# Is intraday data useful for forecasting VaR? The evidence from EUR/PLN exchange rate

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## Abstract

In this paper, we evaluate alternative volatility forecasting methods under Value at Risk (VaR) approach by calculating one-step-ahead forecasts of daily VaR for the EUR/PLN foreign exchange rate within the 4-year period. Using several risk models, including GARCH specifications and realized volatility models as well as hybrid of these two, we examine whether incorporation of intraday data allows to produce better one-step-ahead volatility forecasts in daily horizon than in case of using daily data only. The volatility forecasts are compared within VaR framework in two-step procedure: the statistical accuracy test are conducted as well as the loss functions are obtained. We find that GARCH models produce better backtesting results than models for realized volatility. When the loss functions of the models that passed the first-stage filtering procedure are compared, there is no distinct winner of the race. We also find no evidence that skewed Student *t* distribution assumption within GARCH models provides better VaR forecasts when compared to symmetric Student.

## Keywords

VaR Intraday data Realized volatility GARCH ARFIMA HAR-RV Jumps## Introduction

Since Basel II banks are obligated to calculate Value at Risk (VaR) forecasts of the expected losses. There is a huge tension therefore to improve the existing estimation techniques and backtesting procedures. If it is assumed that returns belong to a location-scale family, and it often is, VaR is a linear function of somehow calculated volatility. There is a variety of approaches possible to choose when estimating unobservable volatility based on different assumptions and different information sets. The availability of intraday prices in recent years contributes to the development of new class of volatility measures using data sampled at different frequencies (Andersen and Bollerslev 1998; Barndorff-Nielsen and Shephard 2004). It is still open to question, which approach, based on daily or intradaily data, generate better volatility forecasts and thus VaR forecasts. On the one hand, the high-frequency data contain more information on the actual changes in prices within a day, but using intraday data both cause higher costs of data gathering and are limited to the assets for which the data are available. On the other hand, the application of daily data requires less computer power, but the data are not as informative.

Although extensive research has been carried out on the comparison of the usefulness of intraday and daily data, the conclusions are not consistent. Giot and Laurent (2004) indicate that in the case of stock indices and exchange rates an adequate ARCH-type model, that account for asymmetry of returns, delivers as good VaR forecasts as the models based on realized variance. Lunde and Hansen (Hansen and Lunde 2005) in the analysis of exchange rates show similar results—they find no evidence that GARCH(1,1) is outperformed by more sophisticated models based on intraday data. Contrary view is presented in McMillan et al. (2008) who show that by using intraday data one improves daily volatility and VaR forecasts relatively to daily data. Also Fuertes and Olmo (2012) show that the ARFIMA models produce better backtesting results than GARCH models. However, they emphasize that the GARCH models prevail in terms of independence of hits sequence. Clements et al. (2008) compare HAR and MIDAS models to simple AR(5) and show that the latter yields satisfactory VaR forecasts. Louzis et al. (2014) found that both the realized variance and the augmented GARCH models with the filtered historical simulation or the extreme value theory quantile estimation methods produce equally good VaR forecasts. Brownless and Gallo (2009) propose the daily range for VaR prediction and shows that this simple volatility measure behaves as good as computationally complicated and costly measures based on ultra-high-frequency data and better than baseline GARCH models. Będowska-Sójka (2015) finds that in the stock market VaR estimates based on daily and intradaily returns give comparable results. However, when loss functions are considered, the models based on daily data allow minimizing regulatory loss function, whereas the models based on realized volatility allow minimizing the opportunity cost of capital. Kambouroudis et al. (2016) find that a preferred model for forecasting volatility is one that combines an asymmetric GARCH model with implied and realized volatility through (asymmetric) ARMA model. Wong et al. (2016) show that although RV models outperform GARCH models for volatility forecasts, in the VaR forecasts framework EGARCH model outperforms other approaches.

The main purpose of the paper is to compare the volatility forecasts’ accuracy of EUR/PLN exchange rate returns within the VaR framework.^{1} The forecasts are based on the different information sets, that consist of daily and intradaily returns. In the comparison of the VaR forecasts, two-stage procedure is undertaken (Sarma et al. 2003; Wong et al. 2016): first, the unconditional and conditional coverage tests are conducted, and then the models that passed the first-stage filtering procedure are compared on the basis of the loss functions.

This paper contributes to the literature in several ways. First, for volatility estimates calculated on the basis of daily data, we consider few GARCH models with different distribution assumption, Gaussian, symmetric Student *t* and skewed Student *t*. Second, for volatility estimates calculated from equally sampled intraday returns, we consider two types of stochastic models with specifications fitted to model-free measures of volatility: ARFIMA models (Andersen et al. 2003; Fuertes and Olmo 2012; Ahoniemi et al. 2016) and the heterogeneous autoregressive realized variance (HAR-RV) models with and without jumps (Andersen et al. 2007; Corsi 2009; Patton and Sheppard 2015). Third, we obtain hybrid VaR forecasts that combine GARCH estimates with these obtained from realized volatility models and examine, if there are the potential advantages of mixing forecasts. Our analysis extends the existing research by focusing on relatively less liquid FX rate and broadening the class of the models.

