Risk measures: Yet another search of a holy grail Dirk Tasche Financial Services Authority 1 dirk.tasche@gmx.net Mathematics of Financial Risk Management Isaac Newton Institute for Mathematical Sciences March 28, 2013 1 The opinions expressed in this presentation are those of the author and do not necessarily reflect views of the Financial Services Authority. Dirk Tasche (FSA) Risk measures 1 / 35
Outline Background Useful properties of risk measures Examples of risk measures Identification of risk concentrations Conclusions References Dirk Tasche (FSA) Risk measures 2 / 35
Background Outline Background Useful properties of risk measures Examples of risk measures Identification of risk concentrations Conclusions References Dirk Tasche (FSA) Risk measures 3 / 35
Types of financial risk Background Credit risk: Borrowers fail to pay due money. Spreads for bonds or other traded securities increase. Market risk: Adverse price changes of traded securities (shares, bonds, currencies, derivatives) and commodities. Liquidity risk: Traded securities or commodities cannot be sold or only be sold at a discount. A financial institution does not obtain sufficient funding for its current obligations. Operational risk: Fire, natural catastrophes, IT incidents, fraud, litigation and others. Business risk: Margins of sold products are smaller than expected. Dirk Tasche (FSA) Risk measures 4 / 35
Background Types of investors Equity only investors: Private investors You? Leveraged investors: Banks (10% equity, 90% liabilities) Pension funds (0% equity, 100% liabilities) Hedge funds (?% equity,?% liabilities) Insurance companies (stochastic liabilities) Many others Equity only investors are free in their choice of a risk measure (or not to measure risk). Leveraged investors are forced by their creditors (sometimes represented by a regulator) to measure risk in a transparent way. Dirk Tasche (FSA) Risk measures 5 / 35
Background The story of risk measures Ruin theory: [Cramér(1930)] Risk-return optimisation: [Markowitz(1952)] Value-at-Risk: [RiskMetrics(1996)] Coherence: [Artzner et al.(1999)] Expected Shortfall: [Acerbi and Tasche(2002), Rockafellar and Uryasev(2002)] Risk contributions: [Litterman(1996), Tasche(1999)] Elicitability: [Gneiting(2011)] Expectiles: [Bellini et al.(2013), Ziegel(2013)] Dirk Tasche (FSA) Risk measures 6 / 35
Useful properties of risk measures Outline Background Useful properties of risk measures Examples of risk measures Identification of risk concentrations Conclusions References Dirk Tasche (FSA) Risk measures 7 / 35
Useful properties of risk measures Economic requirements for risk measures Adapted to the type of investor: Measures of volatility Measures related to potential (monetary) losses. Assign a value to unexpected loss beyond the expected loss. Adapted to the type of risk: Focused on portfolio-wide risk (capture tail risk?) Facilitate portfolio-management (risk contributions) Facilitate scenario-analysis (worst case dependence) Adapted to the data available: Frequently observed long time series (market risk) Low frequency, high impact losses (credit and operational risk) Here we focus on the measurement of tail risks for leveraged investors. Dirk Tasche (FSA) Risk measures 8 / 35
General setting Useful properties of risk measures Generic one-period loss model: L = m L i. (1) i=1 L portfolio-wide loss, m number of risky positions 2 in portfolio, L i loss with i-th position. Losses are positive numbers, gains are negative numbers. Potential future losses L and L 1,..., L m are random variables: Realizations of L and L1,..., L m cannot be predicted. Only probabilities of occurrence and distributions of loss severities can be estimated. 2 Or sub-portfolios or business units etc. Dirk Tasche (FSA) Risk measures 9 / 35
