An Estimation of Risk Measures: Analysis of a Method ^†

Abstract

Extreme value theory comprises a set of techniques for inference at the tail of distributions, where data are scarce or non-existent. The tail index is the main parameter, with risk measures such as value at risk or expected shortfall depending on it. In this study, we will analyze a method for estimating the tail index through a simulation study. This will allow for an application using real data including the estimation of the mentioned risk measures.

1. Introduction

Extreme data behave distributionally different from central data. The extremal types theorem ([1,2,3]) is a result related to the asymptotic distribution of extreme-order statistics, and it plays a role analogous to the famous central limit theorem for averages (sums). Basically, it establishes that the conveniently normalized sample maximum converges to one of three possible distributions, Gumbel, Fréchet, or Max–Weibull. More precisely, let $X_1,X_2,...$ be an independent and identically distributed (i.i.d.) sequence of random variables (r.v.); with distribution function (d.f.) F, we say that F belongs to the max-domain of attraction of $G_{\gamma }\left(x\right)$, with notation $F\in \mathcal{D}\left(G_{\gamma }\right)$ if there exist real constants $a_n>0$ and $b_n$ such that

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle \underset{n\to \infty }{lim}}{F}^n(a_{n}x+b_n)=G_{\gamma }\left(x\right)\end{array}\end{array}$$

for all continuity points of $G_{\gamma }\left(x\right)$. This is called the generalized extreme value (GEV) df, and it is defined by

$$\begin{array}{c}\hfill G_{\gamma }\left(x\right)=exp\left\{-{\left(1+\gamma {\displaystyle \frac{x-\mu }{\sigma }}\right)}_{+}^{-1/\gamma }\right\},\end{array}$$

(1)

where real constants $\mu$, $\sigma >0$, and $\gamma$ correspond to, respectively, location, scale, and shape parameters, which were later called the so-called tail index.

The GEV model summarizes all three possible limit distributions according to the sign of $\gamma$: it is max–Weibull if $\gamma <0$ determines a short tail with a finite right end point, Gumbel if $\gamma =0$ with an exponential-type tail, and Fréchet if $\gamma >0$, corresponding to a heavy tail with an infinite right end point.

The Pickands–Balkema–de Haan theorem ([4,5]) states that if $F\in \mathcal{D}\left(G_{\gamma }\right)$, then the df $F_t\left(x\right)=P(X_t\le x)$ of conditional excesses $X_t=\{X-t|X>t\}$ above a high threshold t approximates a generalized Pareto (GP) df, given by

$$\begin{array}{c}\hfill {\displaystyle H_{\gamma }\left(x\right)=1-{\left(1+\gamma \frac{x}{\psi }\right)}_{+}^{-1/\gamma }}\end{array}$$

(2)

where $\psi >0$ is a scale parameter and $\gamma$ is the tail index.

GP also includes three models, namely beta ($\gamma <0$), exponential ($\gamma =0$), and Pareto ($\gamma >0$). The generalized extreme value (GEV) modeling approach often uses the so-called annual maximum method (AMM), where n data points are divided into m sub-samples, typically corresponding to m years, each with b data points (such that $n=bm$). The GEV model is then fitted to the sample formed by the maxima m, with each maximum representing the random variable $Y=max(X_1,\dots ,X_b)$.

An alternative to the AMM is generalized Pareto (GP) modeling, also known as the peaks over threshold (POT) method. In this approach, only data points that exceed a predetermined high threshold t are considered. The GP distribution is then fitted to the exceedances, specifically to the differences $X_i-t$ for the observations where $X_i>t$, that is, in the set $X_{i1}-t,\dots ,X_{n_t}-t$, where $n_t$ is the number of observations above the threshold t.

Choosing the appropriate block size b or threshold t can be challenging and has a significant impact on the resulting inference.

