TERES - Tail Event Risk Expectile based Shortfall

Transcription

1 TERES - Tail Event Risk Expectile based Shortfall Philipp Gschöpf Wolfgang Karl Härdle Andrija Mihoci Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics HumboldtUniversität zu Berlin

2 Motivation 1-1 Motivation Figure 1: The teres major muscle

3 Motivation 1-2 Tail Risk Figure 2: Nezha (link)

4 Motivation 1-3 VaR and ES Value at Risk (VaR) Basel III Not coherent Coherence Expected Shortfall (ES) Basel Committee (214) Coherent, focus on tail structure ES Denition Example: Deutsche Bank - risk levels {.2,.1,.1}, Kalkbrenner et al. (214)

5 Motivation 1-4 Quantile VaR Figure 3: Distribution of returns, VaR.5 remains unchanged under 25 chang- 21 ing tail structure, clouding the investors risk perception

6 Motivation 1-5 Objectives (i) Understanding Expected Shortfall (ES) Extreme events and associated risk Distributional environments - implications (ii) TERES Tail driven risk assessment, robustness of ES Advantages of expectiles

7 Motivation 1-6 Tail Risk Example: 28 subprime mortgage crisis S&P 5 long position in 28 (261 daily returns) Quantication of the 1% portfolio risk Scenario analysis, µ =.2, σ =.25 Scenario (i) Normal distribution VaR = 5.9%, ES = 6.8% Scenario (ii) Laplace distribution VaR = 1.%, ES = 12.5%

8 Motivation 1-7 Tail Risk Normal distribution VaR ES Laplace distribution VaR ES -2.4 Dynamics ES Estimated VaR and ES in % for S&P 5 index returns atlevel α =

9 Motivation 1-8 Research Questions What are the thrills for ES estimation? What are the robustness properties of ES? Which risk range is expected under dierent tail scenarios?

10 Outline 1. Motivation 2. Expected Shortfall 3. TERES Conclusions

11 Expected Shortfall 2-1 Expected Shortfall Financial returns Y pdf f (y) and cdf F (y) Here: lower tail (downside) risk Expected shortfall s η = E[Y Y < η] Basel: Value at Risk threshold η = q α = F 1 (α)

12 Expected Shortfall 2-2 Expectiles ES estimation Using expectiles, Taylor (28) Expectiles reect the tail structure Loss function ρ α,γ (u) = α I {u < } u γ (1) Expectile e α = arg min E ρ α,2 (Y θ) θ Quantile q α = arg min E ρ α,1 (Y θ) θ

13 Expected Shortfall 2-3 Loss Function u ES 6 Dynamics u Figure 4: Expectile and quantile loss functions at α =.1 (left) and α =.5 (right) LQRcheck

14 Expected Shortfall 2-4 Tail Structure Quantiles and expectiles - one-to-one mapping Goal: e w(α) = q α Find expectile level w (α) Details ES using expectiles, Taylor (28) Proof s qα = e w(α) + e w(α) E[Y ] w(α) 1 2w(α) α

15 Expected Shortfall 2-5 Expectiles and Quantiles Jones (1993), Guo and Härdle (211) Analytical formula for level w(α) Details Assumption: known return distribution F ( ) Example, N(, 1) w(α) = ϕ(q α ) q α α {ϕ(q α ) + q α α} + q α E[Y ]

16 Expected Shortfall 2-6 Sensitivity Analysis For the more general framework given in (1) Proof w(α, γ) = qα y q α γ 1 df (y) y q α γ 1 df (y), γ 1 Quantile case w(α, 1) = α, convergence in γ y q α > 1: exponential convergence towards α asγ 1 y q α < 1: root convergence

17 TERES 3-1 Heavy Tailed Returns Std Return Quantiles Standard Normal Quantiles Figure 5: S&P 5 return quantiles from , standardized using a GARCH(1,1) model TERES_Standardization

18 TERES 3-2 Quantile - Expectile Relation Properties of ES ES depends on the e α to q α distance Other component: VaR Implications of thickening the tail Expectile-quantile relation given a distribution Examples: Normal and Laplace case

19 TERES 3-3 Quantile - Expectile - Normal 3 Quantiles vs. Expectiles qα,eα qα eα α Difference Figure 6: Top: Quantile (blue) and Expectile (red), bottom: dierence α

