Autoregressive Conditional Skewness and Kurtosis on Russian and US Markets

The table below summarizes the results of fitting the model with dynamic asymmetry to our data, it contains estimated values of the coefficients of skewness equation, along with their p-values and log-likelihoods for each of the assets:

Table 3. Model with dynamic asymmetry coefficient

What can be noticed is that the constant term is not significant for all assets. However, slope coefficients are sometimes significant even for the assets which had insignificant asymmetry coefficients in the constant parameters model. This allows us to propose that the model with dynamic skewness parameter captures some of the asymmetry which the constant parameters model did not manage to. And in cases when at least one of the slope coefficients is significant, the overall improvement in the model fit according to the LR test is significant as well. Indeed, the values of the LR statistic for S&P500, RTS, Google, Gazprom and Chevron are respectively 31.32, 42.08, 13.54, 16.294 and 11.6, whereas the critical values for the chi-square distribution with 3 degrees of freedom are 7.815 and 11.345 for 5% and 1% significance levels, respectively. Hence, we can conclude that we have a significant improvement in the fit for the market indices and natural resources companies and no improvement for exchange rates and one of the technological companies - Yandex. Notice that constant parameters model was not able to capture asymmetry in returns of RTS index, Gazprom and Google shares, but it is captured by the dynamic model.

Speaking about coefficients' values, nothing specific can be said: they are individual for each of the assets. It may be nevertheless noticed that coefficient, when significant (the case of market indices and Google corporation), is positive and is substantially higher than 0.5. This, applying the similar logic as to the GARCH analysis, indicates that there is some sort of substantial skewness clustering: periods of high asymmetry tend to be followed by periods with high, and periods with low-by periods with low.

Now we extend our model further and allow for dynamics not only in the asymmetry, but also in the degrees of freedom. The functional form for the degrees of freedom parameter was chosen to be the same as for skewness. Hence, the overall specification of the model is the following:

,

This model will be called Model 3 for the future reference. The following table specifies the results of the estimation:

Table 4. Model with dynamic asymmetry and degrees of freedom

It does not include the results for USD/RUB exchange rate as they could not be computed correctly: the algorithm did not converge which resulted in nonsense values of estimated coefficients and “NA” instead of standard errors and p-values. The problem is probably in the fact, that skewness and kurtosis parameters fell out of the required ranges in the process of estimation. This issue was mentioned briefly in the literature review. To solve it, some bounding transformation (such as logistic) needs to be performed. For now, we will not address this issue, as the model without these restrictions still works fine. Nevertheless, this problem will become prominent in the next chapter.

Apart from that, the model with both dynamic parameters seems to improve the fit for almost all assets compared to the model with only one dynamic parameter with exceptions to Gazprom (LR= 3.244), RTS (LR=6.34) and Yandex (LR=6.982). The critical values are the same here as previously, since we have the same number of restrictions compared to the model with constant degrees of freedom but dynamic asymmetry. We also have the case for USD/CHF exchange rate, for which the model with dynamic skewness is not preferable to the model with constant one. But there is a noticeable improvement in fit if the model with two time-varying parameters is used: the value of LR statistic used is higher than 8. However, the question is, whether this improvement is significant when the restricted model is the one with constant parameters? The number of restrictions in this case is 6 (3 from asymmetry and 3 from degrees of freedom specifications) and the critical values for the chi-square distribution with 6 degrees of freedom is 12.59 and 16.81 for 5% and 1% significance levels respectively. The LR statistic in this case is 20856.24-20949.31=13.86 which indicates that unrestricted model is better than restricted one at 5% significance level. So, apparently, for USD/CHF the skewness parameter is not significant for all days in the studied sample, whereas kurtosis parameter is significant and accounting for its time variation improves the fit of the model.

