Modeling Operational Risk: Estimation and Effects of Dependencies

Transcription

1 Modeling Operational Risk: Estimation and Effects of Dependencies Stefan Mittnik Sandra Paterlini Tina Yener Financial Mathematics Seminar March 4, 2011

2 Outline Outline of the Talk 1 Motivation Operational Risk VaR and Subadditivity 2 The data 3 Modeling Dependence Correlation Copulas Nonparametric Measures of Extremal Dependence 4 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates 5 Conclusion

3 Introduction Operational Risk Operational Risk: Definition The Basel Committee defines operational risk as The risk of loss resulting from inadequate or failed internal processes, people and systems, or from external events. This definition includes legal risk, but excludes strategic and reputational risk.

4 Introduction Operational Risk Operational Risk: Introduction Operational loss events can be very heterogeneous. Event Types Internal fraud External fraud Employment practices and workplace safety Clients, products and business practices Damage to physical assets Business disruption and system failures Execution, delivery and process management Business Lines Corporate Finance Trading and Sales Retail Banking Commercial Banking Payment and Settlement Agency Services Asset Management Retail Brokerage

5 Introduction Operational Risk Operational Risk: Introduction Three distinct approaches to determine minimal capital requirements: 1 Basic Indicator Approach 2 Standardised Approach 3 Advanced Measurement Approach (AMA) Under AMA, the risk measure to be used is the Value at Risk (VaR α ), i.e., the quantile of the loss distribution, with α = 99.9%.

6 Introduction Operational Risk Operational Risk: Heterogeneity of Events The heterogeneity of Operational Risk leads to the regulatory requirement of a separate modeling within 56 event type/business-line combinations.

7 Introduction Operational Risk Operational Risk: Heterogeneity of Events The heterogeneity of Operational Risk leads to the regulatory requirement of a separate modeling within 56 event type/business-line combinations. Business Lines Corporate Finance... Retail Brokerage Event Types Internal Fraud L 1,1... L 1, Execution, Delivery & L 7,1... L 7,8 Process Management

8 Introduction Operational Risk Operational Risk: Total Risk Capital The quantity of interest is ( 56 ) VaR.999 (L) = VaR.999 L i ; (1) clearly, it is influenced by dependencies among cells i and j. i=1

9 Introduction Operational Risk Operational Risk: Total Risk Capital The quantity of interest is ( 56 ) VaR.999 (L) = VaR.999 L i ; (1) clearly, it is influenced by dependencies among cells i and j. However, Basel II prescribes to calculate Total Risk Capital as i=1 56 TRC = VaR.999 (L i ) ; (2) i=1 only under certain qualifying conditions, banks may explicitly model dependencies.

10 Introduction VaR and Subadditivity VaR and Subadditivity For comonotonic risks, VaR co α (L i + L j ) = VaR α (L i ) + VaR α (L j ). (3) Comonotonicity translates into perfect positive correlation in the elliptical (Gaussian) world.

11 Introduction VaR and Subadditivity VaR and Subadditivity For comonotonic risks, VaR co α (L i + L j ) = VaR α (L i ) + VaR α (L j ). (3) Comonotonicity translates into perfect positive correlation in the elliptical (Gaussian) world. For elliptical distributions, the sum of the single VaRs provides an upper bound and thus a worst case scenario for VaR α (L).

12 Introduction VaR and Subadditivity VaR and Subadditivity For comonotonic risks, VaR co α (L i + L j ) = VaR α (L i ) + VaR α (L j ). (3) Comonotonicity translates into perfect positive correlation in the elliptical (Gaussian) world. For elliptical distributions, the sum of the single VaRs provides an upper bound and thus a worst case scenario for VaR α (L). Given that we hardly have perfect positive correlation, the idea of modeling dependencies seems to be appealing in order to reduce capital estimates...

13 Introduction VaR and Subadditivity VaR and Subadditivity However, for non elliptical distributions, it may happen that VaR α (L 1 + L 2 ) > VaR α (L 1 ) + VaR α (L 2 ), (4) the reason being the lack of subadditivity of the VaR measure (Artzner et al., 1999).

