Quantitative Methods for Business and Management - Assignment Example

QUANTITATIVE METHODS FOR BUSINESS AND MANAGEMENT By: Code+ course name: Professor’s name: University name: City, State: Date: Introduction Quantitative methods and techniques are usually used in the identification and evaluation of the statistical relationship between different variables. Some of the quantitative methods that are considered robust in data analysis include regression and correlation analyses. In regression analysis, a statistical model of the relationship between the variables is hypothesized, and estimates of the regression coefficients are used to fit the estimated regression equation. Subsequently, appropriate statistical tests are used to determine and evaluate whether the statistical model is satisfactory. Consequently, if the regression model is found to be satisfactory, the established regression model can be used to predict values of the dependent variable using the given values of the explanatory (independent) variables. On the other hand, correlation analysis seeks to evaluate the measure of the linear association between any two variables. The value of correlation coefficients always lies between -1 and +1 (Green, 2008). If the correlation coefficient is +1, it indicates that the two variables are related perfectly and linearly in a positive sense. However, if a correlation coefficient is -1, it indicates that the two variables are related perfectly but linearly in a negative sense. On the other hand, if the correlation coefficient is 0, it indicates that the selected two variables have no linear relationship. The data set being analyzed relates to the Australian stock market and was retrieved from yahoo finance. The sampling frame consists of two variables, which is a subset of the combined data available in the website. Included in the sampling frame are the daily returns for the Australian Ordinary Index for the period from January, 2001 to April, 2013. Also include in the sample data set are the daily returns for Primary Health Care Ltd, which is currently considered one of the market movers in the Australian stock market. Therefore, this study seeks to establish the relationship between the Australian Ordinary Index, and Primary Health Care Ltd share returns. However, to achieve reliable results from statistical data analysis, appropriate statistical tests should be used and applied correctly (Mogull, 2004). This would entail the identification of the type of the respective variables to be analyzed. Conditions and assumptions required to use a given statistical test should also be evaluated. Data analysis To carry out the appropriate statistical test using the data set identified, assume that the variables in the sample data set are independent. The normal distribution assumption will also suffice based on the central limit theorem since the sample size, n > 30 (Green, 2008). The two continuous variables, which are included, in the data set are: Australian Ordinary index (AOI) daily returns Primary Health Care Ltd’s share daily returns Descriptive Statistics The descriptive statistics for the daily returns of AOI from January 2001 to April 2013 is as indicated in Table 1. Table 1  Descriptive statistics of AOI daily Returns Mean -0.019495964 Standard Error 0.018261725 Median -0.050851767 Mode 0 Standard Deviation 1.016933798 Sample Variance 1.03415435 Kurtosis 5.913452278 Skewness 0.463972845 Range 13.7043559 Minimum -5.506373938 Maximum 8.197981964 Count 3101 As indicated in table 1 above, the average daily return of AOI (M = -0.0195, SD = 1.02), from January 2001 to April 2013 is negative. The median of the AOI daily returns (-0.0509) does not seem to differ significantly with the mean. The difference between the AOI maximum daily return (8.198) and the minimum daily return (-5.506) is 13.704%. The histogram of the daily returns of Australia Ordinary Index (AOI) is as illustrated in figure 1 below; Figure 1 The histogram is approximately symmetrical. This suggests that the daily returns of the Australian Ordinary Index (AOI) are approximately normally distributed. According to the empirical rule, for a normal distribution, 68% of the values fall within one standard deviation from the mean, 95% within two standard deviations from the mean and 99.7% of values fall within 3 standard deviations from the mean. For the data set with the AOI daily returns, the percentage of values, which fall within one, two and three standard deviations from the mean, can be determined, as a way of establishing whether the AOI daily returns are normally distributed. The percentage of values within one standard deviation of the mean is given by; The percentage of values within two standard deviations