The importance of window size: a study on the required window size for optimal-quality market risk models

Mateusz Buczyński and Marcin Chlebus

Abstract

When it comes to market risk models, should we use the full data set that we possess or rather find a sufficient subsample? We conduct a study of different fixed moving-window lengths: moving-window sizes varying from 300 to 2000 are considered for each of the 250 combinations of data and a value-at-risk evaluation method. Three value-at-risk models (historical simulation, a generalized autoregressive conditional heteroscedasticity (GARCH) model and a conditional autoregressive value-at-risk (CAViaR) model) are used for three different indexes (the Warsaw Stock Exchange 20, the Standard & Poor’s 500 and the Financial Times Stock Exchange 100) for the period 2015–19. We also address subjectivity in choosing the window size by testing change point detection algorithms (binary segmentation and pruned exact linear time) to find the best matching cutoff point. Results indicate that a training sample size greater than 900–1000 observations does not increase the quality of the model, while lengths lower than this cutoff provide unsatisfactory results and decrease the model’s predictive power. Change point detection methods provide more accurate models: applying the algorithms to each model’s recalculation on average provides results better by one exceedance. Our recommendation is to use GARCH or CAViaR models with recalculated window sizes.

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Introduction

1 Introduction

According to econometric modeling strategies, there always exists a minimum number of observations that allows us to draw conclusions from estimated covariates. In equations that involve confidence intervals, we notice that the uncertainty decreases as the number of observations increases. In terms of market risk forecasting, this suggests that the more data that is fed into a model, the more certain and accurate its forecasts are. But is this true?

Value-at-risk (VaR) is one of the most popular measures of market risk. It pictures the maximum loss over a target horizon that will not be exceeded with a given confidence level, given normal market conditions (Jorion 2006; Dowd 2010). There are many approaches to VaR modeling, but in general it can be understood as a specific quantile of an assumed distribution of the predicted realized rates of return. Our point of interest is the question of how many observations should be used to build such a distribution. Too few observations and the training sample’s distribution will be very volatile, while too many will expose the model to unnecessary bias that is not observed at the time of modeling. Therefore, the task of determining the size of the training window is not as straightforward as it would seem. In addition, there is no empirical consensus due to the lack of wide-ranging studies in the area.

2 Literature review

The document that regulates approaches to modeling market risk in financial institutions is Basel III (Basel Committee on Banking Supervision 2019), which lays out rules for internal model creation to be followed by all banks and funds (Lee 2014). According to its most recent supplement (which is from 2017 and emerged in response to the global financial crisis of 2007–9), any market risks should be defined by a measure of expected shortfall (also called conditional VaR) at a 2.5% confidence level for at least the next 10 trading days. As for the time series length used to build the model, the necessary minimum number of observations is 250 trading days (approximately one year).

Regardless of the predetermined rules, Basel III does not specify any particular approach to VaR modeling. There are three main families of models to consider: parametric approaches that assume a specific distribution of the realized rates of return and aim to estimate its parameters (eg, generalized autoregressive conditional heteroscedasticity (GARCH) models); nonparametric approaches that do not assume any distributions, with estimates based only on empirical data (eg, historical simulation); and semiparametric approaches that have characteristics of both the other families (eg, the conditional autoregressive value-at-risk (CAViaR) model).

Due to this abundance of approaches, the field of empirical studies devoted to VaR contains many comparisons of different approaches in different scenarios, identifying the best scenarios in which to use particular models (Abad et al 2014). None of these papers claim any particular model to be the best overall, but they specify the characteristics of market conditions in which particular models perform best. Most recent papers are in favor of semiparametric methods, which are both accurate and flexible (Patton et al 2019; Wang and Zhao 2016; Abad and Benito 2013; Martins-Filho et al 2018; Taylor 2019; Abad et al 2016; Nozari et al 2010; Şener et al 2012). However, some researchers have found parametric methods to perform better, claiming that modeling the distribution is more accurate (Buczyński and Chlebus 2018, 2020; Ergün and Jun 2010; Berkowitz and O’Brien 2002; Bao et al 2006; Consigli 2002; Danielsson 2002; Sarma et al 2003). In addition, most of the aforementioned studies compare their favored models with the most popular nonparametric method – historical simulation. These studies tend to find that historical simulation performs worse than the other studied models.

Unfortunately, most of the studies in this field fix the length of the training window beforehand. A very long training sample window is most commonly assumed, with sizes of 1000 observations or more (Bao et al 2006; Danielsson 2002; Sarma et al 2003; Patton et al 2019; Martins-Filho et al 2018; Nozari et al 2010; Buczyński and Chlebus 2018, 2020; Ergün and Jun 2010). Some of the aforementioned studies use a small range of different window lengths, but they do not draw any conclusions as to particular models’ sensitivities to these values. However, Hendricks et al (1996) comment that longer window sizes produced forecasts of better quality. It is rather rare for researchers to try shorter window lengths (Wang and Zhao 2016; Abad and Benito 2013; Şener et al 2012; Berkowitz and O’Brien 2002). Finally, some researchers do not report the length of the training sample at all (Das and Rout 2020; Gerlach et al 2011; Ergün and Jun 2010).