In a semi-parametric context, any parametric model underlying the sample of i.i.d. observations coming from df F is not assumed. We only assume that $F\in \mathcal{D}\left(G_{\gamma }\right)$ for some $\gamma$ and base the estimation of $\gamma$ on the highest k-order statistics of the sample, $X_{n:n}\ge X_{n-1;n}\ge ...\ge X_{n-k+1:n}$, above a random threshold $X_{n-k:n}$, which is considered an intermediate upper order statistic, i.e., with $k\equiv k_n$, an intermediate sequence of integers between 1 and n such that $k\to \infty$ and $k/n\to 0$, as $n\to \infty$. The choice of k is crucial in inferring the tail and undertakes a bias–variance trade-off: a small k means large variance in estimates, and a large k leads to significant bias. This issue has been largely addressed in the literature. See, for instance, Silva Lomba and Fraga Alves [6] (2020), Schneider et al. [7] (2021), and the references therein.

There are many semi-parametric estimators of $\gamma$ in the literature. A survey can be seen in Beirlant et al.’s study [8] (2004). Here we address the semi-parametric maximum likelihood (ML) estimator introduced by Smith [9] (1987). This estimator is associated with the GP approximation for excesses above a high threshold, i.e., with the POT approach. The excesses $E_{ik}:=X_{n-i+1:n}-X_{n-k:n}$, $1\le i\le k$ are approximately the top k-order statistics associated with a k-dimensional sample of a GP with d.f. $H\left(x\right)=1-{(1+\tau x)}^{-1/\gamma }$ with the reparameterization $\tau =\gamma /\psi$. The solution of the associated ML equations gives rise to the ML estimator

$$\begin{array}{c}\hfill {\widehat{\gamma }}_k={\displaystyle \frac{1}{k}}\sum _{i=1}^{k}log(1+\widehat{\tau }{E}_{ik})\end{array}$$

(3)

where $\widehat{\tau }$ is the (implicit) ML estimator of the scale parameter $\tau$, which is also unknown. Drees et al. [10] (2004) present an exhaustive study of the asymptotic properties of the ML estimator in (3) for $\gamma >-1/2$. Weak consistency and asymptotic normality were found by Zhou [11,12] (2009, 2010) in the region $\gamma \in ]-1,-1/2]$. The non-consistency for $\gamma <-1$ is also proved by Zhou [12] (2010).

The purpose of this study is to analyze a heuristic method to choose k with respect to the ML estimator. We conduct a simulation study that allows us to draw guidelines for applying the aforementioned k selection method. An application to the Apple stock price completes this paper.

2. Method

The Reiss and Thomas (RT) method (Reiss and Thomas [13] 1997) is a heuristic procedure to choose the number k of upper-order statistics to consider in the estimation of the tal index. Neves and Fraga Alves ([14] 2004) presented a study concerning other semi-parametric estimators of $\gamma$. Here, we address the RT method within the ML estimator.

The RT method proposes the selection of k, which minimizes

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{k}}\sum _{i\le k}{i}^{\beta }|{\widehat{\gamma }}_i-median({\widehat{\gamma }}_1,...,{\widehat{\gamma }}_k)|\end{array}$$

(4)

or alternatively

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{k-1}}\sum _{i<k}{i}^{\beta }{({\widehat{\gamma }}_i-{\widehat{\gamma }}_k)}^2\end{array}$$

(5)

for $0\le \beta \le 1/2$. The weighting factor aims to penalize the $\gamma$ estimates obtained based on observations further away from the tail. We will denote the criterion (4) RT1 and the criterion (5) RT2.

The value of k selected by the RT method, in addition to being used in the estimation of $\gamma$, will enable us to estimate other tail parameters, such as the risk measures value at risk (VaR) and expected shortfall (ES). The ${\mathrm{VaR}}_p=F^{-1}(1-p)$ is a high quantile exceeded with a small probability p. It also corresponds to a level that is expected to be exceeded once in the period $1/p$. The risk measure ES corresponds to a conditional mean of values beyond ${\mathrm{VaR}}_p$ defined by ${\mathrm{ES}}_p=E\left(X\right|X>Va{r}_p)$. Estimators under the POT approach are given by, respectively,

