Volatility forecasting performance on OMX Stockholm 30 during a financial crisis

A comparison and evaluation of GARCH-family models

Elias Åstrand

Master thesis Economics, 15 hp Spring term 2020

Abstract

Volatility is an important factor in financial economics, investment decisions and risk management. GARCH models are helpful tools to explain uncertainty of financial stock markets in which volatility changes. The purpose of this study is to examine whether symmetric or asymmetric GARCH-family models forecasts volatility better during and post the financial crisis of 2008. Examined models are the symmetric GARCH and IGARCH and asymmetric EGARCH, TGARCH, NGARCH and APARCH. Daily volatility forecasts on return of the OMX Stockholm 30 index are compared against a proxy of realized volatility by four statistical evaluation measures. The result indicating different outcomes depending on period and evaluation method. Post-crisis result proves that the asymmetric EGARCH performs the most accurate forecasts which is evidenced with several other studies. During the crisis, the symmetric models outperforms asymmetric models in general. The result indicating that IGARCH and the asymmetric NGARCH models are superior during a more volatile period. Finally, although differences between models’ forecasts are extremely small, all models tend to underestimate the realized volatility frequently.

1 Table of Contents

1 Introduction ...... 3

2 Literature review ...... 6

3 Theoretical approach...... 9

3.1 Symmetric models ...... 10 3.1.1 GARCH model ...... 10 3.1.2 IGARCH model ...... 11

3.2 Asymmetric models ...... 11 3.2.1 EGARCH model ...... 11 3.2.2 TGARCH model ...... 12 3.2.3 NGARCH model...... 13 3.2.4 APARCH model ...... 13

4 Data and Methodology ...... 14

4.1 out-of-sample Forecast method ...... 15

4.2 Realized volatility ...... 16

4.3 Forecasting evaluation ...... 17

5. Results ...... 19

6 Discussion ...... 23

6.1 Conclusion ...... 25

References ...... 27

Appendix ...... 30

2 1 Introduction

Volatility is an important factor in asset allocation, option pricing and risk management on the financial market. The risk associated with an asset is a key measure in finance and volatility is one of the most if not the most commonly used risk measures. The significance of forecasting and estimating volatility is based on that volatility is not directly noticeable on the market. However, asset derivatives and prices are detectable in practice and therefore an adequate method to estimate volatility from these prices as accurate as possible is crucial. The global financial crisis that occurred late 2007 had its peak fifteenth of September 2008 when the Lehman Brother went bankrupt, had a huge impact on the financial market worldwide. Sweden was no exception when weakened investors' confidence affected the stock market with large losses in securities. Stricter financial regulations and investors disbelief emphasized accurate future volatility predictions. As the world currently encounters a global financial crisis due to COVID-19 an understanding of volatility forecasting in a volatile time could help risk managers and investors minimize losses.

If there was one stock corresponding to all companies listed on the Swedish stock exchange, its price would replicate the value of the entire stock market. Such market stock does not exist, instead market changes are measured in indices, a weighting of shares. These shares combining price movements reflect the Swedish stock markets' movements well. Further, this forms a market value on which options and futures can be traded. OMX Stockholm 30, (OMXS30) is an index consisting of the thirty shares with the highest turnover in Sweden. A table presenting stocks of all companies included in OMXS30 can be found in Appendix A1. Both professional and private investors use OMXS30 derivatives to, among other things construct a portfolio in a cost-effective way, increase leverage in the case of price increases or cover their equity portfolios against fall in prices (Nasdaqomxnordic.com, 2020).

In graph 1 below you can see how the price of OMXS30 decreased from nearly 1300 to under 600 in practically one year. After 15:th of September 2008, the price decreased approximately 20 percent in only six days and maintained unstable for the following year. The importance of accurate volatility forecast during a financial crisis would not only help investors minimize loss but besides understanding the Swedish stock market.

3 Graph 1. Serie of OMX Stockholm 30 index closing price.

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GARCH models are helpful tools to explain uncertainty of financial markets in which volatility changes, being relatively stable and less volatile during periods of balanced economic growth or more volatile during financial crises. As an example, the period before the crisis of 2008 the market and returns might look relatively uniform. However, in the period during the crisis, returns can swing uncontrollably from positive to negative values. A simple homoscedastic regression model does not account for the increased volatility, which could be foretelling of volatility going onward and then return to be more uniform and is therefore not suitable. In comparison GARCH processes, being autoregressive, are of better fit as it includes historical conditional variance and squared observations to model current volatility. The process aims to minimize the difference between predicted volatility and realized volatility by taking past errors into account, thus improving the accuracy of forecast predictions (Hassan et al. 2018).

Even though volatility is not directly noticeable in practice, it has four commonly seen characteristics in asset return. Firstly, volatility jumps are uncommon which are correlated with that volatility progresses over time in a continuous way. Secondly, volatility fluctuates between certain high and low periods, referred to as volatility clusters. Thirdly, large price drops are appeared to have a greater impact on volatility than an equally large price increase. This is defined as the leverage effect. Fourthly, volatility is usually stationary and varies within some static range. These four properties composite an important role in the progress of volatility models (Tsay, 2013, page. 177) .