The estimated coefficients in the equation for asymmetry are similar to those of the previous model. If we take a look at the estimated parameters in the degrees of freedom equation, we may notice that b₁ and b₃ coefficients, when significant, are almost always negative, and b₁ is larger in absolute value than b₃. This indicates us that adverse market events decrease the value of parameter more than favorable ones. And, as we remember, this parameter is in charge of tails' heaviness, which implies that negative shocks have more significant impact on tails than positive ones, which is another source of asymmetry, apart from the direct influence of parameter.

Now we want to summarize the results of all likelihood ratio tests into one single table. This will help to easily tell which of the three models suits best each of the assets. Denote by LR₁ the likelihood ratio statistic when the unrestricted model is the model where only skewness is dynamic and restricted model where both parameters are constant; by LR₂ the statistic where unrestricted model is the one with two time-varying parameters and the restricted one- with time-varying asymmetry; and by LR₃ the one where the unrestricted model is with two dynamic parameters and restricted model without any. Notice that LR₃=LR₁+LR₂

Table 5. The values of likelihood ratio statistic for three tests

What can these results tell us? For most assets inclusion of dynamic degrees of freedom parameter into specification improves the fit of the model and it does so in more cases than inclusion of the dynamic asymmetry parameter. That may be explained by the fact that asymmetry is not present in all assets, while kurtosis certainly does, and accounting for its time variability certainly improves the fit of the model.

As a further robustness check, we also estimated another model with dynamic parameters. The equation for dynamic degrees of freedom is the same as before, however, the equation for conditional skewness parameter now follows Harvey & Siddique (1999) and looks the following way: . The table with estimated results is presented below

Table 6. Model with dynamic asymmetry and degrees of freedom: 2^nd specification

As before, we may notice that dynamic asymmetry specification allows us to detect skewness in some cases when the constant parameter model produced insignificant estimates. It is noticeable in the cases of Google, Gazprom and RTS. Overall, this model specification seems to perform pretty good and can be used as well to model the skewness dynamics. Nevertheless, it is generally inferior to Model 3 in terms of log-likelihood: the latter outperforms it in almost all cases except for Yandex shares (we can directly compare likelihoods in this case as both models have the same number of parameters-12). Hence, further in this paper we will use Model 3 for analysis.

5. Dividend yield, skewness and kurtosis

In the final chapter of this paper I would like to take a look at one of the exogeneous factors that can be potentially correlated with the values of skewness and kurtosis parameters on the Russian stock market. One of the promising candidates are dividends and different metrics associated with them since dividends are not direct factors that affect the distributions of the stock returns (such as overall market fluctuations due to economic cycles), but rather decisions made by individual companies based on some internal processes and policies of the company. From the one hand, higher dividends should be preferred by the investors as additional source of income. From the other, they reduce the company free cash flow which could otherwise be invested into some profitable projects. Thus, dividends payments may substantially affect the form of the distribution of stock prices and returns and, therefore, the asymmetry and the form of the tails, although their effect is not clear from the first glance. In this chapter several questions will be addressed: whether there is an effect of dividends payments on skewness and degrees of freedom parameters and which of the stocks (high or low dividend ones) can potentially be chosen by investors.

5.1 Skewness and kurtosis estimation for two portfolios: constant parameters model

The metric which was taken to measure the effect of dividend payments is dividend yield. It measures the dividend income relative to the stock price and is calculated as expected dividend divided by the current daily stock price. The further analysis is somewhat similar to that implemented by Aggarwal & Rao to measure the effect of institutional ownership on skewness and kurtosis. Except for that instead of forming the quintiles of stocks by their dividend yields, two equally weighted portfolios have been formed: the one with relatively high dividend yield and the portfolio where stocks either paid no dividends at all, or the dividend yield was pretty small (smaller than 4%). Such portfolios were first formed separately for years 2019, 2018 and 2017 and then, the portfolios for 4 years, from 2016 to 2019 has been formed, where the criterion on belonging either to high dividends paying or low dividend paying one was the median dividend yield paid during these three years. Obviously, the portfolios are different from each other as the dividend yield differs from year to year and from company to company, and in one particular year some company shares may pay a lot of dividends and be located in high dividend yield portfolio, while in another year the company may be located in low dividend yield portfolio. There was not any analysis performed separately for the year of 2016 as some of the companies analyzed paid no dividends in that year, having thus 0 dividend yield. Hence, if we included these companies in the low dividend yield portfolio, for instance, the two portfolios would be imbalanced, the low dividend yield one having more companies in it. And as we know, theoretically, more diversified portfolios yield lesser risks and thus can have higher estimated value of the degrees of freedom parameter estimated by our model. I wanted to separate the pure effect of dividend yield on skewness and kurtosis and, therefore, all the portfolios in this study have the same number of companies selected from the fixed list.

