The Return of an Investment on a Stock - Essay Example

A Quantitative Study of Behavior of Stock Returns Affiliated A Quantitative Study of Behavior of Stock Returns An investment in a stock offers two type of returns: dividend and capital gain. A capital gain is a profit one can get from the sales of the investment (assets). The capital gain is also labeled as returns. The return of an investment on a stock over a historical period is a vector that has direction and quantity. This assignment has selected four stocks from the S&P 500 index. The stocks are of companies Microsoft Corporation (MSFT), Apple, Inc. (AAPL), Intel Corporation (INTC), and Hewlett-Packard (HPQ). The objective of this assignment is to study the behavior of return (capital gain) using different statistical tools (Hong, Tu & Zhou, 2007). Behavior of Stock Prices This assignment studies behavior of stock prices at daily interval for the period from July 20, 1988 to July 20, 2009. Figure 1, 2 and 3 illustrate the dynamic character of stock prices respectively of companies Microsoft, Intel, and H&P. The graphical views tell us that these three stocks exhibited exponential growth from the beginning until the middle of 2000. All three stocks later dropped in prices by 30%-% 40% in about 24 moths. They never achieved the previous peak. Microsoft maintained stable prices for the rest of the period while Intel and Hewlett & Packard went through bumpy roads. The dynamic character of Apple stock was different from that of previously mentioned stocks. Figure 4 shows the asymptotic behavior of Apple stock prices from the beginning of the observation period until the middle of 1999. In the latter period, Apple stock price exhibited exponential growth until was hit by the global financial crisis of 2008. Apple’s sudden growth after the mid-1999 can be associated with the release of new products and services. The exponential growth of Microsoft, Intel and H&P stock prices from 1988 until 2000 should be contributed to the development of digital technology of that time. Thus, we can conclude that the innovation and new products influence the rise in stock prices. At the same time, irrespective of innovation, the overall market condition also causes influence on the stock prices. Figure 5 depicts S&P 500 index values from September 26, 2008 to November 21, 2008. In 41 days, the index dropped by 34%. This incident is named as the Global Financial Crisis of 2008. It caused a drop in prices of Microsoft, Apple, Intel and H&P stocks respectively by 27.69%, 35.61%, 31.13%, and 27.55%. Figure 1. Microsoft stock prices for 1988-2009 Figure 2. Intel stock prices for 1988-2009 Figure 3. H&P stock prices for 1988-2009 Figure 4. Apple stock prices for 1988-2009 Figure 5. Global financial crisis 2008 Figure 6. Stock prices during 2008 crisis Asset Return Calculation Method This assignment is using time-series data of stock prices. Let Pt be the price of an asset at a time index t and P t-1 at a time t-1. We assume that the stocks were held for the entire observation period, and they generated compound returns. A continuously compound return is defined as the natural logarithm of the gross simple return and is expressed by the formula rt = ln(1+Rt)=ln(Pt/Pt-1); where Rt is the gross simple interest, and rt is log return of the asset. Descriptive Statistics of Returns Daily returns of stocks are calculated using the formula rt = ln(1+Rt) and the values are expressed in percent. This section presents an analysis of stock returns using descriptive statistics method. Microsoft Corporation The character of the histogram (Figure 7) defines that the returns are normally distributed; out of 5478 observations, 83.5% are within the range of -2% to 4% daily returns. The values of mean and standard deviation imply that 83.5% of data are within -1σ and +2σ of the normal distribution. Descriptive statistics shows negative skewness and excess kurtosis causing asymmetry in the normal distribution. However, it should not be considered as significant. Apple Inc. The character of the histogram (Figure 8) defines that the returns are normally distributed; out of 5478 observations, 92% are within the range of -5% to 5%. The values of mean and standard deviation imply that 92% of data are within -2σ and +2σ. Descriptive statistics shows a substantial negative skewness as well excess kurtosis in the distribution. The asymmetry caused by skewness and kurtosis should be considered as significant. Intel Corporation The character of the histogram (Figure 9) defines that the returns are normally distributed; out of 5478 observations, 72.36% returns are within the range of 0% to 5%. The values of mean and standard deviation imply that 72% of data are within -1σ and +2σ. Descriptive statistics shows a negative skewness and excess kurtosis in the distribution. The asymmetry caused by skewness and kurtosis should not be considered as significant. Hewlett & Packard The character of the histogram (Figure 10) defines that the returns are normally distributed. Out of 5478 observations, 50.42% are within the range of 0% to 5%; and 44.85% are within 5% to 10%. Even though, the data set has a negative skewness, but it is not significant. The distribution should be considered symmetrical despite the existence of excess kurtosis. Zero Return Hypothesis Daily return (capital gain) was calculated using the natural log formula rt=ln(Pt/Pt-1). The mean values are 0.07%, 0.05%, 0.05%, and 0.04% respectively on MSFT, AAPL, INTC, and HPQ stocks. We assume that sample data comes from the independent population. The objective of the analysis is to evaluate if the sample data comes from a population with mean equal to zero. Statistics uses hypothesis test to find the answer to the issue mentioned above. In this case, we will use a one-sample t-test. In this test, null, H0 = t-statistics (x) is based on the premise that the data x comes from a normal distribution with mean equal to zero. The alternative is that the population does not have a mean equal to zero. The test uses the formula; t-statistics = (Xmean-µ)/{Sx/Sqrt (n)}; where µ is the population mean, Sx is the sample standard deviation, and n is the number of observations. We conducted a test for H0: µ = 0 or Ha: µ ≠ 0 for each asset. The result of the test is presented in the table attached below. Table 1. One-sample t-test for H0: µ = 0 or Ha: µ ≠ 0 The results shown in the above table state that the return on MSFT stock