1. Under the assumption that the returns of each asset are drawn from an independently and identically distributed normal distribution, are the expected returns statistically different fromzero for each asset? State clearly the null and alternative hypothesis in each case.
2. Assume the returns of each asset are independent from each other, are the mean returns statistically different from each other?
3. Calculate the correlation matrix of the returns
Normality Test Results
This prices of AAPL asset between July 2008 and July 2009, after which the price rose up at a relatively low rate where and later decrease gently on Dec 2014. The recession and recovery are registered between Sep 2017 and 22010. Due to the regular pattern of recoveries and recession o the price, the series is said to display an upward trend which associated with seasonality, which is the main source of the fluctuation of prices of AAPL asset. The upwards trend, reveals the AAPL asset is performing well in the stock market as no extremely low prices that have been registered between July 2008 and July 2018.
it's clear that the prices of HQP started at a higher level almost at the same but a slightly higher level. This trend has been accompanied by a series of recessions and recoveries. The recessions have been registered in July 2009, April 2012 and September 2016. The recoveries were registered at the between 2010 and 1011, 2015 and 2016, and 2017 and 2018. This pattern reveals that the prices of HQP asset are fluctuating across the years. The pattern is also irregular thus there no stability of prices, even though the HQP asset begins selling at higher prices and ends at higher prices.
These prices of INTC are fluctuation ate very low as revealed by gentle falls and rises. Though there's no regular pattern of prices movements displayed, the prices of INTC are stable. There are few but minor falls of prices. Also, the prices of INTC start at a low level and end at a higher level, this suggests an upward trend of prices.
These prices of INTC are fluctuation ate very low as revealed by gentle falls and rises. Though there's no regular pattern of prices movements displayed, the prices of INTC are stable. There are few but minor falls of prices. Also, the prices of INTC start at a low level and end at a higher level, this suggests an upward trend of prices.
he price s of MSFT between July 2008 and December 2013, were low but increasing at a relatively low rate. The graph shows an upward trend between 2014 and 2018. , implying that the prices of MSFT are stable. Moreover, the graphs reveal that the fluctuation rate of price is very low
Task 2
 Computation of the returns of the four assets was done using the formula:
The computation was done in Microsoft Excel and the results are recorded in the file named Task2 computation of return.
 Interpretation of Kurtosis
According to Petters, Arlie &Xiaoying Dong (2016), the kurtosis of the normally distributed variable is 3. The Kurtosis of AAPL, HQP, INTC, and MSFT are greater than 3, implying that the returns of the four assets are not normally distributed.
Hypothesis test: Are the average returns of the four assets significantly different from 0 at 0.05 significant levels? Assumption: assets are from independently and identically normal distribution.
The ztest for a single sample mean was employed. The hypotheses to be tested are:
Rejection region:
If
The following are the tables showing the results of the tests for the four returns series.
Hypothesis Test Results
AAPL Asset
zTest: OneSample for Mean 

AAPL 

Mean 
0.0857 
Known Variance 
3.53 
Hypothesis test: Are the average returns of the four assets significantly different from 0 at 0.05 significant levels? Assumption: assets are from independently and identically normal distribution.
The ztest for a single sample mean was employed. The hypotheses to be tested are:
Rejection region:
If
The following are the tables showing the results of the tests for the four returns series.
AAPL Asset
zTest: OneSample for Mean 

AAPL 

Mean 
0.0857 
Known Variance 
3.53 
Observations 
2518 
Hypothesized Mean Difference 
0 
z 
2.2814 
P(Z<=z) onetail 
0.0113 
z Critical onetail 
1.6449 
P(Z<=z) twotail 
0.0225 
z Critical twotail 
1.9600 
From table zstatistic (twotailed) is 2.2814, which is greater than the zcritical, 1.96, implying that the null hypothesis ( will be rejected. This suggests that the average returns of AAPL are significantly different from 0. This has also been supported by Pvalue, 0.0225 which is less than 0.05 significant levels, which also allows the rejection of .
HQP Asset
zTest: OneSample for Mean 

HQP 

Mean 
0.006 
Known Variance 
4.53 
Observations 
2518 
Hypothesized Mean Difference 
0 
z 
0.145 
P(Z<=z) onetail 
0.442 
z Critical onetail 
1.645 
P(Z<=z) twotail 
0.885 
z Critical twotail 
1.960 
From table zstatistic (twotailed) is 0.145, which is less than the zcritical, 1.96, implying that null hypothesis ( will not be rejected. This reveals that the average return of HQP is not significantly different from 0.
INTC Asset
zTest: one Sample for Mean 