14 Introduction VaR and Subadditivity VaR and Subadditivity However, for non elliptical distributions, it may happen that VaR α (L 1 + L 2 ) > VaR α (L 1 ) + VaR α (L 2 ), (4) the reason being the lack of subadditivity of the VaR measure (Artzner et al., 1999). Does this mean that banks may not be rewarded for a more realistic dependency modeling by a decrease in risk capital, but instead be punished by an increase?

15 Introduction VaR and Subadditivity VaR and Subadditivity However, for non elliptical distributions, it may happen that VaR α (L 1 + L 2 ) > VaR α (L 1 ) + VaR α (L 2 ), (4) the reason being the lack of subadditivity of the VaR measure (Artzner et al., 1999). Does this mean that banks may not be rewarded for a more realistic dependency modeling by a decrease in risk capital, but instead be punished by an increase? Yes! (see, e.g., Embrechts et al., 2002)

16 Introduction VaR and Subadditivity VaR and Subadditivity However, for non elliptical distributions, it may happen that VaR α (L 1 + L 2 ) > VaR α (L 1 ) + VaR α (L 2 ), (4) the reason being the lack of subadditivity of the VaR measure (Artzner et al., 1999). Does this mean that banks may not be rewarded for a more realistic dependency modeling by a decrease in risk capital, but instead be punished by an increase? Yes! (see, e.g., Embrechts et al., 2002) But is this practically relevant?

17 Introduction Aim Aim of our Analyses Based on n = 60 observations of monthly aggregate losses from the Italian DIPO 1 database, we aim at evaluating VaR (L i + L j ) (VaR (L i ) + VaR (L j )) } {{ } =TRC (5) for different cells i and j of the event type/business line matrix. 1

18 Introduction Aim Aim of our Analyses Based on n = 60 observations of monthly aggregate losses from the Italian DIPO 1 database, we aim at evaluating VaR (L i + L j ) (VaR (L i ) + VaR (L j )) } {{ } =TRC (5) for different cells i and j of the event type/business line matrix. This task is non trivial, because it means analyzing the 99.9% quantile of a distribution estimated from a small sample with extreme data under consideration of dependencies. 1

19 Introduction Data The DIPO data base DIPO (Database Italiano delle Perdite Operative): 31 members (209 different entities) of all sizes and mainly not AMA; about 75% of the Italian banking system in terms of Gross Income and Operating Costs Collection of operational risk data (loss threshold 5,000 EUR), starting from 2003 Data collection twice a year; common definition of gross loss, event type decision tree and business line mapping 47,123 events with a total loss amount of 3,295 million EUR

20 Introduction Data Distribution across business lines (in %) Share in... Business line Number Total loss of events 1 (Corporate Finance) (Trading & Sales) (Retail Banking) (Commercial Banking) (Payment & Settlement) (Agency Services) (Asset Management) (Retail Brokerage)

21 Introduction Data Distribution across event types (in %) We aggregate losses on a monthly basis across event types. Share in... Event type Number Total loss of events 1 (Internal Fraud) (External Fraud) (Employment Practices & Workplace Safety) (Clients, Products & Business Practices) (Damage to Physical Assets) (Business Disruption & System Failures) (Execution, Delivery & Process Management)

22 Introduction Data Distribution across event types (in %) We aggregate on a monthly level and obtain n = 60 observations (aggregate losses) per event type category. Due to data scarcity, we do not further subdivide on the business line level Abs. Frequency Abs. Frequency 5 Abs. Frequency 10 5 Abs. Frequency l1 0 l2 0 l3 0 l Abs. Frequency Abs. Frequency Abs. Frequency l5 0 l6 0 l7

23 Modeling Dependencies Measures of Dependence Measures of Dependence For Operational Risk quantification, it is crucial to model dependencies for small tail probabilities. In our investigation, we consider correlation, copulas, nonparametric measures of extremal dependence.

