Evaluating Value-at-Risk models via Quantile Regression

Transcription

1 Evaluating Value-at-Risk models via Quantile Regression Luiz Renato Lima (University of Tennessee, Knoxville) Wagner Gaglianone, Oliver Linton, Daniel Smith. NASM /31/2009

2 Motivation Recent nancial disasters have emphasized the need for accurate risk measures for nancial institutions; Since 1998, regulatory guidelines have required banks to set aside capital to insure against extreme portfolio losses the size of the set-aside capital requirement is directly related to a measure of portfolio risk. Currently, portfolio risk is measured in terms of its "value-at-risk"

3 Value-at-Risk (VaR): a statistical risk measure of potential losses; Pr (R t < VaR t j F t 1 ) = τ (1) F (VaR t ) = τ (2) VaR t = F 1 (τ ) = Q Rt (τ j F t 1 ) Under the internal model approach of the Basle Accord on banking, nancial institutions have the freedom to specify their own model to compute value at risk!

4 Value-at-Risk models Berkowitz & O Brien (2002)

5 Existing Backtests De ne a "violation" by the hit sequence: H t = 1 ; if Rt < VaR t 0 ; otherwise Berkowitz et al (2009) extended and uni ed the existing tests by noting that the de-meaned violations Hit t = H t τ form a martingale di erence sequence (m.d.s.) with respect to F t 1. This implies that Hit t = H t τ is uncorrelated at all leads and lags. In other words, for any vector X t in F t 1, E [(H t τ ) X t ] = 0

6 Backtests Notice that E [(H t estimation τ ) X t ] = 0 is the basis of GMM

7 Notice that E [(H t estimation τ ) X t ] = 0 is the basis of GMM This framework o ers a natural way to construct a Lagrange Multiplier (LM) test.

8 Notice that E [(H t estimation τ ) X t ] = 0 is the basis of GMM This framework o ers a natural way to construct a Lagrange Multiplier (LM) test. Kupiec (1995) sets X t = 1 and test H o : p = E (H t ) = τ.

9 Notice that E [(H t estimation τ ) X t ] = 0 is the basis of GMM This framework o ers a natural way to construct a Lagrange Multiplier (LM) test. Kupiec (1995) sets X t = 1 and test H o : E (H t ) = τ. Christo ersen (1998) considers X t = [1, H t 1 ] 0 and Engle and Manganelli (2004) set X t = [Hit t 1, Hit t 2,..., VaR t, VaR t 1...]

10 Some Comments The existing Lagrange Multiplier (LM) test has low power in nite samples Berkowitz et al (2009) show that the Dynamic Quantile (DQ) test of Engle and Manganelli (2004) with X t = [1, VaR t ] 0 has the highest power against a variety of misspeci ed models No guidance to why a given VaR is misspeci ed

11 Main Results We show that Quantile regression o ers a natural setup for the development of a Wald type of test. This Wald test has more power than the DQ test test in nite samples Monte Carlo simulations and an empirical exercise corroborate our ndings our framework is useful to show why a given VaR model is misspeci ed

12 Model Random coe cient model representation for the time series R t (Koenker & Xiao, 2002) R t = α 0 (U t ) + α 1 (U t )VaR t (3) (4) = x 0 t β(u t ) (5) where U t iid U(0, 1), α i (U t ), i = 0, 1 are assumed to be comonotonic in U t and β(u t ) = [α 0 (U t ); α 1 (U t )] 0 and xt 0 = [1, VaR t ].

13 On Comonotonicity De nition: Two random variables X, Y are comonotonic if there exists a third random variable Z and increasing functions f and g such that X = f(z) and Y = g(z). X and Y are driven by the same random (uniform) variable From our point of view the crucial property of comonotonic random variables is the behavior of quantile functions of their sums, X, Y comonotonic implies: Q X +Y (τ) = Q X (τ) + Q Y (τ).

14 A Wald-type test to evaluate VaRs Proposition 1 Given the random coe cient model (3) and the comonotonicity assumption of α i (U t ), i = 0, 1, the τth conditional quantile of R t can be written as Q Rt (τ j F t 1 ) = α 0 (τ) + α 1 (τ) VaR t ; for all τ 2 (0, 1). What do we really want to test? VaR t = Q Rt (τ j F t 1 )

15 Mincer-Zarnowitz (quantile) regression: Q Rt (τ j F t 1 ) = α 0 (τ ) + α 1 (τ ) VaR t (2) Ho : α0 (τ ) = 0 α 1 (τ ) = 1 or Ho : θ(τ ) = 0, where θ(τ ) = [α 0 (τ ); (α 1 (τ ) 1)] 0 θ(τ ) can be consistently estimated by using Koenker and Bassett (1978)

16 A Wald-type test to evaluate VaRs CLT of Koenker (2005): Assumption 1: Let x t = (1, VaR t ) 0 be measurable with respect to F t 1 and z t fr t ; x t g be a strictly stationary process; Assumption 2: (Density) Let fr t g have distribution functions F t, with continuous Lebesgue densities f t uniformly bounded away from 0 and at the points Q Rt (τ j x t ) = FR 1 t (τ j x t ); Assumption 3: (Design) There exist positive de nite matrices J and 1 T! T H τ, such that J = lim 1 T! T H τ = lim T x t xt 0 and t=1 T x t xt[f 0 t (Q Rt (τ j x t ))]; t=1 Assumption 4: max t=1,...,t kx t k / p T! 0.

17 A Wald-type test to evaluate VaRs p T ( bθ(τ ) θ(τ )) d! N(0, τ (1 τ )H 1 τ JH 1 τ ) = N(0, Λ τ ) De nition 1: ζ VQR = T [bθ(τ ) 0 (τ (1 τ )H 1 τ JH 1 τ ) 1 bθ(τ )] Proposition (VQR test) Consider the quantile regression (2). Under the null hypothesis, if assumptions (1)-(4) hold, then, the test statistic ζ VQR is asymptotically χ 2 (2).

18 DQ versus VQR test The null hypothesis in the DQ test is based on the orthogonality condition n 1 n Hit t [1 VaR t ] = 0, t=1 In quantile regression we minimize the loss function R(β) = n t=1 ρ τ (R t α 0 (τ) α 1 (τ) VaR) If β = [α 0 (τ), α 1 (τ)] 0 minimizes R(β), then the directional derivative OR(β, w) 0, f or all w 2 R 2 with kwk = 1

19 under the null hypothesis,β 0 = [0, 1] 0 the directional derivative of R(β 0 ) becomes n Hitt [1 VaR OR(β 0, w) = t ] w if u t 6= 0 Hit t=1 t [1 VaR t ] w, if u t = 0 Hit t = I (R t VaR t < 0) τ if u t 6= 0 Hit = I (x 0 t w < 0) τ if u t = 0 f or all w 2 R 2 with kwk = 1. the DQ and VQR tests are asymptotically equivalent under the null and under local alternatives In nite samples, the GMM estimation and the quantile estimation can be quite di erent ) DQ and VQR can yield quite di erent results

20 Monte Carlo DGP is a zero mean, unit unconditional variance normal innovation-based GARCH model: we consider τ = 0.01 and T = f250, 500, 1000, 2500g (i.e., approximately 1, 2, 4, and 10 years of daily data). We computer the size adjusted power: Historical simulation is the misspeci ed model Pérignon and Smith (2006) document that almost 3/4 of banks that disclose their VaR method report using historical simulation. we consider 5% tests: Kupiec (1995), Christo ersen (1998) the DQ test (2004) in which we considered the instruments X t = [1 VaR t ] 0. We look at the one-day ahead forecast and simulate 5,000 sample paths of length T + T e observations, with T e = 250

21 All four tests have small size distortions for T = 1000 and 2500 The VQR test is oversized for small sample sizes, T = 250 and 500 The VQR test is more powerful than any othe existing test for any sample size and level of signi cance τ

22 Table 2: Size-adjusted Power of 5% tests Panel A: τ = 1% Sample Size ζ Kupiec ζ Christ ζ DQ ζ VQR Panel B: τ = 5% Sample Size ζ Kupiec ζ Christ ζ DQ ζ VQR

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24 Figure 4 - S&P500 daily returns 3.00% 2.00% 1.00% 0.00% 1.00% 2.00% 3.00% Oct 03 Aug 04 May 05 Mar 06 Jan 07 Nov Series: SP500 Sample Observations Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Jarque Bera Probability

25 Local Analysis De nition 2 W t fq Ut (eτ) =eτ2 [0, 1] j VaR t = Q Rt (eτ j F t 1 ) g, representing the empirical quantile of the standard iid uniform random variable, U t, such that the equality VaR t = Q Rt (eτ j F t 1 ) holds at period t. In other words, W t is obtained by comparing VaR t with a full range of estimated conditional quantiles evaluated at τ 2 [0, 1]. Note that W t enables us to conduct a local analysis, whereas the proposed VQR test is designed for a global evaluation based on the whole sample. if VaR t is a correctly speci ed VaR model, then W t should be as close as possible to τ for all t. However, if VaR t is misspeci ed, then it will vague away from τ, suggesting that VaR t does not correctly approximate the τ th conditional quantile.

26 Figure 3 - Local Analysis of VaR Models. HS12m - 1% VaR GARCH - 1% VaR HS12m - 5% VaR GARCH - 5% VaR

27 Table 3: Backtesting Value-at-Risk Models Model % of Hits ζ Kupiec ζ Christ. ζ DQ ζ VQR τ = 1% HS12M GARCH(1,1) τ = 5% HS12M GARCH(1,1) Notes: P-values are shown in the ζ s columns

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29 Final Remarks Although the proposed methodology has several appealing properties, it should be viewed as complementary rather than competing with the existing approaches Furthermore, several important topics remain for future research, such as:

30 (i) time aggregation: how to compute and properly evaluate a 10-day regulatory VaR? (ii) Our randomness approach of VaR also deserves an extended treatment and leaves room for weaker conditions; (iii) extension of the analysis for the multivariate quantile regression (see Chaudhuri (1996) and Laine (2001)); (iv) inclusion of other variables to increase the power of VQR test in other directions; (v) improvement of the BIS formula for market required capital; (vi) nonlinear quantile regressions; among many others.

31 The End