Analysis of the Polish stock market indices based on GARCH-in-mean models

Transcription

1 Analysis of the Polish stock market indices based on GARCH-in-mean models Krzysztof DRACHAL Abstract The aim of this research is to present the result of application of GARCH-in-mean (also know as GARCH-M) models. Over seventeen years period, starting in 1998, was analysed. Various frequencies were considered, i.e., daily, weekly and monthly. It was speculated that this kind of model could shed some light on the problem of a size risk premium for the Polish stock exchange. Unfortunately, the selected methodology did not give the expected conclusions. On the other hand, it can be concluded that there is no size risk premium on the Polish stock market, as far as it could be detected by GARCH-M. This is somehow in agreement with other cited researches. Yet, for a few time series some weak evidence of a risk premium was found. Keywords GARCH, risk premium, size premium, stocks, volatility JEL Classification C22, G12, G17 Introduction One of the very important problem for the investor in the stock market is a matter of risk modelling. It is known that a variety of securities has different risk levels. It is assumed that safe securities (such as, for example, bonds) are characterized by a relatively low rate of return. Whereas securities offer relatively high-risk but also a high returns. This regularity is well known in the literature and the theory of economics [19]. In the literature, however, one can also find the so-called concept of premiums for the size of the company. It is expected that small companies generate somewhat higher risk than large capitalization companies [8, 9, 34]. Therefore, it seems that it is worth checking this relationship for the Warsaw Stock Exchange. This research is based on GARCH-M model.

2 Literature review GARCH model was presented in the article of Bollerslev in 1986 as a generalization of the ARCH model [3, 7]. It constitutes a wide class of models. In particular, it is said that the variable x follows GARCH-M process, if x t = a 0 + b. h t + e t, and e t = u t h t, where u t follows the generalized normal distribution and h t = c 0 + c 1. (e t-1 ) 2 + d 1. h t-1. GARCH class models are very useful in finance and economics, especially if the analysed time series are of a high frequency type [1, 11]. GARCH-M models can be used for modelling the risk. Namely, the estimate of a parameter b in the above equation (if it is statistically significant) can be interpreted in the following way. The increase in the conditional variance (i.e., a measure of the risk) results in the increase in the expected rate of return. It is assumed that the variable x is a rate of return. The simplest form of the above model is, if u t follows the standard normal distribution. However, if one is interested in leptokurtic distributions some generalization is preferred. The Reader interested in this topic for Polish stock market and in its efficiency should consult some other literature (for example, [20] and [21]). Of course, these models can have a more complex variance equation. However, higher complication of the model does not lead to relatively better result in many cases [17, 25]. On the other hand, Nelson [26] pointed out that the simple GARCH model has some significant disadvantages. Among other things, its definition excludes the existence of a negative correlation between the future conditional variance and the current value of the modelled rate of return. In addition, the estimated coefficients often do not meet theoretical assumptions (for example, they tend to be negative). GARCH model has therefore been expanded to include various modifications. For time series from the Polish stock exchange GARCH models, taking into the account the asymmetry of the empirical forecast error has been examined, among others, by Małecka [24] and Rozkrut [30]. The problem of limits of parameters for the data from the Polish stock exchange was discussed by Galin [15]. Fiszeder and Kwiatkowski [12] analysed selected 28 companies from the Warsaw Stock Exchange and came to the conclusion that in the case of stock market indices GARCH model describes the variability conditional variance the best. On the other hand Bartkowiak