24 Modeling Dependencies Correlation Linear (Pearson) Correlation The well known fact that linear correlation is prone to extremes is quickly revealed by the data. For example, for event type combination 2/5, as the two most extreme observations drop out of the sample, correlation becomes negative. 01/03 12/07 (entire sample): 01/03 04/06 (2/3 of sample): l5 l5 l 2 ρ 2,5 = ρ 2,5 = l 2

25 Modeling Dependencies Correlation Linear (Pearson) Correlation Similarly, for event type combination 3/4: 01/03 12/07: 01/03 04/06: l4 l4 l 3 ρ 3,4 = ρ 3,4 = If we further remove the most extreme observation between 01/03 and 04/06, the correlation decreases to ρ 3,4 = l 3

26 Modeling Dependencies Correlation Correlation: Results The well known sensitivity of linear correlation with respect to extremes leads to substantial variations, depending on the sample size.

27 Modeling Dependencies Correlation Correlation: Results The well known sensitivity of linear correlation with respect to extremes leads to substantial variations, depending on the sample size. Its inability to capture possible nonlinear dependency structures provides another important reason for discarding linear correlation as a reliable measure of dependency.

28 Modeling Dependencies Correlation Correlation: Results The well known sensitivity of linear correlation with respect to extremes leads to substantial variations, depending on the sample size. Its inability to capture possible nonlinear dependency structures provides another important reason for discarding linear correlation as a reliable measure of dependency. Rank correlations were also considered but not found to lead to considerably more stable results.

29 Modeling Dependencies Copulas Copulas Instead of boiling down dependency into one single number, copulas contain the dependence structure of a joint distribution.

30 Modeling Dependencies Copulas Copulas Instead of boiling down dependency into one single number, copulas contain the dependence structure of a joint distribution. The central theorem of copula theory can be traced back to Sklar (1959) and summed up by C(u1, u2) u u F i,j (l i, l j ) = C(F i (l i ), F } {{ } j (l j )), (6) } {{ } u i u j where C denotes the copula of L i and L j and U i, U j Unif(0, 1).

31 Modeling Dependencies Copulas Copulas and Tail Dependence Tail dependence accounts for possibly nonlinear dependence among extremes and thus overcomes one drawback of correlation.

32 Modeling Dependencies Copulas Copulas and Tail Dependence Tail dependence accounts for possibly nonlinear dependence among extremes and thus overcomes one drawback of correlation. In our context of loss distributions, we only consider the upper tail dependence coefficient λ U = lim t 1 Pr[F i(l i ) > t F j (L j ) > t]. (7)

33 Modeling Dependencies Copulas Copulas and Tail Dependence Tail dependence accounts for possibly nonlinear dependence among extremes and thus overcomes one drawback of correlation. In our context of loss distributions, we only consider the upper tail dependence coefficient λ U = lim t 1 Pr[F i(l i ) > t F j (L j ) > t]. (7) It can also be expressed in terms of the copula of X and Y : 1 2t + C(t, t) λ U = lim. (8) t 1 1 t

34 Modeling Operational Risk: Estimation and Effects of Dependencies Modeling Dependencies Copulas Copulas and Tail Dependence ℓi Gaussian copula: no tail dependence ℓj ℓj ℓj Different copulas imply different tail dependence structures. ℓi Gumbel copula: upper tail dependence ℓi Clayton copula: lower tail dependence

35 Modeling Dependencies Copulas Copulas: Estimation Upper tail dependence coefficient implied by the Gumbel copula parameter estimates: ET 2 ET 3 ET 4 ET 5 ET 6 ET 7 ET ET ET ET ET ET

36 Modeling Dependencies Copulas Copulas: Estimation Upper tail dependence coefficient implied by the Clayton survival copula parameter estimates: ET 2 ET 3 ET 4 ET 5 ET 6 ET 7 ET ET ET ET ET ET

37 Modeling Dependencies Copulas Copulas: Estimation Upper tail dependence coefficient implied by the Student-t copula parameter estimates: ET 2 ET 3 ET 4 ET 5 ET 6 ET 7 ET ET ET ET ET ET

38 Modeling Dependencies Copulas Copulas: How can we find the right one? 1. Goodness of fit testing. We consider two g o f tests (A 2 and A 4 ) found to perform particularly well by Berg (2009).

39 Modeling Dependencies Copulas Copulas: How can we find the right one? 1. Goodness of fit testing. We consider two g o f tests (A 2 and A 4 ) found to perform particularly well by Berg (2009). Both are based on the distance between the estimated copula and the empirical copula of Deheuvels (1979).