20 TERES 3-4 Quantile - Expectile - Laplace 3 Quantiles vs. Expectiles qα,eα qα eα α Difference Figure 7: Top: Quantile (blue) and Expectile (red), bottom: dierence α

21 TERES 3-5 Quantile - Expectile - Equality Is there a distribution such that e α = q α Consider a (very heavy tailed) cdf, Koenker (1993) ( ) x, if x < F (x) = 2 ( ) x, else

22 TERES 3-6 Quantile - Expectile - Equality 3 Quantiles vs. Expectiles qα,eα qα eα α Difference Figure 8: Top: Quantile (blue) and Expectile (red), bottom: dierence α

23 TERES 3-7 TERES Flexible statistical framework - ES tail scenarios Properties of ES in an environment Risk corridor, scenario analysis Family of distributions - environment Distributional families, e.g. exponential Mixtures, e.g. two-component linear mixture

24 TERES 3-8 Example: Contaminated Normal Environment δ environment, Huber (1964) f δ (y) = (1 δ) ϕ (y) + δh (y), δ [, 1] Practice: normality assumption, ndings: heavy tails Financial markets: h( ) is symmetrical and heavy tailed Contamination degree δ

25 TERES 3-9 Example: Normal-Laplace Mixture ES 6 Figure 9: Theoretical ES qα.5 δ 1% 5% 1 1% α ES, S&P5 6 ES Dynamics.5 for dierent contamination δ and risk level α δ 1 1% TERES_ES_Analytical

26 TERES 3-1 Relation to MLE Asymmetric Generalized Error Distribution (AGED), Ayebo and Kozubowski (23) f (x) = γ κ {( σγ( 1 γ ) 1 + κ exp κγ I{x µ } 2 σγ 1 ) } κ γ I{x µ < } x µ γ σγ Γ(x) = x t 1 exp( x)dx Scale σ, skewness κ, location µ and shape γ (Asymmetric Laplace γ = 1 and normal γ = 2)

27 TERES 3-11 M-Quantiles as Location Estimate AGED Log-likelihood ( κ γ log{f (x µ, σ, γ, κ)} =c(γ, σ, κ) + I{x µ } σγ + 1 κ γ I{x µ < } σγ ) x µ γ The location MLE µ OPT is equal to the τ-m-quantile if ( ) 1 τ 2δ κ(δ) = 1 τ

28 4-1 Financial Applications Stock returns, SE 1 and S&P 5 Dierent risk levels Stock Example Intraday Margin Foreign Exchange EUR/UAH exchange rate Not relying on standardization Forex Example Portfolio Selection TEDAS (Tail Event Driven ASset allocation) Small sample size TEDAS Example

29 4-2 Data, SE 1 and S&P 5 daily returns Risk level α:.1,.5 and.1 Varying tail thickness δ Span: (269 trading days) One-year time horizon (25 trading days) - moving window Standardized returns Financial Applications

30 4-3 Returns SE 1 SP Figure 1: Standardized returns of the selected indices from Financial Applications TERES_Standardization

31 4-4 Expected Shortfall δ SE 1 S&P Table 1: Estimated ES qα for selected indices at α =.1, from (25 trading days) Financial Applications TERES_RollingWindow

32 4-5 Expected Shortfall δ SE 1 S&P Table 2: Estimated ES qα for selected indices at α =.1, from (25 trading days) Financial Applications TERES_RollingWindow

33 4-6 Setup Risk level α:.1,.5 and.1 Scenarios: Laplace and normal Empirical Study Rolling window exercise - one-year time horizon (25 days) Stock markets: German, UK, US Financial Applications

34 4-7 SE 1 SP Figure 11: ES qα and VaR at α =.1; δ = (top) and δ = 125 (bottom) 21 Financial Applications TERES_RollingWindow

35 4-8 SE 1 SP Figure 12: ES qα and VaR at α =.5; δ = (top) and δ = 125 (bottom) 21 Financial Applications TERES_RollingWindow

36 4-9 SE 1 SP Figure 13: ES qα and VaR at α =.1; δ = (top) and δ = 125 (bottom) 21 Financial Applications TERES_RollingWindow

37 4-1 Intraday Margin Example: Finance - portfolio exposure An investor enters a 1 Mio USD long position (e.g., S&P 5) on Using the last 25 standardized returns, the rescaled ES is obtained as ES qα in USD α =.5 α =.1 α =.5 δ = -1,358-16,694-19,188 δ = 1-11,88-18,319-2,821 Financial Applications TERES_RollingWindow