3 [2] analysed only the assets related to the option market on the Warsaw Stock Exchange (WSE). Their results are in part consistent with those from the developed markets. On the other hand, received concerns suggest that it is still worth to study Polish market, as it continues to grow and develop. In this sense, even the stylized facts are worth noting. Płuciennik [28] pointed out that the use of models with autoregression results in a considerably better results than the simple GARCH models for the WIG 20 index. GARCH models were also explored in the context of the Polish stock exchange by Filipowicz [10], Karkowska [22], Doman [6], and others. Finally, it seems interesting to reflect on whether the larger companies on the WSE are characterized by slightly lower risk compared to the market average [5, 8, 9]. In this context, certain model of Fama and French is sometimes analysed [8, 9, 13]. This issue is, however, debatable. Foye, Mramor and Pahor [13] analysed the markets of the so-called new European Union members and found that the model of Fama and French has some disadvantages. Thus, they have proposed some modifications. Yet, another modification was proposed by Czapkiewicz and Wojtowicz [4] based on the analysis of the data from WSE for the years using monthly data. On the other hand, Słoński and Kwiatkowski [32] recognized the three-factor model as sufficient to describe Polish companies. They also claimed that the premium for the size is important for the smallest companies. However, Sekula [31] based on the data from the years 2002 to 2010 questioned the relationship between market capitalization of the companies and their rates of return. Moreover, he noted that the premium for the size was reversed, i.e., companies with medium and large capitalization generated higher returns than companies with low capitalization. Similarly, doubts on the existence of a premium for the size appeared during the analysis of the data from developed markets [18]. Van Dijk [33] conducted a fairly extensive comparative analysis and came to the conclusion that the premium for the size fades from approx. 1980s. Similarly, its existence is disputed by Paschall [27]. For the markets of the Central and Eastern Europe doubts to its existence were presented by Konieczka and Zaremba [23]. Methodology The data were obtained from the data base Stooq.pl ( Five indices were analysed: WIG all stock index, WIG20 consisting of blue chips, WIG BANKI consisting

4 of banks, mwig40 representing medium sized companies and swig80 representing small companies. The daily data were obtained from the period between 31/12/1997 and 7/7/2015. Indices were chosen from the available ones as to cover at least ten year period and to have the same number of observations. As a result, these five indices and 4390 observations for each index were able to be obtained. All calculations were done in R [28]. Everywhere 5% significance level was assumed. The aim was to estimate the following, aforementioned, equation x t = a 0 + b. h t + e t, and e t = u t h t, where u t follows the generalized normal distribution and h t = c 0 + c. 1 (e t-1 ) 2 + d. 1 h t-1, where x t denotes the return from the selected index. This can be done in R, for example, with a help of rugarch package [15]. Logarithmic daily returns were computed and ordinary weekly and monthly returns. It was assumed that a week consists of 5 subsequent observations and a month 21 observations. The idea was to estimate the parameter b in the above equation, and if found statistically significant, to compare it for different time series. It was expected that this parameter would be smaller for wig20 returns series and bank return series in comparison to wig returns series. It was also expected that wig80 returns series would give the highest value for estimate of the parameter b, etc. As the risk premium (expressed by the parameter b) should diminish with the average size of listed companies. Results Fig. 1 presents logarithmic daily returns from all the considered indices. A volatility clustering seems to be evident. However, ARCH-LM test was performed for every time series. Its results and that of augmented Dickey-Fuller test for stationarity are presented in Tab.1. It can be seen that for all time series and both tests the null hypotheses should be rejected. Therefore, one can assume that all the return time series are stationary and there are significant ARCH effects. As a result, it is quite reasonable to perform the analysis of the GARCH type models. Unfortunately, out of 15 evaluated models, only in 4 of them the parameter b was statistically significant. In particular, they are the daily and monthly model for swig80 and weekly models for WIG BANKI and mwig40 (see Tab. 2, Tab. 3, Tab. 4 and Tab. 5).

5 Fig. 1: Logarithmic daily returns Source: Own estimation in R Tab.1: ADF and ARCH-LM tests results p-values ADF ARCH-LM WIG WIG WIG BANKI mwig swig Source: Own estimation in R In case of daily logarithmic returns only the model for swig80 gave statistically significant parameter b. Yet, for the same index but ordinary monthly returns this parameter is also statistically significant, however, smaller. Notice, that in case of daily frequency the parameter is much larger than the unit. It means that the reaction form returns