40 Modeling Dependencies Copulas Copulas: How can we find the right one? 1. Goodness of fit testing. We consider two g o f tests (A 2 and A 4 ) found to perform particularly well by Berg (2009). Both are based on the distance between the estimated copula and the empirical copula of Deheuvels (1979). Unfortunately, we cannot discard any of the copula families considered based on these tests.

41 Modeling Dependencies Copulas Copulas: How can we find the right one? 2. Consideration of different sample sizes. Contours of fitted copulas, event type combination 3/4: Empirical Gaussian Student-t Gumbel Gumbel Surv. Clayton Clayton Surv Empirical Gaussian Student-t Gumbel Gumbel Surv. Clayton Clayton Surv. u4 u u3 u3 01/03 12/07 01/03 04/06 Accordingly, parameter estimates vary with the sample size.

42 Modeling Dependencies Copulas Copulas: Results Copulas can be used to detect specific dependence structures, i.e., tail dependence.

43 Modeling Dependencies Copulas Copulas: Results Copulas can be used to detect specific dependence structures, i.e., tail dependence. It seems that there are some event type combinations which are characterized by tail dependence, while others are not.

44 Modeling Dependencies Copulas Copulas: Results Copulas can be used to detect specific dependence structures, i.e., tail dependence. It seems that there are some event type combinations which are characterized by tail dependence, while others are not. However, we do neither find an overall best fitting copula, nor can we exclude any copula family considered.

45 Modeling Dependencies Copulas Copulas: Results Copulas can be used to detect specific dependence structures, i.e., tail dependence. It seems that there are some event type combinations which are characterized by tail dependence, while others are not. However, we do neither find an overall best fitting copula, nor can we exclude any copula family considered. Again, the availability of a small data set strongly affects the stability of estimation results.

46 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Estimators It is also possible to estimate tail dependence without a parametric assumption for the copula, i.e., using the empirical copula of Deheuvels (1979).

47 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Estimators It is also possible to estimate tail dependence without a parametric assumption for the copula, i.e., using the empirical copula of Deheuvels (1979). We consider two common nonparametric estimators of λ U : λ U (t) = 2 ln Ĉn (t, t) ln (t), (Coles et al., 1999)

48 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Estimators It is also possible to estimate tail dependence without a parametric assumption for the copula, i.e., using the empirical copula of Deheuvels (1979). We consider two common nonparametric estimators of λ U : λ U (t) = 2 ln Ĉn (t, t) ln (t), (Coles et al., 1999) χ(t) = 2 1 Ĉn (t, t) 1 t. (Joe et al., 1992)

49 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Estimators However, as pointed out by Coles et al. (1999), asymptotic dependence implies not only that λ U > 0, but also that 2 ln(1 t) χ = lim t 1 ln( C(t, t)) 1 = lim χ(t) = 1, (9) t 1 where C(t, t) = Pr[U > t, V > t].

50 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Estimators However, as pointed out by Coles et al. (1999), asymptotic dependence implies not only that λ U > 0, but also that 2 ln(1 t) χ = lim t 1 ln( C(t, t)) 1 = lim χ(t) = 1, (9) t 1 where C(t, t) = Pr[U > t, V > t]. This quantity can be estimated via 2 ln(1 t) χ(t) = 1. (10) ln( C(t, t))

51 Modeling Dependencies Nonparametric Tail Dependence Asymptotic vs. Quantile Dependence Summing up, asymptotic and quantile dependence are characterized by the following situations. Asymptotic Independence Asymptotic Dependence λ U 0 (0,1] χ [-1,1] 1 For λ U = 0, we have asymptotic independence, but we may still have quantile dependence at t < 1, the strength (and direction) of which is determined by χ.

52 Modeling Dependencies Nonparametric Tail Dependence Asymptotic vs. Quantile Dependence Summing up, asymptotic and quantile dependence are characterized by the following situations. Asymptotic Independence Asymptotic Dependence λ U 0 (0,1] χ [-1,1] 1 For λ U = 0, we have asymptotic independence, but we may still have quantile dependence at t < 1, the strength (and direction) of which is determined by χ. We therefore consider the three estimators λ U (t), χ(t) and χ(t).