To start with, we fit the model with constant parameters. If its power won't be enough to capture the relationship, the model with dynamic parameters will be fitted. As we already know, the dynamic model may capture skewness which the model with constant parameters was not able to detect.

First, the results for the year 2017 are presented:

Table 7. Skewness and degrees of freedom parameters for two portfolios in 2017

***p-value<0.01, **p-value<0.05, *p-value<0.1

The obtained results are quite surprising. The asymmetry coefficient for both portfolios is not significant at 5% level, which is quite usual practice. However, the results for the degrees of freedom parameter are more interesting. We can see, that it is significant for low dividend yield portfolio (at 10% level) and has a quite standard value slightly higher than 8. For high dividend yield portfolio the estimated value is quite high, and the parameter is not significant at all, implying that Student's distribution might be not the best choice for this data. If we perform the Jarque-Bera test for normality, the resulting test statistic will be 1.1 with the p-value of 0.5767. Hence, the null hypothesis about the normality is not rejected, which implies that the data can actually be pretty accurately described by the normal distribution, which is quite unusual phenomenon for financial data. This also means that the distribution of portfolio of high dividend yield stocks has much lighter tails than that of low dividend yield stocks, as t distribution has much heavier tails than normal one, especially for such relatively low value of the degrees of freedom as 8.

Now let us move to the next year, 2018. The results of the estimation are presented in the table below:

Table 8. Skewness and degrees of freedom parameters for two portfolios in 2018

***p-value<0.01, **p-value<0.05, *p-value<0.1

We can see that again skewness parameter is not significant for both portfolios. Concerning the degrees of freedom parameter, it is significant for both portfolios and is higher for the high dividends one, implying more moderate tails and smaller kurtosis. The difference does not seem significant though due to pretty high standard error of DF coefficient for high dividend yield portfolio.

Next, the results for the year 2019 will be discussed. The results of estimation are quite interesting in this case and are similar to those of the year 2017, but are somewhat even more extreme:

Table 9. Skewness and degrees of freedom parameters for two portfolios in 2019

***p-value<0.01, **p-value<0.05, *p-value<0.1

It can be seen that asymmetry parameter is again insignificant for both portfolios. The results for the degrees of freedom parameter are much more surprising though: for both portfolios this parameter is not significant. Further inspection of the histogram and sample statistics for high dividend yield portfolio reveals the value of sample skewness to be around -0.089 and sample kurtosis around 3.204 which implies the Jarque-Bera test statistic to be equal to 0.764 with p-value of 0.683- so the hypothesis about the normality of distribution is not rejected. Taking this result into consideration, it is not surprising that the estimation yielded insignificant result for the degrees of freedom coefficient, as there is no evidence that data follows t-distribution and rather follows the normal one. As a consequence, the tails of this distribution are pretty mild and the kurtosis is small, just like in case of year 2017. Speaking about the results for low dividend yield portfolio, the estimated degrees of freedom in this case is again insignificant. However, the value of Jarque-Bera statistic in this case is 10.154 which implies the p-value of 0.006 meaning that the hypothesis about normality is rejected. This can be mainly attributed to a pretty high negative value of sample skewness (-0.384) which, despite not very high value of excess kurtosis of 0.613, contributes substantially to the value of test statistic. Overall, apparently, in the year 2019 two portfolios did not differ much in term of their tails' heaviness and riskiness.

