Expected Shortfall is not elicitable so what?

Transcription

1 Expected Shortfall is not elicitable so what? Dirk Tasche Bank of England Prudential Regulation Authority 1 dirk.tasche@gmx.net Finance & Stochastics seminar Imperial College, November 20, The opinions expressed in this presentation are those of the author and do not necessarily reflect views of the Bank of England. Dirk Tasche (PRA) ES is not elicitable so what? 1 / 23

2 Outline Background Risk measures Value-at-Risk and Expected Shortfall Elicitability Expected Shortfall and Expectiles References Dirk Tasche (PRA) ES is not elicitable so what? 2 / 23

3 Background Motive of this presentation For more than 10 years, academics have been suggesting Expected Shortfall (ES) as a coherent alternative to Value-at-Risk (VaR). Recently, the Basel Committee (BCBS, 2013) has confirmed that ES will replace VaR for regulatory capital purposes in the trading book. Gneiting (2011) points out that elicitability is a desirable property when it comes to making and evaluating point forecasts. He finds that conditional value-at-risk [ES] is not [elicitable], despite its popularity in quantitative finance. Expectiles are coherent and elicitable. That is why several authors have suggested to drop both VaR and ES and use expectiles instead. Dirk Tasche (PRA) ES is not elicitable so what? 3 / 23

4 Risk measures What risk do we measure? Rockafellar and Uryasev (2013) distinguish 4 approaches to the measurement of risk: Risk measures aggregated values of random cost. Deviation measures deviations from benchmarks or targets. Measures of regret utilities in the context of losses. They generate risk measures. Error measures quantifications of non-zeroness. They generate deviation measures. Risk measures may be understood as measures of solvency Use by creditors and regulators. Deviation measures may be interpreted as measures of uncertainty Use by investors of own funds (no leverage). Dirk Tasche (PRA) ES is not elicitable so what? 4 / 23

5 Solvency measures Risk measures There are many papers on desirable properties of risk measures. Most influential: Artzner et al. (1999) Coherent risk measures: How much capital is needed to make position 2 L acceptable to regulators? Homogeneity ( double exposure double risk ): ρ(h L) = h ρ(l), h 0. (1a) Subadditivity ( reward diversification ): ρ(l 1 + L 2 ) ρ(l 1 ) + ρ(l 2 ). (1b) Monotonicity ( higher losses imply higher risk ): L 1 L 2 ρ(l 1 ) ρ(l 2 ). (1c) Translation invariance ( reserves reduce requirements ) ρ(l a) = ρ(l) a, a R. (1d) 2 Convention: Losses are positive numbers, gains are negative. Dirk Tasche (PRA) ES is not elicitable so what? 5 / 23

6 Risk measures Important and less important properties Characterisation: A risk measure ρ is coherent if and only there is a set of probability measures Q such that ρ(l) = max Q Q E Q [L], for all L. (2) Interpretation of coherent measures as expectations in stress scenarios. Duality: ρ(l) solvency risk measure δ(l) = ρ(l) E[L] deviation measure Homogeneity and subadditivity are preserved in δ. Monotonicity and translation invariance are not preserved. Conclusion: Monotonicity and translation invariance are less important properties. Dirk Tasche (PRA) ES is not elicitable so what? 6 / 23

7 Risk measures Other important properties Comonotonic additivity ( No diversification for total dependence ): L 1 = f 1 X, L 2 = f 2 X ρ(l 1 + L 2 ) = ρ(l 1 ) + ρ(l 2 ). (3a) X common risk factor, f 1, f 2 increasing functions. Law-invariance ( context independence 3 ): P[L 1 l] = P[L 2 l], l R ρ(l 1 ) = ρ(l 2 ). (3b) Proposition: Coherent risk measures ρ that are also law-invariant and comonotonically additive are spectral measures, i.e. there is a convex distribution function F ρ on [0, 1] such that ρ(l) = 1 0 q u (L) F ρ (du), for all L. (3c) q u (L) = min{p[l l] u} denotes the u-quantile of L. 3 Identical observations in a downturn and a recovery imply the same risk. Dirk Tasche (PRA) ES is not elicitable so what? 7 / 23

8 Risk contributions Risk measures Generic one-period loss model: L = m L i. (4) i=1 L portfolio-wide loss, m number of risky positions in portfolio, L i loss with i-th position. Risk sensitivities ρ(l i L) = d ρ(l+h L i) d h h=0 are of interest for risk management and optimisation. ρ homogeneous and differentiable m ρ(l i L) = ρ(l). (5) i=1 Interpretation of sensitivities as risk contributions 4. 4 This approach to contributions is called Euler allocation. Dirk Tasche (PRA) ES is not elicitable so what? 8 / 23