We find that VaR forecasts from GARCH models based on daily data have better backtesting results that these based on ARFIMA or HAR models for realized volatility. However, there is no prevailing advantage in all loss functions when well-fitted GARCH and realized volatility models are compared. EWMA specification for daily data passes all backtesting procedures and obtains the lowest firm loss function, but simultaneously achieves the highest values of binominal and regulatory loss function. Also, contrary to several works (Giot and Laurent 2003; Louzis et al. 2014; Wong et al. 2016), we do not find an evidence that skewed Student *t* distribution assumption in GARCH models provides better VaR forecasts when compared to other distribution, namely Gaussian or symmetric Student. In line with Clements (2008), we find that simple AR(5) model for realized variance performs as good as ARFIMA model showing that long-memory specification is not obligatory. We also show that combining both approaches, based on daily and intradaily returns into hybrid models, offer possibility of obtaining lower regulatory loss functions.

The rest of the paper is as follows: in “Models for volatility forecasting” section competing volatility models used in the study are presented, in “Data” section the data are described, “Empirical results” section presents the methods of VaR evaluation and the empirical results. The last section concludes.

## Models for volatility forecasting

*t*. The models are specified as follows:

*t*of innovations \(\varepsilon_{t}\), based on the information set \(\varOmega_{t - 1}\), whereas \(\,z_{t}\) is an independently and identically distributed (I.I.D.) unit variance random variable that follows one of three different density functions: Gaussian (N.I.D.), symmetric Student

*t*and skewed Student

*t*. The form of the conditional volatility \(\sigma_{t}^{2}\) differs among specifications: \(\sigma_{t}^{2}\) is either a GARGH-type conditional variance of the daily return, or a realized volatility conditional expectation forecasted from ARFIMA or HAR models.

### ARCH-type models

*p*,

*q*) model introduced by Bollerslev (1986) is a plain vanilla model, often standing as a benchmark for volatility specification (Hansen and Lunde 2005). We consider also exponential GARCH (EGARCH) model that allows for asymmetric effects between positive and negative returns, the so-called leverage effect (Nelson 1991), where \(\gamma_{1}\) and \(\gamma_{2}\) are real constants. Both \(\varepsilon_{t}\) and \(\left| {\varepsilon_{t - i} } \right| - E\left| {\varepsilon_{t - i} } \right|\) are zero mean I.I.D. sequences with continuous distributions. Next, IGARCH(p,q) model is included as a representation that is an extension of EWMA. As long memory in volatility is one of the stylized facts known in the literature, the fifth model in this group is FIGARCH(

*p,d,q*), that is capable of accommodating the persistence in volatility (Baillie et al. 1996), where

*L*is the backshift operator,

*d*is a fractionally integrated parameter, \(0 \le d \le 1\), \(\varPhi (L) = 1 - \sum\nolimits_{i = 1}^{q} {\phi_{i} L^{i} }\) and \(B(L) = 1 - \sum\nolimits_{i = 1}^{p} {\beta_{i} L^{i} }\).

GARCH specifications of conditional volatility

Model | Specification | Eqs. |
---|---|---|

EWMA | \(\sigma_{t}^{2} = 0.06r_{t - 1}^{2} + 0.94\sigma_{t - 1}^{2}\) | (3) |

GARCH( | \(\sigma_{t}^{2} = \omega + \sum\limits_{i = 1}^{q} {\alpha_{i} r_{t - i}^{2} } + \sum\limits_{j = 1}^{p} {\beta_{j} \sigma_{t - j}^{2} }\) | (4) |

EGARCH( | \(\log \sigma_{t}^{2} = \alpha_{0} + \sum\limits_{i = 1}^{q} {\alpha_{i} \left\{ {\gamma_{1} \varepsilon_{t - i} + \gamma_{2} \left[ {\left| {\varepsilon_{t - i} } \right| - E\left( {\left| {\varepsilon_{t - i} } \right|} \right)} \right]} \right\}} + \sum\limits_{j = 1}^{p} {\beta_{i} \log \sigma_{t - j}^{2} }\) | (5) |

IGARCH( | \(\sigma_{t}^{2} = \omega + p\sum\limits_{i = 1}^{p} {(1 - \beta_{i} )r_{t - i}^{2} } + \sum\limits_{j = 1}^{q} {\beta_{j} \sigma_{t - j}^{2} }\) | (6) |

FIGARCH( | \((1 - L)^{d} \varPhi (L)r_{t}^{2} = \omega + B(L)(r_{t}^{2} - \sigma_{t}^{2} )\), | (7) |

In the estimation, we use rolling sample of size *T* (*T* equals 537 observations) and set one-step ahead conditional standard deviation forecasts within the underlying specifications (Eqs. (3)–(7)) and VaR forecasts following Eq. (2). On the basis of information criteria, for all GARCH specifications we assume \(p = 1\,\,\) and \(\,q = 1\). In all cases, we test the specifications of the models with respect to autocorrelation of residuals and squares of residuals (Ljung–Box test).

### Models for realized variance and realized bipower variation

*n*-th intraday return on day

*t*. This volatility estimator converges in probability (as the sampling frequency of the return series increases, \(\Delta \to 0\)) to the quadratic variation process that characterizes the latent true variance \(RV_{t} (\Delta ) \to \int_{t - 1}^{t} {\sigma^{2} (s)ds} + \sum\limits_{t - 1 < s \le t}^{{}} {\kappa^{2} (s)}\), where the first term \(\int_{t - 1}^{t} {\sigma^{2} (s){\text{d}}s}\) is called integrated variance, whereas \(\sum\limits_{t - 1 < s \le t}^{{}} {\kappa^{2} (s)}\) is describing the jump process at time

*s*, and \(t - 1 < s \le t\). In the absence of jumps, realized variance will be a consistent estimator of integrated variance. This result is fundamental for modeling and forecasting realized variance (Andersen et al. 2003). However, as jumps are quite common in financial returns series, Barndorff-Nielsen and Shephard (2004) introduced another measure called realized bipower variation, which is an estimate of integrated variance that is robust to jumps:

*J*at zero:

Models for realized volatility and realized bipower variation

Model | Specification | Eqs. |
---|---|---|

ARFIMA-RV | \(\begin{aligned} \varPhi (L)(1 - L)^{\delta } \ln (RV_{t} ) = \varTheta (L)e_{t} \hfill \\ (1 - \varphi L)(1 - L)^{\delta } (s_{t} - \omega ) = e_{t} ,e_{t} \sim {\text{NID}}(0,\sigma_{e}^{2} ) \hfill \\ \end{aligned}\) | (11) |

ARFIMA-RBV | \(\begin{aligned} \varPhi (L)(1 - L)^{\delta } \ln (RV_{t} ) = \varTheta (L)e_{t} \hfill \\ (1 - \varphi L)(1 - L)^{\delta } (s_{t} - \omega ) = e_{t} ,e_{t} \sim {\text{NID}}(0,\sigma_{e}^{2} ) \hfill \\ \end{aligned}\) | (12) |

AR5RV | \(\ln (RV_{t} ) = \varphi_{0} + \sum\nolimits_{i = 1}^{m} {\varphi_{i} \ln (RV_{t - i} )} \,\, + e_{t} ,\,\,\,e_{t} \sim {\text{NID}}(0,\,\,\sigma_{e}^{2} )\,\,\,\,\,\,\,\,\) | (13) |

HAR-RV | \(\begin{aligned} \ln (RV_{t} ) = \, & \beta_{0} + \beta_{d} \ln (RV_{t - 1} ) + \beta_{w} \ln (RV_{t - 1}^{w} ) \\ & \; + \beta_{m} \ln (RV_{t - 1}^{m} ) + e_{t} e_{t} \sim {\text{NID}}(0,\sigma_{e}^{2} ) \\ \end{aligned}\) | (14) |

HAR-RV-J | \(\begin{aligned} \ln (RV_{t} ) =\, & \beta_{0} + \beta_{d} \ln (RV_{t - 1} ) + \beta_{w} \ln (RV_{t - 1}^{w} ) + \beta_{m} \ln (RV_{t - 1}^{m} ) \\ & \; + \beta_{j} \ln (J_{t - 1} + 1) + e_{t} ,e_{t} \sim {\text{NID}}(0,\sigma_{e}^{2} ) \\ \end{aligned}\) | (15) |

First, in the ARFIMA(*m*,*δ*, *s*) process *L* is the backshift operator; \((1 - L)^{\delta }\) is the fractional integration operator; \(\delta\) is a fractional integration parameter, and \(e_{t}\) is a stationary process. Within the study, we use ARFIMA(*m*,*δ*,*s*) for realized volatility and realized bipower variation (hereafter ARFIMA-RV and ARFIMA-RBV, respectively). This allows us to check, if capturing the long memory feature of RV or RBV improves VaR forecasts. Based on information criteria, we end up with ARFIMA(*1*,*d*,*0*) specification both for RV and RBV. As Clements et al. (2008) indicated that simple autoregressive models for RV provide accurate volatility forecasts for majority of currencies, we also consider autoregressive process for the logs of realized volatility with 5 lags (hereafter AR5RV).

Second class of models that take into account the long memory feature used to forecasts RV is the heterogeneous autoregressive model of realized variance (hereafter HAR-RV) (Corsi 2009). This regression models mix information of volatility components from different frequencies. In the HAR-RV specification, \({\text{RV}}_{t}\) is a daily realized volatility (for simplicity we omit \(\Delta\)); \({\text{RV}}_{t}^{w}\) and \({\text{RV}}_{t}^{m}\) are weekly and monthly volatility components, respectively, that are defined as simple averages of daily measures, that is \({\text{RV}}_{t}^{w} = \frac{1}{5}({\text{RV}}_{t} + {\text{RV}}_{t - 1d} + \cdots + {\text{RV}}_{t - 4d} )\) and \({\text{RV}}_{t}^{m} = \frac{1}{22}({\text{RV}}_{t} + {\text{RV}}_{t - 1d} + \cdots + {\text{RV}}_{t - 21d} )\). HAR-RV models are additionally extended as in (Andersen et al. 2007) by including the jump component. The heterogeneous autoregressive model of realized variance with jumps, hereafter HAR-RV-J, is specified in Eq. (15), where *J*_{t} stands for the jumps calculated as in Eq. (10). It is assumed that the all RV models fully capture the conditional volatility through a linear functions. For all realized variance models, we estimate ARFIMA or HAR models and obtain the forecasts of volatility and VaR forecasts (Eq. 3).

## Data

*Dukascopy*. We calculate the logarithmic middle price

*p*as the geometric average of the bid (

*p*

_{bid}) and ask (

*p*

_{ask}) price. As such, it is said to create a good approximation of the true price (Dacorogna et al. 2001). At the time

*t*, the middle price is defined as

*t*(

*i*) is a homogenous sequence of times regularly spaced by intervals of size Δ

_{ t }. We use the logarithmic middle price. Thus, the logarithmic return at time

*t*(

*i*) is defined as

## Empirical application

The risk models are estimated using G@RCH 6.2 package of Ox (GARCH and ARFIMA models) as well as PcGive mode (Doornik and Hendry 2005; Laurent 2010). Then VaR forecasts are computed in Ox. We estimate GARCH-type models for daily returns as well as ARFIMA models for realized variance and realized bivariate variation data and HAR models for realized volatility. All models are tested for possible misspecifications. At the end, we choose one model that is well specified and has the best fit according to information criteria. Then we generate one-step-ahead forecasts of volatility from 2012-10-24 to 2014-09-30 which gives 500 forecasts from each approach. Every model from those presented in “Models for volatility forecasting” section is re-estimated in a moving window consisting of 537 observations.