6 on volatility change is high. In case of monthly frequency the parameter is less than the unit. For mwig40 the estimation is a bit surprising, because the estimated parameter is negative. This would mean that an investor is penalized for holding a risky instrument. Yet, a similar findings were reported by Gabrisch and Orlowski [14]. Finally, for WIG BANKI the parameter is the smallest amongst the presented models, less than the unit and positive. Indeed, comparing the models for swig80 (small companies) and WIG BANKI (banks) it can be stated that small companies are characterised by higher risk premium than banks (which can be assumed as stable companies). Another interesting problem is to consider the structural stability of the parameter b in time. Indeed, a refitting was done for the models reported in Tab. 2, Tab. 3, Tab. 4 and Tab. 5. The first estimation was done based on the period including 750 session days (approximately 3 years). The refits were done after every new 125 session days (approximately half a year). The outcomes are presented in Fig. 2, Fig. 3, Fig. 4 and Fig. 5. For the model reported in Fig. 2 the parameter b is quite stable in time. However, a high uncertainty in its estimation is present at the beginning of the analysed period and around 2007, when the global financial crisis started. The Nyblom stability test gives the individual statistics (under 0.47 critical value). Therefore, it confirms that the parameter is not stable. For the model reported in Fig. 3 the parameter b varies from the values lower than the unit, then exceeds the unit and finally, becomes lower than the unit again. However, it is always positive. The Nyblom stability test gives the individual statistics (under 0.47 critical value). Therefore, it confirms that the parameter is not stable. For the model reported in Fig. 4 the parameter b starts form the positive value, but systematically decreases to the certain negative value. It is negative for the most of the analysed period. The Nyblom stability test gives the individual statistics (under 0.47 critical value). Therefore, it confirms that the parameter is stable. For the model reported in Fig. 5 the parameter b starts from values higher than the unit, but decreases to certain positive value, lower than the unit. The Nyblom stability test gives the individual statistics (under 0.47 critical value). Therefore, it confirms that the parameter is stable. It can be seen that the parameter for swig80 (small companies) is more volatile. For example, the direction of the movements changes in time. The parameter for mwig40 and WIG BANKI (medium companies and banks) the parameter oscillates around certain time

7 trend. Finally, it should be emphasised that ARCH-LM and Ljung-Box tests for standardized residuals provided evidence that the constructed models are not good. All models suffer from the problem of autocorrelation. Fortunately, the models for mwig40 and WIG BANKI have no remaining ARCH effects in residuals. Yet, the models for swig80 still have some remaining ARCH effects. a 0 b c 0 c 1 d 1 Tab.2: Estimation of GARCH-M for daily logarithmic returns from swig80 estimate std. error t value p-value shape Source: Own estimation in rugarh in R a 0 b c 0 c 1 d 1 Tab.3: Estimation of GARCH-M for monthly ordinary returns from swig80 estimate std. error t value p-value shape Source: Own estimation in rugarh in R a 0 b c 0 c 1 d 1 Tab.4: Estimation of GARCH-M for weekly ordinary returns from mwig40 estimate std. error t value p-value shape Source: Own estimation in rugarh in R

8 a 0 b c 0 c 1 d 1 Tab.5: Estimation of GARCH-M for weekly ordinary returns from WIG BANKI estimate std. error t value p-value shape Source: Own estimation in rugarh in R Fig.2: Refitting of parameters for daily logarithmic returns from swig80 Source: Own estimation in rugarh in R Fig.3: Refitting of parameters for monthly ordinary returns from swig80

9 Source: Own estimation in rugarh in R Fig.4: Refitting of parameters for weekly ordinary returns from mwig40 Source: Own estimation in rugarh in R Fig.5: Refitting of parameters for weekly ordinary returns from WIG BANKI

10 Source: Own estimation in rugarh in R Yet, also another variations of the models were evaluated. They were not reported in full details here, however, due to the clarity and simplicity of this short report. The change was to compute weekly and monthly logarithmic returns, but based on aggregated data (to weeks and months, where one month is assumed to be equal to 4 weeks). This resulted in no autocorrelation and ARCH effects in standardized residuals. Yet, the parameter b was significant only for swig80. For weekly aggregation its estimate was According to the Nyblom stability test this parameter is stable. Conclusions Some weak evidence was found that small companies are characterised by the positive risk premium. However, its stability cannot be definitely decided. For medium sized companies this premium occurred to be negative (and stable). Also, some relatively small risk premium (also stable) was found for banks. However, no such a premium was found for all stocks or big companies. Yet, the whole research was based on GARCH-in-mean methodology. Therefore, some other approach can lead to other interpretation. Finally, it should be mentioned that even the models, which had the significant

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