Automatic methods of detecting the appropriate sample size have recently received some attention. The most popular approach is to find the closest change point in the time series in order to train the model on homogeneous (in terms of expected value or volatility) series, assuming normal (stable) market conditions. For example, Smith and Huang (2019) explored two approaches to finding such a point: at-most-one-change and binary segmentation. Their results indicate that these methods might provide models with greater predictive power than those choosing fixed training sample sizes. Číźek et al (2009) argued that by employing change point techniques it is possible to achieve a more accurate and flexible model that works over longer periods of time.

The aim of this paper is to compare different VaR approaches for multiple sample sizes. Primarily, we want to estimate VaR models for window lengths from 50 to 2000 and compare their excess ratios to find out whether there is any level over which increasing the sample size does not provide any benefit in terms of greater quality. Such an analysis can provide a comprehensive overview of sample size selection for a particular model. To create an unconditional environment for these models, we test 15 different time series for each approach (five time periods for three different stock indexes). In addition, we specify a nonsubjective criterion to find the best-fitting sample size. Specifically, we select two change point detection algorithms – binary segmentation and pruned exact linear time (PELT) (Killick et al 2012) – to find the window size with the best fit. These algorithms are used both before each model retraining and during the estimation process.

In the next section, the methodologies of the VaR approaches used in the paper are presented. Section 4 presents the data and discusses the setup of the experiment. Section 5 presents the results, and Section 6 states our main conclusions.

3 Methodology

VaR is defined as a maximum loss over a given time horizon $t$, at a given level of confidence $\alpha$ and assuming normal market conditions. Importantly, VaR is also a quantile of the empirical distribution of gains and losses over a selected time horizon. Jorion (2006) defines VaR via the following equation:

(3.1)

where $r_t$ is the rate of return of the asset under consideration and ${\mathrm{\Omega }}_{t-1}$ is an information set given at time $t-1$.

It is also important to show how VaR models are backtested. One of the simplest approaches is to count the number of occurrences of the realized rate of return exceeding the VaR forecast. Such a measure is called an exceedance (denoted by $I_t$); when the exceedances are expressed in terms of a relation to the whole backtested horizon of length $N$ we may introduce the excess ratio as follows:

$$\widehat{\alpha }=\frac{1}{N}\sum _{t=1}^N{I}_t.$$

(3.2)

3.1 Historical simulation

The simplest nonparametric approach to VaR modeling is historical simulation (Dowd 2010). It is based on the aforementioned fact that VaR is a quantile of historical returns. VaR is summarized by an $\alpha$-quantile, $q_{\alpha }$, of the empirical distribution of the rates of return of the studied asset. In this approach, much depends on the sample size, due to intentional (or not) inclusion of time periods of heterogeneous volatility. In practice, the width of the window is fixed and usually ranges from six months to two years (125–500 observations) (Engle and Manganelli 2001):

$${\mathrm{VaR}}_{\alpha }(t)=q_{\alpha }.$$

(3.3)

3.2 GARCH models

GARCH models are one the most common parametric VaR models used in market risk modeling over the past few years. In this paper the specific GARCH model under consideration is the $\mathrm{GARCH}(1,1)$ model with a skewed Student $t$ distribution. $\mathrm{GARCH}(1,1)$ is one of the simplest of the GARCH family of models and can be described by the following system (Engle 1982; Bollerslev 1986):

(3.4)

where $\mathrm{iid}({\mu }_t,h_t)$ is an independent and identical distribution with conditional mean ${\mu }_t$ and with conditional variance $h_t$ described by the sum of a specific number of lagged squared error terms and conditional variances weighted by two vectors of parameters, $𝜷$ and $𝜸$.

Given these equations, we can define VaR as

$${\mathrm{VaR}}_{\alpha }(t)={\widehat{\mu }}_t+q_{\alpha }\sqrt{{\widehat{h}}_t},$$

(3.5)

where $q_{\alpha }$ is an $\alpha$-quantile of the assumed distribution, while ${\widehat{\mu }}_t$ and ${\widehat{h}}_t$ are the estimated conditional mean and variance for time $t$ (Angelidis et al 2004).

Theoretically, only a normal distribution should be used as the conditional distribution of the model; however, Bollerslev and Wooldridge (1992) have proven that if the model is not conditionally normally distributed, but it specifies the first two conditional moments correctly, then the estimates of the quasilikelihood function will be consistent and asymptotically normal. Therefore, the use of distributions other than the normal for the underlying process is completely correct and desirable. It is a common characteristic of any time series with a financial origin that the distribution of the returns is skewed and has a tendency to have nonzero kurtosis, which drastically lowers the quality of models based on a normal distribution. Most studies in the literature find the Student $t$ distribution – in particular the skewed version (Ergen 2012) – to have the best fit.

3.3 The CAViaR model

One of the most common semiparametric VaR models is the CAViaR model, introduced by Engle and Manganelli (2004). This estimates the quantile of the distribution of the data directly instead of trying to model the whole distribution. It is based on the quantile regression methodology of Koenker and Bassett (1978). The basic formula for the CAViaR model (with one lagged VaR and one lagged observed value) can be expressed as

$${\mathrm{VaR}}_{\alpha }(t)={\beta }_0+{\beta }_1{\mathrm{VaR}}_{\alpha }(t-1)+l({\beta }_2,r_{t-1},{\mathrm{VaR}}_{\alpha }(t-1)),$$

(3.6)

where $l(\cdot )$ is a linking function of the lagged values of the observables and VaR, while $𝜷$ is a vector of parameters. The point of using a linking function in the expression of CAViaR is to link the model’s outcome to the rate of return at time $t-1$.

