Quantitative Methods for Finance

Assigned Index: MSCI Switzerland Index

Table of Contents

Introduction

The aim of this report is to examine if a profitable trading strategy can be formulated by using the historical returns of a given asset. The asset in question is the MSCI Switzerland index for the period between 1^st January 2000 and 31^st December 2017. We will begin by presenting key summary statistics, testing for normality, and testing for seasonality. In addition, we will continue by exhibiting the results of multiple regressions through which we can not only propose our trading strategies, but also examine if they are statistically significant. We will conclude this report by commenting on possible sources of the returns, and how the strategies performed during periods of growth or decline in the market.

Task 1 and 2

We obtained the required data through the Bloomberg platform (2018) for the period stated in the introduction. We initially imported the data into EViews and began to calculate the returns, volatility, skewness, and kurtosis for our data. Using the log return formula:

$R_t= 100*\ln \left(\frac{p_t}{p_{\left(t–1\right)}}\right)$

  (1)

We calculated the monthly returns of the index and named the variable LogR. We then found the mean, standard deviation, skewness, kurtosis, and Jarque-Beta (a.k.a JB) statistic by using the “Quick>Series Statistics>Histogram and Stats” command. Moreover, we chose to annualize the mean and standard deviation in order to make the comparison with the results of Georgopoulou and Wang (2017) easier. To achieve this, we used the following formulas respectively:

$r_{\mathit{annual}}=r*12$

  (2)

${\sigma }_{\mathit{annual}}^2= {\sigma }^2* \sqrt{12}$

  (3)

In the following Table 1, we exhibit our findings and also the results of Georgopoulou and Wang (2017):

Table 1

Source: The results in the above table have been sourced through personal research and the study published in 2017 by Georgopoulou and Wang.

From Table 1 it is apparent that the results obtained by our research are significantly different to the ones obtained in the earlier study. The annualized mean differs by 5.19% which can be explained due to the difference in volatility and length between the two periods. Furthermore, our results show higher skewness and kurtosis meaning that the distribution of our observations is more asymmetric and high-tailed than that of the 2017 study. The higher skewness and kurtosis indicate that our observations are not normally distributed since a normal distribution has a skewness of zero and a kurtosis of 3 (Brown, 2017). Moreover, in order to confirm if our observations are normally distributed, we utilized the JB statistic and conducted an empirical distribution analysis. We tested the null hypothesis of normal distribution existing against the alternative, that the data is not derived from a normal distribution. From our earlier calculations we had found the JB coefficient to be 21.29054 with a p-value of 0.000024. Since our variable JB coefficient is above three and significant at the level of 5%, we were able to reject the null hypothesis and conclude that our returns were not normally distributed. Furthermore, by using the “View>Descriptive Statistics> Empirical Distribution Analysis” command on the LogR variable, we were able to confirm that the JB test result is consistent with results to the Lilliefors, Cramer-von Mises, Watson, and Anderson-Darling tests for normality. Thus, we can conclude with certainty that our distribution is not normally distributed.

We were also asked to test our data against the presence of seasonality. Seasonality is defined as a characteristic of a time series in which the data is exposed to a systematic, although not always regular, event or action that causes predictable movements on a yearly basis (Hylleberg, 1986, p. 241). Furthermore, it is important to state that seasonality differs from cyclical effects since the latter can span long periods of time, whereas seasonal effects are observed in periods spanning no more than one calendar year (1. Investopedia, 2018). Moreover, we initially examined our series against the January effect. This effect was first observed in the monthly returns of the New York stock exchange by Rozeff and Kinney (1976). They reported statistically significant results proving that the returns in the period from 1904 until 1974 are higher in January than any other month. Our data, however, did not show significant results confirming the existence of this seasonal effect as is noticeable in Table 2 below.

Table 2

In order to determine the existence of the January effect, a dummy variable was created that took the value of 1 in the months of January included in our data set. We then regressed the dummy with the LogR variable in order to determine if there was a seasonal effect. Since the probability of the variable is 0.146 it is not significant to the level of 5% and we cannot confirm the presence or absence of seasonality in the data.

