l i ] = π i Lemma: ES aggr [L] = ES[L + Z] ES[L] + E[Z] Consequence: if E[Z] > 0 then ES aggr [L] > ES[L] no matter how conservative the internal model for L is scenario aggregation penalizes conservative internal models Scenario Aggregation in the Swiss Solvency Test 12/48

13 SST Method: Discussion No control on how far F aggr L (x) from F L (x) is No control on how far ES aggr (L) from ES(L) is Confusion among stakeholders about double-counting High degree of subjectivity about auxiliary weights π i Focus is on the mean but not on the tail of the loss distribution Aggregation is on the level of F L (x), not P Scenario Aggregation in the Swiss Solvency Test 13/48

14 Outline Scenario Aggregation in the Swiss Solvency Test Minimum φ-divergence Approach Minimum φ-divergence Scenario Aggregation Robustness of Capital Requirement Solving the Optimization Problem Case Studies Case Study 1 Case Study 2 Minimum φ-divergence Approach 14/48

15 A Moment s Reflection on Stress Tests Stress test: selected states of the world ω i Ω Leads to a maximal insurer specific loss l = max i L(ω i ) Internal model (null hypothesis) P passes the stress test if not rejected on significance level 1 α = 1%. That is, if l VaR α (L) Equivalently, P[S] 1 α for the scenario S = {L l} Minimum φ-divergence Approach 15/48

16 Outline Scenario Aggregation in the Swiss Solvency Test Minimum φ-divergence Approach Minimum φ-divergence Scenario Aggregation Robustness of Capital Requirement Solving the Optimization Problem Case Studies Case Study 1 Case Study 2 Minimum φ-divergence Approach 16/48

17 Definition of Views Given a collection of scenarios S 1,..., S d F Given a vector of target probabilities c = (c 1,..., c d ) Define S 0 = Ω \ d i=1 S i Denote M = set of probability measures on (Ω, F) Views on Q M: Q[S i ] c i, i = 1,..., d (views) Minimum φ-divergence Approach 17/48

18 Views in Terms of Atoms Let U 0,..., U n be the atoms of S = σ(s 1,..., S d ): S 0 = U 0, S i = j J(i) U j, i = 1,..., d Views on Q M: for vector q j = Q[U j ] j J(i) q j c i, i = 1,..., d (views) Views in matrix form: for matrix A ij = 1 J(i) (j) A q c (views) Minimum φ-divergence Approach 18/48

19 Scenario Aggregation Modification of internal model: find minimizer for minimize d(q, P) subject to (views) with domain M d(, P) measures the difference from P on M Minimum φ-divergence Approach 19/48

20 φ-divergence φ-divergence d(q, P) = { E[φ(dQ/dP)], +, if Q P otherwise where φ is convex, and strictly convex at 1 with φ(1) = 0 Standard measure for difference of Q from P in statistics, Csiszar (1963) Fact: d(q, P) is not a metric, but convex in Q Minimum φ-divergence Approach 20/48

21 Examples and Facts Examples: t log t, relative entropy φ(t) = ( t 1) 2, Hellinger distance t 1 p, L p -distance, p 1 Facts: dq/dp 1 1 2d E (Q, P) d H (Q, P) dq/dp 1 1 2d H (Q, P) dq/dp 1 1 = d TV (Q, P) total variation Fact: all but the L 1 -distance are strictly convex in Q Minimum φ-divergence Approach 21/48

22 Outline Scenario Aggregation in the Swiss Solvency Test Minimum φ-divergence Approach Minimum φ-divergence Scenario Aggregation Robustness of Capital Requirement Solving the Optimization Problem Case Studies Case Study 1 Case Study 2 Minimum φ-divergence Approach 22/48

23 Robustness Check Question: is the capital requirement robust under minimum φ-divergence scenario aggregation? In other words: is VaR and ES continuous with respect to d(q, P)? Minimum φ-divergence Approach 23/48

24 Recall Definitions of VaR and ES Value at risk VaR(X ) = q α (X ), left α-quantile Expected shortfall ES(X ) = 1 1 α E [ (X q) +] + q = 1 ( E[X 1{X 1 α >q} ] + q (P[X q] α) ) for any α-quantile q [q α (X ), q + α (X )] Folk theorem: VaR is more robust than ES... Minimum φ-divergence Approach 24/48

