The Use of Penultimate Approximations in Risk Management

Transcription

1 The Use of Penultimate Approximations in Risk Management degen (joint work with P. Embrechts) 6th International Conference on Extreme Value Analysis Fort Collins CO, June 26, 2009 Penultimate approximations in Risk Management 1 / 14

2 Motivation [...] a bank must be able to demonstrate that its approach captures potentially severe tail loss events. Whatever approach is used, a bank must demonstrate that its operational risk measure meets a soundness standard comparable to that of the internal ratings-based approach for credit risk, (i.e. comparable to a one year holding period and a 99.9th percentile confidence interval). BIS (2006) International Convergence of Capital Measurement and Capital Standards: A Revised Framework - Comprehensive Version ( 667, Quantitative Standards for AMA to Operational Risk) Penultimate approximations in Risk Management 2 / 14

3 AMA to Operational Risk Penultimate approximations in Risk Management 3 / 14

4 AMA to Operational Risk few data heavy tails dvar 99.9% (L T+1 i,k )? Penultimate approximations in Risk Management 3 / 14

5 Framework: (second-order) extended regular variation Let U = (1/F) and ξ 0, ρ 0. Then for x > 0, first-order (U ERV ξ ): U(tx) U(t) a(t) second-order (U 2ERV ξ,ρ ): U(tx) U(t) a(t) D ξ (x) = xξ 1, t. (1) ξ D ξ (x) = H ξ,ρ (x), t. A(t) Theory on 2ERV and estimators based on (1) well elaborated Arbitrarily slow convergence possible, i.e. U 2ERV ξ,ρ with ρ = 0 so that A RV 0 Penultimate approximations in Risk Management 4 / 14

6 Framework: (second-order) extended regular variation Let U = (1/F) and ξ 0, ρ 0. Then for x > 0, first-order (U ERV ξ ): U(tx) U(t) a(t) second-order (U 2ERV ξ,ρ ): U(tx) U(t) a(t) D ξ (x) = xξ 1, t. (1) ξ D ξ (x) = H ξ,ρ (x), t. A(t) Theory on 2ERV and estimators based on (1) well elaborated Arbitrarily slow convergence possible, i.e. U 2ERV ξ,ρ with ρ = 0 so that A RV 0 In case of data scarcity: estimating VaR 99.9% (= U(1000)) based on (1) Penultimate approximations in Risk Management 4 / 14

7 Alternative: power norming For ξ 0, U ERV ξ (a) log U ERV 0(b) with b(t) = a(t)/u(t). Hence first-order: second-order: «1/b(t) U(tx) x, t, (2) U(t) ` D U(tx) 1/b(t) ξ D U(t) ξ (x) B(t) for some non-degenerate limit K ξ,ρ K ξ,ρ (x), t, Convergence rates in (1) and (2) depend on choice of the normalizations a( ) and b( ) respectively Penultimate approximations in Risk Management 5 / 14

8 Why power norming? (Hope to) speed up convergence by judicious choice of power norming through b( ) as opposed to standardly used linear normalization through a( ) How to choose b( )? Restriction on choice of b( ): lim b(t) = ξ 0 t Idea: With ξ indicating ultimate heavy-tailedness of U, choose b(t) as indicator of local heavy-tailedness of U at points t Penultimate approximations in Risk Management 6 / 14

9 Local heavy-tailedness Consider U on a log-log scale by setting U(t) = e ϕ(log t) (i.e. (s, ϕ(s)) gives the log-log plot of U) lim t ϕ (log t) = ξ since U ERV ξ Locally, i.e. at any point t 0, the loss model U may be interpreted as an exact Pareto model e U(t) = c t γ with tail-index γ = ϕ (log t 0) and some (unknown) c = c(t 0) Penultimate approximations in Risk Management 7 / 14

10 Local heavy-tailedness Consider U on a log-log scale by setting U(t) = e ϕ(log t) (i.e. (s, ϕ(s)) gives the log-log plot of U) lim t ϕ (log t) = ξ since U ERV ξ Locally, i.e. at any point t 0, the loss model U may be interpreted as an exact Pareto model e U(t) = c t γ with tail-index γ = ϕ (log t 0) and some (unknown) c = c(t 0) local heavy-tailedness log U(t) log t ϕ(log t) ϕ(log t) ϕ (log t 0) as indicator of local heavy-tailedness of U at points t 0 Penultimate approximations in Risk Management 7 / 14

