Business Forecasting: Share Price Evaluation - Essay Example

? Share price evaluation Generally, descriptive statistics tells a lot about the stock returns as well as the indices data used in the study. The main descriptive statistics that helped in description of data are the mean, data skewness, kurtosis, standard deviation, data ranges, as well as the minimum and maximum values. Analysis is based on the following principles, By analyzing the log returns, the key data that stood out was that the max value was generated by FTSE MID 250 (0.580766) and the minimum return generated was by MILAN COMIT GENERAL (-0.006832. The lowest mean value was generated by BRUSSELS ALL SHARE (-.00319). For Skewness, ideally it should be zero indicating symmetry and normality. Positively skewed data are said to be right tailed and it is an indication that stocks carry on a likelihood of have a higher probability of earning positive returns. In this case, only FTSE MID 250 (1.690595) was seen to be positive and this is also because the company prides in having one of the leading share returns in the market over the recent period. On the other hand, negatively skewed data are referred to as left-tailed and this is reflected in the rest of the variable ranging from BRUSSELS ALL SHARE 0 (-0.25139) to MILAN COMIT GENERAL (-0.486624). It is however worth noting these values are relatively close to zero and they are therefore slightly left tailed. For Kurtosis, it is often used to estimate the clustering of scores and as such a value of zero reflects data that are clustered around the mean. On the other hand a value greater than 0 shows clustering around numbers other than the mean. On the other hand, a value less than 0 shows that the data is spread out and as such there is no clustering. Based on findings in Table 1, it can be concluded that the data used in the study are all clustered around numbers other than the mean as shown by the results which are 9.286315, 25.30415, and 8.49093 for MILAN COMIT GENERAL, FTSE MID 250, and BRUSSELS ALL SHARE respectively. It is important to mention that kurtosis is the relative flatness or peakedness of the distribution of the returns as compared to the normal distribution. A normal distribution is said to have a kurtosis of 3. Whenever the tails of a given data are thinner compared to those associated with the normal distribution, they are said to be platykurtic (Harvey, 1990). Nevertheless, market returns are often slightly leptokurtic, implying that dramatic market moves often take place at a frequency larger than that predicted by a normal distribution. (Poon & Granger, 2005) The results obtained can be said to affirm this assumption and hence tend to move towards confirmation of the concept being evaluated in this study. This is due to the fact that a kurtosis larger than 3 implies fatter tails and hence the respective model under-prices both out-of-the-money as well as in-the-money calls/puts. On the other hand, kurtoses lower than 3 imply thinner tails and consequently the market options are overpriced (Mian & Adam, 2001). Such stock valuations have also make way for kurtosis and skewness trading methods. Further test by looking at the histogram also reaffirm this sense of normality in the data as seen from Figure 1. Figure 1: Histograms Figure 2: Stock share price distribution The distribution of the stock share price further illustrates the samples as seen from Figure 2. It is important to note that graphs simply give a presentation of what has already been discussed in with regard to daily data statistics returns. Based on the descriptive statistics as detailed in Table I, the returns show that the mean return is for FTSE MID 250 daily stock return set at 0.000161 which is lower than MILAN COMIT GENERAL. However, MILAN COMIT GENERAL records a lower median returns. BRUSSELS ALL SHARE records lesser minimal returns as compared to the FTSE MID 250. To the contrary, the standard deviation for FTSE MID 250 is much lower as compared to that of MILAN COMIT GENERAL. Table 1: Descriptive Statistics of Daily Data This table shows the descriptive statistics of the daily returns for 3 weekly share prices indices over the period of Jan 2002 to April 2012. The R denotes arithmetic returns .The LOG denotes Log returns. MILAN COMIT