1) The price plots for the given stock prices has been drawn using EXCEL and highlighted as shown below.
With regards to the Apple stock, it is evident that the general trend of the stock is up only with intermittent corrections witnessed during August 2012- August 2013 and also again during August 2015- August 2016. However, during the period, the stock price has grown from about $ 25 to about $ 200 and has doubled in the last one year of the given time period.
The movement of the stock has been more sideways considering the fact that there have been periods of increase followed by periods by decrease and hence the actual returns for the investors is quite minimal over the period. The starting period price is about $ 20 whereas the ending price is about $ 24. It is interesting to note that the highest price during the 10 year period is about $ 25 while the lowest price is about $ 5. Hence, the stock has been quite volatile.
With regards to the Intel stock, it is evident that the general trend of the stock is up only with intermittent corrections witnessed during August 2012- August 2013 and also again during August 2015- August 2016. A significant increase in the stock has taken place during 2017 and the stock price has almost doubled. Over the given period the stock has given returns in excess of 150%.
The upward trend for the Microsoft stock during the given period is quite apparent. The stock movement does not indicate any sharp corrections. The growth in the stock price has been quite rapid from August 2016 to August 2017 as the stock has more than doubled during the given period. The overall returns for the stock are quite spectacular during the period as the starting price is about $ 22 while the ending price is about $ 110 making for an overall return of 400%. The volatility in the stock price also has been quite low.
2) The returns on the stock price has been computed using the formula provided. The plots are indicated as follows.
3) The histogram and descriptive statistics for each of the returns is highlighted as shown below.
It is apparent that the kurtosis for all the stock returns above is non-zero which implies that neither of the above distribution would be considered as normal.
4) The relevant hypothesis is indicated below.
H0: µAAPL= 0
H1: µAAPL≠ 0
The two tail one sample t test has been conducted using the requisite returns for the stock using Excel as the preferred tool. The relevant output is indicated below.
The null hypothesis would be rejected since p value (0.0222) is lesser than significance level (0.05). Hence, the average returns on Apple stock tend to significantly differ from zero.
The relevant hypothesis is indicated below.
H0: µHQP= 0
H1: µHQP≠ 0
The two tail one sample t test has been conducted using the requisite returns for the stock using Excel as the preferred tool. The relevant output is indicated below.
The null hypothesis would not be rejected since p value (0.8844) is greater than significance level (0.05). Hence, the average returns on HQP stock can be assumed to be zero.
The relevant hypothesis is indicated below.
H0: µINTC= 0
H1: µINTC≠ 0
The two tail one sample t test has been conducted using the requisite returns for the stock using Excel as the preferred tool. The relevant output is indicated below.
The null hypothesis would not be rejected since p value (0.3797) is greater than significance level (0.05). Hence, the average returns on INTC stock can be assumed to be zero.
The relevant hypothesis is indicated below.
H0: µMSFT= 0
H1: µMSFT≠ 0
The two tail one sample t test has been conducted using the requisite returns for the stock using Excel as the preferred tool. The relevant output is indicated below.
The null hypothesis would not be rejected since p value (0.0951) is greater than significance level (0.05). Hence, the average returns on MSFT stock can be assumed to be zero.
5) One column ANOVA test would be conducted in the given case. The requisite hypotheses are highlighted below.
H0: The mean return of all the four stocks is the same.
H1: The mean return of atleast one stock tends to differ from other stocks
The test has been performed in Excel and the relevant output is indicated as follows.
The p value has come out as 0.4787 which is higher than the assumed significance level of 5%. Hence, the available evidence is not sufficient to reject the null hypothesis and accept the alternative hypothesis. Hence, it may be concluded that the average returns of all the four stocks can be assumed to be same.
6) The requisite correlation matrix is shown below.
7) Based on the above correlation matrix, it is apparent that the returns of the majority of the stocks seem to be interlinked. As a result, dependent t paired test has been performed for all the possible combinations based on four stocks.
The relevant hypothesis is as indicated below.
H0: µAAPL= µHQP
H1: µAAPL ≠ µHQP
The relevant Excel output is indicated as follows.
Since p value (0.0687) exceeds significance level (0.05), hence null hypothesis cannot be rejected. Thus, there is no significant difference between the mean returns of AAPL stock and HQP Stock.
The relevant hypothesis is as indicated below.
H0: µAAPL= µINTC
H1: µAAPL ≠ µINTC
The relevant Excel output is indicated as follows.
Since p value (0.1361) exceeds significance level (0.05), hence null hypothesis cannot be rejected. Thus, there is no significant difference between the mean returns of AAPL stock and INTC Stock.
The relevant hypothesis is as indicated below.
H0: µAAPL= µMSFT
H1: µAAPL ≠ µMSFT
The relevant Excel output is indicated as follows.
Since p value (0.4377) exceeds significance level (0.05), hence null hypothesis cannot be rejected. Thus, there is no significant difference between the mean returns of AAPL stock and MSFT Stock.
The relevant hypothesis is as indicated below.
H0: µHQP= µINTC
H1: µHQP ≠ µINTC
The relevant Excel output is indicated as follows.
Since p value (0.5129) exceeds significance level (0.05), hence null hypothesis cannot be rejected. Thus, there is no significant difference between the mean returns of HQP stock and INTC Stock.
The relevant hypothesis is as indicated below.
H0: µHQP= µMSFT
H1: µHQP ≠ µMSFT
The relevant Excel output is indicated as follows.
Since p value (0.2010) exceeds significance level (0.05), hence null hypothesis cannot be rejected. Thus, there is no significant difference between the mean returns of HQP stock and MSFT Stock.
The relevant hypothesis is as indicated below.
H0: µINTC= µMSFT
H1: µINTC ≠ µMSFT
The relevant Excel output is indicated as follows.
Since p value (0.8889) exceeds significance level (0.05), hence null hypothesis cannot be rejected. Thus, there is no significant difference between the mean returns of INTC stock and MSFT Stock.
8) A portfolio of two assets needs to be found. In order to find the superior stocks from the given four, the return per unit risk i.e. average return per unit standard deviation has been computed which has come out to be maximum for MSFT and AAPL shares which would be preferred on the INTC and HQP shares.
Let us assume that x is the percentage weight in Apple Stock and y is the percentage weight in Microsoft Stock.
Hence, returns on the portfolio = 0.086x + 0.057y
The portfolio risk is captured by the standard deviation of the portfolio as indicated below.
Portfolio standard deviation = (3.532x2 + 2.956y2 + 2xy*3.532*2.956*0.478)0.5 = (3.532x2 + 2.956y2 + 9.98xy)0.5
The objective is to maximise the return per unit risk.
Hence, the objective function is as follows.
Maximise [(0.086x + 0.057y)/ (3.532x2 + 2.956y2 + 9.98xy)0.5]
Subject to conditions
x+y =100
x, y ≥0
The above LPP has been solved using SOLVER tool in Excel and for the portfolio, the weight of AAPL stock is coming as 100% while that of MSFT is coming as 0%. Hence, it would make sense to invest the complete amount in APPLE stock based on the historical performance during the given period so as to maximise the returns per unit risk.
9) It would be realistic to assume that the return distribution are not normal owing to the fact that the skew is present in all of the return distributions as is apparent from the descriptive statistics. Also the histograms are not symmetric. Further, the kurtosis is also non-zero.
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