25 Lemma for Value at Risk Lemma 2.1. If dp n /dp 1 in L 1 then and sup P n [X x] P[X x] 0 x q α (X ) lim inf n q n lim sup q n q α + (X ) n for any sequence (q n ) of α-quantiles of X, for any X L 0. Minimum φ-divergence Approach 25/48

26 Non-Robustness of Value at Risk: Example Define X = 0 or 1 with P[X = 0] = α Define dp n /dp = VaR does not converge: { 1 + (1 α)( 1) n /(αn), on {X = 0} VaR n (X ) = 1 + ( 1) n+1 /n, on {X = 1} { 0 = q α (X ), for n even 1 = q + α (X ), for n odd Minimum φ-divergence Approach 26/48

27 Robustness of Expected Shortfall Theorem 2.2. Let p [1, ]. If dp n /dp 1 in L p then ES n (X ) ES(X ) for all X L r, where p 1 + r 1 = 1. Proof. Using previous lemma and (1 α) ES n (X ) ES(X ) E [ dp n /dp 1 (X q n ) +] + E [ (X q n ) + (X q) + ] + (1 α) q n q for any converging (sub-)sequence of α-quantiles q n q. Minimum φ-divergence Approach 27/48

28 Robustness of Expected Shortfall Theorem 2.2. Let p [1, ]. If dp n /dp 1 in L p then ES n (X ) ES(X ) for all X L r, where p 1 + r 1 = 1. Proof. Using previous lemma and (1 α) ES n (X ) ES(X ) E [ dp n /dp 1 (X q n ) +] + E [ (X q n ) + (X q) + ] + (1 α) q n q for any converging (sub-)sequence of α-quantiles q n q. Minimum φ-divergence Approach 27/48

29 Outline Scenario Aggregation in the Swiss Solvency Test Minimum φ-divergence Approach Minimum φ-divergence Scenario Aggregation Robustness of Capital Requirement Solving the Optimization Problem Case Studies Case Study 1 Case Study 2 Minimum φ-divergence Approach 28/48

30 Back to the Optimization Problem Modification of internal model: find minimizer for minimize d(q, P) subject to (views) (P) with domain M Lemma 2.3. For every R P satisfying the views there exists a R Q := {Q P dq/dp is S-measurable} satisfying the views and d(r, P) d(r, P). Proof. Set dr /dp = E [dr/dp S] and use Jensen s inequality. Minimum φ-divergence Approach 29/48

31 Existence and Uniqueness Note dim Q = n + 1: identify Q Q with q by q j = Q[U j ], j = 0,..., n Theorem 2.4. There exists a solution of (P) in Q. Moreover, if φ is strictly convex then the solution is unique. Minimum φ-divergence Approach 30/48

32 Solution of Optimization Problem Define p (0, 1) n+1 by p j = P[U j ] The optimization problem (P) reduces to minimize subject to n j=0 p j φ(q j /p j ) A q c 1 q = 1 (P) with domain (0, 1) n+1 Solution via dual problem or Kuhn Tucker (FOC) conditions Reference e.g. Boyd and Vandenberghe (2004) Minimum φ-divergence Approach 31/48

33 Special Case: d = 1 Scenario Corollary 2.5. For d = 1 scenario S 1 a (the) solution to (P) is given by dq dp = 1 max{c 1, p 1 } 1 S0 + max{c 1, p 1 } 1 S1 p 0 p 1 independently of the choice of the (strictly) convex divergence function φ. Proof. Convexity of φ implies that q 1 d(q(q 1 ), P) is non-decreasing in q 1 for q 1 > p 1. Minimum φ-divergence Approach 32/48

34 Special Case: d = 1 Scenario Corollary 2.5. For d = 1 scenario S 1 a (the) solution to (P) is given by dq dp = 1 max{c 1, p 1 } 1 S0 + max{c 1, p 1 } 1 S1 p 0 p 1 independently of the choice of the (strictly) convex divergence function φ. Proof. Convexity of φ implies that q 1 d(q(q 1 ), P) is non-decreasing in q 1 for q 1 > p 1. Minimum φ-divergence Approach 32/48