Useful properties of risk measures Capital and risk measures Economic capital: Capital reserve intended to cover all but truly catastrophic losses. Expected loss E[L] is assumed to be covered by normal cash-flow. Economic capital demand is amount ρ(l) such that the probability P[L E[L] > ρ(l)] of a loss shortfall is small. ρ(l) is then called Unexpected Loss of the portfolio with loss variable L. ρ is called risk measure. Numbers ρ(l 1 L),..., ρ(l m L) are called risk contributions if m ρ(l i L) = ρ(l). (2) i=1 Dirk Tasche (FSA) Risk measures 10 / 35
Useful properties of risk measures Mathematical properties of risk measures Homogeneity ( double exposure double risk ): ρ(h L) = h ρ(l), h 0. (3a) Subadditivity ( reward diversification ): ρ(l 1 + L 2 ) ρ(l 1 ) + ρ(l 2 ). (3b) Comonotonous additivity ( concurrent risks are worst ): L 1 = f 1 X, L 2 = f 2 X ρ(l 1 + L 2 ) = ρ(l 1 ) + ρ(l 2 ). (3c) X common risk factor, f 1, f 2 increasing functions. Monotonicity ( higher losses imply higher risk ): L 1 L 2 ρ(l 1 ) ρ(l 2 ). (3d) Dirk Tasche (FSA) Risk measures 11 / 35
Useful properties of risk measures Statistical properties of risk measures Law-invariance ( risk observable from past observations only ): Pr(L 1 l) = Pr(L 2 l), l R ρ(l 1 ) = ρ(l 2 ). (4a) Elicitability ( accuracy of estimators can be compared ): There is a score function s 0 such that E [ s(l ρ(l)) ] E [ s(l l) ], l R. (4b) Alternative interpretation: ρ(l) can be estimated by means of regression. Sensitivities d ρ(l+h L i ) d h h=0 exist and can be robustly estimated from observations. ρ homogeneous (2) holds with ρ(l i L) = d ρ(l+h L i ) d h h=0. Dirk Tasche (FSA) Risk measures 12 / 35
Examples of risk measures Outline Background Useful properties of risk measures Examples of risk measures Identification of risk concentrations Conclusions References Dirk Tasche (FSA) Risk measures 13 / 35
Standard deviation Examples of risk measures Scaled standard deviation (with constant a > 0): σ a (L) = a var[l] = a E [ (L E[L]) 2]. (5a) By Chebychev s inequality: P[L E[L] > σ a (L)] P [ L E[L] > a var[l] ] a 2. (5b) Hence, choosing a = 1 γ (e.g. γ = 0.001) yields P[L E[L] > σ a (L)] γ. (5c) Very conservative approach! Alternative: Choose a such that (5c) holds for, e.g., normally distributed L. Underestimates risk for skewed loss distributions! Dirk Tasche (FSA) Risk measures 14 / 35
Examples of risk measures Properties of standard deviation Homogeneous, subadditive and law-invariant Not comonotonously additive, but additive for risks with correlation 1 Not monotonous Overly expensive if calibrated (by Chebychev s inequality) to be a shortfall measure Risk contributions (not sensitive to tail risks): σ a (L i L) = a cov(l i, L) var(l). (6) Not directly elicitable but composed of elicitable components E[L 2 ] and E[L]. Dirk Tasche (FSA) Risk measures 15 / 35
Examples of risk measures Value-at-Risk 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: ρ VaR,α (L) = q α (L) E[L]. (7a) By definition ρ VaR,α (L) satisfies P [ L > ρ VaR,α (L) + E[L] ] 1 α. (7b) Dirk Tasche (FSA) Risk measures 16 / 35
Examples of risk measures Properties of Value-at-Risk Homogeneous, comonotonously additive and law-invariant Not subadditive Elicitable Provides least loss in worst case scenario may be misleading. Risk contributions (sensitive to tail risks): ρ VaR,α (L i L) = E [ L i L = q α (L) ] E[L i ]. (8) Estimation of risk contributions is difficult. Dirk Tasche (FSA) Risk measures 17 / 35
Expected Shortfall Examples of risk measures Expected Shortfall (ES, Conditional VaR, superquantile): ES α (L) = 1 1 α 1 α q u (L) du = E[L L q α (L)] (9a) + ( 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-based risk measure: ES α (L) = E[L L q α (L)]. ρ ES,α (L) = ES α (L) E[L]. (9b) ES dominates VaR: ρ ES,α (L) ρ VaR,α (L). Dirk Tasche (FSA) Risk measures 18 / 35