4 This thesis will compare six single regime volatility models during and post the financial crisis of 2008. These models used to forecast volatility is originated from the ARCH-family, Autoregressive Conditional Heteroskedasticity (ARCH) model introduced by Engle (1982). The most well-known model is an extension of the ARCH model, called the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model introduced by Bollerslev (1986). Second model included in this study is the expanded Integrated Generalized Autoregressive Conditional Heteroskedasticity (IGARCH) model. The other four models compared are the Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH) model by Nelson (1991), the Nonsymmetric Generalized Autoregressive Conditional Heteroskedasticity (NGARCH) model by Engle and Ng (1993), the Threshold Generalized Autoregressive Conditional Heteroskedasticity (TGARCH) model by Zakoian (1994) and the Asymmetric Power Autoregressive Conditional Heteroskedasticity (APARCH) model by Ding et al. (1993). The purpose of this study is to examine which GARCH model produces best estimates for future volatility using the historical data during the recent financial crisis.

The thesis is organized as follows: In section 2 literature review and model selection are presented. Section 3 describes the theoretical approach of the examined models. In section 4 data and methodology are introduced. Section 5 presents the out-of-sample forecast result and finally, in section 6 includes conclusion and discussion.

5 2 Literature review

Poon and Granger (2001), reviewed and compared 72 published papers on volatility forecasting findings. The research analyzes forecasted performance of different volatility models combined with the importance of volatility in risk management, investment, and security valuation. All different types of models used in papers are sorted into four groups. The first group consists of simple time series models based on directly observed historical volatility such as exponentially weighted moving average (EWMA). The second group includes all GARCH-family models. Option pricing models such as implied volatility cover the third group. The fourth group is the stochastic volatility (SV). The comparison of which method and group provide the most accurate forecast of volatility is found very problematic and difficult. The result depends on characteristics of the market investigated and the quantity of several different periods.

As mentioned, this thesis will compare models from the second group describes by Poon and Granger (2001). Tsay (2013) and Alexander (2001) presents a framework of several different GARCH models and six of these frequently used models is examined in this study. The first model and the foundation of the other models is the simple Generalized ARCH model. The model uses a function of past conditional variance, long-run average variance rate, and past shocks to forecast volatility. If the variance process is stationary i.e. the volatility does not change to average over time, the model is called integrated GARCH. This second examined model with unit root indicates the past squared shocks in the function is being persistent (Tsay, 2013). These two models are symmetric types of models which means they can deal with the second property of volatility clustering but lack in capturing the leverage effect. The other four models examined belongs to asymmetric GARCH models since they can capture the leverage effect (Alexander, 2001). By different functions, these models indicating negative returns tend to have a higher impact on volatility than an increase in return, leverage effects. The first model examined in this class is the exponential GARCH which is a function of lagged past shocks and the exponential character, ensuring a variance always being positive. The second model called threshold GARCH uses instead a dummy variable in the function to capture negative shocks having larger impact on volatility. The third model includes a leverage parameter and is called the non-symmetric GARCH model. The final asymmetric power ARCH model uses a power function (Tsay, 2013). All models examined belong to the GARCH family and are developed from the standard function using different specifications.

6 Awartani and Corradi (2005) studied the predictability of different GARCH models, (including GARCH, IGARCH, EGARCH, and TGARCH), especially focusing on the asymmetric component. The forecast method is one day ahead predictions over the years of 1900 to 2001 on stock index S&P500. Models forecasted volatility is compared against squared return, a proxy of realized volatility. The result of the study indicates different GARCH models allowing for asymmetries and leverage effects produce more precise volatility predictions. However, GARCH is superior compared to other models that do not allow for leverage effects.

Evans and McMillan (2007) studied in parallel the forecasting performance of nine different models of daily volatility, utilizing stock returns in 33 countries. The out-of-sample forecasted period stretches from 1 of January 2004 to 22 of April 2005. The evaluation method used to compare models is RMSE. The study demonstrated the best forecast predictions by the GARCH models allowing for leverage. Furthermore, the empirical study presented EGARCH outperforming all other models in 8 countries including Sweden.

Chuang et al. (2007) analyzed forecasting performance of GARCH models based on distributional assumptions across exchange rate returns and stock market indices. They established the GARCH model combined with the student's t distribution is one of the preferred solutions on stock market indices. Although, complex distributions are not always superior over simpler ones. The correct ranking between distributions and models depends on the underlying asset, however, they concluded student´s t distribution being one of the most consistent in general. Furthermore, Gokcan (2000) proves that small lag models like (1,1) are suitable for handling changing volatility. In this thesis, all models will follow the simple (1,1) lag order and student´s t distribution to forecast volatility.

Chong et al. (1999) compared the stationary GARCH, integrated GARCH, exponential GARCH, among other GARCH models on the Kuala Lumpur Stock market. They found EGARCH performing best out-of-sample one step ahead forecasts although the models' in- sample goodness of fit was not the greatest. The model was also proved to explaining the skewness in indices the best. Besides, they found that the IGARCH model performed the worst forecast predictions and poor in-sample goodness of fit statistics.

Lim and Sek (2013) analyzed the performance of the standard GARCH model against the asymmetric EGARCH and TGARCH, to model volatility during the crisis in Malaysia 1997. They compared one month ahead forecast based on three evaluation measures i.e. MSE, RMSE

7 and MAPE. The out-of-sample forecasted result is showing the TGARCH outperforming the other models during the pre- and post-crisis period however the standard GARCH model is superior during the period of crisis. In addition, the study revealed the performance of models having variation across evaluation methods and periods.