$$\begin{array}{c}\hfill \begin{array}{ccc}{\widehat{VaR}}_p=\left\{\begin{array}{cc}{X}_{n-k:n}+{\displaystyle \frac{1}{\widehat{\tau }}}\left({\left({\displaystyle \frac{np}{k}}\right)}^{-\widehat{\gamma }}-1\right)\hfill & \widehat{\gamma }\ne 0\hfill \\ X_{n-k:n}-{\displaystyle \frac{\widehat{\gamma }}{\widehat{\tau }}}log\left({\displaystyle \frac{np}{k}}\right)\hfill & \widehat{\gamma }=0\hfill \end{array}\right.& \mathrm{and}& {\widehat{ES}}_p={\displaystyle \frac{\widehat{\gamma }/\widehat{\tau }+{\widehat{VaR}}_p-\widehat{\gamma }{X}_{n-k:n}}{1-\widehat{\gamma }}}.\end{array}\end{array}$$

(6)

3. Simulation Study

We simulate 1000 replicas of samples of size $n=1000$ from GEV models in (1) and GP in (2), with $\mu =0$, $\sigma =1$, and $\psi =1$, respectively. We take $\gamma =-0.75\left(0.25\right)2$ and $\beta =0\left(0.1\right)0.5$. In addition to an analysis of the methodology in an application of the ML estimator, our goal is to find which $\beta$ suits better in each case. The results are placed in

Table A1. Simulation results from GP models by applying criterion RT1 in (4).

GP - RT1	$\mathit{\beta }$	bias	rmse	sd		$\mathit{\beta }$	bias	rmse	sd
$\gamma =-0.75$	0	−0.0185	0.0460	0.0422	$\gamma =0.75$	0	−0.0039	0.0528	0.0527
	0.1	−0.0185	0.0460	0.0422		0.1	−0.0039	0.0528	0.0527
	0.2	−0.0466	0.0960	0.0840		0.2	−0.0053	0.0986	0.0985
	0.3	−0.0258	0.0857	0.0817		0.3	−0.0931	0.3500	0.3376
	0.4	−0.0249	0.0887	0.0852		0.4	−0.0931	0.3500	0.3376
	0.5	0.0009	0.1209	0.1210		0.5	−0.0931	0.3500	0.3376
$\gamma =-0.5$	0	−0.0062	0.0264	0.0257	$\gamma =1$	0	0.0011	0.0734	0.0734
	0.1	−0.0050	0.0283	0.0278		0.1	0.0011	0.0734	0.0734
	0.2	−0.0050	0.0283	0.0278		0.2	0.0011	0.0734	0.0734
	0.3	−0.0050	0.0283	0.0278		0.3	0.0011	0.0734	0.0734
	0.4	−0.0627	0.1267	0.1102		0.4	−0.1050	0.4479	0.4356
	0.5	−0.1716	0.2840	0.2264		0.5	−0.1050	0.4479	0.4356
$\gamma =-0.25$	0	−0.0100	0.0408	0.0396	$\gamma =1.25$	0	−0.0005	0.0733	0.0734
	0.1	−0.0397	0.1124	0.1053		0.1	−0.0005	0.0733	0.0734
	0.2	−0.0425	0.1164	0.1084		0.2	−0.0005	0.0733	0.0734
	0.3	−0.0425	0.1171	0.1092		0.3	−0.0975	0.5014	0.4921
	0.4	−0.0442	0.1205	0.1121		0.4	−0.0975	0.5014	0.4921
	0.5	−0.0442	0.1205	0.1121		0.5	−0.0975	0.5014	0.4921
$\gamma =0$	0	−0.0030	0.0321	0.0320	$\gamma =1.5$	0	0.0016	0.0784	0.0784
	0.1	−0.0030	0.0321	0.0320		0.1	0.0016	0.0784	0.0784
	0.2	−0.0031	0.0324	0.0323		0.2	0.0016	0.0784	0.0784
	0.3	−0.1611	0.3625	0.3249		0.3	0.0016	0.0784	0.0784
	0.4	−0.1611	0.3625	0.3249		0.4	−0.0956	0.4717	0.4621
	0.5	−0.1611	0.3625	0.3249		0.5	−0.1099	0.4995	0.4876
$\gamma =0.25$	0	−0.0017	0.0391	0.0391	$\gamma =1.75$	0	−0.0051	0.1237	0.1237
	0.1	−0.0017	0.0391	0.0391		0.1	−0.0054	0.1250	0.1250
	0.2	−0.0067	0.0630	0.0627		0.2	−0.0052	0.1257	0.1257
	0.3	−0.0068	0.0630	0.0627		0.3	−0.0059	0.1267	0.1266
	0.4	−0.0066	0.0640	0.0637		0.4	−0.0062	0.1265	0.1264
	0.5	−0.1218	0.3444	0.3223		0.5	−0.1160	0.6424	0.6322
$\gamma =0.5$	0	−0.0042	0.0460	0.0459	$\gamma =2$	0	−0.0069	0.0957	0.0955
	0.1	−0.0042	0.0460	0.0459		0.1	−0.0069	0.0957	0.0955
	0.2	−0.0042	0.0460	0.0459		0.2	−0.0069	0.0957	0.0955
	0.3	−0.0741	0.2971	0.2879		0.3	−0.1186	0.6578	0.6474
	0.4	−0.0761	0.3055	0.2960		0.4	−0.1186	0.6578	0.6474
	0.5	−0.0761	0.3055	0.2960		0.5	−0.1186	0.6578	0.6474