9 Risk measures Some properties of risk contributions ρ(l) positively homogeneous ρ(l i L) ρ(l i ) ρ subadditive For subadditive risk measures, the risk contributions of positions do never exceed their stand-alone risks. ρ(l) positively homogeneous and subadditive ρ(l) ρ(l L i ) ρ(l i L) (6) So-called with without risk contributions underestimate the Euler contributions. ρ spectral risk measure, smooth loss distribution ρ(l i L) = 1 0 E [ L i L = q u (L) ] F ρ (du). (7) Dirk Tasche (PRA) ES is not elicitable so what? 9 / 23

10 Value-at-Risk and Expected Shortfall Shortfall probability risk measures Special case of solvency risk measures. Construction principle: For a given confidence level γ, the risk measure ρ(l) specifies a level of loss that is exceeded only with probability less than 1 γ. Formally, ρ(l) should satisfy P[L > ρ(l)] 1 γ. (8) γ is often chosen on the basis of a target rating, for example for a target A rating with long-run average default rate 5 of 0.07%: 1 γ = 0.07% Popular examples: (Scaled) standard deviation, Value-at-Risk (VaR), Expected Shortfall (ES). 5 Source: S&P (2013), table 21. Dirk Tasche (PRA) ES is not elicitable so what? 10 / 23

11 Standard deviation Value-at-Risk and Expected Shortfall Scaled standard deviation (with constant a > 0): σ a (L) = E[L] + a var[l] = E[L] + a E [ (L E[L]) 2]. (9a) By Chebychev s inequality: P[L > σ a (L)] P [ L E[L] > a var[l] ] a 2. (9b) Hence, choosing a = 1 γ (e.g. γ = 0.001) yields P[L > σ a (L)] γ. (9c) Alternative: Choose a such that (9c) holds for, e.g., normally distributed L. Underestimates risk for skewed loss distributions. Dirk Tasche (PRA) ES is not elicitable so what? 11 / 23

12 Value-at-Risk and Expected Shortfall Properties of standard deviation Homogeneous, subadditive and law-invariant Not comonotonically additive, but additive for risks with correlation 1 Not monotonic, hence not coherent Easy to estimate moderately sensitive to outliers in sample Overly expensive if calibrated (by Chebychev s inequality) to be a shortfall measure Risk contributions: σ a (L i L) = a cov(l i, L) var(l) + E[L i ]. (10) Dirk Tasche (PRA) ES is not elicitable so what? 12 / 23

13 Value-at-Risk Value-at-Risk and Expected Shortfall For α (0, 1): α-quantile q α (L) = min{l : P[L l] α}. In finance, q α (L) is called Value-at-Risk (VaR). If L has a continuous distribution (i.e. P[L = l] = 0, l R), then q α (L) is a solution of P[L l] = α. Quantile / VaR-based risk measure: VaR α (L) = q α (L). (11a) By definition VaR α (L) satisfies P [ L > VaR α (L) ] 1 α. (11b) Dirk Tasche (PRA) ES is not elicitable so what? 13 / 23

14 Value-at-Risk and Expected Shortfall Properties of Value-at-Risk Homogeneous, comonotonically additive and law-invariant Not subadditive, hence not coherent Easy to estimate by sorting sample not sensitive to extreme outliers Provides least loss in worst case scenario may be misleading. Risk contributions: VaR α (L i L) = E [ L i L = q α (L) ]. (12) Estimation of risk contributions is difficult in continuous case. Dirk Tasche (PRA) ES is not elicitable so what? 14 / 23

15 Expected Shortfall Value-at-Risk and Expected Shortfall Expected Shortfall (ES, Conditional VaR, superquantile). Spectral risk measure with F ρ (u) = (1 α) 1 max(u, α): ES α (L) = 1 1 α 1 α q u (L) du = E[L L q α (L)] (13) + ( E[L L q α (L)] q α (L) ) ( P[L q α(l)] 1 α 1 ). If P[L = q α (L)] = 0 (in particular, if L has a density), ES α (L) = E[L L q α (L)]. ES dominates VaR: ES α (L) VaR α (L). Dirk Tasche (PRA) ES is not elicitable so what? 15 / 23

16 Value-at-Risk and Expected Shortfall Properties of Expected Shortfall Coherent, comonotonically additive and law-invariant Easy to estimate by sorting. Provides average loss in worst case scenario Least coherent law-invariant risk measure that dominates VaR Risk contributions (continuous case): ES α (L i L) = E [ L i L q α (L) ]. (14) Very sensitive to extreme outliers. For same accuracy, many more observations than for VaR at same confidence level might be required. Big gap between VaR and ES indicates heavy tail loss distribution. Dirk Tasche (PRA) ES is not elicitable so what? 16 / 23

17 Elicitability Related definitions A scoring function is a function s : R R [0, ), (x, y) s(x, y), (15a) where x and y are the point forecasts and observations respectively. Let ν be a functional on a class of probability measures P on R: ν : P 2 R, P ν(p) R. A scoring function s : R R [0, ) is consistent for the functional ν relative to P if and only if E P [s(t, L)] E P [s(x, L)] (15b) for all P P, t ν(p) and x R. s is strictly consistent if it is consistent and E P [s(t, L)] = E P [s(x, L)] x ν(p). (15c) Dirk Tasche (PRA) ES is not elicitable so what? 17 / 23