As Poon and Granger (2001), highlighted, it is difficult and problematic to compare studies of volatility forecasting. Different markets and time periods can generate different results. Few researches have been made on forecasting volatility performance of the Swedish stock market during the financial crisis. The purpose of this paper is to follow up on the previous literature testing if asymmetric GARCH models still outperform the symmetric GARCH models during a more volatile time. More precisely, analyzing which single regime GARCH model performs the most accurate volatility prediction during and post the financial crisis of 2008. These forecasts are made on rate of return utilizing OMXS30 which reflects the Swedish stock market.

8 3 Theoretical approach

The main feature of all selected GARCH-family models is that they are using a conditional variance of return estimated by a maximum likelihood function. Unlike, other time series models using historical standard deviation, the conditional variance models use historical observations of returns and will, therefore, change at any point in time (Poon & Granger, 2003, p. 484).

퐸[휀2|퐼 푡−1 ] = 휎 푡−1 (1) where

휎푡 = 휎푡(퐼푡−1) (2)

This implies returns at any point in a period is conditional on the collected existing information up to that point. This is called the information set, 퐼푡−1 which is concluding all historical observations on the return up to time t-1 (Alexander, 2008, p. 132).

The first model and the basis of further volatility forecast models is the basic idea of Autoregressive Conditional Heteroskedasticity, 퐴푅퐶퐻 model, by Engel (1982). The idea of the ARCH model is utilizing a simple quadratic function of its lagged values describing the reliant of the shock. As well, the shock of an asset return is serially uncorrelated but dependent. Two weaknesses of the ARCH model are the model is being likely to over predict the volatility and assuming both negative and positive shocks on volatility have the same effect. The simple ARCH model for conditional variance is specified as:

푛 2 2 휎푡 = 훾푉 + ∑ 훼푡휀푡−1 (3) 푡=1

2 Where σ is the volatility of the index at day t and 휎푡 is the variance rate., V is a long-run average variance rate (Tsay 2013, page 187). Even though the ARCH model is simple, it frequently requires several parameters to describe the volatility process of an asset return. Therefore, some alternative models were sought, and below the six GARCH models studied in this thesis are specified. These models are divided into two separate sections depending on if the model captures the leverage effect, the third proposition described for volatility models.

9 3.1 Symmetric models

These symmetric GARCH models places a non-negative constraint on parameters to avoid causing negative volatility. This may restrain the dynamic of the model and these models like the ARCH model assume that negative and positive shocks have the equal effect on volatility. Even though it is known that the return of an asset reacts differently to negative and positive shocks (Tsay 2013, page 187). The standard GARCH and IGARCH belongs to symmetric models.

3.1.1 GARCH model

Bollerslev (1986) proposed an extended method for ARCH and introduced the Generalized ARCH model, GARCH. Strengths of GARCH models are the ability to capture volatility clustering and the have a heavy tail distribution of models. Similar to, ARCH model the generalized model is simple and provides a convenient function used to describe volatility evolution. The simple GARCH(1,1) model is defined as:

2 2 2 휎푡 = 훼0 + 훽1휎푡−1 + 훼1휀푡−1 (4)

Where 훼0 = 훾푉 and restricted on:

훼0 > 0 훽1,훼1 ≥ 0 훾 + 훽1 + 훼1 = 1

The restrictions guarantee positive estimates of conditional variance which is contingent on long-run average variance rate and the previous conditional variance and market shock. The error parameter 훼1 measure the effect of a shock on conditional variance and the lag parameter

훽1 measures the persistence in volatility (Alexander 2008, page 137). Furthermore, the rate of return is restricted on the conditional mean equation and a constant c, defined as:

푟푡 = 푐 + 휀푡 (5)

All following models' rate of return is restricted on the same conditional mean equation 5 above. Discussed issues with the symmetric GARCH(1,1) model is the lack of capturing asymmetries in returns and also, reacts too slow on fluctuations.

10 3.1.2 IGARCH model

Most financial markets have volatility forecasts tending to move to the average over time, mean aversion. However, some assets may have volatility, which is not as mean-reverting and thus don't need convergence in the term structure. During these circumstances, the standard GARCH models do not apply. When α+β=1 the variance process is stationary and a GARCH model with a unit root becomes an Integrated GARCH, IGARCH model (Alexandes, 2001, p. 75). A main feature of the model is the effect of past squared shocks being persistent. An IGARCH(1,1) model for conditional variance can be written as:

2 2 2 휎푡 = 훼0 + 훽1휎푡−1 + (1 − 훽1)휀푡−1 (6)

The key difference between GARCH and IGARCH is the unconditional variance of a shock not being defined under the equation above (Tsay 2013, p. 211). Besides, this model has similar weaknesses as the standard GARCH.

3.2 Asymmetric models

Many models have been developed to overcome the weaknesses of symmetric models and especially the leverage effect. Different specifications of the standard GARCH function have evolved in literature and practice to adapt an asymmetric response (Alexander 2008, page 79). As assets return may react differently on negative and positive shocks, these asymmetric EGARCH, TGARCH, NGARCH and APARCH models are evaluated.