We also conducted a similar test in order to determine if there was a significant Halloween effect. The effect was observed by Jacobsen and Visaltanachoti (2009) in their paper “The Halloween Effect in US Sectors”. They report significant findings that prove that stock market returns are significantly higher in the period of October until May, in specific sectors, when compared to the rest of the year. In order to test our data set against this effect, we created a dummy variable called Halloween, which took the value of 1 in the months from October until May in our data set. We then regressed this variable against our LogR variable and found that again the results were not significant at the 5% level. Thus, we concluded that there is no evidence of the January or Halloween effect in our index.

Task 3

As mentioned in the introduction, the aim of this report is to propose profitable trading strategies based on the results of a regression between the returns of our index a constant and the 1-month lagged returns of the index. We used the following formula in order to run the regression:

$\mathit{LogR}=c+\mathit{LogR}(–1)$

(4)

The regression results were as follows:

Table 3

The p-value of the regressor ‘LogR(-1)’ is 0.0052. It is known that the p-value for statistical significance is p < 0.05. Therefore, in this regression, the 1-month-lagged return is statistically significant at the 5% level. This means that the 1-month-lagged return has significant explanatory power for the variability of the current returns of the MSCI Switzerland Index. This result is economically relevant as well because it suggests that the previous month’s return has some predictive power on the current month’s returns, and investors can use these returns in a bid to gain in the following month. Moreover, in this regression, the  = 0.139 and the  = 0.1904. These estimate values can be used to construct a trading strategy. In this case, since the sign and coefficient of the slope parameter is positive, a profitable trading strategy would be to hold a long position for the following month. The direction of the slope suggests that in the next month, the returns are likely to be increasing, thus a long position would be a profitable position to be in.

Task 4

Using a similar method as the one used for the completion of task 3, we obtained the following results from the regression of the current returns with the 2^nd– until 12^th-lagged-month returns:

Table 4

Regressing the current returns against the 2^nd– until 12^th-month-lagged returns gives a Regressor p-value, as can be seen in the table above. None of them are statistically significant at the 5% level and suggest that the monthly lagged returns do not have significant explanatory power for the variability of the current returns. However, the 9, 10 & 11 month lagged variables show that the  have negative values, and this suggests that if an investor were to consider these variables and take on a short position, he would be profitable. For the remaining months, taking a long position would be profitable to the investor.

Task 5

In this part of the report, we will be testing the 11 different strategies in order to determine if the mean returns are significant and also to determine if there is evidence of autocorrelation in our data set.

Below are the key summary statistics for each strategy:

Table 5

Each trading strategy gives a p-value that is higher than our significance level of 5% (0.05). As a result, we conclude that none of the average returns from the 11 trading strategies are statistically significant.

Moreover, in order to test for autocorrelation using the following regression formula:

$\mathit{Lagreturns}\left(x\right)=c$

(5)

We used the Durbin Watson test for autocorrelation and the Breusch Godfrey test in order to determine serial correlation. In the table below, we have analyzed the Durbin Watson Test statistic and the p-value of the F-test and Chi-Square.

Table 6

The Durbin Watson test for each of the returns lagged by 2-12 months gives us a value between 0  DW  4 which is closer to 2. Hence, there is little evidence of autocorrelation.

We further our research by conducting the Breusch Godfrey test using the following hypothesis:

H₀ = no serial correlation at up to 12 lags

We observed that the p-values of both the F-test and the Chi-Square are higher than our significance level of 0.05. As a result, we do not reject the null hypothesis for all 11 trading strategies. To conclude, since the residuals of our trading strategies lack evidence of autocorrelation, the parameter estimates and the standard errors are unbiased and accurate.

Task 6

In this part of our report, we will present the key statistics of two subsamples created from our original data. The two subsamples range from January 2000 until December 2008 and January 2009 until December 2017 respectively. The aim of this division is to compare the returns between the two subsamples in order to compare our trading strategies over time. The summary statistics for the two samples are as follows:

Table 7

As is visible in the above table, the mean returns of the two subsamples are significantly different. However, we formally tested the following hypothesis, in order to determine if the average returns are significantly different.

Hypothesis: Ho: mean of subsample 2 = -0.183871 against the alternative.