Finally, we proceed to the analysis of the distribution of returns for the whole period of 2016-2019.

Table 10. Skewness and degrees of freedom parameters for two portfolios in 2016-2019

***p-value<0.01, **p-value<0.05, *p-value<0.1

First thing that can be noticed is that asymmetry coefficients are negative and significant for both portfolios which is an interesting result, as in any of the periods before skewness was not significant for either of the two portfolios. This might be explained by the smaller sample size on which the model was estimated before, which was insufficient to construct the representative density function and correctly estimate parameters of the distribution. As for the degrees of freedom parameter, now the estimated value of it of high dividend portfolio is significantly different from that of low dividend yield one, with latter being smaller. This suggests that low dividend yield portfolio has fatter tails and, accounting for the similar value of skewness, may thus be considered as a riskier one.

To further distinguish the effect of dividend payments, the high dividend yield portfolio was divided into two portfolios based on the dividend stability index (DSI). This index helps to outline how regularly the company pays dividends and increases them. It makes sense accounting for DSI as dividend paying stocks can be divided into two kinds: those which steadily pay dividends and do not plan to decrease them and those which only occasionally pay them, but in rather high amounts, so as a result, these companies “on average” pay high dividends. In the former case investors may rely on incoming stream of dividend payments and thus buy the stocks for receiving these payments, while in the latter case this is not true. So, from investor's point of view there is a big difference. The DSI is calculated as follows: , where is the number of years out of the last 7 when dividends were paid, - the number of years out of the last 7 when the size of the dividend was not smaller than the last year one. This index takes values between 0 and 1. For the sample of high dividend paying stocks in this study the smallest value of this index is 0.36, while the largest is 1. The threshold value of division is 0.5: stocks with higher DSI go to “high DSI” portfolio, while stocks with lower or equal DSI- to “low DSI” portfolio. All the data for the values of this index along with its formula was obtained from the website dohod.ru. https://www.dohod.ru/ik/analytics/dividend/pdf/dsi.pdf.

The table below presents the results of estimation of the constant parameters model on the portfolios with high and low DSI for the whole period of 2016-2019:

Table 11. Skewness and degrees of freedom parameters for two portfolios based on the value of DSI in 2016-2019

***p-value<0.01, **p-value<0.05, *p-value<0.1

Now the asymmetry is not significant for both portfolios, whereas in the case when these stocks were in the same portfolio, asymmetry was present. Quite interesting results are revealed by the estimated degrees of freedom parameter: it is higher for the portfolio consisting of the stocks with low DSI, those, which dividend payments are not very reliable and are rather sporadic. The implication is that for such portfolio the probability of extreme events is smaller than for a portfolio with high DSI. The potential explanation may be that companies which pay dividends nor regularly, when anticipating not very good financial results due to market conditions or some internal problems, prefer not to pay dividends at all, or pay little. Thus, they are left with more free cash flow which can be spent on protective measures against losses. Hence, their shares price fluctuates less in times of distress.

Overall, it was found out that the distribution of returns for the high dividend yield portfolios tend to have larger values of estimated degrees of freedom than that of low dividend yield. However, nothing specific was found about the skewness, which in most cases is simply not significant. The dynamic model may capture some of the asymmetry which the constant model cannot. Hence, now we will discuss the results of the estimation of the dynamic model for all time periods, namely the model with dynamic skewness and constant degrees of freedom.

5.2 Skewness and kurtosis estimation for two portfolios: dynamic model

There was a problem in the process of estimation of dynamic model, however. This problem is mentioned in all of the literature concerning the modelling of higher order moments and deals with the fact that estimated parameters of the distribution fall out of the required range (absolute value of skewness higher than 1, or degrees of freedom lower than 2). In order to prevent this, new parameters should be created, which are the bounded functions (like logistic, hyperbolic tangent and others) of the old parameters. These new constrained parameters are then plugged into the likelihood function which is then maximized with respect to them. Previously this problem did not appear in this study (except for USD/RUB exchange rate), so not much attention has been paid to it. However, it came into place when the model was run on the data for portfolios of Russian stocks. To resolve it, the logistic transformation on the asymmetry parameter was performed, as in Hansen (1994): if we want to bound a certain parameter between the values L and R, we can create a new parameter ( related to the old parameter ( as . In this case, even if is allowed to vary over the entire number line, will be constrained to lie between L and U. We then let the unconstrained parameter to have an autoregressive structure as before, thus obtaining a relationship between the constrained parameter and coefficients in the equation.