3.2.1 EGARCH model

Several GARCH models need to place non-negative constraints on the parameters to escape causing negative variance. Nelson (1991) presented the Exponential GARCH model which removes the requirement for such constraints. In order to catch the Leverage effect, an asymmetry distinguishing a lot of financial time series is applied a logarithmic term on the conditional function is applied. The exponential character guarantee that even if the values are negative the contingent variance is always positive. A function of the lagged error terms allows the natural logarithm of contingent variance to vary over time in the model. An EGARCH(1,1) model conditional variance equation can be defined as:

11 2 2 푙푛(휎푡 ) = 훼0+ 푔(푧푡) + 훽푙푛(휎푡−1 ) (7)

Where:

g(푧푡) = 휃푧푡 + 훾(|푧푡| − √2/휋) (8)

The asymmetric response function, equation (8), captures the leverage effect, where the term

휃푧푡 determines the sign and the term 훾(|푧푡| − √2/휋) the size of the effect. The property of asymmetry implies that the parameter 휃 is characteristically negative and 훾 positive (Alexander 2001, page 79-80). Several studies are demonstrating the logarithmic specification of exponential GARCH fits financial data extremely well. The exponential character seems to have substantial advantages and outperforms often other GARCH models, even without substantial leverage effects.

3.2.2 TGARCH model

Another modified model of GARCH is the Threshold GARCH model presented by Glosten et al. (1993), using dummy variables to capture asymmetries. A multiplicative dummy variable adds into the variance equation to control if a negative shock indicates a statistical significance difference. The TGARCH(1,1) model of conditional variance is defined as:

2 2 2 휎푡 = 훼0 + 훽1휎푡−1 + (훼1 + 훾1푁푡−1)휀푡−1 (9)

Where a, b, y are nonnegative parameters and 푁푡−1 is a dummy variable for a negative shock, 휀푡−1:

1 푖푓 휀푡−1 < 0 푁푡−1 = { (10) 0 푖푓 휀푡−1 ≥ 0

The model signifies a negative shock having a larger impact with y>0 than a positive shock and using zero as a Threshold to divide the effects of past shocks (Tsay 2013, p. 222).

12 3.2.3 NGARCH model

Engle and Ng (1993), proposed an alternative model, which could catch leverage effects of past negative and positive shocks called Nonsymmetric GARCH model.

2 2 2 휎푡 = 훼0 + 훽1휎푡−1 + 훽2(휀푡−1 − 휃휎푡−1) (11)

Where θ is the leverage parameter and β is a non-negative parameters. The model is indicating positive returns having a smaller effect on future volatility than negative returns of the same extent. If the leverage parameter θ = 0 the model is diminishing to a standard GARCH model (Tsay 2013, p. 226).

3.2.4 APARCH model

An interesting extension and the last model included in this study is the asymmetric power autoregressive conditional heteroskedasticity (APARCH) model of Ding et al. (1993). Like standard GARCH models the APARCH(1,1) is mostly used in practice and can be written as:

훿 훿 훿 휎푡 = 훼0 + 훽1휎푡−1 + 훼1(|휀푡−1| + 훾1휀푡−1) (12)

Where the coefficients 훼0, 훼1, 훾1 푎푛푑 훽1 fulfils the regularity conditions that volatility is positive and 훿 is a positive real number. The interesting part characterizing the APARCH model is some special cases of δ. If δ = 0 the model reduces to an EGARCH model and takes as the limit of δ → 0. In the case of δ ≅ 2 the parameter can be fixed, and the model becomes the TGARCH model. The final interesting case is if δ = 1, then the model uses volatility straight in the equation. The power function is used to progress the model's goodness of fit. Further, it is difficult to find a good interpretation of δ, except for the special cases (Ruey S 2013, p. 224- 225).

13 4 Data and Methodology

The data used to estimate all single regime GARCH models in this paper consist of the daily closing-, high- and low price of omxs30 collected from Thomas Reuters Datastream. The dataset contains of 1434 observations from 2005-01-03 to 2010-09-15. The closing price is the main data for rate of return which is specified as:

푐 푐 푟푡 = 100(ln(푝푡 ) − ln(푝푡−1)) (13)

푐 Where 푝푡 is the closing price of omxs30 at time t. The daily return of the index is then divided into an in-sample and out-of-sample for forecasts during the crisis. The in-sample or historical data, used for estimation and forecasting volatility, covers the period from 2005-01-03 to 2008- 09-15. The out-of-sample covers the period during the financial crisis and stretches from 2008- 09-15 to 2009-09-15. Furthermore, for the post-crisis forecast, both the above-mentioned samples are combined to cover a new in-sample and the second out-of-sample data covers the period from 2009-09-16 to 2010-09-15. In table 1 the descriptive statistics of the initial in- sample daily rate of return from omxs30 are summarized.

Table 1. Descriptive statistics of OMX Stockholm 30 in-sample rate of return

In-sample min max mean skewness kurtosis ARCHLM (12) Q2 2005–2008 -4,765 5,495 0,019 -0,186 1,812 112,4* 9,21 2005–2009 -7,237 10,368 0,032 0,32 4,769 172,2* 0,005*** Note: *** significant at 1% level, ** significant at 5% level, * significant at 10% level. ARCLM (12) presents the statistics from the Lagrange Multiplier test and Q2 statistics from the Box-ljung test.

There is a large difference between the minimum and maximum rate of return for both in- samples, which is a common feature of an index. The average standard deviation is also significantly higher on the second in-sample, indicating an increase of fluctuations in rate of return. The skewness between -0,5 and 0.5 implies the data is being fairly symmetrical. The skewness changing from negative to positive in the second sample, implying that the size of the deviation from average return is often above positive values. The increase in kurtosis implies that the tails of distribution becoming heavier and the index seems to follow a leptokurtic distribution.