53 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Estimation Nonparametric tail dependence estimation, event type combination 2/5: λ U (t) χ(t) χ(t) λ U (t) χ(t) χ(t) λ2,5(t) 0.0 λ2,5(t) t t 01/03 12/07 01/03 04/06

54 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Estimation Nonparametric tail dependence estimation, event type combination 3/4: λ U (t) χ(t) χ(t) λ U (t) χ(t) χ(t) λ3,4(t) 0.0 λ3,4(t) t t 01/03 12/07 01/03 04/06

55 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Results We do not find clear cut evidence of tail dependence.

56 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Results We do not find clear cut evidence of tail dependence. A reliable estimation of tail dependence is complicated by the fact that the relevant area (t 1) is characterized by few observations.

57 Modeling Dependencies Nonparametric Tail Dependence Nonparametric Tail Dependence: Results We do not find clear cut evidence of tail dependence. A reliable estimation of tail dependence is complicated by the fact that the relevant area (t 1) is characterized by few observations. With respect to quantile dependence, results differ among event types.

58 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates Now, we want to assess the effects of realistic dependency structures on risk capital estimates.

59 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates Now, we want to assess the effects of realistic dependency structures on risk capital estimates. To this end, we estimate % VaR figures per model and event type combination, using different numbers of replications.

60 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates Now, we want to assess the effects of realistic dependency structures on risk capital estimates. To this end, we estimate % VaR figures per model and event type combination, using different numbers of replications. For each event type combination, we use the copula parameter values obtained from Maximum Likelihood estimation.

61 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates Now, we want to assess the effects of realistic dependency structures on risk capital estimates. To this end, we estimate % VaR figures per model and event type combination, using different numbers of replications. For each event type combination, we use the copula parameter values obtained from Maximum Likelihood estimation. For the margins, a lognormal distribution was fitted and is used here to derive risk capital estimates.

62 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates We consider VaR.999 (L i + L j ) (VaR.999 (L i ) + VaR.999 (L j )) (VaR.999 (L i ) + VaR.999 (L j )) (11)

63 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates We consider VaR.999 (L i + L j ) (VaR.999 (L i ) + VaR.999 (L j )) (VaR.999 (L i ) + VaR.999 (L j )) (11) under two different assumptions: 1 the Gaussian copula for all event type combinations,

64 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates We consider VaR.999 (L i + L j ) (VaR.999 (L i ) + VaR.999 (L j )) (VaR.999 (L i ) + VaR.999 (L j )) (11) under two different assumptions: 1 the Gaussian copula for all event type combinations, 2 the worst case copula, i.e., that copula yielding the highest tail dependence coefficient for the respective event type combination.

65 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates: Boxplots Gaussian copula, lognormal margins, risk capital estimates based on 10,000 simulated losses: 50 B rc = 10, Rel. Diff. (%) /2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7 Event Type Combination

66 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates: Boxplots Worst case copula, lognormal margins, risk capital estimates based on 10,000 simulated losses: 50 B rc = 10, Rel. Diff. (%) /2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7 Event Type Combination

67 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates: Boxplots Gaussian copula, lognormal margins, risk capital estimates based on 50,000 simulated losses: 25 B rc = 50,000 Rel. Diff. (%) /2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7 Event Type Combination

68 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates: Boxplots Worst case copula, lognormal margins, risk capital estimates based on 50,000 simulated losses: 25 B rc = 50,000 Rel. Diff. (%) /2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7 Event Type Combination

69 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates: Boxplots Gaussian copula, lognormal margins, risk capital estimates based on 100,000 simulated losses: 20 B rc = 100, Rel. Diff. (%) /2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7 Event Type Combination

70 Effects on Risk Capital Range of Risk Capital Estimates Range of Risk Capital Estimates: Boxplots Worst case copula, lognormal margins, risk capital estimates based on 100,000 simulated losses: 20 B rc = 100, Rel. Diff. (%) /2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7 Event Type Combination

71 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates Obviously, increasing the number of replications for VaR calculations narrows the range of possible risk capital estimates.

72 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates Obviously, increasing the number of replications for VaR calculations narrows the range of possible risk capital estimates. It is, thus, not clear which part of a change is due to the subadditivity problem, and which one is due to computational issues.