GENERAL, FTSE MID 250, BRUSSELS ALL SHARE, Max stands for Maximum. Min stands for Minimum. Std. Dev stands for Standard Deviation. JB stands for Jarque bera. Obs stands for Observations Using Perron’s procedure conduct a unit root test for each of the stock market indices. Explain carefully each step of the procedure. Critically comment on your results. The Conventional Unit Root Tests In testing market indices, studies often begin by running conventional unit root tests, namely the Augmented Dickey-Fuller (Dickey and Fuller 1979) as well as the Phillips-Perron (Philips and Peron, 1988) tests. The model based on Augmented Dickey-Fuller (ADF) test is as shown below: Whereby refers the stock price index at a given time t. The first equation model constitutes a constant term (, a given trend term ( a k value denoting the lagged number of times, and which represents the noise disturbance term. On the other hand, the second model includes a constant term. The third model that does not include an intercept or a trend term and hence the stationerity hypothesis for all the specifications is ?=0. The autoregressive term is included to make sure that the residual is auto-correlated. However, Philips-Perron (PP) test introduced a non-parametric technique to go through the problem of auto-correlation in the error term. In most instances, the PP test provides similar results compared to the ADF test. The test has the following specifications; This equation is estimated using the OLS method. Unit Root Tests Series ADF Test Phillips-Perron Test Constant Constant + Trend Constant Constant + Trend Level FTSE MID 250 -1.447(0) -0.986(0) -1.484 -1.068 MILAN COMIT GENERAL -1.365(1) -1.234(1) -1.379 -1.313 BRUSSELS ALL SHARE -1.087(0) -1.430(0) -1.081 -1.712 The paper examines whether the stock markets are pair-wise co-integrated with each other. As mentioned above, this paper employed Johansen co-integration approach to test for the interdependence between these stock markets. Table above reports the results of the pair-wise Co-integration tests. All the models show that the stock share prices have a unit root and follow a random walk. This is in consistence with the weak-form of the efficient market hypothesis suggesting that past movements in stock prices cannot be used to predict their future movements. The presence of random walk in the share price data has important implications for issuers of equity and portfolio investors. GARCH Model GARCH model, as already mentioned is a useful tool in modeling and forecasting of variability. Often, it is utilized in analysis asset holding risk or an option’s value. Additionally, considering that forecasting confidence intervals may vary with time, increasingly accurate intervals can be obtained error variance modeling. Lastly, it is important to mention that handling heteroskedasticity appropriately can help in generation of more accurate estimators (Silvennoinen & Terasvirta, 2008). The GARCH models for each of the indices are presented in the tables above with t statistics estimating constant of random walk each returning a critical value lesser than 1% level of significance. Basically, results show that the means for the variables studied are significantly different from zero, which is consistent with the random walk hypothesis. The spectrum of tests reveals that all p values are non-close to zero suggesting non-existence of hidden structures in the data evaluated. The ability of the model to explain series data in the light of constant terms displays some level of weak form efficiency validity. Generally, there is a revelation that the stock returns of varying stocks influence efficiency. The evidence in support of EMH is clear for all the two indices. As displayed in the table, the ARCH as well as the GARCH coefficients given by ?