14 In Figures B1-B3 and B4-B6 in appendix the in-sample observations were plotted against three different distributions i.e. normal, student t and ged distribution. As predicted the students t distribution is the best fit, both for the in-sample data pre-crisis and the whole in-sample including the period during the crisis. This follows the earlier mentioned study by Chuang, Lu and Lee (2007), which showed that the GARCH model with the student's t distribution often being the preferred solution. All GARCH models will, therefore, follow the students t distribution for predicting one day ahead volatility.

Serial correlation in the residuals or squared residuals of the predicted conditional mean equation induces volatility clustering (Zivot, 2008). The Lagrange Multiplier test (ARCHLM) by Engle (1982) and the Box-ljung test by Ljung and Box (1978) are used to test volatility clustering and ARCH effects. These tests are calculated on estimations of OMXS30 daily rate of return for lags of residuals and squared returns. The result is presented in the table by

ARCHLM (12) and Q2 parameters. Both tests show statistically significant parameters that indicate serially correlated heteroskedasticity and independence in the data and residuals. The only coefficient that is insignificant is the Box-ljung test for the first in-sample which implies that the time series is dependent and shows weak autocorrelation.

4.1 Out-of-sample forecast method

The method to forecast volatility used for all models is one day ahead rolling forecast with a refit every trading day. For the first out-of-sample this means that the first forecast and estimate start on September 15, 2008, with a forecast length of 251 trading days due to holidays. The first forecast is derived from the in-sample up to t = 0. Accordingly, the method with refit, the second forecast is calculated based on the true rate of return at t = 1 to predict as accurate predictions as possible. The one step ahead rolling forecast method of GARCH(1,1) is calculated by:

2 ̂ 2 2 휎̂푡+1 = 훼̂0 + 훽1휎푡 + 훼̂1휀푡 (14)

2 Where 휎̂푡+1is the predicted volatility at time t+1 using the historical data from previous in-sample up to t = 0. By allowing the refit window this equation is then used again to 2 calculate 휎̂푡+1 using the observed true data from the previous period up to t+1, t = 2 (Alexander, 2008, p. 142). In the same way, the one step ahead forecast of the other five models

15 is calculated and presented in appendix C1. These forecasting methods generate one step ahead series of conditional variance past the in-sample to later be compared against the observed realized volatility.

All GARCH-family models are estimated by a maximum likelihood function in the program R. The method is typically used to fit the parameters and the estimator is almost constantly consistent. The log-likelihood function allows the conditional variance and the rate of return to be jointly estimated. In presence of time-varying volatility and mean the standard GARCH likelihood can be defined as:

2 1 푛 2 1 푛 휀푡 푙푛퐿(휃) = − ∑푡=1 ln (휎푡 ) − ∑푡=1 ( ) (15) 2 2 휎푡

where 휃 = (훼0, 훼, 훽) and n the number of observations. This procedure minimizes convergence problems for most simple univariate GARCH likelihood functions (Alexander, 2008, pp. 137-138). Accurate forecast by maximum likelihood is restricted on several years and well-behaved daily data. In this study, due to daily data of 5 years and significant tests, these restrictions are fulfilled for estimates and results. Furthermore, one estimation is calculated for each daily forecast.

4.2 Realized volatility

Comparing different models' predicted volatility, the "true" or realized daily volatility is calculated. Collecting the optimal high-frequency price data is not an easy task as it requires short interval data including hundreds of observations. Such data of OMXS30 is not available for everyone, instead, this study adapts a method called Realized Range estimate developed by Parkinson (1980). The Realized Range method uses the daily logged high and low prices and is defined as:

1 252 푛 퐻푡 2 휎퐻퐿 = √ ∑푡=1 푙n ( ) (16) 4ln (2) 푛 퐿푡

Where 퐻푡 equals the daily high price at time t

퐿푡 equals the daily low price at time t The daily realized volatility is the difference between high and low logged prices and shown as:

16 퐻푡 휎푟 = ln ( ) (17) 퐿푡

Andersson and Bollerslev (1998) indicating that Parkinson realized range estimator achieves as good as high-frequency intra-daily volatility based on intervals between 2-3 hours. Hence, in this study, the range estimated realized volatility is used as a proxy for actual daily volatility and compared to all models predicted volatility.

4.3 Forecasting evaluation

Evaluating forecast performance is a widely discussed and fairly criticized topic concerning volatility. A forecast error is the difference between the realized volatility and its forecast and is defined as the unpredictable amount of an observation. Forecast errors can be written as:

푒푡+1 = 휎푟,푡+1 − 휎̂푡+1 (18)

By summarizing forecast errors in different methods forecast accuracy can be measured (Hyndman & Athanasopoulos, 2008). Several loss functions and evaluation methods are existing, and results can differ depending on which function is used so, therefore, choosing the most reasonable loss function is not that straight forward. This study follows four different evaluation measures commonly used in literature and highlighted by Poon and Granger (2003). These measures are Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percent Error (MAPE). The statistical evaluation measurements are defined as:

푛 −1 2 푀푆퐸 = 푛 ∑(휎푟,푡+1 − 휎̂푡+1) (19) 푡=1

푛 −1 2 2 2 푅푀푆퐸 = √푛 ∑(휎푟,푡+1 − 휎̂푡+1) (20) 푡=1

푛 −1 푀퐴퐸 = 푛 ∑|휎푟,푡+1 − 휎̂푡+1| (21) 푡=1

17 푛 −1 2 2 2 푀퐴푃퐸 = 100푛 ∑|(휎푟,푡+1 − 휎̂푡+1)/휎푟,푡+1| (22) 푡=1

Where 휎̂푡+1 represents the estimated volatility, 휎푟,푡+1 is the realized volatility at time t, and n is the number of forecast observations. MSE is a quadratic loss function that is suitable when large differences between realized and estimated volatility are more severe. (Hansen and Lunde, 2005, p. 877). The reason is that it delegates larger weight to large forecasts. If estimated forecast errors are particularly large the RMSE method is appropriated, which also offers a quadratic loss function. MAE measures the average absolute error and is more robust to outliers than MSE and RMSE. The forecast model presenting the lowest value of MSE, MAE and RMSE performs the best forecasts. The final evaluation method MAPE measures the average percentage of the forecast error and is popular to compare performances between different data sets due to the benefit of being unit-free. The MAPE is scale sensitive implying that when the realized value is near zero the MAPE can take on extreme values (Hyndman & Athanasopoulos, 2008). These four evaluation methods are then ranked between all forecast models to present the best model to predict volatility during and post the financial crisis.

18 5. Results

The out-sample forecasted performance by the different GARCH models is estimated by comparing each model's predicted daily volatility against the calculated proxy realized volatility. Accurate forecasts are not necessarily explained by good in-sample fit. The focus of this study is set on accurate forecasts rather than good in-sample fit. In appendix A2 and A3 models, in-sample coefficients are presented. Forecast errors are then evaluated by four selected measures which are Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percent Error (MAPE). The best model forecasting the volatility of OMXS30 daily return is the one with the smallest loss value. The smallest evaluated loss value indicates that the model predicted volatility is closest to the realized daily range volatility. In graph 1 out-of-sample forecasts for all models are plotted against realized volatility during the crisis, respectively after the crisis in graph 2. Tables 2 and 3 present performance by evaluation measures.

Table 2. Out-of-sample forecast evaluation during the crisis Models MSE MAE RMSE MAPE GARCH 0,00028375 0,01166469 0,00170731 61,8228716

IGARCH 0,00027947 0,01160543 0,00170165 62,1929271

EGARCH 0,00035216 0,01331165 0,00180352 56,3174409

TGARCH 0,00032513 0,01284994 0,00174406 56,243747

NGARCH 0,00028155 0,01156129 0,0017057 61,4191251

APARCH 0,00033 0,01300641 0,00175078 56,8450197

Note: forecasts calculated against the proxy of realized volatility. Shaded value equals the smallest evaluated loss estimate.

The result in table 1 shows very small evaluated loss values which indicate that forecasted volatility is on average close to the realized volatility. Comparing all models' error measurements shows that the difference between models' results is exceptionally small. For example, the difference between the best fitted model, IGARCH, and worst model EGARCH, by the MSE measure comes down to 0,00007269. As seen in table 1 the IGARCH model shows the smallest statistical error values of MSE and RMSE. The MAE result evinces that the NGARCH model has the smallest loss value and considered as the best model. In contradiction to other evaluation measurements, the result by MAPE implying that TGARCH model on average generates best predictions.

19

Graph 1. Out-of-sample forecast series during the crisis

Note: The transparent light blue line RV shows the proxy realized volatility of OMX Stockholm 30.

As expected, forecast errors are higher at the beginning of the period where all models have some problems with predicting the fluctuating high volatility. Also, notable are that models react a little bit too slow to fluctuations but follow the trend of the realized volatility series. Correspondingly, all models tempt to underestimate the volatility at the beginning of the crisis, however, yields more accurate forecasts at the end of the period. Finally, it is clear that three models stand out and produce better forecasts overall during the crisis. These are IGARCH, NGARCH and GARCH which on average predicts the realized volatility better.

Table 3. Out-of-sample forecast evaluation post-crisis Models MSE MAE RMSE MAPE GARCH 6,38098E-05 0,00576707 0,000343644 66,9871731

IGARCH 6,3304E-05 0,00574768 0,000342605 67,4632474

EGARCH 6,06756E-05 0,00562928 0,000338543 63,408827

TGARCH 6,21708E-05 0,00568244 0,000341609 62,61814

NGARCH 6,34971E-05 0,00575081 0,000343483 67,3768382

APARCH 6,2518E-05 0,00569444 0,000343068 62,729274

Note: forecasts calculated against the proxy of realized volatility. Shaded value equals the smallest evaluated loss estimate.

20

The difference between the evaluation values in table 2 is still very small and on average closer to each other. Compared to the result during the crisis, statistically evaluated errors are lower post-crisis indicating that all models` forecasts are closer to the realized volatility. The EGARCH model has the best predictions followed by the TGARCH. Ranking the other models shows different results depending on which evaluation method is used. The MAPE method is still inconsistent with other measures and as a result, the TGARCH model generates best predictions followed by APARCH.

Graph 2. Out-of-sample forecast post-crisis

Note: The transparent light blue line RV shows the proxy realized volatility of OMX Stockholm 30.