The equation for dynamic asymmetry (unconstrained) remains the same as before and looks the following way:

Now we can discuss the estimation results, starting with the year 2017

Table 12. Dynamic Skewness model estimation for two portfolios in 2017

***p-value<0.01, **p-value<0.05, *p-value<0.1

The results mainly remain insignificant, as they were in the model with constant parameters. The coefficient however, is highly significant for high dividend yield portfolio. This indicates that for this portfolio high degree of skewness clustering persists: high asymmetry tends to be followed by high, and low-by low. This also implies that for high dividend yield portfolio skewness tend to stay relatively constant over time, as coefficient is really close to 1.

Next the results for the year 2018 shall be discussed.

Table 13. Dynamic Skewness model estimation for two portfolios in 2018

***p-value<0.01, **p-value<0.05, *p-value<0.1

Intercepts for both portfolios are negative and significant, for high dividend one being higher in absolute value. The coefficient shows the sensitivity of the portfolio asymmetry to “negative news”. For high dividend yield portfolio it is negative and significant, while for low dividend yield one-insignificant. This indicates that high dividend yield portfolio becomes much more negatively skewed on average when the return goes down, which in this case tells us that it is less attractive than low dividend one. This is further proven by the values of the coefficient which indicates the reaction of skewness on “positive news”. In case of high dividend portfolio the value is insignificant, so there is basically no reaction, whereas in the case of low dividend yield portfolio the value is positive and significant which tells us that when the return of the portfolio increases, the asymmetry goes in the positive direction which is desirable for investors.

Now let us proceed to the analysis of the results for the year 2019.

Table 14. Dynamic Skewness model estimation for two portfolios in 2019

***p-value<0.01, **p-value<0.05, *p-value<0.1

The intercept is insignificant for both portfolios at 5% level. However, for high dividend yield one it is significant at 10% and is negative. For this year we also have being positive and significant for low dividend yield portfolio, implying that skewness increases in the positive direction when we have negative returns on the portfolio which is an interesting result, as we suspected it to be negative, in case of significance. The second slope coefficient is significant for both portfolios. However, it has different signs: for high dividend yield portfolio it is positive, implying that higher past values of skewness are associated with higher present values, while for low dividend portfolio this coefficient is negative, hence, there is a reversion- if past value of asymmetry coefficient is negative, present value will increase and vice versa. Finally, the third slope coefficient which shows the reaction towards positive news is significant only for high dividend yield portfolio.

In the final part of this section the dynamic skewness model is applied to the portfolios over the whole period of 2016-2019. The results of the estimation are presented in the table below.

Table 15. Dynamic Skewness model estimation for two portfolios in 2016-2019

***p-value<0.01, **p-value<0.05, *p-value<0.1

We can notice that none of the coefficients is significant for low dividend yield portfolio. This indicates that either there is no skewness in the distribution of portfolio returns over the whole period of 2016-2019, or, as it was shown by the constant parameters model, asymmetry is present, but it is constant and does not react to any market events. For high dividend portfolio results are more interesting: all coefficients except the first slope one are significant. The intercept is negative while two of the slope coefficients are positive, which is generally consistent with the previous results.

