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# BUS5SBF Statistics for Business and Finance-Statistical Metrics

Create the line charts for each of S&P, Yahoo and Google series using Close prices against time in Excel and comment on your observations (focusing time series features).
a. Calculate returns for these three series in Excel using the transformation: rt = 100 ln[Pt/Pt-1]
Hints:
• We performed a similar task in Tutorial 01.
• These numbers would represent percentages after multiplication with 100 in the formula above. However,you would not put a percentage sign in your data. For example, returns for two periods are 0.35% and 0.41% but we omit % sign in our excel worksheet and use 0.35 and 0.41.
b. Obtain the summary statistics for your sample and briefly discuss the risk and average return relationship in each stock. Which stock (Google or Yahoo) is relatively riskier than the other
c. Perform the Jarque-Berra test of normally distributed returns for each of Yahoo, Google and S&P.

Write down the sampling distribution of sample means for both stocks when a sample of 36 months is taken. Use the sampling distributions of part (a) above to answer the following parts.
b. What is the probability of observing average return of at least 4% in both stocks
c. What is the likelihood of loss in a sample of 36 periods for both stocks

Before investing in one of the two stocks based on higher risk-return relationship, you further want to determine whether both stocks have same population average return. Perform an appropriate hypothesis test using information in your actual sample of 56 observations and report your findings. Also, which stock will you prefer and why

Create two new columns in Excel for excess return on your preferred stock (yt) and excess market return (xt) by subtracting the 10-year T-Bill rate from both series as follows.
Excess return on your preferred stock: yt = rt - rf,t
Excess return on market: xt = rM,t - rf,t

a. Estimate the CAPM using linear regression where the dependent variable is excess return on your preferred stock while the independent variable is excess market return (computed as return on S&P 500 minus the risk fee rate) and report your results.
b. Interpret the estimated coefficients in relation to the profitability of the Stock and its riskiness in comparison with the market.
c. Interpret the value of R2
d. Interpret 95% confidence interval for the slope coefficient.

The above line graph shows the standard and poor index for 500 index time series for the time between October 2009 and May 2014. There is a generally increasing trend showing that companies' index improved significantly between these two time frames. However, the time between April 2012 and May 2014 seems to have increased in larger rate compared to the previous time. The index data did not have a lot of seasonal effects because there is no a lot of variations as observed in the time series chart.

The line graph in figure 2 shows that yahoo stock has an approximate unchanging growth between May 2009 and August 2012. This indicates there might be business problems that persisted during this time frame, hence affecting the growth of the Yahoo stock index. During all this time, the index was revolving around 15 USD, which then increased after August to a maximum of around 40 in January 2014. Therefore, between September 2012 and January 2014, there was an increasing trend for Yahoo Index and this might have attracted a lot of cus

tomers to invest. After January 2014, Yahoo Index seems to have experienced a decreasing trend, hence affecting the profitability of the business to the investment.

Between October 2009 and August 2010, the Google Price index experienced a decreasing trend from around 570 US dollars to approximately 420 USD dollar. This was a great drop in the investment which might have reduced the rate of investment for the stock. After August 2010, the stock price improved to around 600 USD in October 2010, which approximately remained in that level until February 2011. The price reduced to around 500 USD and it experienced seasonality in around June and July 2011. However, the seasonal effects persisted until June 2012, where the stock price increased significantly from June 2012 to February 2014, marking a maximum stock price of around 1200 US dollars between October 2009 and May 2014.

In comparison between the three stock prices, S & P seems to be have experienced minimum seasonal effects for the trend. In addition, its trend has been constantly increasing from October 2009 to May 2014, although it experienced some significant drop between April 2012 and October the same year. After this challenging period for S & P stock price index, it improved significantly from around 1100 US dollars to 1900 US dollars in May 2014. Google and Yahoo stock prices experienced constant and unchanging prices for longer periods, indicating that they experienced slowed growth in the market and investment.