(The value of -0.183871 is the monthly mean of subsample 1)

The single t-test we conducted returned a t-statistic of 2.113606 with a probability of 0.0369, meaning the results are significant. The test we conducted is a two-tailed test at the 5% significance level and therefore we consider a t-statistic against the value of

${\tau }_{0.025}$

derived from the T-table of critical values of the t distribution. The condition we use to determine if we will accept or reject the null hypothesis is the following:

Accept Ho if

$–{\tau }_{0.025}<2.113603<{\tau }_{0.025}$

Reject Ho if

${\tau }_{0.025}<2.113603\mathit{ or }{–\tau }_{0.025}>2.113606$

$\mathit{with }{\pm \tau }_{0.025}=\pm 1.980$

Therefore, we reject the null hypothesis and conclude that the average returns of our two subsamples are significantly different at the 5% level.

This observed difference in returns can easily be attributed to the significant political and economic events that took place within the two subperiods. For example, the first period includes key events that shook the Swiss and world economy such as the banking crisis of 2008, which caused markets around the world to collapse and changed the financial climate of the world (Mathiason, 2008). Another example is the bankruptcy of Swiss Air in 2001, due to a risky expansionary policy and the events of 9/11 in the USA (Kirby, 2001). Furthermore, there was a small correction of prices in the Swiss stock market around February 2003 that caused prices to plummet. However, after 2009, the Swiss economy recovered from the effects of the 2008 crisis and began to experience steady growth. Therefore, the difference between our two subsamples mean can be explained due to these events.

Task 7

In the closing part of this report, we will attempt to give a possible source for our returns, examine the relationship between our strategies and the buy and hold strategy, and determine how our strategies performed during periods of boom and bust.

The returns predicted through our trading strategies can be attributed to a number of factors. However, the main source of returns arising from our suggested trading strategies is the correlation between the returns and the lagged returns. The notion of returns from time series momentum trading strategies, being driven by correlation between the returns of an asset today and the lagged returns of the same asset, is heavily reinforced in relevant literature, as Moskowitz, Ooi, and Pedersen explain in their journal article “Time Series Momentum” (2011).

In our data set, in order to examine the relationship between our strategies and the buy and hold strategy, we decided to construct a correlation matrix in order to determine if multicollinearity existed within our data set. Multicollinearity is present when there is high intercorrelation between independent variables in models of multiple regressions. Furthermore, its existence can raise issues such as unreliable p-values and wider confidence intervals (2. Investopedia, 2018). As is visible in Table 8 below, the variables we used to obtain our trading strategies are not highly correlated since the highest value in the table is 0.22, which is low enough to be considered as non-existent. Without the existence of multicollinearity, we are able to trust the results of our research and safely recommend the trading strategies in task 3 and 4 above.

Table 8: Correlation Matrix

Finally, we would like to conclude this report by presenting some results concerning the performance of our strategies against historic periods of boom and bust. Graph 1 below shows the movements of our assigned MSCI index in the period beginning in January 2000 until February 2017. By visually inspecting the graph and taking into account the financial events presented in the last paragraph of task 6, we identified periods of boom and bust.

Graph 1

Source: Thomson Reuters, 2018

We then created dummy variables in EViews, in order to find out how our trading strategy performed. We only included the trading strategy derived from the one month lagged returns since this is the only statistically significant strategy we obtained and is the one most investors would follow. Table 9 below includes the results we obtained.

Table 9

From the data in table 9, we are able to deduce that the trading strategy was unable to outperform the buy and hold strategy. This is intuitive since in the periods where we derived significant results the buy and hold strategy either earned more during a boom or lost less during a bust. This result is not a surprise, since we explained earlier that results in trading strategies derived from time series momentum, serial correlation is one of the main drives behind results. Our data set was not highly correlated and therefore could not produce a trading strategy yielding high returns.

Conclusion

To conclude, in this report we examined the trading strategies that could be derived by analyzing the returns of an MSCI index, specifically that of Switzerland. We began by comparing the results to those of an earlier research, examining if they were normally distributed, and testing if the January or Halloween seasonal effects were present in the set. Moreover, we proposed trading strategies derived from multiple regressions between the indexes returns and its lagged returns, of various time scales. Furthermore, we tested the returns for autocorrelation, compared the performance of the strategies, and concluded by commenting on the possible sources of our returns.

Appendix

Bibliography

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