,

Table A2. Simulation results from GP models by applying criterion RT2 in (5).

GP - RT2	$\mathit{\beta }$	bias	rmse	sd		$\mathit{\beta }$	bias	rmse	sd
$\gamma =-0.75$	0	−0.0280	0.0588	0.0518	$\gamma =0.75$	0	−0.0043	0.0628	0.0627
	0.1	−0.0280	0.0588	0.0518		0.1	−0.0043	0.0628	0.0627
	0.2	−0.0185	0.0460	0.0422		0.2	−0.0076	0.1031	0.1029
	0.3	−0.0185	0.0460	0.0422		0.3	−0.0076	0.1031	0.1029
	0.4	−0.0185	0.0460	0.0422		0.4	−0.0076	0.1031	0.1029
	0.5	−0.0048	0.0939	0.0938		0.5	−0.0064	0.1011	0.1010
$\gamma =-0.5$	0	−0.0062	0.0264	0.0257	$\gamma =1$	0	0.0011	0.0734	0.0734
	0.1	−0.0062	0.0264	0.0257		0.1	0.0011	0.0734	0.0734
	0.2	−0.0062	0.0264	0.0257		0.2	0.0011	0.0734	0.0734
	0.3	−0.0062	0.0264	0.0257		0.3	0.0011	0.0734	0.0734
	0.4	−0.0050	0.0283	0.0278		0.4	0.0011	0.0734	0.0734
	0.5	−0.0036	0.0256	0.0253		0.5	0.0011	0.0734	0.0734
$\gamma =-0.25$	0	−0.0050	0.0270	0.0266	$\gamma =1.25$	0	−0.0003	0.0733	0.0733
	0.1	−0.0050	0.0270	0.0266		0.1	−0.0003	0.0733	0.0733
	0.2	−0.0137	0.0524	0.0506		0.2	−0.0005	0.0733	0.0734
	0.3	−0.0399	0.1132	0.1059		0.3	−0.0817	0.4503	0.4431
	0.4	−0.0399	0.1132	0.1059		0.4	−0.0817	0.4503	0.4431
	0.5	−0.0425	0.1171	0.1092		0.5	−0.0975	0.5014	0.4921
$\gamma =0$	0	−0.0040	0.0340	0.0337	$\gamma =1.5$	0	0.0018	0.0787	0.0788
	0.1	−0.0040	0.0340	0.0337		0.1	0.0018	0.0787	0.0788
	0.2	−0.0040	0.0340	0.0337		0.2	0.0019	0.0786	0.0787
	0.3	−0.0040	0.0340	0.0337		0.3	0.0019	0.0786	0.0787
	0.4	−0.0040	0.0340	0.0337		0.4	0.0019	0.0786	0.0787
	0.5	−0.0040	0.0340	0.0337		0.5	0.0018	0.0786	0.0787
$\gamma =0.25$	0	−0.0017	0.0391	0.0391	$\gamma =1.75$	0	0.0136	0.5459	0.5460
	0.1	−0.0017	0.0399	0.0399		0.1	0.0136	0.5459	0.5460
	0.2	−0.0017	0.0399	0.0399		0.2	0.0136	0.5459	0.5460
	0.3	−0.0017	0.0399	0.0399		0.3	−0.0046	0.1027	0.1026
	0.4	−0.0017	0.0399	0.0399		0.4	−0.0046	0.1027	0.1026
	0.5	−0.0017	0.0404	0.0404		0.5	−0.0062	0.1272	0.1271
$\gamma =0.5$	0	−0.0044	0.0469	0.0467	$\gamma =2$	0	−0.0080	0.1007	0.1004
	0.1	−0.0044	0.0469	0.0467		0.1	−0.0080	0.1007	0.1004
	0.2	−0.0785	0.3103	0.3003		0.2	−0.0068	0.0998	0.0996
	0.3	−0.0785	0.3103	0.3003		0.3	−0.0068	0.0998	0.0996
	0.4	−0.0785	0.3103	0.3003		0.4	−0.0068	0.0998	0.0996
	0.5	−0.0785	0.3103	0.3003		0.5	−0.0068	0.0998	0.0996