statistically is different from zero; however, the returns on AAPL, INTC, and HPQ stocks statistically are not different from zero. Independence Test of Stock Returns If we assume that, the stock returns on two assets are independent of one other; it implies that they come from two different underlying populations that do not have same means. We can find authenticity of it using the following hypothesis test: Null hypothesis, H0: μ1 - μ2 = 0; An alternative hypothesis, Ha: μ1 - μ2 ≠ 0. If the above-stated assumption is true, then the hypothesis test will reject the null. We conducted the test using two-sample t-test unequal variance method. We do not reject Null if t-stat < t-critical and p-value > α = 0.05. The following table shows the decisions matrix obtained from the test. Table 2. Two-sample t-test unequal variance for H0: μ1 - μ2 = 0 or Ha: μ1 - μ2 ≠ 0 The result shows that mean returns of selected stocks are not statistically different from each other. Based on the above analysis, we may conclude that returns on stocks from the same sector do not exhibit independent behavior. Correlation Matrix of Stock Returns Over a period, two assets can move in the same or opposite directions. The correlation coefficient can describe this movement. The coefficient obtains a value in between minus 1 and plus 1. If the sign of the coefficient is positive, it implies that two assets are moving in the same direction; if negative, it informs that they are moving in the opposite directions. The value of the coefficient also describes the strength of the movement. Investors use this phenomenon to construct a portfolio. We used Excel built-in function CORREL to find the values of the coefficient. The results and analysis of the calculations are shown in the following table. Table 3. Correlation matrix and its analysis Revisit Independence Assumption of Stocks Returns This assignment selected four stocks from the Technological sector. We assumed that these stocks come from independent populations. Therefore, returns on any two stocks must be independent of one another. However, hypothesis test determined that returns on these stocks statistically do not exhibit independent behavior. Therefore, we cannot conclude that the samples come from the independent population. That is why; we may assume that the sample selected from one population is related to the corresponding sample from the other population. In statistics, it is called paired samples. In order to conduct a hypothesis test, first, we find the differences of returns of two stocks, and then we calculate the Average Return and Standard deviation of the differences. It is assumed that the differences (di) of returns of two stocks follow a normal distribution. We define the following hypothesis: Null hypothesis, H0: μd = 0; An alternative hypothesis, Ha: μd ≠ 0. The t-statistics for paired sample is calculated using formula, t = Mean of di / standard deviation of di * sqrt (number of observations). The t-statistics value is compared with the t-critical value for the two-tail test. The t-critical value is calculated for probability = 5% and degree of freedom = number of observations – 1. If t-statistics < t-critical, we do not reject the null. The results are shown in the table attached below. Table 4. T-test for paired samples At 95% level of confidence, we can state that the return on stocks of the same sector does not depend on the selection of the stock. The comparison of results of independent and parried sample produced the same results. Portfolio Construction The assignment offers four stocks. The correlation matrix defines six different choices, which can be used to construct a two-asset portfolio. Out of six options, we selected three variants using the highest (0.56), second highest (0.48) and lowest (0.36) correlation values. The pairs are Microsoft and Intel; Intel and HP, and HP and Apple. The objective is to pick up the selection that provides the best Risk-Return relationship. We calculated the return and standard deviation of each pair by assigning same weights to the securities. The results of calculations are shown in the following attached tables. The graphical presentations between risk and return are illustrated in Figures 11, 12, 13. Table 5. Quantitative values of MSFT and INTC portfolio Table 6. Quantitative values of HPQ and AAPL portfolio Table 7. Quantitative values of INTC and HPQ Figure 11. Risk vs. Return of portfolio MSFT and INTC Figure 12. Risk vs. Return of portfolio HPA and AAPL Figure 13. Risk vs. Return of portfolio INTC and HPQ We created Table 8 using the results of Figures 8, 9, and 10. Table 8 shows optimal weights and the optimal expected returns of three choices. Among them, the best one is the portfolio of the stocks of companies Microsoft and Intel. Table 8. Optimal weights and expected returns Character of Stock Price Distribution We used historical time series data and converted them into continuously compounded daily return using log transformation. Is it proper to assume that the calculated returns of these stocks are normally distributed? The histograms and values of descriptive statistics do not show that the returns on these stocks are normally distributed. The analysis of values of skewness and kurtosis can support the above statement. The parameter skewness measures the symmetry in a distribution. For a normally distributed data set, it is equal to zero. The values of skewness of MSFT, AAPL, INTC and HPQ of stocks respectively are -0.04, -2.0, -0.41, and -0.02. Distributions of all four stocks are skewed to the left; however, Apple’s data are heavily clustered to the right with a long tail extending to the left. The parameter Kurtosis measures the peakedness of the normally distributed curve; it is equal to 3.0 for a normally distributed data. The values of kurtosis of MSFT, AAPL, INTC and HPQ of stocks respectively are 5.19, 57.0, 5.38, and 6.23. The curves of all four stocks are not bell-shaped; they have peaks, especially the returns on Apple stock. Thus, it is not realistic to assume that the selected stock prices followed a normal distribution. Conclusion The above quantitative analysis demonstrated how various tools of statistics could interpret the behavior of stock return. Reference Hong, Y., Tu, J., & Zhou, G. (2007). Asymmetries in stock returns: Statistical test and economic evaluation. Retrieved from http://apps.olin.wustl.edu/faculty/zhou/HTZ_RFS_W07.pdf Read More