### VaR evaluation

As in Sarma et al. (2003), we use a two-stage model selection and evaluation procedure. In the first stage, models are tested for statistical accuracy, whereas in the second, three various subjective loss functions are used. In evaluation of VaR forecasts, we consider the test for unconditional coverage (Kupiec 1995), for conditional coverage (Christoffersen 1998) and Dynamic Quantile Test, DQT (Engle and Manganelli 2004). The tests and loss functions used in the study are described below.

#### VaR specification tests

*N*is the number of exceedances (hits) of the forecasted VaR, and

*T*is the number of observations.

Christoffersen’s (1998) exceedance independence test (henceforth CEI) is a likelihood ratio test that looks for unusually frequent consecutive exceedances. Defining the indicator function \(I_{t + 1} (\alpha )\) for the exceedances associated with the VaR forecasts, \(I_{t + 1} (\alpha ) \equiv 1(r_{t + 1} \le VaR_{t + 1} \left| {\varOmega_{t - 1} } \right.)\), we have \(q_{0}^{*} = \Pr (I_{t} = 0\left| {I_{t - 1} = 0)} \right.\) and \(q_{1}^{*} = \Pr (I_{t} = 0\left| {I_{t - 1} = 1)} \right.\). These are the VaR measure’s conditional coverages—its actual probabilities of not experiencing an exceedance, given that it did not (in the case of \(q_{0}^{*}\)) or did (in the case of \(q_{1}^{*}\)) experience an exceedance in the previous period. Our null hypothesis *H*_{0} is that \(q_{0}^{*} = q_{1}^{*} = q^{*}\). If a VaR measure is observed for \(\pi + 1\) periods, there will be \(\pi\) pairs of consecutive observations (\(I_{t - 1}\),\(I_{t}\)). Disaggregate these as follows: \(\pi_{00} + \pi_{01} + \pi_{10} + \pi_{11} = \pi\), where \(\pi_{00}\) is the number of pairs (\(I_{t - 1}\),\(I_{t}\)) of the form (0, 0); \(\pi_{01}\) is the number of the form (0, 1); etc. To test if \(\pi_{00} /(\pi_{00} + \pi_{01} ) \approx \pi_{10} /(\pi_{10} + \pi_{11} )\), we estimate the following: \(\hat{q}_{0}^{*} = \frac{{\pi_{00} }}{{\pi_{00} + \pi_{01} }}\), \(\hat{q}_{1}^{*} = \frac{{\pi_{10} }}{{\pi_{10} + \pi_{11} }}\), and \(\hat{q}_{{}}^{*} = \frac{{\pi_{00} + \pi_{10} }}{{\pi_{00} + \pi_{01} + \pi_{10} + \pi_{11} }}\).

The above CIT test is based on a first-order Markov chain only and thus works properly if exceedances are actually consecutive. Therefore, we also use the Dynamic Quantile Test, DQ (Engle and Manganelli 2004), that allows to examine if the present exceedances of the VaR measure are not correlated with the past ones. Taking previously defined the indicator function, \(I_{t} (\alpha )\), hit variables are obtained as follows: \({\text{Hit}}_{t} (\alpha ) = I_{t} (r_{t} \le {\text{VaR}}_{t}^{{}} (\alpha )) - \alpha\). The two hypotheses are tested jointly: \({\text{H}}_{01} :E({\text{Hit}}_{t} (\alpha )) = 0\) and \({\text{H}}_{02} :{\text{Hit}}_{t} (\alpha )\) is uncorrelated with the variables included in the information set. Testing both hypotheses can be done jointly within an artificial regression: \({\text{Hit}}_{t} (\alpha ) = {\mathbf{Z}}\lambda + \varepsilon_{t}\), where **Z** is a \(T\, \times \,k\) matrix with the first column consisting of ones, and the next \(p\) columns consisting of past exceedances \({\text{Hit}}_{t - 1} , \ldots ,{\text{Hit}}_{t - p}\). In the \(k - p - 1\) remaining columns, additional independent variables are included (e.g., past returns, the squared past returns, VaR itself). Engle and Manganelli (2004) showed that the DQ test statistic satisfies the following relation:\(DQ = \frac{{\hat{\lambda }^{T} {\mathbf{X}}^{T} {\mathbf{X}}\hat{\lambda }}}{\alpha (1 - \alpha )}\sim \chi^{2} (k),\) where \(\hat{\lambda }\) is OLS estimate of \(\lambda\).

#### The loss functions

In the second stage of the VaR forecasts evaluation process, we use the so-called ‘loss functions’ (Lopez 1999; Sarma et al. 2003). These functions are defined on the basis of risk managers’ utility functions. Each loss function reflects the different approaches of the risk manager, but they all are defined in negative orientation—they give higher scores when the failure occurs. Consequently, the VaR model that minimizes the value of the loss functions is considered to be attractive. The loss functions should be compared under the condition that the VaR specification passes all statistical accuracy tests.

Assuming that VaR is correctly specified, the fewer failures observed, the lower the binominal loss is, and thus the better the model.

The square term that is calculated when the failure occurs, penalizes more large failures than the small ones. The higher the exceedance is, the higher the loss function value will be. For two models with the same BL, RL might be decisive.