Graph 2 clarifies that forecast errors are smaller during the post-crisis period, however graphically it is difficult to notice which model performs closest predictions against realized volatility. The difference between all models is significantly small and overlaps each other in certain periods. Adapting the evaluation measurement result and scrutinize the graph, both EGARCH and TGARCH models are predicting on average the best volatility forecasts during the post-crisis period. The two green lines are overlapping each other and reacting better to changes between higher and lower volatility. Even post-crisis all models tend to underestimate the realized volatility yet still follow the trend in the series.

21 To summarize the result, symmetric models outperform in general asymmetric models in forecasting volatility during the financial crisis, with exception of the non-symmetric GARCH. By ranking the evaluation values between models and methods prove the IGARCH model performing best predictions on average during the financial crisis followed by the NGARCH and GARCH models. The result of post-crisis analysis proves the asymmetric EGARCH and TGARCH models performing the most accurate volatility forecasts and outperforms symmetric GARCH models. Investigating both graphs provides evidence that all models on average tend to underestimate the volatility however seems to follow the trend quite well.

22 6 Discussion

As proved in the result the difference between models evaluated loss functions is quite small. This was not surprising since the method is one step ahead forecast with a rolling refit of true data every day. On average for both time periods, all models predict volatility close to the proxy realized volatility.

Overall the result follows economic theory and previous studies to a large extent. Lim and Sek (2013), established that the standard GARCH model were outperforming asymmetric TGARCH and EGARCH models during the financial crisis in Malaysia 1997. Furthermore, asymmetric models were superior pre- and post-crisis periods which signified that the model's performance varies across evaluation methods and periods. Similar to Lim and Sek (2013 findings, in this study symmetric models, generated most accurate volatility forecasts during the financial in Sweden 2008. Likewise, asymmetric models were superior post-crisis period.

Why IGARCH generated best forecasts during the crisis can be connected to Box-ljung tests described in section 4. The result from Box-ljung test of the first in-sample indicating tendencies of stationarity and might prove why IGARCH is performing better predictions. A main feature of the IGARCH model is that the effect of past squared shocks is persistent, and the unconditional variance of a shock is not defined in the function. Another notable finding can be seen in appendix A2 where standard GARCH models α1 and β1 parameters together equal 0,993. When α + β = 1 the variance process in the standard models is stationary and becomes an Integrated GARCH and therefore both models have similar results during the crisis. This shows that leverage effects are not as significant during a more volatile period and therefore symmetric models seem to produce better predictions on average.

However, an unexpected finding is that the non-symmetric GARCH model performed best forecasts according to the MAE and second-best by MSE and RMSE. NGARCH is the only model from the asymmetric group that outperformed the standard GARCH model during the crisis. As mentioned, if the leverage parameter 휃 = 0 the model reduces to a GARCH model but as you can see in appendix A2 the parameter is significantly greater than zero.

The mean average percentage error was another finding that is inconsistent with economic theory and other evaluated loss functions during the crisis. According to MAPE all asymmetric

23 models generate on average better predictions than symmetric models. The result implies that the TGARCH model generates most accurate forecasts. Contradictory to MAE the result of MAPE seems to value higher measures during post-crisis even though the MAE measure is much lower. The result MAPE is known to be scale sensitive and can take extreme values if estimates are close to zero or in presence of large outliers. This is a possible reason why MAPE provides the opposite result during the crisis period when volatility changes a lot between values quite close to zero. Therefore, MAPE is not concluded as an suitable statistical evaluation method for the data.

Four of the reviewed studies in section 2 establishes that in general asymmetric models outperform symmetric models in forecasting future volatility. These studies examine different markets and periods. Evans and McMillan (2013) proved that GARCH models allowing for leverage effects produce the best forecast predictions. Further, the empirical result shows that EGARCH outperforms all other models in 8 countries including Sweden. These findings are equal to the result in this study where asymmetric models were best fitted post-crisis period. As well EGARCH was the best model followed by TGARCH which implies that leverage effects are substantial in volatility forecasts.

GARCH models are fairly criticized for only using historical data in forecast functions, few variables used and does not account for future events. Several studies have proven that this is not always a problem for all time series. Nwogugu (2006) listed further critique against GARCH volatility models and stated that mean-variance functions are insufficient for accurate predictions and analyzes. Correspondingly, models are even for short forecast periods essentially static and incomplete because they don't characterize actual decision making and pattern of risk on the financial market. Poon and Grangers (2003) study that reviewed and compared 72 published papers on volatility forecasting findings is one of many studies that state the opposite. They point out that volatility is not a precise substitute for risk. However, accurate volatility forecasts of asset return over a period is a good approximation for valuing investment risk. This uncertainty is a main input in asset allocation and investment decisions. Frennberg and Hansson (1996) established that for smaller markets including Sweden, implied volatility that uses future expectations in the model was not better than historical autoregressive models.

24 6.1 Conclusion

The purpose of this thesis was to examine volatility forecast performance of some of the most popular symmetric and asymmetric GARCH models during and post the financial crisis of 2008. Predictions are based on the OMX Stockholm 30 index with one day ahead horizon. These single regime models' predictions were evaluated against the realized daily volatility using four statistical evaluation measures. In this study the proxy realized volatility uses the daily logged high and low prices, a method developed by Parkinson (1980). The statistical evaluation measures are Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percent Error (MAPE). The importance of accurate volatility forecast during a financial crisis would not only help investors minimize loss but would also help understanding the Swedish market.