from the mean is given by; The percentage of values within three standard deviation of the mean is given by; The results of the proportion of values within one, two and three standard deviations from the mean also indicate that AOI daily returns are approximately normally distributed. Therefore, based on the empirical rule and histogram in figure 1 above, the AOI daily returns, can be said to have an approximately normal distribution. For a normal distribution skewness = 0, while Kurtosis = 3. The distribution of AOI daily returns has a skewness of 0.46, which is greater than zero (0) and a kurtosis of 5.91, which is greater than 3. Therefore, based on the skewness and the kurtosis of the distribution of the AOI daily returns, we can conclude that the distribution of the daily returns of AOI is approximately symmetrical and highly peaked. Using the descriptive statistics (mean and standard deviation), of the AOI daily returns, simulated daily returns can generated. Subsequently, the descriptive statistics of the simulated daily returns can be compared with that of the actual AOI daily returns. Table 2 shows the descriptive statistics of the simulated returns for the AOI; Table 2 Descriptive statistics: Simulated returns Mean -0.03892149 Standard Error 0.01860091 Median -0.029962537 Mode 1.165631468 Standard Deviation 1.035821854 Sample Variance 1.072926914 Kurtosis -0.139708814 Skewness 0.014534773 Range 7.057979004 Minimum -3.643195695 Maximum 3.41478331 Sum -120.6955413 Count 3101 The mean of the simulated returns (M = -0.0389, SD = 1.04), is also negative. The median of the simulated daily returns (-0.03) does not also seem to be significantly different with the mean of the simulated returns. However, the range between the maximum and minimum value of the simulated returns is 7.06%, which is smaller than the range of the actual AOI daily returns (13.704%). The histogram of the simulated returns using the sample size, mean and standard deviation of the AOI returns is illustrated in figure 2 below; Figure 2 As illustrated in figure 2, the histogram of the simulated returns is less peaked than the histogram of the actual returns of AOI, which is illustrated in figure 1. As indicated in table 2, the skewness value of the simulated daily returns is 0.0145, while the kurtosis value is -0.1397. Therefore, for the distribution of the simulated returns, the skewness is almost equal to zero (0), while the Kurtosis is less than 3. Therefore, the distribution of the simulated returns has a smaller skewness value and a lower kurtosis than the distribution of the actual returns. However, the distributions of both the actual and simulated returns are approximately symmetrical, but the distribution for the actual returns is more highly peaked than the distribution of the simulated returns. Further analysis of the data set shows that the sample of the actual AOI daily returns contains both negative and positive values. Out of a total of 3101 values of AOI daily returns, 1633 values were negative. Therefore, the probability that the next day’s return will be negative is given by; As established, the mean of the actual AOI daily returns was -0.0195. Therefore, to test whether the average daily return differs from zero, formulate a null hypothesis; H0 and test it against an alternative hypothesis; H1 at the level of significance 5%. Null hypothesis; H0: U = 0 (Mean of the AOI daily return does not differ significantly from zero) Alternative hypothesis; H1: U ≠ 0 (Mean of the AOI daily return does not differ significantly from zero) Average return = - 0.0195, Standard error of the mean = 0.0182 The p-value = 0.279, hence, greater than the level of significance (0.05). Therefore, the null hypothesis should be accepted. Thus, the average return of AOI does not differ significantly from zero. The other variable included in the sample data set is the daily returns for Primary Health Care Ltd share portfolio. In the recent past, the Primary Health Care Ltd port folio has been considered as one of the market movers in the Australian Stock Market. For this case study, the daily returns for Primary Health Care Ltd, for a one year period (February, 2012 to February, 2013) were analyzed. Table 3 Descriptive statistics: Returns (Primary Health Care Limited) Mean 0.247639327 Standard Error 0.110706852 Median 0.194552529 Mode 0 Standard Deviation 1.725749036 Sample Variance 2.978209736 Kurtosis 8.705351769 Skewness 1.297674482 Range 17.30744337 Minimum -5.333333333 Maximum 11.97411003 Sum 60.1763565 Count 243 As indicated in table 3, the average daily return of Primary Health Care Ltd portfolio for the 12- month period is 0.248% with a standard deviation of 1.726. The range of the daily returns was 17.307, with