11 Choice of power normalization For U 2ERV ξ,ρ (a, A) with ξ 0 and ρ 0, consider i) b(t) ξ (in cases ξ > 0) (ultimate slope) ii) b(t) = ϕ (log t) = tu (t)/u(t) (local/penultimate slope) iii) b(t) = ϕ(log t) 1/t R t ϕ(log s)ds (local pseudo slope) 1 Main theoretical result For the case ρ = 0 (and under some technical assumptions), applying a power normalization b( ) as in ii) or iii), improves the convergence rate in (2) over to the rate in (1), i.e. B(t)/A(t) 0 Implications for risk management practice Hope to improve high-quantile/var estimation using framework of power norming Penultimate approximations in Risk Management 8 / 14

12 Scaling of high-quantile estimators linear normalization: U(tx) U(t) a(t) xξ 1 ξ b ξ Nu n(1 α) 1 dvar α = u + bσ bξ POT-MLE estimates b ξ, bσ vs. power normalization: «U(tx) 1/b(t) x U(t) dvar α = «1 α b(t) d dvar α 1 α estimates d b(t), t = 1/(1 α) with N u exceedances over threshold u and quantile levels 0 < α α < 1 Penultimate approximations in Risk Management 9 / 14

13 Comparison of four different estimators of 99.9% quantile: EVT I: POT-MLE estimates for ξ, σ, based on N u = 10% upper order statistics EVT II: b 1(t) ξ: POT-MLE estimate, based on N u = 10% upper order statistics EVT III: b 2(t) = ϕ (log t): estimate local slope of ϕ using local (quadratic) regression ( locfit package of S-Plus) EVT IV: b 3(t) = log U(t) 1/t R t log U(s)ds: empirical log-quantiles & 1 composite trapezoidal rule for R (for estimators II-IV we take d VaR α to be the empirical quantile) Penultimate approximations in Risk Management 10 / 14

14 Simulation results Table: Bias and SRMSE (in %) of four EVT-based estimators for VaR at the 99.9% level based on 200 datasets of 1000, 500 and 250 observations from 4 (OR-typical) loss models. n = 1000, eα = 0.99 n = 500, eα = 0.98 n = 250, eα = 0.96 Bias SRMSE Bias SRMSE Bias SRMSE Loggamma data (α = 1.25, β = 1.25) EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) g-and-h data (a = b = 1.5, g = 0.8, h = 0.6) EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) Pareto data (x 0 = 1, ξ = 0.85) EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) Burr data (α = 0.75, κ = 1.5, τ = 1.5) EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) Penultimate approximations in Risk Management 11 / 14

15 Simulation results Table: Bias and SRMSE (in %) of four EVT-based estimators for VaR at the 99.9% level based on 200 datasets of 1000, 500 and 250 observations from 4 (OR-typical) loss models. n = 1000, eα = 0.99 n = 500, eα = 0.98 n = 250, eα = 0.96 Bias SRMSE Bias SRMSE Bias SRMSE Loggamma data (α = 1.25, β = 1.25) EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) g-and-h data (a = b = 1.5, g = 0.8, h = 0.6) EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) Pareto data (x 0 = 1, ξ = 0.85) EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) Burr data (α = 0.75, κ = 1.5, τ = 1.5) EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) Penultimate approximations in Risk Management 11 / 14

16 Conclusion Application of power normalizations for quantiles (instead of a linear normalization) may improve the rate of convergence in the respective limit results In situations of very heavy tails together with data scarcity standardly used EVT-based estimators hit the wall Estimation methods based on concept of power norming/local heavy-tailedness might provide viable supplement Penultimate approximations in Risk Management 12 / 14

17 References Degen, M. and Embrechts, P. (2009) Scaling of High Quantile Estimators. Preprint. Degen, M. and Embrechts, P. (2008) EVT-based Estimation of Risk Capital and Convergence of High Quantiles. Advances in Applied Probability 40(3), Penultimate approximations in Risk Management 13 / 14

18 locfit package: Local regression For every smoothing point x: 1) choose bandwidth h(x), e.g. nearest neighborhood fraction of x 2) choose weight kernel 3) choose degree of local polynomial to be fitted in smoothing window (x h(x), x + h(x)) 4) estimate coefficients of local polynomial of degree k, say, via weighted LS (â 0, â 1,..., â k ) 5) local slope at x: given by â 1 (derivative of local ploynomial evaluated at x, provided k 1) the typcally used Kernel smoothing thus corresponds to local regression with polynomials of degree 0 Penultimate approximations in Risk Management 14 / 14