In this study we use one of the CAViaR specifications presented by Engle and Manganelli (2004), indirect GARCH:

$${\mathrm{VaR}}_{\alpha }(t)=\sqrt{{\beta }_0+{\beta }_1{\mathrm{VaR}}_{\alpha }(t-1)+{\beta }_2{(r_{t-1})}^2}.$$

(3.7)

The indirect GARCH approach to the CAViaR model is very similar to GARCH modeling. In fact, it would be a correctly specified model if the underlying data were a $\mathrm{GARCH}(1,1)$ process with an independent and identical distribution.

3.4 Change point detection

Change point detection algorithms aim to find an observation that determines an influential change in the time series. The main objective of these algorithms is to build a nonoverlapping segmentation of the underlying model of the time series, based on the shifts detected in its characteristics. The area covered by the research into this topic is very broad, as these techniques are widely used in signal processing and have many applications in finance, bioinformatics, medicine and many other fields (Aminikhanghahi and Cook 2017).

To provide a theoretical background, let us consider a nonstationary random process $y=\{y_1,\mathrm{\dots },y_t\}$. This process is also assumed to be piecewise stationary, ie, there are $K$ unknown instants at which some characteristics of this process change. The aim of change point detection algorithms is to find the best possible segmentation $\tau$ of the series, according to some general cost function $V(\tau ,y):={\sum }_{k=1}^{K}c(y_{t_k},\mathrm{\dots },y_{t_{k+1}})$, which is a summation over cost functions for particular segments. In our scenario we do not determine the number of segments $K$ beforehand, and hence the general cost function gets an additional penalty for the complexity of segmentation $\tau$. Therefore, following Truong et al (2020), the optimization problem can be determined as

$$\underset{\tau }{\min }V(\tau )+\mathrm{pen}(\tau ),$$

(3.8)

where $\mathrm{pen}(\tau )$ is the penalty, ie, a measure of the complexity of the segmentation $\tau$.

The cost function under consideration in this study is based on kernel methods. The original series is mapped onto the Hilbert space $\mathscr{H}$. In such a setting the cost function of a particular segment can be defined as

$$c(y_{t_k},\mathrm{\dots },y_{t_{k+1}}):=\sum _{t=t_k+1}^{t_{k+1}}{\parallel \varphi (y_t)-{\overline{\mu }}_{t_k,\mathrm{\dots },t_{k+1}}\parallel }_{\mathscr{H}}^2,$$

(3.9)

where ${\overline{\mu }}_{t_k,\mathrm{\dots },t_{k+1}}$ is the empirical mean of the process over the subsample from $t_k$ to $t_{k+1}$ and $\parallel \cdot {\parallel }_{\mathscr{H}}$ is a norm in the Hilbert space. The choice of the kernel function is, of course, unlimited, though the most commonly used kernel for numerical data is either Gaussian or linear. In the Gaussian scenario we can define

$$k(x,y)=\exp (-\gamma {\parallel x-y\parallel }^2),$$

(3.10)

where $x,y\in {ℝ}^d$ and $\gamma >0$ is called the bandwidth parameter. The cost function is therefore defined by (Truong et al 2020)

$$c(y_{t_k},\mathrm{\dots },y_{t_{k+1}}):=(t_{k+1}-t_k)-\frac{1}{t_{k+1}-t_k}\sum _{s,t=t_k+1}^{t_{k+1}}\exp (-\gamma {\parallel y_s-y_t\parallel }^2).$$

(3.11)

Regardless of the chosen cost function, there are several search methods, which are procedures for discrete optimization processes aimed at minimization of the formulated cost function. We have considered two such approaches: an optimal segmentation using the PELT algorithm (explained immediately below) and approximation by binary segmentation. Although control charts are also very simple and easy to implement for change point detection, we have not considered them in this paper as their formulation would require us to define a fixed window of control.

To find the optimal number of segments $K$, we could run the optimization for each $K$ and select the minimum. Fortunately, in the case of linear penalties for the number of segments, high computational costs can be avoided by the use of the PELT algorithm. This considers the series sequentially and, based on the pruning rule, may or may not include particular segments in the set of potential change points. The pruning rule may be determined as follows. If

$$\left[\underset{\tau }{\min }V(\tau ,y_{0,t})+\mathrm{pen}(\tau )\right]+c(y_{t,s})\ge \left[\underset{\tau }{\min }V(\tau ,y_{0,s})+\mathrm{pen}(\tau )\right],$$

where $y_{0,t}$ denotes a segment of observations from index $0$ to index $t$, then $t$ cannot be the last change point prior to $T$. In the literature, several studies make use of the PELT algorithm, for example, in applications to DNA sequences and oceanographic data (Hocking et al 2013; Killick et al 2012; Maidstone et al 2017).

