The Fundamental Review of the Trading Book: from VaR to ES

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The Fundamental Review of the Trading Book: from VaR to ES Chiara Benazzoli Simon Rabanser Francesco Cordoni Marcus Cordi Gennaro Cibelli University of Verona Ph. D. Modelling Week Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 1 / 28

Table of contents 1 Introduction 2 Value at Risk and Expected Shortfall 3 Sensitivity Analysis 4 Back-Testing Procedures Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 2 / 28

Introduction The Context After the nancial crisis in 2008-2010, the Basel Committee tried to sort the unsolved issues in the Basel II. One of the main changes concerned the general market risk requirement, i.e. how we have to measure the unexpected loss of our portfolio. During the past decade, Value-at-Risk (commonly known as VaR) has become one of the most popular risk measurement techniques in nance. The Basel III Committee establishes that in risk measurement VaR has to be replaced by the Expected Shortfall (ES). VaR and ES are used to determine the capital charge of a bank, that is the amount of money which the bank must save to cover unexpected losses. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 3 / 28

Introduction The Context After the nancial crisis in 2008-2010, the Basel Committee tried to sort the unsolved issues in the Basel II. One of the main changes concerned the general market risk requirement, i.e. how we have to measure the unexpected loss of our portfolio. During the past decade, Value-at-Risk (commonly known as VaR) has become one of the most popular risk measurement techniques in nance. The Basel III Committee establishes that in risk measurement VaR has to be replaced by the Expected Shortfall (ES). VaR and ES are used to determine the capital charge of a bank, that is the amount of money which the bank must save to cover unexpected losses. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 3 / 28

Introduction Prices, Returns and Portfolio Generally, we consider a time horizon t [0, M] and a portfolio w = (w 1,..., w N ) made of N assets, the weights w i stand for how much money we invest in the asset i. Denoting by S t,i the market price of the asset i at time t the portfolio value at time t is V t = i w i S t,i and the portfolio Prot-Loss in time [0, M] is PL = V M V 0 = i w i (S t,i S 0,i ). If we denote by R t,i = log(s t,i /S t 1,i ) we can also rewrite the portfolio Prot-Loss as follows PL t = V 0 θ i R t,i, θ i = w is 0,i. V 0 i Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 4 / 28

Introduction Risk Measures The Value at Risk of order α of the portfolio w, for the time horizon M, is dened as VaR α (PL) where VaR α (PL) = q α (PL) PL is a random variable where q α (PL) is the quantile of order α of the probability function F (x) of PL. The Expected Shortfall of order α of the portfolio w, for the time horizon M, is dened as ES α (PL) where ES α (PL) = E[PL PL VaR α ] = 1 α α 0 VaR u (PL)du Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 5 / 28

Introduction Risk Measures Remark (Square Root Rule) We point out that the time window refers to daily data, then the VaR is daily. If we want to compute VaR T for a dierent time horizon T we use the approximation VaR T = VaR T. Such approximation holds exactly as equality when the price process of our Portfolio is driven by a Geometric Brownian Motion such as in the Black-Scholes dynamic. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 6 / 28

Introduction The Approaches VaR and ES computation were performed by using four dierent approaches. We now list the dierent techniques. Parametric models: - -Normal approach; - Exponential Weighted Moving Average (EWMA) risk metrics, Non-parametric models: - Historical Simulation; - Weighted Historical Simulation. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 7 / 28

Introduction -Normal approach Within the -Normal Approach we assume that the vector of the returns R has a multivariate normal distribution, then the vector PL is normally distributed with mean zero and variance σ 2 PL = w T Σ w where Σ ij = Cov(R i, R j ). In this model it is simple to compute the VaR and the ES of our portfolio VaR = σ PL q α (Z) Z N (0; 1); ϕ(q α ) ES = σ PL ϕ gaussian density. α Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 8 / 28

Introduction EWMA Risk Metrics The EWMA approach makes use of the same formulas of the -Normal Approach for the VaR and ES. It requires to build a weighted Covariance Matrix Σ = ( σ ij ) of the returns according to the following denition σ ij = 1 λ M 1 λ M λ m 1 R M m,i R M m,j and replace Σ with Σ. Remark m=1 The EWMA model is similar to GARCH: we can estimate the variance by using a regressive method σ 2 k+1 = λσ2 k + (1 λ)r 2 k, σ 0 xed where R k denotes the Prot and Loss of our Portfolio at time k. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 9 / 28

Introduction Historical Simulation Historical simulation is a widely used method because we do not have to specify any probability model; we do not have to estimate any parameter. We create the order statistics of PL (from the lowest to the highest) PLs = (PL (1),..., PL (N) ) and we assume that the theoretical distribution is exactly the empirical one. Under this Hypothesis it is very simple to compute the VaR α. Denoting by β = [α N] (the integer part of α N), we have VaR α = PLs β Remark The Historical Simulation Method has a high variance estimator. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 10 / 28

Introduction Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 11 / 28

Introduction Weighted Historical Simulation We x some decreasing weights p m, with 1 < m < N, meaning that p m (0, 1), m p m = 1 and p m+1 < p m. For example p m = 1 λ 1 λ M λm 1, λ (0, 1) and we sort the weight p m in accordance with the new position of the element PL m in PLs, we create a vector p sort = (p (1),..., p (N) ) and nally the vector PLw = (p (1) PL (1),..., p (N) PL (N) ). For a xed α we determine m as follows we set VaR α = PLw m. PLw i α < m i=1 m +1 i=1 PLw i Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 12 / 28