38 4-11 Intraday Margin Example: Finance - portfolio exposure An investor enters a 1 Mio USD long position (e.g., S&P 5) at the height of the nancial crisis (27111). Using the last 25 standardized returns, the rescaled ES is obtained as ES qα in USD α =.5 α =.1 α =.5 δ = -143, , ,738 δ = 1-162,529-23,8-21,432 Financial Applications TERES_RollingWindow

39 4-12 Exchange Rate Application EUR/UAH Returns Figure 14: Returns of the EUR/UAH exchange rate from the 2513 to Stand. Returns Financial Applications

40 4-13 EUR UAH ES corridor Figure 15: ES q.1 Normal-Laplace Corridor risk extrema Ratios (corridor). Normal δ = 1.2 and "worst case", i.e. δ = 1 scenarios using a rolling window of 25 obs. 3 Financial Applications 1.15 TERES_RollingWindow TERES - Tail Event 25Risk Expectile based 21Shortfall

41 EUR UAH ES corridor Corridor Ratios Figure 16: Ratio of the minimal and maximal risk indications in a Normal- Laplace environment using risk levels α =.1 (blue) and α =.5 (black) Financial Applications TERES_RollingWindow

42 4-15 Portfolio (TEDAS) Application.1.1 SP5 1/31/28 1/31/213 Figure 17: 73 return observations of a globally selected TEDAS portfolio (blue) versus the benchmark S&P5 index (red), Härdle et al. (214) Financial Applications TERES_RollingWindow

43 4-16 Small Sample Quantile based ES qα sqα.4 Figure 18: VaR α (black) andexpectile quantile based ES ES qα qα using a normal scenario (blue) for the globally selected TEDAS portfolio.4 sewα.8 Financial Applications α TERES_RollingWindow

44 sq Small Sample α Expectile based ES qα sewα Figure 19: Expectile based ES qα using normal (blue) and 1 Laplace contamination (red) scenario for the globally selected TEDAS 3 portfolio Financial Applications α TERES_RollingWindow

45 δ Risk Corridor Ratio of Maximal ES Variation 1.9 α Figure 2: Ratio of the lowest and highest risk indication, i.e. α maximal variation ratio, in a Normal-Laplace environment using the TEDAS sample Financial Applications TERES_RollingWindow

46 Conclusions 5-1 Conclusions (i) Understanding Expected Shortfall (ES) Expectiles are successfully used for ES estimation Distributional families, mixtures (ii) TERES ES qα for dierent risk levels α and scenarios Robustness of ES in a realistic nancial setting

47 TERES - Tail Event 25 Risk Expectile 21 based 25 Shortfall Philipp Gschöpf Wolfgang Karl Härdle Andrija Mihoci Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics HumboldtUniversität zu Berlin Figure 9: ESq α and VaR at α = rolling window of 25 observation TERES - Tail Event Risk Expected

48 References 6-1 References Ayebo, A. and Kozubowski, T. J. An asymmetric generalization of Gaussian and Laplace laws Journal of Probability and Statistical Science 1 (2), , 23 Bellini, F., Klar, B., Muller, A. and Gianin, E. R. Generalized quantiles as risk measures Insurance: Mathematics and Economics 54, 41-48, 214, ISSN: Guo M. and Härdle, W. K. Simultaneous Condence Band for Expectile Function Advances in Statistical Analysis, 211 DOI: 1.17/s

49 References 6-2 References Breckling, J. and Chambers, R. M-quantiles Biometrica 75(4): , 1988 DOI: 1.193/biomet/ Fishburn, P. C. Mean-Risk Analysis with Risk Associated with Bellow-Target Returns The American Economic Review 37(2): , 1977 Härdle, W. K., Nasekin, S., Lee, D. K. C. and Phoon, K. F. TEDAS - Tail Event Driven ASset Allocation Sonderforschungsbereich 649: Ökonomisches Risiko (SFB Papers) 214-(32)

50 References 6-3 References Huber, P. J. Robust Estimation of a Location Parameter The Annals of Mathematical Statistics 35(1): 73-11, 1964 DOI: /aoms/ Huber, P. J. and Ronchetti, E.M. Robust Statistics Second Edition, 29, ISBN: Jones, M.C. Expectiles and M-quantiles are quantiles Statistics & Probability letters 2(2): , 1993, DOI:

51 References 6-4 References Kalkbrenner, M., Lotter, H. and Overbeck, L. Sensible and ecient capital allocation for credit portfolios RISK 19-24, 214, Jan Koenker, R. When are expectiles percentiles? Economic Theory 9(3): , 1993 DOI: Newey, W. K., Powell J.L. Asymmetric Least Squares Estimation and Testing Econometrica 55(4): , 1987 DOI: 1.237/

52 References 6-5 References Taylor, J. W Estimating value at risk and expected shortfall using expectiles Journal of Financial Econometrics (6), 2, 28 Yao, Q. and Tong, H. Asymmetric least squares regression estimation: A nonparametric approach Journal of Nonparametric Statistics (6), 2-3, 1996 Yee, T. W. The VGAM Package for Categorical Data Analysis R reference manual

53 Appendix 7-1 Coherence Coherent risk measure ρ ( ) of real-valued r.v.'s which model the returns Subadditivity, ρ(x + y) ρ(x) + ρ(y) Details Translation invariance, ρ(x + c) = ρ(x) for a constant c Monotonicity, ρ(x) < ρ(y), x < y Positive homogeneity, ρ(kx) = kρ (x), k > VaR and ES

54 Appendix 7-2 Subadditivity Coherence ρ (x + y) ρ (x) + ρ (y) Diversication never increases risk Quantiles are not subadditive Expected shortfall is subadditive, Delbaen (1998) VaR and ES

55 Appendix 7-3 ES using expectiles Expectile (see also 7-4 and 7-5) First order condition e τ = arg min θ E ρ τ,2 (Y θ) ρ τ,2 (u) = τ I {u < } u 2 s (1 τ) (y s)f (y)dy τ s (y s)f (y)dy = Tail Structure

56 Appendix 7-4 ES using expectiles Extension and reformulation (1 τ) Rearranging s (y s)f (y)dy τ e τ E(Y ) = 1 2τ τ = τ eτ s (y s)f (y)dy (y s)f (y)dy (y e τ )f (y)dy Tail Structure

57 Appendix 7-5 ES using expectiles Expected shortfall e τ E[Y ] = 1 2τ E [(Y e τ ) I{Y < e τ }] τ E[Y Y < e τ ] = e τ + τ(e τ E[Y ]) (2τ 1)F (e τ ) Use e w(α) = q α E[Y Y < q α ] = e w(α) + (e w(α) E[Y ])w(α) (2w(α) 1)α Tail Structure

58 Appendix 7-6 Expectiles and Quantiles, w (α) Relation of expectiles and quantiles (proof 7-7 and 7-8) LPM ew(α) (y) e w(α) α w(α) = { } 2 LPM ew(α) (y) e w(α) α + e w(α) E[Y ] With the lower partial moment Tail Structure LPM u (y) = Expectiles and Quantiles u yf (y)dy

59 Appendix 7-7 Expectiles and Quantiles, w (α) Expectile (AND location estimate) solves {α 1} eα Rearrange { α e α 2 (y e α )f (y)dy = α (y e α )f (y)dy e } α {{} eα { =α yf (y)dy 2 Tail Structure Expectiles and Quantiles } eα e α f (y)dy + eα + α eα (y e α)f (y)dy yf (y)dy e α f (y)dy } eα + yf (y)dy

60 Appendix 7-8 Expectiles and Quantiles, w (α) Ordering terms { ( eα eα ) } α 2 yf (y)dy e α f (y)dy + e α E[Y ] eα eα = yf (y)dy e α f (y)dy Solving for the risk level, F (e w(α) ) = α LPM ew(α) (y) e w(α) α w(α) = { } 2 LPM ew(α) (y) e w(α) α + e w(α) E[Y ] Tail Structure Expectiles and Quantiles

61 Appendix 7-9 Expected Shortfall and Value at Risk Value at Risk (VaR) For a cdf F ( ) of an r.v. Y VaR α = q α = F (α) 1 Expected Shortfall (ES) Basel: VaR threshold η = q α ES η = E[Y Y < η] VaR and ES

62 Appendix 7-1 General Quantile Relation Level Solution z α to (1) α 1 α = zα y z α γ 1 df (y) z α y z α γ 1 df (y) Ordering terms α = zα y z α γ 1 df (y) y z α γ 1 df (y) Sensitivity Analysis