Binary segmentation, on the other hand, is a more greedy sequential algorithm. The first change point ${\widehat{t}}_1$ is given by

$${\widehat{t}}_1:=\underset{1\le t\le T-1}{\arg \min }c(y_0,\mathrm{\dots },y_t)+c(y_t,\mathrm{\dots },y_T),$$

(3.12)

which means that the algorithm searches for the change point $t$ that minimizes the sum of costs. The series is then split into two at the instant ${\widehat{t}}_1$ and the same operation is repeated on the resulting subsamples until no further improvement can be made to the cost function. The solution is only an approximation of a perfect segmentation, since the detection of a change point is not based on a homogeneous sample and each detection is based on all previous ones. However, that does not diminish the quality of the algorithm, as it has been used in many applications in finance (Lavielle and Teyssière 2007; Bai 1997; Fryzlewicz 2014), as well as in bioinformatics and DNA subsampling (Niu and Zhang 2012; Olshen et al 2004).

4 Data and experiment setup

The experiment was conducted on a set of major stock indexes from three countries: Poland (the Warsaw Stock Exchange 20 (WIG20)), the United States (Standard & Poor’s 500 (S&P 500)) and the United Kingdom (Financial Times Stock Exchange 100 (FTSE 100)). There were five distinctive testing samples prepared, each consisting of 250 observations and set to end with the ends of the calendar years 2015, 2016, 2017, 2018 and 2019. Therefore, for each VaR approach studied (historical simulation, the GARCH model with a skewed Student $t$ distribution and the CAViaR model), we tested three different indexes on five different testing samples, resulting in 15 different time series. Each of the models was trained 250 times to generate a one-day-ahead forecast using a sliding-window technique with the length determined as described below.

The training sample was our point of interest for the experiment. In this paper we compare VaR models in terms of out-of-sample one-step-ahead predictive ability. Each of the testing samples consisted of 250 observations, while the remaining observations made prior to the beginning of the testing samples were used to build the training sample. We tested different lengths of the training sample, ie, the number of observations to be included, from the set $\{50,100,\mathrm{\dots },1250\}$. We also tested very large samples from the set $\{1500,1750,2000\}$. For each of the models, time series and training sample lengths, we calculated the number of exceedances, the results of which are presented in Tables 1–3.¹¹ 1 In several cases in Tables 1–3 the GARCH model could not reach convergence, and hence the results are not reported. The same applies to the CAViaR model, for which a minimum of 300 observations has been suggested by Engle and Manganelli (2004) to avoid a lack of convergence; the mean for cases below this minimum is also not reported. In addition, we have calculated the mean number of exceedances, which is an average over the tested years for a particular model and training sample size. All the resulting numbers are compared with the assumed number of exceedances, which for VaR at a 2.5% confidence level and a testing sample of 250 observations should be equal to $6.25\pm 4.06$ (with a 90% confidence interval).

We have also created a mechanism for automatically selecting the training sample’s length by applying the change point detection techniques mentioned in Section 3.4. Table 4 shows the results for change point detection applied for all the forecasts in the testing sample before any model fitting has taken place. Table 5 shows results for change point detection applied to recalculate the sample size each time a model is trained for another one-day-ahead forecast, and hence the algorithm was also applied 250 times. For both methods we have used the aforementioned PELT and binary segmentation algorithms. To create an even more controlled environment we propose two approaches to the change point detection algorithms: a liberal one and a conservative one. Both approaches are based on all the change points detected in the range from 500 to 1000, but for the liberal approach we select the change point closest to the 500th observation and for the conservative approach we select the change point closest to the 1000th observation. Therefore, the liberal approach takes the shortest sample size and the conservative one takes the longest. This results in four different scenarios for training sample lengths in each setup: conservative PELT and binary segmentation and liberal PELT and binary segmentation.

Table 1: Number of exceedances for particular training sample sizes in the tested years (WIG20).