To summarize the findings, the dynamic skewness model indeed detected the asymmetry in some periods, where the constant skewness one was not able to. Overall, it appears that portfolio consisting of high dividend paying stocks possesses higher degree of asymmetry which is, moreover, more sensitive to market movements, than asymmetry of the portfolio which consists of low dividend paying stocks. Moreover, the overall skewness is negative which is captured by the negative estimated values of the intercept, when it is significant, and generally positive value of the slope coefficient . The value of the coefficient is also positive when it is significant, which indicates that “positive” market news make the distribution of portfolio returns more positively skewed. For no clear pattern is seen, besides, it is insignificant in the estimation for the portfolios over the whole period of 2016-2019. This may suggest that significant results in some of the years were due to relatively small sample size which was not enough to accurately capture the distribution of portfolio returns.

5.3 High and low dividend portfolios and investors' preferences

In the final part of this chapter (and this study) I would like to elaborate on which portfolio, high or low dividend one, should be chosen by an investor who cares only about capital gains and does not take into account dividend payments.

The model is simple: investor has utility function of the form where u takes values from the interval [0;1] and k is some scaling constant. The future price of the portfolio is modelled in the following way: , where S is the random variable, which is distributed according to skewed t-distribution with mean 0, while is the standard deviation of portfolio returns. Thus, . Investors wish to maximize their expected utility which is in this case equal to:

Here f(S) is the probability density function of the random variable S. In this case u is the parameter unique for each investor, which shows a degree at which this particular investor is concerned about the future value of the security: the higher it is, the more the future price of an asset will be valued.

We want to distinguish, which portfolio should be chosen by an investor based solely on the expected future value of the portfolio, taking into account its distributional characteristics. Hence, the random variable S is given to have the degrees of freedom parameter to be equal to the value estimated by the constant parameters model over the period of 2016-2019, and parameter is equal to the square root of unconditional variance, calculated from the GARCH (1,1) model. The skewness parameter is assumed to be 0 in this case, firstly, for simplifying reasons, and secondly, due to the fact that constant parameters model usually was not able to capture any asymmetry in returns. Thus, we have DF=5.751 for low dividend portfolio and 8.643 for high dividend portfolio; is equal to 0.00727 and to 0.0076 for low and high dividend portfolios respectively. The scaling parameter k was chosen to be equal to 10 for estimated expected utilities to differ from each other more substantially in terms of numbers. The parameter u is then varied from 0 to 1 (by increments of 0.01), and the difference between expected utility of holding a low dividend portfolio and holding a high dividend portfolio is computed for each of the values of u. The calculations were performed via RStudio.

The graph below shows the plotted difference in utilities against the value of the parameter u.

Picture 1. Difference in expected utilities from holding high dividend yield and low dividend yield portfolios as a function of u

We can see that the difference grows exponentially, meaning that as the u parameter increases, it becomes increasingly more attractive to hold the portfolio of low dividend paying stocks than the portfolio of high dividend stocks. The difference, however, becomes noticeable only for the values of u around 0.5 and higher.

All in all, if we take into account only the capital gains from investments, the low dividend yield portfolio provides higher expected utility for all values of the “preferences parameter” u. Nevertheless, if we also consider the dividend payments and additional utility which they bring, the picture won't be so unambiguous: for low values of u, that is, for investors who care less about the future value of the portfolio, the benefits of higher dividends will likely outweigh the lower expected future payoff, so these investors will likely benefit more from holding a high dividend yield portfolio.

Conclusion

This paper had two major goals: firstly, its aim was to estimate skewness and kurtosis parameters for different financial assets on two different markets and to answer the question, whether there are substantial differences in these parameters across the types of assets or markets. The second goal was to look at stocks' prices on one specific market and figure out, whether skewness and kurtosis depend on the amount of dividend paid, and if they do, which stocks, high or low dividend ones, are more attractive for investors. For both these goals skewness and kurtosis parameters were computed using an autoregressive conditional density model, which is estimated via the maximum likelihood method. The major reason for choosing such complex model instead of calculating sample statistics is its ability to incorporate a dynamic nature of skewness and kurtosis. The second reason is that skewness and kurtosis parameters computed in such ways take into account the whole distribution of returns, are less prone to outliers and, consequently, should be more accurate measures of true asymmetry and kurtosis.