+? is close to 1 and hence indicates persistence of volatility shocks, this emphasizes the existence of weak form efficiency. The log for returns from share prices data show + ve coefficients and hence reveals a positive relationship to the indices with regards to the market trends. This affirms the ability to use stock return logs in prediction of the indices logs and hence further confirming existence of weak form efficiency. Another interesting finding is that in both cases, both R-squared as well as the adjusted R squared values return positive values and hence emphasize the ability of the model to predict market conditions. In general, R and adjusted R squared show a perfect model that takes consideration of the dependent variables around it to come up with conclusive evidence of weak hypothesis markets. The positive values affirm this. The graphs provided below further confirm the model’s prediction ability and hence re-affirming the weak efficiency concept. As already mentioned, the ARCH (?) and GARCH (?) coefficients are significant at 1% level, with a sum (0.888823+0.093576) and (0.911074+0.076027) for case 1 and case 2 respectively, which is close to unity and indicates that shocks to the conditional variance are persistent over future horizons. The sum of and is also an estimation of the rate at which the response function decays on daily basis. Since the rates are quite high, the response functions to shocks are likely to die slowly. Moreover, GARCH (1, 1) captures all the volatility clustering (higher z statistic values). This suggests the use of a distribution with a fatter tail to fit our data. These are reflected in table below:   OLS   Garch (1,1)     Garch (1,1)     Garch (1,1)   Garch (1,1)         with Normal Error Distribution with Student-t Error Distribution with GED Error Distribution with Double Exp. Error Distribution Coefficients Value Std.Error P-value Value Std.Error P-value Value Std.Error P-value Value Std.Error P-value Value Std.Error P-value ? 0.0000 0.007116 0.7772 0.08106 0.037515 0.0292** 0.130037 0.000701 0.00122 0.00016 0.008376 0.0095 0.091994 0.035841 0.0103 ?     0.007326 0.00012 0.0000** 0.0006 0.008814 0.0000** 0.13693 0.00012 0.0000** 0.104773 0.02134 0.0000 ?     0.077759 0.00445 0.0000** 0.0927 0.026388 0.0000** 0.060809 0.023295 0.0000** 0.874363 0.00556 0.0000 ?     0.922241 0.00276 0.0000** 0.867901 0.023315 0.0000** 0.87523 0.020918 0.0000** 0.00153 0.01921 0.0000                   Jarque-Bera 2492 0.0000** 2052 0.0000** 2292 0.0000** 2277 0.0000** 2229 0.0000** Ljung-Box (20) 0.04 1.0000 9.77 0.9706 19.20 0.5025 24.20 0.2295 21.62 0.0475 Ljung-Box (20) 0.12 1.0000 12.72 0.9974 20.96 0.7792 27.49 0.5972 27.06 0.1754 Ljung-Box (40) 1.47 1.0000 19.77 0.9967 27.57 0.9217 24.47 0.7165 45.49 0.2551 Ljung-Box (50) 7.22 1.0000 24.14 0.9992 21.77 0.9792 27.56 0.7707 49.52 0.4926 Ljung-Box (100) 61.66 0.9991 75.27 0.9691 72.21 0.9021 77.72 0.7707 100.47 0.4695 LM Test 694.12 0.0001** 15.77 0.2026 16.27 0.1744 16.42 0.1726 16.47 0.1712 F-stat 1.65 0.0066** 2.220680 0.2569 1.559072 0.2217 1.565834 0.2202 1.598365 0.2276 Log Likelihood     2355.972   2553.672   2551.685   2554.944   AIC     1279   917   927   1020   BIC       1705     1242     1252     1227     Figure 3: GARCH (1, 1): Where Rt is the daily return at time t, ? is a constant term, ?, are the ARCH and GARCH parameters respectively. * Statistically significant References Dickey, D. A. & Wayne, A. F. (1979). Distribution of the estimators for Auto-regressive time series with a unit root. Journal of American Statistical Association, 74, pp. 427-431 Harvey, A. C. (1990). The Econometric Analysis of Time Series, 2nd edition, Cambridge, MA: MIT Press. Lo, A. W. & MacKinlay, C. (1988). Stock market prices do not follow random walks: evidence from a simple specification test, Review of Financial Studies, 1, 41 - 66. Martikainen, T. et al. (1994). The linear and non-linear Dependence of stock returns and trading volume in the Finnish stock market. Applied Financial Economics, 4: 159-169. McInish, T. & Wood, R. (1992). An Analysis of Intraday Patterns in Bid-Ask Spreads for NYSE Stocks.” Journal of Finance, 47, 753–764. Mian, G. M. & Adam, C. M. (2001). Volatility dynamics in high frequency financial data: an empirical investigation of the Australian equity returns. Applied Financial Economics, 11, 341 – 352. Pagan, A. (1996). The Econometrics of Financial Markets. Journal of Empirical Finance, 3, 15-102. Phillips, P. C. B. & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75, pp. 335-346. Poon, S.-H., & Granger, C. (2005). Practical Issues in Forecasting Volatility. Financial Analysts Journal, 61(1), 45-56. Silvennoinen, A. & Terasvirta, T. (2008). Multivariate GARCH, Handbook of Financial Time Series. New York, Springer. Read More