73 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates Obviously, increasing the number of replications for VaR calculations narrows the range of possible risk capital estimates. It is, thus, not clear which part of a change is due to the subadditivity problem, and which one is due to computational issues. A natural question is then: What could be the worst capital estimate? Statistically, this means to study whether there are theoretical bounds on VaR. This topic has been treated, for example, by Makarov (1981) and Frank et al. (1987), and recently by Embrechts and Puccetti (2006).

74 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Starting Point The Fréchet Höffding bounds (Fréchet, 1951; Höffding, 1940) apply to any n dimensional copula, i.e., max(u u n n + 1, 0) } {{ } C l (u) C(u) min(u). (12) } {{ } C u(u) C(u1, u2) 0.5 C(u1, u2) 0.5 C(u1, u2) u u u u u u 1 lower bound C l (u 1, u 2) Gaussian copula upper bound C u(u 1, u 2)

75 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Bounds on VaR The tightness of the bounds on VaR depends on the dependence assumption.

76 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Bounds on VaR The tightness of the bounds on VaR depends on the dependence assumption. If we assume that the copula C of X 1,..., X n satisfies C C 0 and C d C1 d with C d (u, v) = u + v C(u, v),

77 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Bounds on VaR The tightness of the bounds on VaR depends on the dependence assumption. If we assume that the copula C of X 1,..., X n satisfies C C 0 and C d C1 d with C d (u, v) = u + v C(u, v), we obtain the bounds F 1 ( min (α) = inf F 1 1 (u 1 ) + F 1 C 0 (u 1,...,u n)=α Fmax(α) 1 = sup C1 d (u 1,...,u n)=α n (u n ) ), (13) ( F 1 1 (u 1 ) + F 1 n (u n ) ). (14)

78 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Bounds on VaR The tightness of the bounds on VaR depends on the dependence assumption. If we assume that the copula C of X 1,..., X n satisfies C C 0 and C d C1 d with C d (u, v) = u + v C(u, v), we obtain the bounds F 1 ( min (α) = inf F 1 1 (u 1 ) + F 1 C 0 (u 1,...,u n)=α Fmax(α) 1 = sup C1 d (u 1,...,u n)=α n (u n ) ), (13) ( F 1 1 (u 1 ) + F 1 n (u n ) ). (14) For the two dimensional case, C C 0 implies that C d C d 0, so that one can take C 0 = C 1.

79 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Assumptions We evaluate upper and lower bounds for three scenarios. 1 C 0 = C 1 = C l : We do not use any restriction on the dependence structure and thus use the lower Fréchet bound, C l.

80 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Assumptions We evaluate upper and lower bounds for three scenarios. 1 C 0 = C 1 = C l : We do not use any restriction on the dependence structure and thus use the lower Fréchet bound, C l. 2 C 0 = C 1 = u i u j : We assume that C u i u j, that is, we have positive quadrant dependence (PQD).

81 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Assumptions We evaluate upper and lower bounds for three scenarios. 1 C 0 = C 1 = C l : We do not use any restriction on the dependence structure and thus use the lower Fréchet bound, C l. 2 C 0 = C 1 = u i u j : We assume that C u i u j, that is, we have positive quadrant dependence (PQD). 3 C 0 = C θ i,j CS, Ĉ1 = C θ i,j C : We take the Clayton Survival copula as lower bound, using the parameter values estimated for the DIPO data. For the survival copula of C 1, we accordingly assume the Clayton copula with respective parameter values.

82 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Boxplots Worst case copula, lognormal margins, risk capital estimates based on 10,000 simulated losses: B rc = 10,000 C 0 = C L C 0 = C I C 0 = C CS, Ĉ 1 = C C Rel. Diff. (%) /2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7 Event Type Combination

83 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Boxplots Worst case copula, lognormal margins, risk capital estimates based on 50,000 simulated losses: 25 B rc = 50,000 C 0 = C L C 0 = C I C 0 = C CS, Ĉ 1 = C C Rel. Diff. (%) /2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7 Event Type Combination

84 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Boxplots Worst case copula, lognormal margins, risk capital estimates based on 100,000 simulated losses: B rc = 100, C 0 = C L C 0 = C I C 0 = C CS, Ĉ 1 = C C Rel. Diff. (%) /2 1/3 1/4 1/5 1/6 1/7 2/3 2/4 2/5 2/6 2/7 3/4 3/5 3/6 3/7 4/5 4/6 4/7 5/6 5/7 6/7 Event Type Combination

85 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Results Risk capital estimates may increase when departing from the comonotonicity assumption.