Examples of risk measures Properties of Expected Shortfall Homogeneous, subadditive, comonotonously additive and law-invariant Provides average loss in worst case scenario Least coherent law-invariant risk measure that dominates VaR Risk contributions (continuous case): ρ ES,α (L i L) = E [ L i L q α (L) ] E[L i ]. (10) Not directly elicitable but composed of elicitable components l = q α (L), E[L L l], Pr[L < l]. 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 (FSA) Risk measures 19 / 35
Expectiles Examples of risk measures Similar to quantiles and Expected Shortfall but both subadditive and elicitable. For 0 < τ < 1 and square-integrable L the τ-expectile e τ (L) is defined as e τ (L) = arg min l R E[ τ max(l l, 0) 2 +(1 τ) max(l L, 0) 2]. (11a) e τ (L) is the unique solution l of the equation τ E[max(L l, 0)] = (1 τ) E[max(l L, 0)]. e τ (L) satisfies the equation (equivalent to (11b)) e τ (L) = τ E[L 1 {L e τ (L)}] + (1 τ) E[L 1 {L<eτ (L)}] τ Pr[L e τ (L)] + (1 τ) Pr[L < e τ (L)]. Expectile-based risk measure: ρ e,τ (L) = e τ (L) E[L]. (11b) (11c) (11d) Dirk Tasche (FSA) Risk measures 20 / 35
Examples of risk measures Properties of expectiles Homogeneous and law-invariant Subadditive for 1/2 τ < 1, superadditive for 1/2 τ > 0 Elicitable Not comonotonously additive for 1/2 < τ < 1: If e τ were comonotonously additive then by Theorem 3.6 of [Tasche(2002)] it would be a spectral measure. By Corollary 4.3 of [Ziegel(2013)] spectral measures are not elicitable. Not clear how to best choose parameter τ. 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)}] τ Pr[L e τ (L)] + (1 τ) Pr[L < e τ (L)]. (12) Dirk Tasche (FSA) Risk measures 21 / 35
Examples of risk measures Quantiles, expectiles, and superquantiles of N(0,1) Standard normal distribution 3 2 1 0 1 2 3 Quantiles Superquantiles Expectiles 0.0 0.2 0.4 0.6 0.8 1.0 Parameter Dirk Tasche (FSA) Risk measures 22 / 35
Identification of risk concentrations Outline Background Useful properties of risk measures Examples of risk measures Identification of risk concentrations Conclusions References Dirk Tasche (FSA) Risk measures 23 / 35
Identification of risk concentrations Portfolio-level diversification index Idea: Compare economic capital demand (EC) for portfolio to EC demand for worst case portfolio. Worst case by assuming co-monotonicity for all loss variables. L portfolio loss, L i loss with i-th position, i.e. L = m i=1 L i. EC demand given by measure ρ of Unexpected Loss, e.g. ρ(l) = VaR α (L) E[L] ρ(l) = ES α (L) E[L] ρ(l) = e τ (L) E[L] or or Diversification index for portfolio: DI ρ (L) = ρ(l) m i=1 ρ(l i) (13) Dirk Tasche (FSA) Risk measures 24 / 35
Identification of risk concentrations Find drivers of risk concentrations ρ sub-additive DI ρ (L) 1 ρ additive for comonotonic loss variables DI ρ (L) = 1 for worst case scenario. DI ρ (L) close to one Portfolio not well diversified Need to define indices DI ρ (L i L) at position / sub-portfolio / business unit level in order to identify drivers of risk concentrations. Desirable: DI ρ (L i L) > DI ρ (L) Reducing i-th position improves portfolio. Dirk Tasche (FSA) Risk measures 25 / 35
Identification of risk concentrations Subportfolio-level diversification index ρ positively homogeneous ρ(h L) = h ρ(l), h > 0 Recall Euler-decomposition (2) of EC demand: ρ(l) positively homogeneous ρ(l) = m ρ(l i L), i=1 ρ(l i L) = d dh ρ(h L i + L) h=0 See (8), (10), and (12) for examples of risk contributions. Diversification index for single position (or sub-portfolio): DI ρ (L i L) = ρ(l i L) ρ(l i ) (14) Dirk Tasche (FSA) Risk measures 26 / 35