Value at Risk and Expected Shortfall Our Portfolio We created a panel including 38 assets, of dierent nature and came from dierent geographical areas. We downloaded the time series from Yahoo!Finance. The observed window data starts from 6/9/2014 up to 6/9/2016. Here the panel composition: index: NYA, DAX, IPSA, etc... equities: Apple, Ford, Bank of America, ENI, Telecom, etc... We construct a portfolio w = (w 1,..., w 38 ) by picking w i = 1 38. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 13 / 28

Value at Risk and Expected Shortfall Cleaning and Matching of the data When we compose a diversied portfolio including for instance index, stocks, bonds, equities, founds and we merge the data in order to made up the matrix of the prices, we can nd missing values (NA). For example in our case we have 624 NA over 20710 values. We adopt the following criterium As concern the data from 6/9/2014 to 6/8/2016 we split the procedure into two cases - If there are more than 15% of NA we erase the row; - If there are less then 15% of NA we interpolate the missing data S t,i S t,i = S t 1,i + S t+1,i (1 + σ i N (0, 1)) 2 where σ i is the variance of the returns of the asset i. As concerns the period from 6/8/2016 to 6/9/2016 we always interpolate the not available data. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 14 / 28

Value at Risk and Expected Shortfall Portfolio Risk Computation We create the matrix of the log-return R and we compute VaR and ES by using the previous 4 methods with α = 0.01 -Normal EMWA (λ = 0.99) EMWA (λ = 0.94) VaR 3.59% 3.74% 2.47% Historical W-Historical (λ = 0.99) W-Historical (λ = 0.94) VaR 4.27% 4.52% 3.19% -Normal EMWA (λ = 0.99) EMWA (λ = 0.94) ES 4.12% 4.29% 2.83% Historical W-Historical (λ = 0.99) W-Historical (λ = 0.94) ES 4.81% 5.10% 3.93% Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 15 / 28

Value at Risk and Expected Shortfall Histogram of returns σ G = 1, 5% σ RM = 1, 6% We performed the Jarque-Bera test which said that the PL distribution is Normal. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 16 / 28

Value at Risk and Expected Shortfall Historical Vs W-Historical Figure: Empirical cumulative function (Historical) Figure: Empirical cumulative function (W-Historical) Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 17 / 28

Sensitivity Analysis Perturbation of Weights in Portfolio This analysis help us to understand which instruments are more/less risky. We use the following notations w = (w 1, w 2,..., w N ), and we start by picking uniform weights: i - index of instrument, r - factor change, w = (w 1,...,w i r,...,w N ) (w 1,...,w i r,...,w N ), w 1 = w 2 =... = w N, Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 18 / 28

Sensitivity Analysis Perturbation analysis of VaR applied to a portfolio Change in VaR 0.004 0.003 0.002 0.001 0.000 0.001 0.002 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Index of instrument Figure: Perturbation Analysis Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 19 / 28

Sensitivity Analysis Wiener Process Approximation in the VaR Estimation over Time Horizon h VaR(α, h) h - time horizon (h = 1), Wiener process approximation (Brownian motion), h 1 VaR(α, h) VaR(α, 1) h, Alternative: divide data into h long intervals and estimate VaR(α, h). Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 20 / 28

Sensitivity Analysis Investigation of the accuracy of the Wiener approximation abs(var) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Wiener approximation Periodisation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 h The gure conrm the small deviations of empiric data from the square root rule, then we can conclude that the price process of our Portfolio can be modeled by a Geometric Brownian Motion. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 21 / 28

Back-Testing Procedures Backtesting of VaR In order to use a quantitative method for the risk measurement, the banks must satisfy some backtesting requirements, that means the model must show to be predictive when it is used in the day-by-day process. Consider the event that the loss on a portfolio at time t + 1 exceeds its reported VaR, VaR t (α) { 1 if R t+1 VaR t (α) I t+1 (α) = e.g. (0, 0, 1, 0,..., 0, 1) 0 if R t+1 > VaR t (α) For being an accurate risk measure Christoerson(1998) stated two properties for the hit sequence I t : Unconditional Coverage Property Independence Property This results in the assumption I t i.i.d B(α). Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 22 / 28

Back-Testing Procedures Kupiec Test The Kupiec test focuses on the unconditional property. It concerns whether VaR is violated more than α 100% of the time. Using a sample of T observations we dene ˆα = 1 T T I t (α) t=1 Null Hypothesis H 0 : ˆα = α Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 23 / 28

Back-Testing Procedures Figure: Value at Risk Number of Failures: 7, 4, 2, 5. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 24 / 28

Back-Testing Procedures Figure: Expected Shortfall Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 25 / 28

Back-Testing Procedures Towards the ES Back-Test The discovery in 2011 that the ES is not elicitable as emphasized by Gneiting Marking and Evaluating Point Forecasts, diused the erroneous belief that it could not be Back-Tested. It is a fact that the absence of a convincing back-test has long been the last obstacle for ES on its way to Basel. The migration from VaR to ES was criticized. Only in 2014 Acerbi and Szakely found three dierent tests to beck-test the ES. We refer to their paper Backtesting Expected Shortfall for further details. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 26 / 28

Back-Testing Procedures Towards the ES Back-Test Acerbi-Szakely Statistics where Z = 1 N 1 M t=1 1 t R t ES t + 1 1 t is the indicator function of the event {R t < VaR t } N 1 = t 1 t The estimator is itself a Random Variable and its distribution is unknown. We perform the test by using a Monte Carlo approximation of the distribution under the H 0 hypothesis. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 27 / 28

Back-Testing Procedures Towards the ES Back-Test The Back-Testing shows good gures as the observed is strictly below of the blue rejection level, but the key point is that we achieved an eective ES backtestability. Finance Group (UniVr) The FRTB: from VaR to ES September 10, 2016 28 / 28

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