Obs. in	Historical simulation						GARCH with skewed Student $t$						CAViaR
training
window	2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean
2000	01	01	00	08	5	03.0	08	08	1	08	5	6.0	09	07	0	07	5	5.6
1750	01	04	00	08	5	03.6	08	07	1	08	4	5.6	09	07	0	08	5	5.8
1500	05	07	00	08	5	05.0	08	09	1	08	4	6.0	09	08	1	08	6	6.4
1250	06	08	00	08	5	05.4	08	08	1	08	5	6.0	09	06	1	09	4	5.8
1200	06	08	00	08	5	05.4	08	08	1	08	5	6.0	09	07	1	07	4	5.6
1150	06	07	00	08	5	05.2	08	07	1	08	4	5.6	09	07	1	08	4	5.8
1100	06	09	00	08	5	05.6	08	08	1	08	4	5.8	09	08	1	08	4	6.0
1050	07	09	00	08	5	05.8	08	09	1	08	4	6.0	09	08	1	08	4	6.0
1000	08	09	00	08	5	06.0	08	08	2	08	4	6.0	09	09	1	08	5	6.4
0950	10	10	00	08	5	06.6	08	08	2	08	4	6.0	09	06	1	08	6	6.0
0900	10	10	00	08	6	06.8	08	08	2	08	5	6.2	10	10	2	08	7	7.4
0850	11	09	00	08	6	06.8	08	09	1	08	5	6.2	09	11	1	08	5	6.8
0800	12	09	00	08	6	07.0	08	10	1	08	5	6.4	11	09	1	08	6	7.0
0750	13	09	00	08	6	07.2	08	10	1	08	5	6.4	11	10	1	11	5	7.6
0700	12	09	00	08	6	07.0	08	11	1	08	5	6.6	12	10	1	13	6	8.4
0650	11	09	00	09	6	07.0	08	10	0	08	5	6.2	11	10	1	16	6	8.8
0600	13	09	00	11	6	07.8	09	10	2	09	—	—	11	10	4	12	6	8.6
0550	14	10	00	12	6	08.4	10	11	1	09	—	—	13	08	7	13	4	9.0
0500	15	08	00	13	6	08.4	10	10	2	10	—	—	12	09	6	14	6	9.4
0450	16	08	01	15	5	09.0	10	08	2	—	—	—	13	08	4	16	5	9.2
0400	17	08	02	15	5	09.4	11	09	2	—	—	—	11	06	4	15	5	8.2
0350	16	08	02	19	5	10.0	09	06	2	—	—	—	13	06	4	19	6	9.6
0300	15	08	02	18	5	09.6	11	06	2	—	—	—	13	05	2	17	5	8.4
0250	14	08	02	17	5	09.2	—	06	—	—	—	—	—	—	—	—	—	—
0200	14	06	02	15	6	08.6	—	06	—	—	—	—	—	—	—	—	—	—
0150	11	06	03	12	5	07.4	—	—	—	—	—	—	—	—	—	—	—	—
0100	12	06	07	09	4	07.6	—	—	—	—	—	—	—	—	—	—	—	—
0050	16	10	11	09	9	11.0	—	—	—	—	—	—	—	—	—	—	—	—

Table 2: Number of exceedances for particular training sample sizes in the tested years (S&P 500).

Obs. in	Historical simulation						GARCH with skewed Student $t$						CAViaR
training
window	2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean
2000	03	1	00	15	4	04.6	07	3	4	08	7	5.8	4	2	3	07	6	4.4
1750	03	3	00	19	5	06.0	07	4	4	09	7	6.2	4	3	3	09	6	5.0
1500	05	5	00	20	5	07.0	07	4	4	09	7	6.2	6	3	3	09	6	5.4
1250	06	5	01	18	4	06.8	07	4	4	08	7	6.0	6	4	3	09	6	5.6
1200	06	5	01	19	4	07.0	07	4	4	08	7	6.0	7	4	3	09	6	5.8
1150	06	6	01	19	4	07.2	07	4	4	08	7	6.0	7	4	3	09	7	6.0
1100	06	6	01	17	4	06.8	08	4	4	08	7	6.2	7	4	3	09	7	6.0
1050	07	8	01	17	4	07.4	08	4	4	08	7	6.2	7	4	3	10	7	6.2
1000	08	8	01	17	4	07.6	09	4	4	08	7	6.4	7	4	3	09	8	6.2
0950	08	8	01	16	4	07.4	10	4	4	08	7	6.6	9	4	3	09	8	6.6
0900	09	8	01	15	4	07.4	10	4	4	08	7	6.6	9	4	3	09	8	6.6
0850	10	7	01	15	4	07.4	10	4	4	08	7	6.6	9	4	3	10	8	6.8
0800	10	8	01	17	4	08.0	10	4	4	09	7	6.8	8	4	3	10	7	6.4
0750	10	8	01	17	4	08.0	10	4	4	09	6	6.6	9	4	4	09	8	6.8
0700	10	7	01	20	4	08.4	10	4	4	09	6	6.6	9	4	3	11	7	6.8
0650	10	7	01	19	4	08.2	10	4	4	09	6	6.6	9	4	3	10	6	6.4
0600	10	6	01	20	4	08.2	10	4	4	09	6	6.6	9	4	3	11	7	6.8
0550	10	5	00	21	4	08.0	10	4	4	09	6	6.6	9	4	4	12	6	7.0
0500	10	5	00	24	4	08.6	10	4	4	10	6	6.8	9	4	4	13	6	7.2
0450	10	5	00	22	4	08.2	10	—	4	10	6	—	9	4	5	12	6	7.2
0400	10	5	03	19	4	08.2	—	—	4	09	6	—	9	5	5	11	7	7.4
0350	09	5	03	18	4	07.8	09	—	4	09	6	—	9	5	5	12	6	7.4
0300	10	5	03	18	4	08.0	09	—	4	11	6	—	8	5	5	12	6	7.2
0250	09	5	06	17	4	08.2	—	—	6	11	6	—	—	—	—	—	—	—
0200	08	4	05	15	4	07.2	—	—	7	10	7	—	—	—	—	—	—	—
0150	08	5	07	16	3	07.8	—	—	8	—	7	—	—	—	—	—	—	—
0100	08	4	08	13	4	07.4	—	—	8	—	—	—	—	—	—	—	—	—
0050	13	8	12	11	6	10.0	—	—	—	—	—	—	—	—	—	—	—	—

Table 3: Number of exceedances for particular training sample sizes in the tested years (FTSE 100).

