The Use of Penultimate Approximations in Risk Management

The Use of Penultimate Approximations in Risk Management www.math.ethz.ch/ 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

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

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

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

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

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

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

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

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

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) 4 2 0 2 4 log t ϕ(log t) ϕ(log t) 0 2 4 6 ϕ (log t 0) as indicator of local heavy-tailedness of U at points t 0 Penultimate approximations in Risk Management 7 / 14

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

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

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

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 ) 17.90 87.55 16.48 89.71 2.39 35.38 9.01 52.43 g-and-h data (a = b = 1.5, g = 0.8, h = 0.6) 36.43 134.34 24.94 142.52-3.47 41.78 11.65 57.18 161.05 1334.11 226.97 1797.60-9.70 50.64 3.38 65.92 EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) 17.84 83.00 14.56 90.49 17.64 53.40 32.13 76.87 Pareto data (x 0 = 1, ξ = 0.85) 33.96 127.23 33.28 140.16 22.87 65.03 40.73 93.25 86.70 347.00 121.31 503.82 38.15 106.75 47.26 133.02 EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) 17.56 73.70 18.00 84.33-10.43 26.70 7.98 45.06 Burr data (α = 0.75, κ = 1.5, τ = 1.5) 47.36 161.63 53.52 180.64-22.62 36.11 2.02 59.71 51.87 332.34 76.11 509.14-27.09 44.19-3.41 54.66 EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) 16.44 103.48 14.24 107.85 0.32 46.44 11.23 62.08 37.90 157.97 35.72 167.87 10.01 59.06 21.85 71.82 109.26 572.75 123.88 601.16 36.74 110.38 37.69 103.68 Penultimate approximations in Risk Management 11 / 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 ) 17.90 87.55 16.48 89.71 2.39 35.38 9.01 52.43 g-and-h data (a = b = 1.5, g = 0.8, h = 0.6) 36.43 134.34 24.94 142.52-3.47 41.78 11.65 57.18 161.05 1334.11 226.97 1797.60-9.70 50.64 3.38 65.92 EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) 17.84 83.00 14.56 90.49 17.64 53.40 32.13 76.87 Pareto data (x 0 = 1, ξ = 0.85) 33.96 127.23 33.28 140.16 22.87 65.03 40.73 93.25 86.70 347.00 121.31 503.82 38.15 106.75 47.26 133.02 EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) 17.56 73.70 18.00 84.33-10.43 26.70 7.98 45.06 Burr data (α = 0.75, κ = 1.5, τ = 1.5) 47.36 161.63 53.52 180.64-22.62 36.11 2.02 59.71 51.87 332.34 76.11 509.14-27.09 44.19-3.41 54.66 EVT I (POT) EVT II (b 1 ) EVT III (b 2 ) EVT IV (b 3 ) 16.44 103.48 14.24 107.85 0.32 46.44 11.23 62.08 37.90 157.97 35.72 167.87 10.01 59.06 21.85 71.82 109.26 572.75 123.88 601.16 36.74 110.38 37.69 103.68 Penultimate approximations in Risk Management 11 / 14

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

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), 696 715. Penultimate approximations in Risk Management 13 / 14

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