Evaluated forecast performance in this study differs between time periods and evaluation methods. During the crisis, IGARCH(1,1) generates the most accurate forecasts on average according to MSE and RMSE. Second best predictions are generated by NGARCH(1,1) followed by GARCH(1,1). Likewise, MAE ranks these three models in top with NGARCH(1,1) as the best model. This demonstrates symmetric models outperforming asymmetric models during a more volatile period, with exception of the NGARCH model.

The post-crisis result provides evidence that the asymmetric models generating more accurate forecasts than symmetric models. The EGARCH(1,1) is the best fitted model according to MSE, RMSE and MAE. In second place came TGARCH(1,1) which furthermore showed the best result according to MAPE. This result is persistent with previous research proving that asymmetric models that can account for the leverage effect are preferable during a more uniform time period.

In this study, due to time restrictions, only a fraction of all GARCH family models were compared. For future studies, it would be interesting to include also other models on the Swedish market. Likewise, GARCH models are just one group that can be used to forecast volatility. In Poon and Grangers (2001) study, four groups with different forecast characteristics where introduced. It would be interesting to compare GARCH models against the other three

25 sections including exponentially weighted moving average (EWMA), stochastic volatility (SV), or implied volatility.

This study investigated forecast performance on the Swedish stock market during the recent financial crisis. Therefore, it is difficult to compare the result with other studies on different periods and stock markets. Further, a similar process on the Swedish market and OMX Stockholm 30 during the ongoing financial crisis would be a good complement for comparing if the result is consistent.

26 References

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29 Appendix

Table A1. Underlying companies’ stocks in OMXS30 Companies 2019 Companies 2008 ABB ABB

Alfa Laval AB Alfa Laval AB

ASSA ABLOY AB ASSA ABLOY B AstraZeneca AstraZeneca Atlas Copco A Atlas Copco A Atlas Copco B Atlas Copco B Autoliv Inc. SDB Boliden

Boliden Electrolux B

Electrolux B Eniro Essity B Ericsson B Getinge B Handelsbanken A H&M B Hennes & Mauritz B Hexagon B Investor B Investor B Lundin Petroleum

Kinnevik B Nokia

LM Ericsson B Nordea Bank Nordea Bank Sandvik Sandvik AB Scania B SCA B SCA B SEB A SEB A Securitas B Securitas B

Skanska B Skanska B SKF B SKF B SSAB A SSAB A Svenska Swedbank Handelsbanken A Swedbank A Swedish Match

Swedish Match Tele2 B

Tele2 AB TeliaSonera Telia Company Volvo B Volvo B Vostok Gas SDB

Note: Total of 30 companies’ stocks.

30 Table A2. In-sample goodness of fit for 2005 - 2008 Model 훼0 훼1 훽1 훾1 휃 훿 GARCH 0.000 0.094* 0.899*** IGARCH 0.000 0.096*** EGARCH -0.247*** -0.143*** 0.973*** 0.079*** TGARCH 0.000*** 0.071*** 0.915*** 1.000*** NGARCH 0.000*** 0.101*** 0.900*** 1.735*** APARCH 0.001*** 0.072*** 0.919*** 1.000*** 0.831*** Note: Note: *** significant at 1% level, ** significant at 5% level, * significant at 10%level. The coefficient 훼0 for GARCH and IGARCH is the only insignificant estimate.

Table A3. In-sample goodness of fit for 2005 - 2009 Model 훼0 훼1 훽1 훾1 휃 훿 GARCH 0.000 0.089 0.910*** IGARCH 0.000 0.089 EGARCH -0.101*** -0.105*** 0.989*** 0.123*** TGARCH 0.000*** 0.059*** 0.942*** 0.944*** NGARCH 0.000 0.095*** 0.914*** 1.598*** APARCH 0.000 0.062*** 0.943*** 0.930*** 0.857*** Note: *** significant at 1% level, ** significant at 5% level, * significant at 10%level.

31

Graph B1- B3. Distribution plots for first in-sample

Graph B4 – B6. Distribution plots the second in-sample

Graph B6. OMX Stockholm 30 rate of return.

32 Appendix C1. Equations

The one step ahead forecast of IGARCH(1,1) is calculated by:

2 ̂ 2 ̂ 2 휎̂푡 = 훼̂0 + 훽1휎푡−1 + (1 − 훽1)휀푡−1 (23)

2 Where 휎̂푡+1is the estimated volatility at time t+1 using the historical data from previous period. By allowing for the refit window this equation is then used again to calculate 2 휎̂푡+1 by the observed true data from previous period up to t+1 when t=2.

The one step ahead forecast of EGARCH(1,1) is calculated by:

2 2훽̂ 휎̂푡+1 = exp(훼̂0) exp (푔̂(푧푡))휎̂푡 (24)

The one step ahead forecast of TGARCH(1,1) is calculated by

2 ̂ 2 2 휎̂푡 = 훼̂0 + 훽1휎푡−1 + (훼̂1 + 훾̂1푁푡−1)휀푡−1 (25)

The one step ahead forecast of NGARCH(1,1) is calculated by:

2 ̂ 2 ̂ 2 휎̂푡 = 훼̂0 + 훽1휎푡−1 + 훽2(휀푡−1 − 휃̂휎푡−1) (26)

The one step ahead forecast of APARCH(1,1) is calculated by:

훿 ̂ 훿 훿 휎̂푡 = 훼̂0 + 훽1휎푡−1 + 훼̂1(|휀푡−1| + 훾̂1휀푡−1) (27)

33