Obs. in	Historical simulation						GARCH with skewed Student $t$						CAViaR
training
window	2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean
2000	04	03	01	03	4	03.0	08	4	4	07	7	6.0	08	4	2	06	5	5.0
1750	04	05	01	04	5	03.8	08	4	4	07	7	6.0	09	4	2	06	6	5.4
1500	09	06	01	04	4	04.8	09	4	4	06	7	6.0	09	4	2	06	5	5.2
1250	10	06	01	04	4	05.0	09	4	4	06	6	5.8	10	4	2	05	6	5.4
1200	10	06	01	04	4	05.0	09	4	4	06	6	5.8	10	4	2	05	5	5.2
1150	10	07	01	04	4	05.2	09	4	4	06	7	6.0	10	4	2	06	6	5.6
1100	10	07	01	04	4	05.2	09	4	4	06	7	6.0	10	4	2	06	6	5.6
1050	10	08	01	03	5	05.4	09	4	4	06	7	6.0	10	4	2	06	6	5.6
1000	11	08	01	03	5	05.6	09	4	4	06	7	6.0	10	4	3	06	7	6.0
0950	13	07	01	04	5	06.0	09	4	4	06	7	6.0	10	4	2	07	7	6.0
0900	13	09	01	04	5	06.4	09	4	4	06	7	6.0	10	4	3	06	6	5.8
0850	13	09	01	04	6	06.6	11	5	4	06	7	6.6	10	4	3	08	7	6.4
0800	14	07	01	04	6	06.4	11	4	4	06	7	6.4	10	4	4	09	7	6.8
0750	16	07	01	04	6	06.8	11	4	4	07	7	6.6	10	5	4	09	7	7.0
0700	15	06	01	05	6	06.6	11	4	4	06	7	6.4	10	3	3	10	7	6.6
0650	13	07	01	06	6	06.6	11	4	4	09	7	7.0	10	2	4	12	8	7.2
0600	14	06	01	08	6	07.0	11	4	4	09	7	7.0	10	3	4	15	8	8.0
0550	15	06	01	10	6	07.6	11	4	4	11	7	7.4	09	3	6	15	9	8.4
0500	14	06	01	13	6	08.0	11	5	4	10	7	7.4	10	3	6	17	9	9.0
0450	13	05	01	14	6	07.8	—	5	4	11	7	—	11	3	6	15	8	8.6
0400	12	04	01	15	6	07.6	11	5	4	11	7	7.6	10	3	4	—	9	—
0350	11	04	02	14	6	07.4	11	4	4	12	7	7.6	12	3	4	—	8	—
0300	11	04	02	14	6	07.4	12	—	4	11	7	—	09	4	5	—	8	—
0250	08	03	02	14	6	06.6	12	—	5	11	7	—	—	—	—	—	—	—
0200	07	04	05	13	6	07.0	10	—	3	13	7	—	—	—	—	—	—	—
0150	08	03	08	12	6	07.4	—	—	—	—	7	—	—	—	—	—	—	—
0100	09	06	09	14	6	08.8	—	—	—	—	—	—	—	—	—	—	—	—
0050	13	12	12	15	7	11.8	—	—	—	—	—	—	—	—	—	—	—	—

Table 4: Number of exceedances using automatic change point detection for predetermined sample sizes, for the WIG20, S&P 500 and FTSE 100. [“BinSeg” refers to the binary segmentation algorithm, while “Lib.” and “Con.” refer to liberal and conservative approaches, which are explained in the text.]

								GARCH
		Historical simulation						with skewed Student $t$						CAViaR
		2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean
(a) WIG20
PELT	Lib.	14	9	0	13	6	8.4	10	10	1	09	5	7.0	11	11	2	12	6	8.4
	Con.	11	9	0	08	5	6.6	08	10	2	08	4	6.4	08	09	2	08	5	6.4
BinSeg	Lib.	13	9	0	10	6	7.6	09	11	0	08	5	6.6	11	10	1	11	3	7.2
	Con.	12	9	0	08	5	6.8	08	08	0	08	4	5.6	09	10	2	08	6	7.0
(b) S&P 500
PELT	Lib.	10	7	1	24	4	9.2	10	00	4	10	7	6.2	09	04	3	12	6	6.8
	Con.	08	8	1	18	4	7.8	10	04	4	08	7	6.6	08	04	3	09	8	6.4
BinSeg	Lib.	10	7	0	24	4	9.0	10	00	4	10	7	6.2	09	04	3	12	6	6.8
	Con.	08	8	1	18	4	7.8	10	04	4	08	7	6.6	08	04	3	09	7	6.2
(c) FTSE 100
PELT	Lib.	15	6	1	11	6	7.8	11	04	4	10	7	7.2	10	03	3	14	9	7.8
	Con.	13	7	1	04	5	6.0	09	04	4	06	7	6.0	10	05	3	09	7	6.8
BinSeg	Lib.	15	6	1	14	6	8.4	11	04	4	10	7	7.2	10	03	3	14	9	7.8
	Con.	13	7	1	04	5	6.0	09	04	4	06	7	6.0	10	05	3	08	6	6.4

Table 5: Number of exceedances using automatic change point detection for recalculated sample sizes, for the WIG20, S&P 500 and FTSE 100. [“BinSeg” refers to the binary segmentation algorithm, while “Lib.” and “Con.” refer to liberal and conservative approaches, which are explained in the text.]