,

Table A3. Simulation results from GEV models by applying criterion RT1 in (4).

GEV - RT1	$\mathit{\beta }$	bias	rmse	sd		$\mathit{\beta }$	bias	rmse	sd
$\gamma =-0.75$	0	−0.2680	0.2874	0.1038	$\gamma =0.75$	0	−0.1104	0.4349	0.4209
	0.1	−0.2591	0.2831	0.1143		0.1	−0.1104	0.4349	0.4209
	0.2	−0.2388	0.2628	0.1097		0.2	−0.1104	0.4349	0.4209
	0.3	−0.1187	0.1301	0.0533		0.3	−0.1104	0.4349	0.4209
	0.4	−0.1154	0.1259	0.0505		0.4	−0.1104	0.4349	0.4209
	0.5	−0.1105	0.1212	0.0500		0.5	−0.1104	0.4349	0.4209
$\gamma =-0.5$	0	−0.2084	0.2284	0.0937	$\gamma =1$	0	−0.1337	0.1467	0.0604
	0.1	−0.2057	0.2266	0.0950		0.1	−0.1073	0.1240	0.0622
	0.2	−0.2017	0.2216	0.0921		0.2	−0.0251	0.0829	0.0790
	0.3	−0.1825	0.2007	0.0835		0.3	−0.0133	0.1707	0.1703
	0.4	−0.1825	0.2007	0.0835		0.4	−0.1002	0.4764	0.4660
	0.5	−0.1736	0.1928	0.0840		0.5	−0.1002	0.4764	0.4660
$\gamma =-0.25$	0	−0.0812	0.0920	0.0434	$\gamma =1.25$	0	0.0021	0.0818	0.0819
	0.1	−0.0791	0.0901	0.0432		0.1	0.0028	0.0826	0.0826
	0.2	−0.0549	0.0851	0.0650		0.2	0.0067	0.0847	0.0845
	0.3	−0.1261	0.2512	0.2174		0.3	0.0067	0.0850	0.0847
	0.4	−0.1261	0.2512	0.2174		0.4	−0.1076	0.5224	0.5115
	0.5	−0.1301	0.2588	0.2238		0.5	−0.1076	0.5224	0.5115
$\gamma =0$	0	−0.1229	0.1292	0.0397	$\gamma =1.5$	0	−0.0974	0.5369	0.5282
	0.1	−0.1217	0.1281	0.0399		0.1	−0.0974	0.5369	0.5282
	0.2	−0.1207	0.1272	0.0401		0.2	−0.0974	0.5369	0.5282
	0.3	−0.0773	0.0916	0.0493		0.3	−0.0974	0.5369	0.5282
	0.4	−0.0771	0.0915	0.0493		0.4	−0.1210	0.5587	0.5457
	0.5	−0.0716	0.1795	0.1647		0.5	−0.1210	0.5587	0.5457
$\gamma =0.25$	0	−0.1328	0.1399	0.0441	$\gamma =1.75$	0	0.1192	0.1557	0.1002
	0.1	−0.1046	0.1152	0.0484		0.1	0.1165	0.1551	0.1024
	0.2	−0.0902	0.1038	0.0514		0.2	0.1099	0.1553	0.1098
	0.3	−0.1401	0.3654	0.3376		0.3	−0.0395	0.4020	0.4003
	0.4	−0.1401	0.3654	0.3376		0.4	−0.0395	0.4020	0.4003
	0.5	−0.1401	0.3654	0.3376		0.5	−0.0395	0.4020	0.4003
$\gamma =0.5$	0	−0.0351	0.1847	0.1815	$\gamma =2$	0	0.9712	25.4133	25.4075
	0.1	−0.0351	0.1847	0.1815		0.1	−0.0415	0.5035	0.5020
	0.2	−0.0351	0.1847	0.1815		0.2	−0.0367	0.6239	0.6231
	0.3	−0.0352	0.1867	0.1835		0.3	−0.0367	0.6239	0.6231
	0.4	−0.0349	0.1872	0.1840		0.4	−0.0367	0.6239	0.6231
	0.5	−0.0510	0.2386	0.2332		0.5	−0.0401	0.7818	0.7812