*c*(Sarma et al. 2003):

In the study, we assume that *c *= 1. From the risk manager point of view, not only is it important, how large the failures are, but also how conservative the VaR measure is. Additionally, to what is reported in the RL function, the FL includes also a penalty for being cautious and ‘freezing’ too much capital. BL and RL usually indicate the same models as they are focused on the minimization of the number of exceedances and minimization of the size of the failure, respectively. The third loss function, FL, is aimed to find the balance between safety and profit maximization and thus obtain the lowest values for different models.

### Empirical results

#### VaR forecasts from singular models

*p*-values of the following tests: Kupiec (1995) unconditional coverage test (KUC), Christoffersen (1998) exceedance independence test (CEI), and Engle and Manganelli (2004) dynamic quantile test (DQ) for two VaR levels. As shown in Table 3, in all approaches in

*α*-quantile 0.95, and eight out of fifteen in

*α*-quantile 0.99 the statistical accuracy tests are passed. The poor performance of various models in the former case results from the clustering of the exceedances detected by the DQ test. In those cases, VaR models are not sufficiently responsive to changing market circumstances.

Statistical accuracy tests for the short position

KUC | CEI | DQ | KUC | CEI | DQ | |
---|---|---|---|---|---|---|

Quantile 0.95 | Quantile 0.99 | |||||

EWMA (N) | 0.84 | 0.86 | 0.97 | 0.05 | 0.38 | 0.05 |

GARCH (N) | 0.05 | 0.67 | 0.25 | | | |

GARCH (St) | 0.20 | 0.81 | 0.85 | | | |

EGARCH (St) | 0.84 | 0.95 | 0.91 | 1.00 | 0.83 | 1.00 |

IGARCH (St) | 0.84 | 0.95 | 0.96 | | | |

FIGARCH (St) | 0.13 | 0.77 | 0.61 | | | |

GARCH (skSt) | 0.05 | 0.67 | 0.25 | 0.33 | 0.90 | 0.96 |

EGARCH (skSt) | 0.20 | 0.43 | 0.48 | 0.33 | 0.90 | 0.96 |

IGARCH (skSt) | 0.29 | 0.86 | 0.95 | | | |

FIGARCH (skSt) | 0.08 | 0.72 | 0.44 | 0.33 | 0.90 | 0.96 |

HAR-RV | 0.53 | 0.34 | 0.70 | 0.13 | 0.93 | 0.60 |

HAR-J-RV | 0.40 | 0.91 | 0.57 | 0.13 | 0.93 | 0.60 |

ARFIMA-RV | 0.13 | 0.77 | 0.14 | | | |

ARFIMA-RBV | 0.40 | 0.67 | 0.38 | | | |

AR5RV | 0.08 | 0.72 | 0.44 | 0.13 | 0.93 | 0.60 |

*α*-quantile 0.95, the lowest values of binominal and regulatory loss function are observed for GARCH model with Gaussian and skewed Student

*t*distribution, whereas the lowest firm loss function is observed for EWMA that at the same time generates the highest values of binominal and regulatory loss function among all methods. Within the RV and RBV model class simple autoregressive model for RV (AR5RV) obtains third rank both in binominal and regulatory loss function. When

*α*-quantile 0.99 is considered, the lowest value of binominal and regulatory loss function are observed in three models for RV: HAR-RV, HAR-J-RV, and AR5RV. As the firm loss function is considered, EWMA obtains the lowest value, followed by EGARCH with Student distribution and AR5RV. Summing up, in the short position, there is no distinct winner of the race as both GARCH-class models and RV models are among these with the lowest loss functions values.

The values of loss functions for the short position

BL | RL | FL | BL | RL | FL | |
---|---|---|---|---|---|---|

Quantile 0.95 | Quantile 0.99 | |||||

EWMA (N) | 26 | 28.47 | 310.85 (1) | 10 | 10.78 | 421.56 (1) |

GARCH (N) | 16 (1) | 17.66 (1) | 342.25 | | | |

GARCH (St) | 19 | 20.89 | 330.16 | | | |

EGARCH (St) | 19 | 26.05 | 332.29 | 6 | 6.57 | 466.44 (2) |

IGARCH (St) | 24 | 26.06 | 323.00 (2) | | | |

FIGARCH (St) | 18 | 19.93 | 327.80 | | | |

GARCH (skSt) | 16 (1) | 17.70 (2) | 340.88 | 3 | 3.26 | 525.39 |

EGARCH (skSt) | 19 | 20.83 | 343.75 | 3 | 3.39 | 519.74 |

IGARCH (skSt) | 20 | 21.84 | 333.54 | | | |

FIGARCH (skSt) | 17 (3) | 18.73 | 339.48 | 3 | 3.26 | 521.30 |

HAR-RV | 22 | 23.34 | 363.87 | 2 (1) | 2.49 (3) | 501.84 |

HAR-J-RV | 21 | 22.26 | 365.05 | 2 (1) | 2.43 (2) | 504.05 |

ARFIMA-RV | 18 | 19.28 | 342.14 | | | |

ARFIMA-RBV | 21 | 22.65 | 327.70 (3) | | | |

AR5RV | 17 (3) | 18.16 (3) | 346.55 | 2 (1) | 2.25 (1) | 480.14 (3) |

*α*-quantile 0.05 and 0.01. In the former quantile, two GARCH-class models and four RV or RBV out of five models are rejected at the first stage. In most cases, two tests are violated: neither the number of exceedances is proper, nor VaR models are responsive enough to market dynamics. For the

*α*-quantile 0.01, all considered models are specified correctly.