the minimum return recorded over the one year period being -5.333% while the maximum return was 11.974%. Figure 3 The histogram of the Primary Health Care Ltd daily returns is as illustrated in figure 3 above. The histogram is approximately symmetrical. This suggests that the distribution of the daily returns of the Primary Health Care Ltd share portfolio is approximately symmetrical. However, the distribution has a higher skewness and kurtosis value compared to that of the normal distribution. This indicates that the distribution of the Primary Health Care Ltd daily returns is skewed and more highly peaked than the normal distribution. Correlation analysis The share portfolio of the Primary Health Care Ltd is currently considered one of the market movers in the Australian stock market. Consequently, it would be expected that the returns of the Primary Health Care Ltd share portfolio greatly influence the Australian Ordinary Index (AOI) returns. Therefore, to establish the degree of association between AOI and Primary Health Care Ltd daily returns, correlation analysis would be appropriate to use. The correlation coefficient between the Australian Ordinary Index (AOI) and Primary Health Care Ltd daily returns is -0.2613. Subsequently, to test whether the correlation between the AOI and Primary Health Care Ltd daily returns is significant, formulate and test a null hypothesis; H0, against an alternative hypothesis; H1. Null hypothesis; H0: ρ = 0 (The correlation between AOI and Primary Health Care Ltd daily returns is not statistically significant). Alternative hypothesis; H1: ρ ≠ 0 (The correlation between AOI and Primary Health Care Ltd daily returns is statistically significant). Decision rule: Reject the null hypothesis; H0, if the p-value of the correlation analysis is less than the level of significance, a = 0.05. At 5% level of significance, p-value < 0.05, therefore, reject the null hypothesis H0 and conclude that the correlation between AOI and Primary Health Care Ltd daily returns is statistically significant. Figure 4 Figure 4 above, illustrates the scatter plot of the Primary Health Care Ltd daily returns against the AOI daily returns. The scatter plot also illustrates a strong correlation between the daily returns of the AOI and Primary Health Care Ltd. Regression analysis The simple linear regression model showing the relationship between the two continuous variables: Primary Health Care Ltd and AOI daily returns are as shown below in the regression equation. The dependent variable, in the model, is Primary Health Care Ltd’s daily share returns. The Australian Ordinary Index (AOI) is the independent variable used in the simple regression model. Therefore, the regression model analysis would evaluate by how much the Australian Ordinary Index explains variability in of Primary Health Care Ltd’s share portfolio daily returns. Primary Health Care Ltd’s share daily returns = B0 + B1Australia Ordinary Index The estimated regression model for Primary Health Care Ltd returns and AOI returns is given as shown in the regression model below. The return for Primary Health Care Ltd is the dependent variable while the return for AOI is the independent variable. Figure 5 Using the estimated regression coefficients, the simple regression model of two variables is as follows; Primary Health Care Ltd’s share daily returns = 0.2236 – 0.6199 Australia Ordinary Index The intercept is 0.2236, while the regression coefficient for the returns of AOI is –0.6199. The regression between the dependent and independent variable had a p-value which is less than the 5% significance level, which indicates that the relationship between the returns of the AOI and Primary Health Care Ltd is statistically significant. The coefficient of determination, R2 = 0.0683, hence, AOI returns only explain 6.83% of the variability in the returns of Primary Health Care Ltd. This further indicates that the AOI’s daily return is a statistically significant independent variable for the Primary Health Care Ltd returns. Thus, the Primary Health Care Limited represents a market mover in the Australian Stock Market. References List Frank, R., 2008, Microeconomics and Behaviour (7th ed.), McGraw-Hill. Hallerbach, G.W., 2003. Holding Period Return-Risk Modeling: Ambiguity in Estimation. Green, S. B., 2008, Using SPSS for Window and Macintosh: Analyzing and Understanding data (5th ed.), Upper Saddle River, NJ: Pearson Prentice Hall. Mahdavi D. B, 2013, "The Non-Misleading Value of Inferred Correlation: An Introduction to the Cointelation Model". Wilmott Magazine. Mogull, Robert G, 2004, Second-Semester Applied Statistics, Kendall/Hunt Publishing Company. Yahoo finance, 2014, Retrieved from; http://finance.yahoo.com/ Read More