and

Table A4. Simulation results from GEV models by applying criterion RT2 in (5).

GEV - RT2	$\mathit{\beta }$	bias	rmse	sd		$\mathit{\beta }$	bias	rmse	sd
$\gamma =-0.75$	0	−0.2920	0.3135	0.1140	$\gamma =0.75$	0	−0.1104	0.4349	0.4209
	0.1	−0.2920	0.3135	0.1140		0.1	−0.1104	0.4349	0.4209
	0.2	−0.2920	0.3135	0.1140		0.2	−0.1104	0.4349	0.4209
	0.3	−0.2920	0.3135	0.1140		0.3	−0.1104	0.4349	0.4209
	0.4	−0.2920	0.3135	0.1140		0.4	−0.1104	0.4349	0.4209
	0.5	−0.2383	0.2637	0.1130		0.5	−0.1104	0.4349	0.4209
$\gamma =-0.5$	0	−0.2084	0.2284	0.0937	$\gamma =1$	0	−0.0682	0.0949	0.0660
	0.1	−0.2057	0.2266	0.0950		0.1	−0.0682	0.0949	0.0660
	0.2	−0.2017	0.2216	0.0921		0.2	−0.0682	0.0949	0.0660
	0.3	−0.2017	0.2216	0.0921		0.3	−0.0689	0.0954	0.0661
	0.4	−0.1991	0.2185	0.0900		0.4	−0.0689	0.0954	0.0661
	0.5	−0.1991	0.2185	0.0900		0.5	−0.0689	0.0954	0.0661
$\gamma =-0.25$	0	−0.0779	0.0895	0.0441	$\gamma =1.25$	0	0.0072	0.0853	0.0851
	0.1	−0.0779	0.0895	0.0441		0.1	0.0077	0.0879	0.0876
	0.2	−0.0779	0.0895	0.0441		0.2	0.0077	0.0879	0.0876
	0.3	−0.0779	0.0895	0.0441		0.3	0.0077	0.0879	0.0876
	0.4	−0.0779	0.0895	0.0441		0.4	0.0077	0.0879	0.0876
	0.5	−0.1316	0.2671	0.2326		0.5	0.0077	0.0879	0.0876
$\gamma =0$	0	−0.1215	0.1279	0.0400	$\gamma =1.5$	0	−0.1210	0.5587	0.5457
	0.1	−0.1204	0.1269	0.0401		0.1	−0.1210	0.5587	0.5457
	0.2	−0.1204	0.1269	0.0401		0.2	−0.1210	0.5587	0.5457
	0.3	−0.1175	0.1243	0.0406		0.3	−0.1210	0.5587	0.5457
	0.4	−0.1175	0.1243	0.0406		0.4	−0.1210	0.5587	0.5457
	0.5	−0.1131	0.1203	0.0412		0.5	−0.1210	0.5587	0.5457
$\gamma =0.25$	0	−0.0771	0.0946	0.0548	$\gamma =1.75$	0	0.0505	0.1010	0.0875
	0.1	−0.0771	0.0946	0.0548		0.1	0.0505	0.1010	0.0875
	0.2	−0.0771	0.0946	0.0548		0.2	0.0505	0.1010	0.0875
	0.3	−0.0771	0.0946	0.0548		0.3	0.1001	0.1499	0.1117
	0.4	−0.0771	0.0946	0.0548		0.4	0.0985	0.1489	0.1118
	0.5	−0.0771	0.0946	0.0548		0.5	0.0985	0.1489	0.1118
$\gamma =0.5$	0	−0.0352	0.1889	0.1857	$\gamma =2$	0	0.9712	25.4133	25.4075
	0.1	−0.0352	0.1889	0.1857		0.1	−0.0439	0.5346	0.5331
	0.2	−0.0361	0.1917	0.1884		0.2	−0.0367	0.6239	0.6231
	0.3	−0.0361	0.1917	0.1884		0.3	−0.0367	0.6239	0.6231
	0.4	−0.0361	0.1917	0.1884		0.4	−0.0367	0.6239	0.6231
	0.5	−0.0361	0.1917	0.1884		0.5	−0.0367	0.6239	0.6231