								GARCH
		Historical simulation						with skewed Student $t$						CAViaR
		2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean	2015	2016	2017	2018	2019	Mean
(a) WIG20
PELT	Lib.	12	9	0	08	6	7.0	08	8	2	8	5	6.2	11	7	2	08	7	7.0
	Con.	06	8	0	08	5	5.4	08	8	1	8	5	6.0	09	8	1	09	4	6.2
BinSeg	Lib.	09	9	0	08	5	6.2	08	8	2	8	4	6.0	10	6	2	09	5	6.4
	Con.	06	8	0	08	5	5.4	08	7	1	8	5	5.8	09	7	1	08	5	6.0
(b) S&P 500
PELT	Lib.	09	8	1	15	4	7.4	10	4	4	9	7	6.8	09	4	3	10	7	6.6
	Con.	06	5	1	19	4	7.0	07	4	4	8	7	6.0	08	4	3	09	6	6.0
BinSeg	Lib.	10	8	1	15	4	7.6	10	4	4	9	7	6.8	09	4	3	10	8	6.8
	Con.	06	5	1	19	4	7.0	07	4	4	8	7	6.0	07	4	3	09	6	5.8
(c) FTSE 100
PELT	Lib.	15	9	1	04	5	6.8	11	4	4	6	7	6.4	10	5	3	09	7	6.8
	Con.	10	6	1	04	4	5.0	09	4	4	6	6	5.8	10	4	2	06	6	5.6
BinSeg	Lib.	14	8	1	04	6	6.6	10	4	4	6	7	6.2	10	5	3	09	7	6.8
	Con.	10	6	1	04	4	5.0	09	4	4	6	7	6.0	10	4	2	06	7	5.8

5 Results

5.1 Fixed training sample size

For the WIG 20, most of the exceedances are within the confidence bands regardless of the model. The results are rather stable over the different sample sizes, whereas they are not stable between years. For the S&P 500, the results are very similar; however, the average number of exceedances is lower than in the case of the WIG20. The results here are also much more stable along the window size axis. For the FTSE 100, the results are the least stable in terms of the window size. However, even for this index the results are lower than the upper confidence band.

The results indicate that for each of the tested indexes and models there is a downward tendency of mean exceedances with increasing length of the training sample. For the smallest numbers of observations in the training sample, we can observe widely varying, often high numbers of exceedances over the studied years. The stabilization of the number of exceedances, regardless of the underlying time series and model, starts at around 900–1000 observations. It is a subjectively chosen point; but having studied all the cases separately, we draw the conclusion that, beyond this point, if we increase the number of observations in the sample, we do not see any significant increase in the number of exceedances. There is also a small, but noteworthy, upward trend of exceedances for the smallest learning sample sizes (excluding the cases where the algorithms did not converge).

In this study we also researched very long spans for the training samples (up to 2000 observations) to prove that the information added to the model with such long spans is not reflected in an improvement in results. For each of the studied indexes, we see very small improvements in the number of exceedances (apart from the historical simulation results, which tend to score minimums for lengths of 1500–2000 observations) and we believe that such a tendency would apply for even larger training samples.

The historical simulation results tend to agree with the literature. The lowest score for each of the indexes belongs to historical simulation with a very long span for the learning sample (2000 observations); however, such a score is below the assumed number of exceedances. For shorter training samples, we observe much worse results for historical simulation than we do for the other models. In addition, we do not observe the “convergence” in the number of exceedances that is clearly seen in the results of the GARCH and CAViaR models. We conclude that the assumed significance level of a VaR model is not important at all, because it can easily be affected by the number of observations in the training sample. Nevertheless, the assumed significance level is met for most of the series at 600–800 observations, although for some it is never met and for others it is met even with very small training samples.

The GARCH and CAViaR results are very similar to each other, with CAViaR’s results having a tendency to be slightly worse in the number of the exceedances for the years that were more volatile. The results show that they correctly estimate the risk to be more or less the assumed level for at least 500 observations in the training sample for the GARCH model and 700–900 observations (depending on the testing sample) in the training sample for the CAViaR model. Unsurprisingly, the higher the number of observations in the training sample is for these models, the lower the estimated number of exceedances; but we can also observe that the benefit gained from increasing the sample size diminishes with progressively larger samples. The only flaw of both of these models is a problem with convergence for very small samples, which is to be expected, but it gives an advantage to historical simulation if training needs to be done on a small training sample. In addition, both of these models are better in terms of mean exceedances when compared at the same level of training sample size for all the tested sizes up to around 1000.

The downward tendency in the number of exceedances that we observe can also be subjectively divided into two distinct ranges of training sample lengths: the liberal one (from around 500–600 observations to the threshold of convergence, around 900–1000 observations) and the conservative one (the remaining studied training sample sizes above the threshold of convergence). The division is based on the differing results for the number of exceedances in each range for particular years. For the liberal sample size lengths we can observe large differences for 2015 and 2018, which are characterized by several excessive volatility shocks. As we increase the number of observations in the training sample, the differences tend to diminish and the distribution of exceedances for particular years starts to become more uniform, which in our opinion is a characteristic of conservative models.

5.2 Automatic training sample length selection

Table 6: Calculated number of observations using change point detection algorithms for the WIG20, S&P 500 and FTSE 100. [“BinSeg” refers to the binary segmentation algorithm, while “Lib.” and “Con.” refer to liberal and conservative approaches, which are explained in the text.]

	WIG20				S&P 500				FTSE 100
	PELT		BinSeg		PELT		BinSeg		PELT		BinSeg
	Lib.	Con.	Lib.	Con.	Lib.	Con.	Lib.	Con.	Lib.	Con.	Lib.	Con.
2015	550	850	585	855	605	975	605	975	535	965	535	965
2016	680	800	695	835	625	945	625	955	590	955	590	955
2017	685	930	675	675	575	975	520	950	580	915	580	915
2018	530	945	635	945	525	995	525	995	520	825	510	835
2019	725	980	615	980	625	950	625	985	535	895	540	915
Mean	634	901	641	858	591	968	580	972	552	911	551	917

The results for the automatically chosen lengths of the training sample for each studied index are presented in Tables 4 and 5. In addition, we report the calculated window sizes for the initial application of change point detection algorithms before model fitting in Table 6. The results indicate that the automatically chosen span is in accordance with the previous conclusions about liberal and conservative samples. Our method to determine the best point is more or less stable and fluctuates at around 500–600 observations for the liberal method and around 900–1000 observations for the conservative one. Note also that there is not much difference in the training sample sizes determined by both of these models. We cannot determine which one of them is better, given that this was not the aim of this research.

The results for the automatically chosen lengths of the training sample indicate that the length can be predetermined using an objective method to produce estimates of good quality and in accordance with previous conclusions. The mean number of exceedances for both methods falls very close to the assumed number of exceedances for the CAViaR and GARCH models, whereas for historical simulation the number of exceedances is slightly higher (7–8 on average). This difference should be attributed to the earlier conclusion that historical simulation needs a longer training sample to produce estimates of the same quality as the other models in the comparison.

Much better results are obtained if the length of the training window is not predetermined but is recalculated with each model fitting. The mean number of exceedances is lower for almost all of the methods in each setting (liberal or conservative). Even though the training sample did not exceed 1000 observations, the mean number of exceedances lowered by 1–2 exceedances. For the GARCH model, the mean number of exceedances is around 4–5, with a maximum of 9 exceedances. The CAViaR model obtained slightly worse results, with around 5–6 exceedances, but it is mostly in line with the assumed excess level. The maximum number of exceedances for CAViaR was 11. The historical simulation results are still worse than those of the other two models, with 6–7 mean exceedances.

6 Conclusions

In this study we researched the impact of the training sample’s size on the results of several VaR models: historical simulation, the GARCH model with a skewed Student $t$ distribution and the CAViaR model at a 2.5% confidence level. Each of these approaches was tested 250 times for a span of five years: 2015–19. The training sample sizes that we tested range from 50 to 1250, in increments of 50 (ie, $50,100,\mathrm{\dots },1250$), along with three additional sample sizes: 1500, 1750 and 2000. In addition, we created a setup for the automatic detection of the necessary sample size to provide sufficient results by utilization of change point detection algorithms: PELT and binary segmentation in liberal and conservative settings. These methods were applied to determine the number of observations in the training sample, either before the whole fitting process or with each one-day-ahead forecast.

Based on our results, our recommendation would be to use the GARCH or CAViaR models with the proposed automatic training sample length selection that is recalculated with every model’s refitting. Using this approach, we can get much better results than when using a predefined window length while still being in the range of 500–1000 observations. We emphasize that the mean number of exceedances for this method is below the assumed level, and hence for shorter training samples it would reach the assumed level. We recommend rejecting models based on the historical simulation approach, as these results are much worse than those of other models trained on the same number of observations. In addition, it is easy to manipulate the assumed excess level just by increasing the number of observations, which, in our opinion, is not in accordance with Basel rules. However, these simpler models have one advantage: they do not require a minimum number of observations. In cases where data is poor, institutions might be prone to considering a model type that will provide any reasonable VaR estimates, regardless of our findings.

In cases where automatically chosen lengths cannot be used or determined, we recommend the use of at least 900 observations, as we have proven that, above a sample size of around 900–1000, the number of exceedances converges to the assumed excess level. In addition, we recommend using a training sample size in line with the subjective division that we have created: in cases where risk management tends toward liberal solutions, the number of exceedances should be lower than the threshold we have set; if more conservative models are preferred, the number of exceedances should be raised to levels above the threshold.

Finally, we highlight that the model’s accuracy depends on the information it is trained on. It is obvious that the more shocks the model “observes” while in training, the more biased toward conservatism it will be, while the fewer shocks it “observes” in training, the more it will be biased toward liberalism. Therefore, each decision about the length of the training sample should be based on a thorough study of the underlying time series, if applicable.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

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• Keywords
• Generalised autoregressive conditional heteroscedasticity (Garch)
• Historical simulation
• Value-at-risk (VAR)
• Original research
• Risk model validation
• Forecasting