18 Elicitability Elicitability The functional ν is elicitable relative to P if and only if there is a scoring function s which is strictly consistent for ν relative to P. Examples: Expectation: ν(p) = x P(dx), s(x, y) = (y x) 2. (16a) Quantiles: ν(p) = { x : P[(, x)] α P[(, x]] }, Interpretation: s(x, y) = α 1 α max(y x, 0) + max(x y, 0). (16b) Point estimates of elicitable functionals can be determined by means of regression: ν(p) = arg min x E P [s(x, Y )], Y P. (16c) Point estimation methods of elicitable functionals can be compared by means of the related scoring functions (interesting for backtesting). Dirk Tasche (PRA) ES is not elicitable so what? 18 / 23

19 Elicitability Standard deviation and ES are not elicitable Necessary for ν being elicitable ( convex level sets ): 0 < π < 1, t ν(p 1 ) ν(p 2 ) t ν(π P 1 + (1 π) P 2 ) (17a) By counter-examples: Standard deviation and ES violate (17a). Standard deviation and ES are not elicitable. But standard deviation and ES can be calculated by means of regression, with s as in (16a) and (16b): [ var(p) = min E ] P (Y x) 2 (17b) x { [ ES α (P) = min E α P x 1 α max(y x, 0) (17c) + max(x Y, 0) ] } + E P [Y ]. Dirk Tasche (PRA) ES is not elicitable so what? 19 / 23

20 Expected Shortfall and Expectiles Expectiles For 0 < τ < 1 the τ-expectile of square-integrable Y is defined by e τ (Y ) = arg min E [ τ max(y x, 0) 2 +(1 τ) max(x Y, 0) 2] (18a) x e τ is elicitable with scoring function s(x, y) = τ max(y x, 0) 2 + (1 τ) max(x y, 0) 2. (18b) e τ (Y ) is the unique solution of τ E[max(Y x, 0)] = (1 τ) E[max(x Y, 0)] (18c) e τ is law-invariant and coherent for τ 1/2 (Bellini et al., 2013). Dirk Tasche (PRA) ES is not elicitable so what? 20 / 23

21 Expected Shortfall and Expectiles Properties of expectiles e 1/2 [Y ] = E[Y ]. e τ is sensitive to extreme outliers. corr[y 1, Y 2 ] = 1 e τ (Y 1 + Y 2 ) = e τ (Y 1 ) + e τ (Y 2 ) But e τ is not comonotonically additive for τ > 1/2. If e τ were comonotonically additive then it would be a spectral measure. By Corollary 4.3 of Ziegel (2013) the only elicitable spectral measure is the expectation. Hence τ = 1/2 contradiction! Hence, for non-linear dependence expectiles may see diversification where there is none. Risk contributions (conceptually easy to estimate): e τ (L i L) = τ E[L i 1 {L eτ (L)}] + (1 τ) E[L i 1 {L<eτ (L)}]. (19) τ P[L e τ (L)] + (1 τ) P[L < e τ (L)] Dirk Tasche (PRA) ES is not elicitable so what? 21 / 23

22 Expected Shortfall and Expectiles Comparison Expectiles: Coherent, law-invariant and elicitable. No obvious interpretation in terms of solvency. May see diversification where there is none. Expected Shortfall: Coherent, law-invariant and comonotonically additive. Clearly related to solvency probability (via confidence level). Not elicitable but composition of elicitable conditional expectation and quantile. From (13): ES γ (L) 1/4 (q γ (L) + q 0.75 γ+0.25 (L) + q 0.5 γ+0.5 (L) + q 0.25 γ+0.75 (L)) (20) Hence backtest q γ (L), q 0.75 γ+0.25 (L), q 0.5 γ+0.5 (L), and q 0.25 γ+0.75 (L) to backtest ES. Dirk Tasche (PRA) ES is not elicitable so what? 22 / 23

23 References P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9(3): , BCBS. Fundamental review of the trading book: A revised market risk framework. Basel Committee on Banking Supervision, October F. Bellini, B. Klar, A. Müller, and E. Rosazza Gianin. Generalized quantiles as risk measures. Preprint, URL T. Gneiting. Making and evaluating point forecasts. Journal of the American Statistical Association, 106(494): , R.T. Rockafellar and S. Uryasev. The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surveys in Operations Research and Management Science, 18(1):33 53, S&P. Default, Transition, and Recovery: 2012 Annual Global Corporate Default Study And Rating Transitions. Report, Standard & Poor s, March J. F. Ziegel. Coherence and elicitability. ArXiv e-prints, March Dirk Tasche (PRA) ES is not elicitable so what? 23 / 23