in the Appendix A. We present the bias, the root mean squared error (rmse), and the standard deviation (sd).

We observe better results within GP models, which are the natural framework of the ML estimator application. In these models, better results are achieved for $\beta \le 0.1$ except for $\gamma =1.75$ in the RT2 criterion, where a better option is $\beta =0.3,0.4$. In GEV models, in the RT1 criterion, if $\gamma \le 0$, we recommend $\beta =0.3,0.4$; otherwise, if $\gamma >0$, take $\beta =0.1,0.2$. According to the RT2 criterion, if $\gamma \le 0$, the choice $\beta =0.5$ is advisable (except in the case of $\gamma =-0.25$, where $\beta =0$ works better); otherwise, if $\gamma >0$, then $\beta =0.1$ is a better option. However, we do not recommend this methodology in GEV models whenever $\gamma =0.75,1.5,2$ (the rmse approximates $0.5$).

4. Application

Our dataset consists of the daily maximum log returns of Apple Inc. (AAPL) in the stock market obtained from https://finance.yahoo.com/quote/AAPL/ (accessed on 20 January 2025), in the period of 2020–2024. We apply a GARCH filter to remove heteroscedasticity. In Figure 1, we can see the filtered data. We will analyze both gains (right tail) and losses (left tail). The trajectory of the ${\widehat{\gamma }}_k$ estimates is plotted in Figure 2 for the left tail and for the right tail. The horizontal lines correspond to estimates obtained by applying the RT1 and RT2 criteria, with $\beta =0.1$. Looking at gains, we obtain $k=290$ and ${\widehat{\gamma }}_{290}=0.1028$ using the RT1 method and $k=288$ with ${\widehat{\gamma }}_{288}=0.1061$ by the RT2 method. Both values are quite close. Regarding losses, the RT1 method leads to $k=444$ and ${\widehat{\gamma }}_{444}=-0.0442$, while RT2 leads to $k=516$ and ${\widehat{\gamma }}_{516}=-0.0459$. These two values are quite close to 0. See also Figure 3, where we can observe that the tail empirical d.f. is quite close to the tail modeling through the POT approach based on the ML estimates obtained from the RT method. The case $\gamma =0$ (middle panel of Figure 3) seems plausible too. ${\mathrm{VaR}}_p$ estimation is plotted in Figure 4 for periods $1/\left(250p\right)$, with $p=0.1,...,0.00001$ in years (we consider $\sim 250$ observations per year). ${\mathrm{ES}}_p$ estimation can be seen in Figure 5. Both Var and ES estimates also appear to corroborate the obtained RT estimates. Based on these results, gains are more substantial than losses, as the former presents a heavier tail, which is a good sign for investors.