Statistical accuracy tests for the long position

KUC | CEI | DQ | KUC | CEI | DQ | |
---|---|---|---|---|---|---|

Quantile 0.05 | Quantile 0.01 | |||||

EWMA (N) | 0.20 | 0.81 | 0.08 | 0.11 | 0.71 | 0.31 |

GARCH (N) | | | | 0.64 | 0.87 | 1.00 |

GARCH (St) | 0.05 | 0.50 | 0.06 | 0.33 | 0.90 | 0.96 |

EGARCH (St) | 0.05 | 0.50 | 0.06 | 0.64 | 0.87 | 1.00 |

IGARCH (St) | 0.13 | 0.77 | 0.24 | 0.33 | 0.90 | 0.96 |

FIGARCH (St) | 0.13 | 0.45 | 0.23 | 0.13 | 0.93 | 0.60 |

GARCH (skSt) | 0.08 | 0.48 | 0.13 | 0.64 | 0.87 | 1.00 |

EGARCH (skSt) | | | | 0.64 | 0.87 | 1.00 |

IGARCH (skSt) | 0.13 | 0.77 | 0.24 | 0.64 | 0.87 | 1.00 |

FIGARCH (skSt) | 0.20 | 0.43 | 0.33 | 0.64 | 0.87 | 1.00 |

HAR-RV | | | | 0.33 | 0.90 | 0.96 |

HAR-J-RV | | | | 0.33 | 0.90 | 0.96 |

ARFIMA-RV | | | | 0.13 | 0.93 | 0.60 |

ARFIMA-RBV | 0.08 | 0.59 | 0.15 | 0.64 | 0.80 | 1.00 |

AR5RV | 0.00 | 0.62 | 0.00 | 0.13 | 0.93 | 0.60 |

*α*-quantile 0.05, EGARCH and GARCH with Student distribution have the lowest values of BL and RL functions, while the lowest firm loss function is observed for IGARCH model with skewed Student

*t*distribution. In

*α*-quantile 0.01, VaR from FIGARCH model obtains the lowest value of the regulatory loss, whereas EWMA again gains the lowest firm loss function, followed by EGARCH with skewed Student distribution and ARFIMA-RBV model.

The values of loss functions for the long position

BL | RL | FL | BL | RL | FL | |
---|---|---|---|---|---|---|

Quantile 0.05 | Quantile 0.01 | |||||

EWMA (N) | 19 | 20.25 | 306.15 (2) | 9 | 9.21 | 421.62 (1) |

GARCH (N) | | | | 4 | 4.05 | 473.31 |

GARCH (St) | 16 (2) | 16.84 (1) | 328.67 | 3 (3) | 3.03 | 500.90 |

EGARCH (St) | 12 (1) | 16.99 (2) | 316.60 | 4 | 4.10 | 465.60 |

IGARCH (St) | 18 | 18.94 | 319.80 | 3 (3) | 3.03 | 490.82 |

FIGARCH (St) | 16 (2) | 18.74 | 327.13 | 2 (1) | 2.01 (1) | 506.51 |

GARCH (skSt) | 17 | 18.02 | 315.27 | 4 | 4.06 | 471.53 |

EGARCH (skSt) | | | | 4 | 4.15 | 447.48 (2) |

IGARCH (skSt) | 18 | 19.16 | 305.74 (1) | 4 | 4.07 | 460.03 |

FIGARCH (skSt) | 19 | 19.93 | 313.46 (3) | 4 | 4.03 | 466.09 |

HAR-RV | | | | 3 (3) | 3.01 | 501.39 |

HAR-J-RV | | | | 3 (3) | 3.02 | 503.75 |

ARFIMA-RV | | | | 2 (1) | 2.06 (3) | 472.91 |

ARFIMA-RBV | 17 | 17.62 (3) | 324.28 | 4 | 4.07 | 449.85 (3) |

AR5RV | 12 | 12.43 | 343.99 | 2 | 2.05 (2) | 480.29 |

Summarizing, in the GARCH models group three of them have been always qualified after the first stage: GARCH and FIGARCH with skewed Student *t* distribution and EGARCH with symmetric Student *t* distribution. Within the VaR forecasts based on RV and RBV models, each has been rejected at least once. Thus, we confirm the result of Fuertes and Olmo (2012) that GARCH models prevail in terms of independence of hits sequence. When the loss functions are taken into account, the scoring obtained for GARCH models is ambiguous, for realized volatility models the best results are definitely obtained for AR5RV followed by HAR-RV and HAR-J-RV models. Taking into account these two categories, statistical accuracy tests and loss functions, for both short and long position, there is no distinct winner in the single model race. Thus, we move toward a hybrid approach.

#### Forecast combinations: hybrid models

Mixing forecasts in our study allow to use different information sets and different methodological approaches: one coming from daily volatility obtained from non-linear ARCH-type models, and the other based on intraday information from models using intraday data. The combination of forecasts might improve forecast accuracy if one properly uses the advantages of the single approach. We examine fifteen hybrid models that mix two broad classes. The selection is based on the already-obtained results and take into account the best models among these using daily data: from GARCH class we choose three models that pass all tests in both long and short position (GARCH with skewed Student *t* distribution, EGARCH with Student *t* distribution and FIGARCH with skewed Student *t* distribution). As for the models that use intradaily data no single model passes all tests, we consider all possible specifications. Thus, the hybrid models are ARFIMA-RV and GARCH (hereafter RV-GARCH), EGARCH (RV-EGARCH) or FIGARCH (RV-FIGARCH), and by analogy the following models combined with these three GARCH specifications: AR5RV (AR5RV-GARCH, AR5RV-EGARCH, AR5RV-FIGARCH), HAR-RV (HAR-GARCH, HAR-EGARCH, HAR-FIGARCH), HAR-J-RV (HARJ-GARCH, HARJ-EGARCH, HARJ-FIGARCH), and ARFIMA-BV (RBV-GARCH, RBV-EGARCH, RBV-FIGARCH).