Identification of risk concentrations Properties of the diversification index ρ(l) positively homogeneous ρ(l i L) ρ(l i ) ρ sub-additive ρ sub-additive DI ρ (L i L) 1 Portfolio best diversified (i.e. DI ρ (L) minimal) DI ρ (L i L) = DI ρ (L) for all i (15) It can be shown [Tasche(1999)] that property (15) uniquely determines DI ρ (L i L). max i=1,...,m DI ρ (L i L) DI ρ (L) is indicator of lack of diversification. Dirk Tasche (FSA) Risk measures 27 / 35
Identification of risk concentrations Diversification indices in action Diversification index 0.6 0.7 0.8 0.9 1.0 Diversification indices: correlated systematic factors Portfolio 1st sector 2nd sector 0.0 0.2 0.4 0.6 0.8 1.0 Exposure to 1st sector as fraction of total exposure Dirk Tasche (FSA) Risk measures 28 / 35
Identification of risk concentrations Diversification and optimization Minimizing the portfolio diversification index (13) can be interpreted as a kind of robust performance optimization. Stand-alone Unexpected Loss ρ(l i ) of i-th position should roughly be proportionate to margin / risk premium r i. 1 Hence, DI ρ(l) can be regarded as approximation to portfolio RORAC (return on risk-adjusted capital). Conclusion: Minimizing DI ρ (L) yields an optimum in diversification and a nearly RORAC-optimum at the same time. Dirk Tasche (FSA) Risk measures 29 / 35
Conclusions Outline Background Useful properties of risk measures Examples of risk measures Identification of risk concentrations Conclusions References Dirk Tasche (FSA) Risk measures 30 / 35
Conclusions Some answers, more questions There is no universally most appropriate financial risk measure. Users have to make choices of the properties of risk measures that are most important for their purposes. All risk measures discussed have strengths and weaknesses: For instance, VaR is not subadditive but elicitable and comonotonously additive. ES is subadditive and comonotonously additive but not elicitable. Expectiles are subadditive and elicitable but not comonotonously additive. Not discussed: Trade-off between sensitivity to tail risks and robustness of the estimator [Cont et al.(2010)]. Questions (for instance): What are the worst case scenarios for expectiles in the subadditivity inequality? How to best choose the parameter τ for expectiles? Dirk Tasche (FSA) Risk measures 31 / 35
References Outline Background Useful properties of risk measures Examples of risk measures Identification of risk concentrations Conclusions References Dirk Tasche (FSA) Risk measures 32 / 35
References C. Acerbi and D. Tasche. On the coherence of expected shortfall. Journal of Banking & Finance, 26(7):1487 1503, 2002. P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9(3):203 228, 1999. F. Bellini, B. Klar, A. Müller, and E. Rosazza Gianin. Generalized quantiles as risk measures. Preprint, 2013. URL http://ssrn.com/abstract=2225751. R. Cont, R. Deguest, and G. Scandolo. Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance, 10(6):593 606, 2010. H. Cramér. On the Mathematical Theory of Risk. Skandia Jubilee Volume, Stockholm, 1930. Dirk Tasche (FSA) Risk measures 33 / 35
References T. Gneiting. Making and evaluating point forecasts. Journal of the American Statistical Association, 106(494):746 762, 2011. R. Litterman. Hot spots TM and hedges. The Journal of Portfolio Management, 22:52 75, 1996. H. Markowitz. Portfolio Selection. The Journal of Finance, 7(1):77 91, 1952. RiskMetrics. RiskMetrics TM Technical Document (4th ed.). J. P. Morgan/Reuters, December 1996. R.T. Rockafellar and S. Uryasev. Conditional Value-at-Risk for general loss distributions. Journal of Banking & Finance, 26(7):1443 1471, 2002. Dirk Tasche (FSA) Risk measures 34 / 35
References D. Tasche. Risk contributions and performance measurement. Working paper, Technische Universität München, 1999. D. Tasche. Expected Shortfall and Beyond. Journal of Banking and Finance, 26(7):1519 1533, 2002. J. F. Ziegel. Coherence and elicitability. ArXiv e-prints, March 2013. Dirk Tasche (FSA) Risk measures 35 / 35