35 Special Case: Stress Testing (d = 1 Scenario) Recall stress testing is equivalent to d = 1 views Q[S 1 ] 1 α on the scenario S 1 = {L l} with l = max i L(ω i ) In this case we obtain a closed form expression for ES: Corollary 2.6. The expected shortfall under Q given in Corollary 2.5 satisfies ES Q, α(l) = ES P, max{p[l<l],α} (L). Minimum φ-divergence Approach 33/48

36 Outline Scenario Aggregation in the Swiss Solvency Test Minimum φ-divergence Approach Minimum φ-divergence Scenario Aggregation Robustness of Capital Requirement Solving the Optimization Problem Case Studies Case Study 1 Case Study 2 Case Studies 34/48

37 Outline Scenario Aggregation in the Swiss Solvency Test Minimum φ-divergence Approach Minimum φ-divergence Scenario Aggregation Robustness of Capital Requirement Solving the Optimization Problem Case Studies Case Study 1 Case Study 2 Case Studies 35/48

38 Case Study 1: Setup Loss L N (0, 10.2) Compare scenario aggregation using SST and minimum entropy method Case Studies 36/48

39 Case Study 1: SST Method d = 1 scenario with probability c and extra-ordinary loss l Recall: F aggr L (x) = (1 c) F L (x) + c F L (x l) 6.0 ES c l Case Studies Figure : Sensitivity of ES aggr 37/48 [L] with respect to c and l

40 Case Study 1: Minimum Entropy Method d = 1 scenario S 1 = {L l} View: Q[L l] c for some auxiliary level c ES c l Case Studies 38/48 Figure : Sensitivity of ES Q [L] with respect to c and l

41 Case Study 1: Difference SST - Minimum Entropy Method 0.8 Difference in ES c l Figure : ES aggr [L] ES Q [L] as function of c and l Case Studies 39/48

42 Case Study 1 (Stress Test): Minimum Entropy Method CDF after Aggregation Figure : Impact on cumulative distribution function for l = VaR α (L) with α = 0.99 (black), α = (grey), α = (blue) Case Studies 40/48 x

43 Outline Scenario Aggregation in the Swiss Solvency Test Minimum φ-divergence Approach Minimum φ-divergence Scenario Aggregation Robustness of Capital Requirement Solving the Optimization Problem Case Studies Case Study 1 Case Study 2 Case Studies 41/48

44 Case Study 2: Setup Two risk factors (X 1, X 2 ) normal with mean 0, and var(x 1 ) = 1, var(x 2 ) = 4, and corr(x 1, X 2 ) = 0.5 X 1 : change in interest rates X 2 : risk factor related to CAT events with reinsurance Loss L = max{x 1, 1} + max{min{x 2, 5}, 1} is capped in X 2 (reinsurance), and gains are capped at 1 d = 2 scenarios S 1 = {X 1 1, X 2 1} and S 2 = {X 1 < 2} Case Studies 42/48

45 Case Study 2: Shortfall Region Shortfall region W = {L > VaR 0.99 (L)} overlaps with S 1, but not with S 2, W S 1 W S 2 = Conditional expected losses given S 1 and S 2 are positive l 1 = E[L S 1 ] = 3.2, l 2 = E[L S 2 ] = 1.32 SST aggregation of S 2 leads to a capital increase even though S 2 does not intersect with the shortfall region W Case Studies 43/48

46 Case Study 2: Scenarios and Shortfall Region X S 2 S 1 W X 1 Case Studies Figure : Scenarios S 1, S 2, shortfall region W 44/48

47 Case Study 2: SST Method ES p 2 c p c Figure : Sensitivity of ES aggr [L] with respect to c 1 and c 2 Case Studies 45/48

48 Case Study 2: Minimum Entropy Method ES p 2 c p c Figure : Sensitivity of ES Q [L] with respect to c 1 and c 2 Case Studies 46/48

49 Case Study 2: Difference SST - Minimum Entropy Method 1.0 Difference in ES c 2 p p c Figure : ES aggr [L] ES Q [L] as function of c 1 and c 2 Case Studies 47/48

50 Conclusion Scenario aggregation is vital part of risk-based solvency regulation internal risk modelling Current (SST) methods subject to critical review Minimum φ-divergence approach is a coherent scenario aggregation method: No penalty for conservative internal models Focus on tail events Control over distance from internal model Robustness of capital requirement Highly tractable (closed form solutions sometimes) Conclusion 48/48