Statistical accuracy tests for the hybrid models: short position

KUC | CEI | DQ | KUC | CEI | DQ | |
---|---|---|---|---|---|---|

Quantile 0.95 | Quantile 0.99 | |||||

RV-GARCH | | | | 0.13 | 0.93 | 0.60 |

RV-EGARCH | | | | 0.33 | 0.90 | 0.96 |

RV-FIGARCH | | | | | | |

AR5RV-GARCH | 0.05 | 0.67 | 0.25 | | | |

AR5RV-EGARCH | 0.08 | 0.72 | 0.50 | | | |

AR5RV-FIGARCH | 0.08 | 0.72 | 0.44 | 0.33 | 0.90 | 0.96 |

HAR-GARCH | | | | 0.13 | 0.93 | 0.60 |

HAR-EGARCH | 0.05 | 0.50 | 0.26 | 0.33 | 0.90 | 0.96 |

HAR-FIGARCH | | | | 0.13 | 0.93 | 0.60 |

HAR-J-GARCH | | | | 0.33 | 0.90 | 0.96 |

HAR-J-EGARCH | | | | 0.33 | 0.90 | 0.96 |

HAR-J-FIGARCH | | | | 0.13 | 0.93 | 0.60 |

RBV-GARCH | | | | 0.33 | 0.62 | 0.96 |

RBV-EGARCH | | | | 1.00 | 0.83 | 1.00 |

RBV-FIGARCH | | | | 0.33 | 0.90 | 0.96 |

The values of loss functions for the hybrid models: short position

BL | RL | FL | BL | RL | FL | |
---|---|---|---|---|---|---|

Quantile 0.95 | Quantile 0.99 | |||||

RV-GARCH | | | | 2 (1) | 2.14 (1) | 535.39 |

RV-EGARCH | | | | 3 | 3.20 | 517.80 |

RV-FIGARCH | | | | | | |

AR5RV-GARCH | 16 (1) | 17.02 (1) | 362.79 (3) | | | |

AR5RV-EGARCH | 17 (2) | 18.52 | 337.48 (1) | | | |

AR5RV-FIGARCH | 17 (2) | 18.41 (3) | 343.13 (2) | 3 | 3.24 | 500.62 |

HAR-GARCH | | | | 2 (1) | 2.25 (2) | 541.34 |

HAR-EGARCH | 16 (1) | 17.10 (2) | 377.42 | 3 | 3.32 | 523.72 |

HAR-FIGARCH | | | | 2 (1) | 2.27 | 539.24 |

HAR-J-GARCH | | | | 3 | 3.40 | 475.89 (1) |

HAR-J-EGARCH | | | | 3 | 3.30 | 524.97 |

HAR-J-FIGARCH | | | | 2 (1) | 2.25 (2) | 540.47 |

RBV-GARCH | | | | 3 | 3.19 | 502.18 |

RBV-EGARCH | | | | 5 | 5.26 | 484.73 (2) |

RBV-FIGARCH | | | | 3 | 3.19 | 500.02 (3) |

Statistical accuracy tests for the hybrid models: long position

KUC | CEI | DQ | KUC | CEI | DQ | |
---|---|---|---|---|---|---|

Quantile 0.05 | Quantile0.01 | |||||

RV-GARCH | | | | 0.13 | 0.93 | 0.60 |

RV-EGARCH | | | | 0.13 | 0.93 | 0.60 |

RV-FIGARCH | | | | 0.13 | 0.93 | 0.60 |

AR5RV-GARCH | | | | 0.13 | 0.93 | 0.60 |

AR5RV-EGARCH | | | | 0.13 | 0.93 | 0.60 |

AR5RV-FIGARCH | 0.05 | 0.50 | 0.24 | 0.13 | 0.93 | 0.60 |

HAR-GARCH | | | | 0.33 | 0.90 | 0.96 |

HAR-EGARCH | | | | 0.33 | 0.90 | 0.96 |

HAR-FIGARCH | | | | 0.33 | 0.90 | 0.96 |

HAR-J-GARCH | | | | 0.33 | 0.90 | 0.96 |

HAR-J-EGARCH | | | | 0.33 | 0.90 | 0.96 |

HAR-J-FIGARCH | | | | 0.33 | 0.90 | 0.96 |

RBV-GARCH | | | | 0.13 | 0.93 | 0.60 |

RBV-EGARCH | | | | 0.13 | 0.93 | 0.60 |

RBV-FIGARCH | | | | 0.13 | 0.93 | 0.60 |

The values of loss functions for the hybrid models: long position

BL | RL | FL | BL | RL | FL | |
---|---|---|---|---|---|---|

Quantile 0.05 | Quantile 0.01 | |||||

RV-GARCH | | | | 2 (1) | 2.01 (1) | 508.85 |

RV-EGARCH | | | | 2 (1) | 2.02 (2) | 513.07 |

RV-FIGARCH | | | | 2 (1) | 2.02 (2) | 506.21 |

AR5RV-GARCH | | | | 2 (1) | 2.04 | 475.86 (2) |

AR5RV-EGARCH | | | | 2 (1) | 2.05 | 480.08 (3) |

AR5RV-FIGARCH | 16 (1) | 16.58 (1) | 328.07 (1) | 2 (1) | 2.04 | 473.22 (1) |

HAR-GARCH | | | | 3 | 3.00 | 514.30 |

HAR-EGARCH | | | | 3 | 3.01 | 518.59 |

HAR-FIGARCH | | | | 3 | 3.00 | 511.66 |

HAR-J-GARCH | | | | 3 | 3.01 | 515.61 |

HAR-J-EGARCH | | | | 3 | 3.01 | 519.90 |

HAR-J-FIGARCH | | | | 3 | 3.00 | 512.97 |

RBV-GARCH | | | | 2 (1) | 2.02 (2) | 495.56 |

RBV-EGARCH | | | | 2 (1) | 2.02 (2) | 499.78 |

RBV-FIGARCH | | | | 2 (1) | 2.02 (2) | 492.91 |

*j*over model

*k*and vice versa. In order to save space, we only present the values of standardized sign statistics for each pair of competitors. According to the sign tests there is no statistically significant difference between the models with respect to the binominal or regulatory loss functions. Table 11 presents the results of the comparison of pairs of models.

The comparison of the competing VaR models

Model | Short position | Long position | ||
---|---|---|---|---|

Quantile 0.95 | Quantile 0.99 | Quantile 0.05 | Quantile 0.01 | |

GARCH-EGARCH | 7.33 | 12.25 | − 1.16 | |

EGARCH-GARCH | | | 1.16 | 3.40 |

GARCH-FIGARCH | 2.24 | 3.58 | 3.58 | 5.01 |

FIGARCH-GARCH | | | | |

EGARCH-FIGARCH | | | 3.67 | 5.10 |

FIGARCH-EGARCH | 2.06 | 5.99 | | |

AR5RV-GARCH | 0.89 | 9.66 | | 1.52 |

GARCH-AR5RV | − 0.89 | | 5.46 | |

EGARCH-AR5RV | |
| | 2.86 |

AR5RV-EGARCH | 2.77 | 0.18 | 2.24 | |

FIGARCH-AR5RV | − 0.18 | | 8.85 | 0.98 |

AR5RV-FIGARCH | 0.18 | 11.72 | − 8.85 | − 0.98 |