It was found out that skewness is generally not significant if we use the model with constant parameters, whereas degrees of freedom are, suggesting that the choice of Student's t-distribution was a reasonable approximation. The model revealed that there was no substantial difference in degrees of freedom parameter between Russian and US markets, consequently, these markets did not differ much in terms of their riskiness. However, there is a substantial difference in kurtosis across different assets: the lowest value of estimated degrees of freedom and, thus, the heaviest tails are possessed by the returns of technological companies. Natural resources companies have the lightest tails and, hence, are the least risky. Market indices which consist of many companies with various natures of businesses are somewhere in the middle. Speaking about exchange rates, all depends on the currencies: in case of US Dollar and Swiss Franc, two reserve currencies from developed countries, the estimated value of degrees of freedom and thus overall riskiness is similar to the market indices. For USD/RUB pair, on the other hand, the estimated value of the kurtosis parameter is the smallest among all assets, consequently, its returns possess the heaviest tails.

Moreover, even if constant parameters model revealed no significant skewness in most assets, the estimates of the model with dynamic parameters were significant in some cases with respect to skewness modelling and in most cases with respect to kurtosis modelling. Performed likelihood ratio tests concluded that in most cases inclusion of the dynamic degrees of freedom parameter improves the fit of the model. Inclusion of the dynamic skewness into the model also sometimes improves the fit, which indicates that dynamic skewness modelling allows to sometimes capture the asymmetry, which simple constant parameters model was not able to. The dynamic asymmetry model was not able to improve the fit in case of exchange rates and shares of one of the technological companies, Yandex, suggesting that these assets possess no asymmetry at all.

Speaking about the second part of the study, it was found out that on the Russian stock market portfolios consisting of stocks that pay high dividends generally possess fewer extreme outcomes than portfolios consisting of stocks that pay little or no dividends. Despite of this, the division of high dividend yield portfolio by two based on the dividend stability index revealed that the estimated value of the degrees of freedom parameter is larger for portfolio consisting of stocks that do not pay dividends regularly. Hence, regular dividend payments are associated not only with certain additional income, but also with higher risk, although stocks paying substantial dividends tend to have lower kurtosis than the stocks that do not pay them, or pay little.

Further estimation of the dynamic skewness model also revealed that high dividends portfolios tend to have more pronounced negative skewness, and this skewness is more variable in nature: it reacts more significantly to market changes than skewness of low dividends portfolio. This indicates that although the distribution of returns for high dividend yield portfolio is less heavy tailed, it is actually more negatively skewed, which is not a desirable thing for investors. This result could not be achieved by simple constant parameter model, which showed practically equal skewness coefficients for both portfolios.

The simple model of investors' preferences, demonstrated in the final section of the paper, showed that, on average, portfolio consisting of low dividend paying stocks yields higher expected utility in terms of future portfolio value than the portfolio of high dividend yield stocks. However, the difference in expected utilities becomes significant only for relatively high values of parameter responsible for “careness” for future value of the portfolio. For low values of this parameter the difference is practically undistinguishable, which implies that higher dividends will likely outweigh lower potential capital gains for such investors, and high dividends portfolio will be a better choice for them. Of course this model is quite limited and can potentially be made much more accurate by including, for instance, dividends payments into investors' utility functions, or considering dynamic skewness and degrees of freedom, thus making the model of intertemporal kind. In such model the decision of an investor of which portfolio to choose would likely change over time with changing values of skewness and kurtosis parameters.

All in all, the autoregressive conditional density model proved itself to be quite useful in modelling the dynamics of higher order moments by its ability to detect and predict asymmetry in returns, as well as the implied riskiness by estimating the degrees of freedom parameter. All this can be very important for questions related to portfolio choice, asset pricing and risk management.

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Appendix

This appendix contains results of the estimation of GARCH(1,1) model for all the data used in this paper. The equation for the model is

Table 16. GARCH(1,1) coefficients for various Russian and American assets

Table 17. GARCH(1,1) coefficients for high and low dividend yield portfolios

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