### 1. Summary Statistics

#### a. Determining the stock that has more risk

 S&P 500 Index Google Stock Price Yahoo stock Price US TN (10 years) Rt - S & P Rt - Google Rt - Yahoo Mean 1382.69 691.15 19.93 2.56 1.09 1.25 1.48 Standard Error 32.60 26.79 1.04 0.09 0.53 1.05 1.02 Median 1345.20 613.40 16.40 2.61 1.76 1.28 1.69 Standard Deviation 241.76 198.70 7.69 0.68 3.90 7.76 7.57 Sample Variance 58445.77 39482.73 59.13 0.46 15.18 60.15 57.29 Kurtosis -0.59 0.51 0.81 -1.12 0.23 -0.50 -0.32 Skewness 0.57 1.19 1.46 0.14 -0.40 0.06 0.17 Range 853.24 770.70 27.34 2.35 18.78 33.27 33.95 Minimum 1030.71 444.95 13.10 1.49 -8.55 -15.69 -13.81 Maximum 1883.95 1215.65 40.44 3.84 10.23 17.58 20.14 Sum 76048.09 38013.41 1096.17 140.71 59.78 68.98 81.58 Count 55 55 55 55 55 55 55

The table above shows the summary statistics of three stock prices, S & P, Google and Yahoo together with their adjacent returns. The returns can be used to determine the riskier stock and the best within the three. Yahoo stock price has the highest average returns of 1.48% followed by Google stock that has 1.25% and finally S & P stock price that has an average of 1.09%. Comparing the three stocks, Google has the highest variance of 60.15% against S & P stock that has 15% variation. Therefore, it can be concluded that Google stock price has the most investment risk followed by Yahoo and finally S & P Index.  S & P is the best stock to invest for individuals who does not wish to be involved in high-risk business, although stock with great risk has the highest returns on average. Comparing Google and Yahoo Index, the former is riskier than Yahoo because it has more variation.

### b. Normality test

Testing normality for the three returns; S & P, Google and Yahoo, the test statistics will be obtained and the Q-Q plot will be plotted to determine whether the data is normally distributed.

 Rt - S & P Rt - Google Rt - Yahoo W 0.978788 0.979012 0.990743 p-value 0.437831 0.446812 0.94758 alpha 0.05 0.05 0.05 normal yes yes yes

Based on the table above, the three data for the stock price returns are normally distributed. This test is based on the following hypothesis.

Null hypothesis: The data is normally distributed

Since the observed p-values are greater than the significance level 0.05, we fail to reject the null hypothesis and conclude that the data is normally distributed. The argument can also be observed on the Q-Q plot plots shown below for the three variables.

The points are concentrated within the straight line indicating that they are normally distributed.After standardising the data, most of the points are concentrated in the straight line, which is a clear indication that the data is normally distributed which also applies to Yahoo stock returns data.

### 3. Testing whether Google and Yahoo have the same average returns

 Mean 1.2542 Known Variance 60.15 Observations 55 Hypothesized Mean Difference 0 z -0.1568 P(Z<=z) one-tail 0.4377 z Critical one-tail 1.6449 P(Z<=z) two-tail 0.8754 z Critical two-tail 1.96

Null hypothesis: The average of Google and Yahoo are different

Alternative hypothesis: Google and Yahoo returns averages are the same

Based on the two-sample test shown above, the p-value is greater than the significance level, hence failing to reject the null hypothesis. Therefore, I would prefer to use Google stock as the investment stand because it has a greater chance of making at least 4% returns and a lower chance of making losses.