When the firm loss function is considered, in the short position EGARCH model is the winner, as it is better than other three specifications in all but one quantiles considered. Then FIGARCH model is better than GARCH and AR5-FIGARCH in both quantiles and 0.99 quantile, respectively, whereas AR5-FIGARCH itself is better than GARCH in terms of 0.99 quantile only. For the long position, we find that FIGARCH model is superior over GARCH and EGARCH in both quantiles, then GARCH model is better than AR5-FIGARCH and EGARCH (0.01 quantile only). Finally, AR5-FIGARCH model is superior over EGARCH (0.01 quantile). Although the sign test does not indicate one distinct winner of the race, it seems that a single GARCH class model performs better than our hybrid.

## Conclusions

In the paper, we consider the evaluation of alternative volatility forecasting methods within the risk management framework, namely under Value at Risk approach. We compare one-step ahead VaR forecasts, where volatility is estimated from the GARCH models based on daily data and models that use realized variance or realized bipower variation, calculated from intradaily returns. The alternative VaR measures obtained from differently sampled data give different VaR forecasts. We find that only three GARCH models from all specifications considered in the study pass the statistical accuracy tests for both positions and both VaR quantiles. VaR forecasts based on ARFIMA or HAR models for RV have worse backtesting results. When the loss functions are compared at the second stage, there is no distinct winner of the race. Surprisingly, in the case of firm loss function only, EWMA method seems to perform the best, but it is simultaneously characterized by the highest values of binominal and regulatory loss functions. Within GARCH class models, we find no evidence that skewed Student *t* distribution assumption provides better VaR forecasts when compared to other distribution such as Gaussian and symmetric Student.

This study shows that although the informational content of daily data seems to be sufficient to predict VaR properly, models based on intradaily data sometimes offer better result in searching for the best VaR forecasts. However, the gains must be carefully assessed by taking into account the effort required to gather and manage intradaily datasets. We also consider the hybrid models, that account for a balance between the different speeds of reaction to the market changes observed in the models based on daily or intradaily sampling frequencies. Although the combination of these two sets of information seem to open promising fields in the research, in our study the hybrids do not offer more efficient capital allocations. Only one out of fifteen hybrid models passes the statistical verification stage. In the comparison to single models, the hybrid obtains lower value of regulatory loss function, but loses on the firm loss function.

The results presented here should be of interest to firms that search for the optimal risk management model. If the statistical accuracy tests could be disregarded, the probably choice of the models might be directed to ARFIMA or HAR as they offer lower regulatory and firm loss functions, and thus allow for better capital allocation. However, if statistical accuracy tests are to be taken into account, GARCH approach is more advisable. Results should be of interest also for regulators, as they could help to recommend models that allow for achieving better regulatory profile and mitigate overall systematic risk. Realized volatility models could be recommended, especially if one controls for validity of VaR estimates, as these models often allow to diminish regulatory costs. The introduction of realized volatility, however, could be recommended only under circumstances of sufficient availability of intraday data.

## Footnotes

- 1.
The Polish zloty currency (PLN) is obviously not one of the most liquid in FX market and the issue of entering Eurozone is still questionable. About 80 per cent of Polish international trade is accountable in Euro.

## Notes

### Acknowledgements

I gratefully acknowledge the comments by anonymous referees as well as conference participants at the International Risk Management Conference 2016 organized by University of Florence, NYU Stern Salomon Center and Hebrew University of Jerusalem. All remaining errors are mine.

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