86 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Results Risk capital estimates may increase when departing from the comonotonicity assumption. However, this effect depends on the presence of extremal (tail/quantile) dependence;

87 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Results Risk capital estimates may increase when departing from the comonotonicity assumption. However, this effect depends on the presence of extremal (tail/quantile) dependence; such an increase may as well be caused by an insufficient number of replications in the simulation of losses.

88 Effects on Risk Capital Range of Risk Capital Estimates Bounds on Risk Capital Estimates: Results Risk capital estimates may increase when departing from the comonotonicity assumption. However, this effect depends on the presence of extremal (tail/quantile) dependence; such an increase may as well be caused by an insufficient number of replications in the simulation of losses. Theoretical bounds may help to assess which part of the change in risk capital is due to computational effects.

89 Conclusion Conclusion Contrary to expectations, risk capital may not always decrease when modeling dependencies. Measuring risk capital for operational risk is a challenging task due to the scarcity of data, high variability, presence of extremes and the need to compute the VaR at a very high confidence level. Simple methods, as correlations, may not lead to a complete and/or reliable picture of dependencies in operational risk losses. More sophisticated methods, such as copulas, could provide relevant information about dependencies and their effect on risk capital estimates.

90 Conclusion Conclusion The question whether risk capital estimates may increase compared to the comonotonicity case crucially depends on the presence of extremal (tail/quantile) dependence and the ellipticity of the multivariate distribution.

91 Conclusion Conclusion The question whether risk capital estimates may increase compared to the comonotonicity case crucially depends on the presence of extremal (tail/quantile) dependence and the ellipticity of the multivariate distribution. Therefore, the estimation of extremal dependence is of paramount importance for an assessment of the relevance of this effect.

92 Conclusion Conclusion The question whether risk capital estimates may increase compared to the comonotonicity case crucially depends on the presence of extremal (tail/quantile) dependence and the ellipticity of the multivariate distribution. Therefore, the estimation of extremal dependence is of paramount importance for an assessment of the relevance of this effect. For small sample sizes, parametric (copula) and nonparametric methods should both be used in order to gain a picture as comprehensive as possible.

93 Conclusion References I Artzner, P., Delbaen, F., Eber, J., and Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, 9: Berg, D. (2009). Copula goodness-of-fit testing: An overview and power comparison. European Journal of Finance, 15: Coles, S. G., Heffernan, J. E., and Tawn, J. A. (1999). Dependence measures for extreme value analyses. Extremes, 2: Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés. un test non paramétrique d independance. Bulletin de la Classe des Sciences, 65: Embrechts, P., McNeil, A., and Straumann, D. (2002). Correlation and dependence in risk management: properties and pitfalls. In Dempster, M., editor, Risk Management: Value at Risk and Beyond. Cambridge University Press.

94 Conclusion References II Embrechts, P. and Puccetti, G. (2006). Bounds for functions of multivariate risks. Journal of Multivariate Analysis, 97(2): Frank, M. J., Nelsen, R. B., and Schweizer, B. (1987). Best possible bounds for the distribution of a sum a problem of kolmogorov. Probability Theory and Related Fields, 74(2): Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont donnés. Annales de l Université de Lyon, 3(14): Höffding, W. (1940). Masstabinvariante korrelationstheorie. Schriften des Mathematischen Instituts und des Instituts fur Angewandte Mathematik der Universität Berlin, 5:

95 Conclusion References III Joe, H., Smith, R. L., and Weissman, I. (1992). Bivariate threshold models for extremes. Journal of the Royal Statistical Society, Series B, 54: Makarov, G. (1981). Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed. Theory of Probability and its Applications, 26: Sklar, A. (1959). Fonctions de répartition a n dimensions et leurs marges. Publications de l Institut de Statistique de L Universitá de Paris, 8: