Empirical Tail Index and VaR Analysis

Transcription

1 International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo. June 28-30, 2006 Empirical Tail Index and VaR Analysis M. Ivette Gomes Universidade de Lisboa, DEIO, CEAUL and FCUL. Lígia Rodrigues Instituto Politécnico de Tomar CEAUL, Universidade de Lisboa. Clara Viseu ISCA, Coimbra CEAUL, Universidade de Lisboa. Abstract In many areas of application, lie for instance statistical quality control, insurance and finance, a typical requirement is to estimate a high quantile, i.e., the Value at Ris at a level p (VaR p), high enough, so that the chance of an exceedance of that value is equal to p, small. In this paper we provide an empirical data analysis of logreturns associated to four sets of financial data, through the use of reduced bias tail index and associated high quantile estimators. These tail index estimators depend on two second order parameters, and in order to achieve a reduction in bias without any inflation of the asymptotic variance, the second order parameters in the bias are both estimated at a higher level than the one used for the estimation of the tail index. A percentile method for quantile estimation is also considered and a heuristic adaptive choice of the threshold for reduced bias estimators is provided, being their finite sample properties studied through a small-scale Monte Carlo simulation. Key Words: Heavy tails, High quantiles, Value at Ris, Semi-parametric estimation, Percentile estimation, Statistics of Extremes. 1 Introduction In ris management it is crucial to evaluate adequately the ris of a big loss that occurs very rarely. The ris is generally expressed as the Value at Ris (VaR), the size of the loss occurred with fixed small probability, p. We are then dealing with a high quantile χ 1 p := F (1 p) of a probability distribution function (d.f.) F, with F (y) := inf {x : F (x) y}, the generalized inverse function of F. Let us denote U(t) the inverse function of 1/(1 F ). Then, for small p, we want to estimate the parameter VaR p = U (1/p), p = p n 0, n p n 1. Correspondence to: M. Ivette Gomes. DEIO, Faculdade de Ciências de Lisboa, Bloco C6, Piso 4, Campo Grande, Lisboa, Portugal.

2 2 M. I. Gomes, L. Rodrigues and C. Viseu Usually, we even have n p n < 1, i.e., we want to extrapolate beyond the sample. Since we are dealing with a small probability, we may confine ourselves to modeling the tail. Moreover, since in financial applications we find generally heavy tails, we shall assume that the d.f. underlying the data satisfies, for some positive constant C, ( x ) 1/γ 1 F (x), as x, with γ > 0. (1.1) C Weissman (1978) proposed the following semi-parametric estimator of a high quantile (i.e., the Value-at-Ris): ( ) Q (p) bγ bγ () := X n +1:n, (1.2) np where X n +1:n is the -th top order statistic (o.s.), γ any consistent estimator for γ and Q stands for quantile function. Further details on semiparametric estimation of extremely high quantiles for a general tail index γ R may be found in de Haan and Rootzén (1993) and, more recently, in Ferreira et al. (2003) and Matthys and Beirlant (2003). Gomes and Figueiredo (2003), Matthys and Beirlant (2003), Mathys et al. (2004), Gomes and Pestana (2005) and Beirlant et al. (2006) deal with reduced bias quantile estimation. Interesting references on parametric quantile estimation, also used in this paper, are Castillo and Hadi (1994; 1995). The estimator in (1.2) provides useful estimates for large sample sizes, n. Also, and as usual in semi-parametric estimation of parameters of extreme events, we need that [1, n) : = n, = o(n) as n. (1.3) We then say that is an intermediate sequence of integers. For heavy tails, the classical semi-parametric tail index estimator, usually the one which is used in (1.2) for a semi-parametric quantile estimation, is the Hill estimator γ = γ() =: H() (Hill, 1975), with the functional expression, H() := 1 U i, U i := i (ln X n i+1:n ln X n i:n ), 1 i. (1.4) i=1 If we insert in (1.2) the Hill estimator, H(), we get the so-called classical quantile estimator, with the notation, Q (p)(). H

3 Empirical Tail Index and VaR Analysis 3 In order to be able to study the asymptotic non-degenerate behavior of Q (p)(), as well as of alternative V ar H p-estimators, it is useful to impose a second order expansion on the tail function 1 F or on the quantile function U. Here we shall assume that we are woring in Hall-Welsh class of models (Hall and Welsh, 1985), where ( U(t) = Ct γ 1 + γ β ) tρ + o (t ρ ), as t, (1.5) ρ with C, γ > 0, ρ < 0 and β non-zero. We shall further use the notation A(t) := γ β t ρ, γ > 0, β 0, ρ < 0. (1.6) The class in (1.5) is a wide class of models, that contains most of the heavy-tailed parents useful in applications, lie the Fréchet, the Generalized Pareto and the Student-t. From the results of de Haan and Peng (1998), we may say that in Hall- Welsh class of models in (1.5) and for intermediate, H() γ d = γ P + A(n/) 1 ρ (1 + o p(1)), (1.7) with A the function in (1.6) and P an asymptotically standard normal r.v. Regarding semi-parametric quantile estimation: under condition (1.5), the asymptotic behavior of Q (p) () is also well-nown: H ln(/(np)) ( Q (p) () H 1 VaR p ) ( ) d λ Normal n 1 ρ, γ2, provided the sequence = n is intermediate, ln(/(np)) = o( ) and lim n A(n/) = λ R, finite, with A the function in (1.6). On the basis of the results in Gomes and Figueiredo (2003), Gomes and Pestana (2005) and Beirlant et al. (2006), we here proceed to the use in (1.2) of specific reduced bias tail index estimators. Those tail index estimators and associated quantile estimators are described in section 3, after a brief review, in section 2, of a possible methodology to build asymptotic confidence intervals for the tail index and the V ar on the basis of classical

4 4 M. I. Gomes, L. Rodrigues and C. Viseu semi-parametric estimators. Section 4 is dedicated to the description, in an algorithmic way, of procedures dedicated to the semi-parametric estimation of the second order parameters (β, ρ), of the tail index γ and of an extreme quantile or V ar. In section 5, we perform a small-scale Monte Carlo simulation to illustrate the performance of the percentile method, devised in a parametric set-up, and to present some simulated properties of the heuristic adaptive choice considered for the tail estimation through reduced bias estimators. Finally, in section 6, we provide an empirical data analysis of log-returns associated to four sets of financial data, collected over the same period, from January 4, 1999, until November 17, 2005: the Euro-USA Dolar (EUSD) and the EU-UK Pound (EGBP) daily exchange rates, as well as the daily close values of Dow Jones Industrial Average In (DJI) and International Business Machines Corp. (IBM) stocs. 2 Asymptotic confidence intervals for the tail index and the Value-at-Ris Since from (1.7), together with the definition of A(t) in (1.6), we may guarantee that {H()/γ 1 β(n/) ρ /(1 ρ)} Normal(0, 1), provided that (n/) ρ λ, finite, we may get approximate 95% confidence intervals for γ, given by H() H(), =: (LCL 1 + β(n/)ρ 1 ρ β(n/)ρ 1 ρ 1.96 H (), UCL H ()). (2.1) If λ = 0, we may replace in (2.1) the bias summand β(n/) ρ /(1 ρ) by 0. Moreover, since for models in (1.5) the optimal -value for the tail index estimation through the Hill estimator, in (1.4), is well approximated by { 1 0 H 0 H (n; β, ρ) := arg min + β2 (n/) 2ρ } (1 ρ) 2 ( ) (1 ρ) n ρ 2/(1 2ρ) = β, (2.2) 2ρ ( ) we get 0 H n/ H ρ ( ) 0 (1 ρ)/ β 2ρ, and consequently, the r.v. ( ( ) ) 0 H H H 0 /γ 1 1/ 2ρ is approximately standard normal. If we

5 Empirical Tail Index and VaR Analysis 5 use H(0 H ) to build a confidence interval for γ, we may thus replace in (2.1) the bias summand β(n/) ρ /(1 ρ) by 1/ 2ρ. Similarly to what has been done for the Hill estimator, we may easily estimate, now numerically, the optimal threshold for the VaR estimation through Q H, i.e., the level Q H 0 Q H 0 (n, p; β, ρ) := arg min { ( ) ( 1 ln 2 np + β2 (n/) 2ρ )} (1 ρ) 2. (2.3) We may also find approximate confidence intervals for VaR p on the basis of Q (p)() and for any level such that A(n/) λ, finite. We get a H 95% confidence interval, dependent on γ, and given by ( ) γ β(n/)ρ ( ) 1 ρ γ 1.96 β(n/)ρ 1 ρ Q H (), Q np H (). np Let us introduce the notation a := β(n/)ρ 1 ρ, b := 1.96 β(n/)ρ 1 ρ. In order to have a guarantee of a coverage probability at least equal to 95%, and with LCL H () and UCL H () given in (2.1), we shall wor with ( ( ) a LCL H () ( ) ) a UCL H () LCL QH () = Q H () min, (2.4) np np and UCL QH () = Q H () max ( ( ) b LCL H () ( ) ) b UCL H (),. (2.5) np np 3 Reduced bias tail index and quantile estimators 3.1 Tail index estimation We shall wor here with the reduced bias tail index estimators in Caeiro et al. (2005) and Gomes et al. (2005). The estimator in Caeiro et al. (2005), also used in Gomes and Pestana (2005) for quantile estimation, has the functional expression, H ˆβ,ˆρ () := H() ( 1 ˆβ 1 ˆρ ( n )ˆρ ), (3.1)

6 6 M. I. Gomes, L. Rodrigues and C. Viseu where ( ˆβ, ˆρ) is an adequate consistent estimator of (β, ρ), with both ˆβ and ˆρ based on a number of top o.s. 1 of a larger order than the number of top o.s. used for the tail index estimation. Note that the estimator in (3.1) is a bias-corrected Hill estimator: the dominant component of the bias of Hill s estimator, provided in (1.7) and given by γ β(n/) ρ /(1 ρ), is estimated through H() ˆβ (n/)ˆρ /(1 ˆρ) and directly removed from the Hill estimator in (1.4). Apart from the class of estimators in (3.1), we shall also consider the reduced bias class of estimators in Gomes et al. (2005). Such a class is of the same type of the one in (3.1), but it has been inspired in the tail index estimator provided in Gomes and Martins (2002), i.e., it is based on an approximate maximum lielihood approach associated to the scaled log-spacings U i in (1.4). With the same notation as before, we shall thus wor with the tail index estimator MLˆβ,ˆρ () := H() ˆβ ( n )ˆρ D (ˆρ), D (α) := 1 i=1 ( ) i α U i. (3.2) This is another example of a bias-corrected Hill estimator, where we are using D (ˆρ) as an estimator of γ/(1 ρ). We may state the following: Proposition 3.1 (Caeiro et al., 2005; Gomes et al., 2005). For models in (1.5), let us assume that (1.3) holds and that, with A(t) given in (1.6), A(n/) λ, finite and non necessarily null, as n. Then, with T denoting either H in (3.1) or ML in (3.2), (Tβ,ρ () γ) d n Normal ( 0, γ 2). This same limiting behaviour holds if we replace T β,ρ by T ˆβ,ˆρ, provided that we consider a ρ-estimator ˆρ, such that ˆρ ρ = o p (1/ ln n), and we choose ˆβ, consistent for the estimation of β. More specifically, and with V and W asymptotically standard normal r.v. s, we may write H ˆβ,ˆρ () MLˆβ,ˆρ () } { d γ V / = γ + γ W / } + o p (A(n/)).

7 Empirical Tail Index and VaR Analysis 7 Remar 3.1. Notice that, contrarily to what happens in Drees class of functionals (Drees, 1998), where the minimal asymptotic variance of a reduced bias tail index estimator is given by (γ(1 ρ)/ρ) 2, we have been here able to obtain a reduced bias tail index estimator with an asymptotic variance equal to γ 2, the asymptotic variance of Hill s estimator, which is indeed the maximum lielihood estimator of γ for the strict Pareto model F γ (x) = 1 x 1/γ, x 1, γ > 0. Here, for illustration, we shall also initially wor with a representative of the reduced bias statistics in Drees (1998) class of functionals, the estimator with H ˆβ,ˆρ () given in (3.1) and ˆβˆρ () := ( n ( 1 )ˆρ {( 1 i=1 ( i i=1 Ĥˆρ () := H ˆβˆρ (), ˆρ (), (3.3) ( i ) ˆρ ) ( 1 ) ˆρ ) ( 1 i=1 ( i i=1 ) ( 1 U i ) ˆρ Ui ) ( 1 )} ( i ) ˆρ Ui ), (3.4) ( i ) 2ˆρ Ui with U i, 1 i, the scaled log-spacings in (1.4). The asymptotic variance of the estimator in (3.3) is minimal in Drees class of functionals, and consequently given by (γ(1 ρ)/ρ) 2 (Caeiro et al., 2005). i=1 i=1 3.2 Asymptotic confidence intervals for γ based on second order reduced bias tail index estimation On the basis of the statistics H and ML in (3.1) and (3.2), respectively, and for levels such that (n/) ρ λ, finite, possibly different from zero, Proposition 3.1 enables us to get the following 95% approximate confidence interval for γ, ( (LCL T (), UCL T ()) = T () /, T () / ), (3.5) again with T denoting any of the estimators in either (3.1) or (3.2).

8 8 M. I. Gomes, L. Rodrigues and C. Viseu 3.3 An adaptive choice of the level for reduced bias estimators Here, we have decided to use the heuristic adaptive choice of suggested in Gomes and Pestana (2005). Up to now, we do not have simple techniques to estimate the optimal threshold of second order reduced bias estimators, but we now that such a should be larger than 0 H in (2.2). If we plot the 95% approximate confidence region in (2.1), as a function of, the Hill estimate is sooner or later going to cross this region. We have thus decided to use such a -value for the tail index estimation through the second order reduced bias tail index estimator H() and ML() in (3.1) and (3.2), respectively, as well as for the associated VaR estimation. Such a crossing level is solution of the equation β (n/) ρ /(1 ρ) = 1.96/, i.e., we get ( ) 1.96(1 ρ)n ρ 2/(1 2ρ) (n; β, ρ) =. (3.6) β Levels of this type are still levels such that A(n/) λ, finite, and are not yet optimal for the tail index estimation through second order reduced bias tail index estimators. However, with a tail index estimator of the type of the ones in (3.1) and (3.2) we are safe for all, and the level in (3.6) has revealed to provide an interesting adaptive choice of the threshold for reduced bias estimation. We anyway thin that extra investment is needed on the adaptive estimation of the optimal threshold for this type of estimators, but such a research overpasses the scope of this paper. 3.4 Extreme quantile or VaR estimation We shall here consider the alternative VaR p estimators Q (p) and Q(p) H ML, with Q (p) bγ, H and ML given in (1.2), (3.1) and (3.2), respectively. Under the same conditions as before, i.e., under conditions (1.3), (1.5) and provided that ln(/(np)) = o( ) and lim n A(n/) = λ, finite, if we wor with any of the reduced bias tail index estimators H or ML, generally denoted T, we get ln(/(np)) ( Q (p) () ) T 1 VaR p d n Normal ( 0, γ 2), even when λ 0. Note that with a classical reduced bias tail index estimator lie Ĥˆρ() in (3.3), i.e., a tail index estimator of the type of the

9 Empirical Tail Index and VaR Analysis 9 ones used, for quantile estimation, in Gomes and Figueiredo (2003), we were able to get also a null bias, but at expenses of a higher asymptotic variance. 3.5 Asymptotic confidence intervals for VaR p on the basis of reduced bias estimators Whenever woring with Q T, with T denoting either H in (3.1) or ML in (3.2), and levels such that A(n/) λ, finite, we shall use the confidence interval, ) ( ) 1.96 UCL (LCL QT, UCL QT = Q T T, np with UCL T given in (3.5). ( ) 1.96 UCL T, (3.7) np 4 An algorithm for semi-parametric tail estimation We propose the following: Algorithm: 1. Given a sample (X 1, X 2,, X n ), plot, for τ = 0 and τ = 1, the estimates ( ) 3(T n (τ) () 1) ˆρ τ () := min 0, T n (τ), (4.1) () 3 where, with M n (j) () := 1 {ln X n i+1:n ln X n :n } j, j = 1, 2, 3, i=1 and the notation a bτ = b ln a whenever τ = 0, T (τ) n () := ( ( M (1) n M (2) n ) τ ( () ) τ/2 ( ()/2 M (2) n M (3) n ) τ/2 ()/2 ) τ/3, τ R. ()/6

10 10 M. I. Gomes, L. Rodrigues and C. Viseu 2. Consider {ˆρ τ ()} K, for integer values K = ([ n 0.995], [ n 0.999]), and compute their median, denoted ρ τ. Next choose the tuning parameter τ := arg min τ K (ˆρ τ () ρ τ ) Wor then with ˆρ τ = ˆρ τ ( 1 ) and ˆβ τ := ˆβˆρτ ( 1 ), where 1 = [ n 0.995], (4.2) being ˆβˆρ () and ˆρ τ () given in (3.4) and (4.1), respectively. 4. Plot the classical Hill estimates H(), given in (1.4), and adaptively consider H(ˆ 0 H), ˆH 0 = 0 H(n; ˆβ τ, ˆρ τ ), 0 H (n; β, ρ) given in (2.2), together with the 95% approximate confidence interval, (LCL H (ˆ 0 H), UCL (ˆ H H 0 )), given in (2.1) for a general. 5. Plot also the reduced bias tail index estimates H H τ () and ML ML τ (), associated to the estimates (ˆρ τ, ˆβτ ) obtained in step 3. Adaptively consider H(ˆ 01 ) and ML(ˆ 01 ), ˆ 01 = 01 (n; ˆβ τ, ˆρ τ ), 01 (n; β, ρ) given in (3.6), together with the 95% confidence intervals (LCL T (ˆ 01 ), UCL T (ˆ 01 )), provided in (3.5) for a general, and with T standing either for H τ or ML τ. 6. Choose the tail index estimate providing the smallest 95% confidence size. Let us denote Γ the associated estimator (either H, H or ML). 7. Plot the classical VaR estimates Q H (), with Q bγ () and H given in (1.2) and (1.4), respectively, and adaptively consider Q H (ˆ Q H 0 ), where ˆ Q H 0 = Q H 0 (n, p; ˆβ τ, ˆρ τ ), with Q H 0 (n, p; β, ρ) given in (2.3). Consider also the approximate confidence interval, (LCL QH (ˆ Q H 0 ), UCL QH (ˆ Q H 0 )), given in (2.4) and in (2.5) for a general ; 8. If the estimator Γ chosen in step 6. is one of the reduced bias estimators, plot Q (p) Γ () and adaptively consider Q(p) Γ (ˆ 01 ), ˆ01 = 01 (n; ˆβ τ, ˆρ τ ), with 01 (n; β, ρ) given in (3.6), together with the confidence interval in (3.7) for = ˆ 01. Remar 4.1. For asymptotic and finite sample details on the estimators of ρ in (4.1), see Fraga Alves et al. (2003). The estimator of β in (3.4) has been introduced in Gomes and Martins (2002), where conditions that enable its asymptotic normality have been set, whenever ρ is estimated at

11 Empirical Tail Index and VaR Analysis 11 a level 1 of a larger order than the level used for the estimation of β. Details on the asymptotic distribution of ˆβˆρ() () may be found in Gomes et al. (2004b). Remar 4.2. Steps 1. and 2. of the algorithm lead in almost all situations to the tuning parameter τ = 0 whenever ρ 1 and τ = 1, otherwise. Such an educated guess usually provides better results than a possibly noisy estimation of τ, and it is highly recommended in practice. For details on this and similar algorithms for the ρ-estimation, see Gomes and Pestana (2004). 5 The use of a parametric quantile method in tail index and quantile estimation. The heuristic adaptive choice of the threshold in reduced bias estimation. Castillo and Hadi (1994; 1995) introduced a percentile (P ) method for the estimation of parameters and quantiles of the Extreme-Value (EV ) d.f., given by EV γ (x) := exp( (1 + γx) 1/γ ), 1 + γx > 0, γ R. Whenever γ > 0, i.e., whenever we are dealing with heavy-tailed models, the EV distribution is usually called a Fréchet distribution. Here, we shall wor with a standard Fréchet γ d.f., F (x) = exp ( x 1/γ), x 0. Apart from the Fréchet, we shall wor with Generalized Pareto GP γ parents, with d.f. F (x) = 1 (1 + γx) 1/γ, x 0, and with Burr γ,ρ parents, with d.f. F (x) = 1 (1 + x ρ/γ ) 1/ρ, x 0, with γ > 0. The estimation procedure considered by Castillo and Hadi consists of a two-stage P -method. In a first stage we obtain initial estimates ˆγ ijr of γ, applying the P -method, through the use of the o.s., x i:n, x j:n and x r:n, the quotient (x j:n x r:n )/(x i:n x j:n ) and the bisection method. We fix the levels i = 1 and r = n and, in a second stage, compute the median of the (n 2)-vector of estimates ˆγ 1jn, with j = 2,, n 1. The final estimate is denoted ˆγ P. The estimate of the quantile of probability 1 p is then given by ˆχ (p) := ( ln(1 p)) ˆγ P for Fréchet models, by ˆχ (p) := (p ˆγ P 1)/ˆγ P P P for GP γ models and by ˆχ (p) := (p bρ i 1) ˆγ P /bρ i for B P γ,ρ parents, with i = 0 or 1 according as ρ 1 or ρ > 1. We shall proceed here to a small-scale Monte Carlo comparison of the tail index P -estimators with Hill s estimator computed at the estimated

12 12 M. I. Gomes, L. Rodrigues and C. Viseu value of 0 H in (2.2), as well as with the reduced bias estimators H and M L, in (3.1) and (3.2), respectively, both computed at the estimated value of 01 in (3.6). The simulation results were obtained on the basis of 1000 runs, from the above mentioned parents with γ = {0.2, 0.5, 0.8, 1 and 2} and samples of size n = The estimates obtained are summarized in Table 1. The values in bracets are the simulated mean squared errors (MSE) for each estimator and each value of γ. The underlined values are the ones providing minimum bias, that correspond in all cases to minimum MSE. γ Fréchet γ parent: (ρ = 1) ˆγ P ( ) ( ) ( ) ( ) ( ) H(ˆ 0 H ) ( ) (0.0164) (0.0171) (0.0203) (0.0488) H(ˆ 01) ( ) (0.0061) (0.0029) ( ) (0.0047) ML(ˆ 01) ( ) (0.0058) (0.0028) ( ) (0.0028) GP γ parent: (ρ = γ) ˆγ P (0.0313) (0.0129) (0.0043) (0.0018) (0.0027) H(ˆ 0 H ) (0.0572) (0.0041) (0.0099) (0.0060) (0.1247) H(ˆ 01) (0.0413) (0.0028) (0.0055) ( ) (0.0025) ML(ˆ 01) (0.0395) (0.0023) (0.0040) ( ) (0.0016) Burr γ,ρ parent with ρ = 1 ˆγ P (0.0145) ( ) ( ) (0.0018) (0.0096) H(ˆ 0 H ) (6.57E-4) (0.0023) (0.0133) (0.0790) (0.0378) H(ˆ 01) ( ) ( ) ( ) (0.0058) (0.0184) ML(ˆ 01) ( ) ( ) ( ) (0.0057) (0.0193) Table 1: Simulated mean values and mean squared errors of the γ-estimators under comparison.

13 Empirical Tail Index and VaR Analysis 13 The ln-v ar p -estimates presented in Table 2 have been obtained again on the basis of 1000 runs, for the same parents as before, for samples with size n = 1000 and p = Once again the values in bracets are the simulated mean squared errors for each estimator and each value of γ and the underlined values are the ones leading to minimum bias, again correspondent to minimum M SE. γ Fréchet γ parent: (ρ = 1) ln-v ar p ln ˆχ (p) P ( ) ( ) ( ) ( ) ln Q H (ˆ Q H 0 ) (0.0240) (0.3965) (0.1050) (0.3756) (1.1157) ln Q H (ˆ 01 ) (0.0019) (0.1092) (7.66E-4) (0.0183) (0.0058) ln Q ML (ˆ 01 ) (0.0018) (0.1029) ( ) (0.0140) ( ) GP γ parent: (ρ = γ) ln-v ar p ln ˆχ (p) P (0.6449) (0.3581) (0.1423) (0.0636) (0.1097) ln Q H (ˆ Q H 0 ) (0.0242) (0.4270) (0.1559) (0.0500) (2.2713) ln Q H (ˆ 01 ) (0.0050) (0.0105) (0.0015) (0.0180) (0.0807) ln Q ML (ˆ 01 ) (0.0024) (0.0099) ( ) (0.0325) (0.0507) Burr γ,ρ parent with ρ = 1 ln-v ar p ln ˆχ (p) P (0.6908) (0.0270) (0.0219) (0.0870) (0.4581) ln Q H (ˆ Q H 0 ) (0.0344) (0.1352) (0.9261) (1.4985) (1.2981) ln Q H (ˆ 01 ) ( ) ( ) (0.0017) (0.0671) (1.1177) ln Q ML (ˆ 01 ) ( ) ( ) (0.0011) (0.0663) (1.1567) Table 2: Simulated mean values and MSE s for the different estimators of V ar p under comparison, p = Tables 1 and 2 clearly show that the P -estimators of tail index and high quantiles are overall better than any of the adaptive semi-parametric estimators, for Fréchet parents. Indeed, apart from quantile estimation for

14 14 M. I. Gomes, L. Rodrigues and C. Viseu γ = 2, they exhibit simultaneously minimum bias and minimum MSE. For the other parents, things may wor differently. For instance, for the GP γ parents, the reduced bias estimators have an overall better performance. A similar remar applies to Burr γ,ρ models. Also, we cannot forget that we are comparing two different approaches on quantile and tail index estimation, and we are not being totally fair to the semi-parametric approach. Indeed in a parametric context, lie the one in this simulation study, we would expect to observe a even higher performance of the P -estimators, specifically devised for a specific model. Note further that, both in terms of minimum bias and minimum mean squared error, the choice of either H(ˆ 01 ) or ML(ˆ 01 ) has always led to a performance better than the one achieved by H(ˆ 0 H ). This is thus a point in favor of the reduced bias estimators here proposed. 6 Financial data analysis 6.1 Second order parameter estimation The sample paths of the ρ-estimates associated to τ = 0 and τ = 1 lead us to choose for any of the series and on the basis of any stability criterion for large values of, lie the one suggested in step 2. of the Algorithm, the estimate associated to τ = 0. The estimates obtained are summarized in Table 3, where we use the notation n 0 for the number of positive log-returns in any of the series. Table 3: Estimates of the second order parameters associated to the four data sets under study. n 0, ˆρ 0, ˆβ 0 DJI EGBP EUSD IBM (867, 0.724, 1.018) (835, 0.724, 1.023) (867, 0.697, 1.030) (881, 0.744, 1.016) In Figure 1, woring with the positive log-returns of the EUSD data, as an illustration, we picture the sample paths of the estimators of the second order parameters ρ (left) and β (right).

15 Empirical Tail Index and VaR Analysis ˆ 0 () ˆ " = # " ˆ # ˆ 0 () -2 ˆ 1 () 1 ˆ " = Figure 1: Estimates of ρ, through ˆρ τ () in (4.1), τ = 0 and 1 (left) and of β, through ˆβˆρ0 () in (3.4) (right), for the positive log-returns on EUSD3 data. 6.2 Tail index and V ar p estimation The sample paths of the classical Hill estimator H in (1.4), the second 1 order reduced bias tail index estimators H 0 = H ˆβ0,ˆρ 0, ML 0 = MLˆβ0,ˆρ 0 and Ĥ 0 = Ĥˆρ 0, provided in (3.1), (3.2) and (3.3), respectively, as well as the associated Var p estimators, for p = 0.001, 0 are pictured in Figure 2, again for the EUSD data ˆ H 0 ˆ " H = 0.27 ML 0 ˆ " ML = H H Q ˆ H 0 Q H ˆ " H = 3.58 Q H ˆ " ML = 3.27 Figure 2: Estimates provided through H, H 0, H and b H 0 in (1.4), (3.1), (3.2) and (3.3)(left) and the associated ln-var p estimates, for the positive log-returns on EUSD data and p = Q ML0 C=

16 16 M. I. Gomes, L. Rodrigues and C. Viseu From a theoretical point a view the chosen estimate in step 6. of the Algorithm should be H 0 (ˆ 01 ) or ML 0 (ˆ 01 ). Indeed, for any of the data sets considered, we have been always led to the choice ML 0 (ˆ 01 ). The reduction in the size of the the new confidence intervals is provided in Table 4, together with other relevant characteristics. Table 4: Tail index estimates and associated 95% confidence intervals for the four data sets under study DJI EGBP EUSD IBM (ˆ 0 H, H(ˆ 0 H )) (73, ) (71, 0.303) (68, ) (76, 0.386) (LCL H, UCL H ) (0.203,0.311) (0.227, 0.350) (0.200, 0.311) (0.292,0.445) ˆ H 0 (ˆ 01 ) (LCL H, UCL H ) (0.260, 0.361) (0.255, 0.356) (0.216, 0.305) (0.324, 0.447) Reduction in size 93.81% 82.44% 79.24% 80.65% ML 0 (ˆ 01 ) (LCL ML, UCL ML ) (0.254, 0.354) (0.251, 0.350) (0.213, 0.300) (0.319, 0.440) Reduction in size 91.99% 81.06% 77.99% 79.47% Table 5 is equivalent to Table 4, now related to VaR estimation, and only with the quantile estimates associated to ML 0 as suggested in step 8. of the Algorithm. Table 5: Estimates of the different parameters characterising the tail behaviour of the four data sets. DJI EGBP EUSA IBM (ˆ Q H 0, Q H (ˆ Q H 0 )) (48, 6,679) (47, 2.804) (44, 3.579) (52, ) (LCL QH, UCL QH ) (4.067, 8.953) (1.726, 3.732) (2.237, 4.721) (9.994, ) Q ML (ˆ 01 ) (LCL QML, UCL QH ) (4.918, 8.864) (2.104, 3.804) (2.537, 4.221) (11.822, ) 6.3 Graphical illustration of the adaptive threshold choice for tail index and V ar estimation In Figures 3 and 4 we illustrate graphically steps from 4. until 6. in the Algorithm, for two of the data sets under analysis.

17 17 Empirical Tail Index and VaR Analysis 0.45 H UCL H UCL H UCLML 0.35 H0 "ˆ = LCLML LCL H ML0 LCL H ˆ0H = Figure 3: ˆ01 = Confidence intervals for γ (GBPdata). 0.5 H UCL H UCLML UCL H 0.4 "ˆ = H0 LCL H 0.3 ML0 LCL H ML LCL Figure 4: ˆ0H = 76 ˆ01 = Confidence intervals for γ (IBM data). Figures 5 and 6 exhibit now the steps 7. and 8., in the Algorithm. Again, the VaR0.001 estimate is denoted χ b0.999 and is associated to M L0.

18 18 M. I. Gomes, L. Rodrigues and C. Viseu Q H Q ML ˆ " = Figure 5: Confidence intervals for VaR p, p = (GBP data) Q H ˆ " = Q ML0 Figure 6: Confidence intervals for VaR p, p = (IBM data). 6.4 The use of a percentile method in quantile estimation The direct application of the P -method in section 5 to the four data sets has not been successful, in the sense that we could not adequately fit to

19 Empirical Tail Index and VaR Analysis 19 the data a model with a closed form for the quantile function. The better fitting has been provided by Student-t models, and for such models the P -method is computationally expensive and not at all promising. We have here used a mixed technique based upon the percentile P - method in Castillo and Hadi, but estimating γ only on the basis of high quantiles of probability 1 p = 0.9, 0.95, 0.975, and On the basis of the approximation in (1.1), or equivalently the approximation U(t) = Ct γ, the use of any pair of p-values, (p 1, p 2 ), in the above mentioned set, enables us to get γ-estimates, ( ) γ := ln X [n(1 p1 )]+1/X [n(1 p2 )]+1. P ln (p 1 /p 2 ) The median of these estimates, denoted γ, was then considered as the overall estimate of γ provided by this technique. The estimation of C through P the percentile method led us to very large estimates of C, based upon estimates ĈP := X [n(1 p 1 )]+1(p 1 ) bγ P, very large high quantile estimates, and was consequently discarded. We have then used a least square technique based on the points ( q i = ln(1 i/(n + 1)), (ln X n i+1:n ln C)/ γ ) P, for large i, getting an estimate ĈLS. A high quantile of probability 1 p is then estimated through χ = P ĈLS p bγ P. The estimates of γ, C and χ provided by this method are summarized in Table 6, where we also place, for comparison, the estimates obtained through the Hill estimator H and the ML 0 estimator. Table 6: Estimates of the tail index γ and χ 0.999, provided through a mixed percentile/leastsquares method for the tail. DJI EGBP EUSA IBM (bγ H, bγ ML0, bγ ) P (0.27, 0.30, 0.32) (0.30, 0.29, 0.32) (0.27, 0.25, 0.27) (0.39, 0.37, 0.41) bc LS ( bχ H, bχ ML0 ) (6.68, 6.60) (2.80, 2.83) (3.58, 3.27) (19.53, 16.95) bχ P From Table 6 we see that the percentile method seems to be in this case providing a slight over estimation of the tail index and under estimation of high quantiles. When we compare the methodologies according to the size of associated confidence intervals, the method based on reduced bias

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