### 4. Calculating Excess returns

 US TN (10 years) Rt - S & P Rt - Google Rt - Yahoo Yt Xt 3.392 0.000 0.000 0.000 -3.392 -3.392 3.201 5.578 8.383 -6.027 5.182 2.377 3.843 1.761 6.150 11.414 2.307 -2.082 3.609 -3.768 -15.692 -11.147 -19.301 -7.377 3.595 2.811 -0.594 1.979 -4.189 -0.784 3.833 5.713 7.375 7.667 3.542 1.880 3.663 1.465 -7.584 0.000 -11.247 -2.198 3.301 -8.553 -7.928 -7.471 -11.229 -11.854 2.951 -5.539 -8.749 -10.290 -11.700 -8.490 2.907 6.652 8.588 0.289 5.681 3.745 2.477 -4.861 -7.455 -5.707 -9.932 -7.338 2.517 8.393 15.561 7.775 13.044 5.876 2.612 3.619 15.460 15.163 12.848 1.007 2.797 -0.229 -9.926 -4.148 -12.723 -3.026 3.305 6.326 6.658 4.993 3.353 3.021 3.378 2.239 1.070 -3.115 -2.308 -1.139 3.414 3.146 2.149 1.722 -1.265 -0.268 3.454 -0.105 -4.440 1.693 -7.894 -3.559 3.296 2.810 -7.548 5.935 -10.844 -0.486 3.050 -1.359 -2.811 -6.718 -5.861 -4.409 3.158 -1.843 -4.374 -9.567 -7.532 -5.001 2.805 -2.171 17.577 -13.810 14.772 -4.976 2.218 -5.847 -10.972 3.819 -13.190 -8.065 1.924 -7.447 -4.910 -3.286 -6.834 -9.371 2.175 10.231 14.034 17.189 11.859 8.056 2.068 -0.507 1.133 0.447 -0.935 -2.575 1.871 0.850 7.473 2.638 5.602 -1.021 1.799 4.266 -10.743 -4.178 -12.542 2.467 1.977 3.979 6.368 -4.225 4.391 2.002 2.216 3.085 3.651 2.596 1.435 0.869 1.915 -0.753 -5.842 2.081 -7.757 -2.668 1.581 -6.470 -4.047 -1.949 -5.628 -8.051 1.659 3.879 -0.136 3.798 -1.795 2.220 1.492 1.252 8.727 0.063 7.235 -0.240 1.562 1.957 7.913 -7.810 6.351 0.395 1.637 2.395 9.651 8.690 8.014 0.758 1.686 -1.999 -10.352 5.242 -12.038 -3.685 1.606 0.284 2.622 10.850 1.016 -1.322 1.756 0.704 1.282 5.846 -0.474 -1.052 1.985 4.920 6.606 -1.366 4.621 2.935 1.888 1.100 5.848 8.212 3.960 -0.788 1.852 3.536 -0.879 9.910 -2.731 1.684 1.675 1.792 3.754 4.974 2.079 0.117 2.164 2.055 5.503 6.155 3.339 -0.109 2.478 -1.511 1.045 -4.551 -1.433 -3.989 2.593 4.828 0.835 11.135 -1.758 2.235 2.749 -3.180 -4.711 -3.514 -7.460 -5.929 2.615 2.932 3.368 20.137 0.753 0.317 2.542 4.363 16.261 -0.696 13.719 1.821 2.741 2.766 2.776 11.569 0.035 0.025 3.026 2.329 5.608 8.944 2.582 -0.697 2.668 -3.623 5.237 -11.602 2.569 -6.291 2.658 4.221 2.894 7.127 0.236 1.563 2.723 0.691 -8.686 -7.433 -11.409 -2.032 2.648 0.618 -4.198 0.139 -6.846 -2.030
1. ### Estimating the CAPM linear regression

 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 0.5121 0.9015 0.5680 0.5724 -1.2961 2.3203 Xt 1.2343 0.2163 5.7064 0.0000 0.8005 1.6682

The intercept coefficient indicates that in cases where the excess market return is zero, the mean excess return on Google stock will be 0.5121%. Therefore, increasing the excess market return by 1% percent improves Google’s excess returns by 1.2343. Therefore, the market return more risky than Google returns because it contributes to the increment of excess returns for Google stock.

1. ### R2interpretation

The predictor variable (market return) was significant at 95% confidence level. Therefore, it contributes 38.06% of the total variation experienced on the Google’s excess returns. This can be observed in the table below, which the regression summary.

 Regression Statistics Multiple R 0.6169 R Square 0.3806 Adjusted R Square 0.3689 Standard Error 6.2552 Observations 55

### Slope coefficient

The confidence interval for the slope is [0.8005, 1.668], which means that at 95% confidence level, the slope coefficient will always be contained in the interval provided that the data meets the same standards.

### 6. Determining whether Google stock was neutral

The table above shows the lower and upper bounds of the confidence interval for testing whether Google stock returns are neutral. In this case, the test determines if the Google stock returns that are greater than 0 accounts for 50% of the data. The obtained bounds are [0.4497, 0.7140], indicating that at 95% confidence interval, the proportion of Google returns above 0 will always be with the interval.

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