INTC 

Mean 
0.0313 
Known Variance 
3.2 
Observations 
2518 
Hypothesized Mean Difference 
0 
z 
0.8751 
P(Z<=z) onetail 
0.1908 
z Critical onetail 
1.6449 
P(Z<=z) twotail 
0.3815 
z Critical twotail 
1.9600 
From the above table zstatistic (twotailed) is 0.8751, which is less than the zcritical, 1.96, implying that the null hypothesis ( will not be rejected. This reveals that the average return of INTC is not significantly different from 0.
MSFT Asset
zTest: OneSample for Mean 

MSFT 

Mean 
0.0572 
Known Variance 
2.96 
Observations 
2518 
Hypothesized Mean Difference 
0 
z 
1.6615 
P(Z<=z) onetail 
0.0483 
z Critical onetail 
1.6449 
P(Z<=z) twotail 
0.0966 
z Critical twotail 
1.9600 
From the above table zstatistic (twotailed) is 1.662, which is less than the zcritical, 1.96, implying that the null hypothesis ( will not be rejected. This reveals that the average return of MSFT is not significantly different from 0.
Hypothesis Test: Are the mean returns significantly different from each other at 0.05 significant levels? Assumption: assets are independent of each other. The hypotheses to be tested are
Rejection region:
If or
To determine this, one way ANOVA test was conducted.
Anova: Single Factor 

SUMMARY 

Groups 
Count 
Sum 
Average 
Variance 

AAPL 
2518 
215.853 
0.086 
3.532 

HQP 
2518 
15.527 
0.006 
4.530 

INTC 
2518 
78.858 
0.031 
3.199 

MSFT 
2518 
144.049 
0.057 
2.956 

ANOVA 

Source of Variation 
SS 
df 
MS 
F 
Pvalue 
F crit 
Between Groups 
8.8197443 
3 
2.9399 
0.8271 
0.4787 
2.6058 
Within Groups 
35785.925 
10068 
3.5544 

Total 
35794.745 
10071 
From the table above, the Fstatistics (0.8271) is less than Fcritical (2.6058). Also, the pvalue is greater than 0.05. These results suggest that the null hypothesis will be accepted. Therefore, the mean returns of the four assets are not significantly different from each other.
Task 6
Correlation Matrix of the Returns
The following table shows the correlation between different pairs of the four assets.
AAPL 
HQP 
INTC 
MSFT 

AAPL 
1 

HQP 
0.4069 
1 

INTC 
0.5024 
0.5266 
1 

MSFT 
0.4780 
0.4749 
0.6312 
1 
All the correlations between returns of different assets are positive. This reveals that returns of different assets are positively correlated.
Task 7
The four assets are not independent as the correlation between different pairs of assets are not 0. This has been clearly revealed in task 6 above.
Hypothesis Test: Are the mean returns significantly different from each other at 0.05 significant levels? Assumption: assets are independent of each other. The hypotheses to be tested are
The Paired ttest as conducted on different pairs of the four return series. Results are shown in the table below
tTest: Paired Two Sample for Means 

AAPL 
HQP 

Mean 
0.085724153 
0.006166509 
Variance 
3.53180602 
4.530370335 
Observations 
2518 
2518 
Pearson Correlation 
0.406902996 

Hypothesized Mean Difference 
0 

df 
2517 

t Stat 
1.820860402 

P(T<=t) onetail 
0.034373379 

t Critical onetail 
1.645459242 

P(T<=t) twotail 
0.068746758 

t Critical twotail 
1.96090693 
t Statistic=1.821, which is less than criticalt(1.96), this implies that the mean difference 0 between AAPL and HQP
tTest: Paired Two Sample for Means 

AAPL 
INTC 

Mean 
0.085724153 
0.03131784 
Variance 
3.53180602 
3.199130054 
Observations 
2518 
2518 
Pearson Correlation 
0.502435483 

Hypothesized Mean Difference 
0 

df 
2517 

t Stat 
1.490894465 

P(T<=t) onetail 
0.068057255 

t Critical onetail 
1.645459242 

P(T<=t) twotail 
0.13611451 

t Critical twotail 
1.96090693 
The tstatistics is 1.49, which is less than the criticalt (1.96), suggesting that the mean difference between APPL and INTC is 0.
tTest: Paired Two Sample for Means 

AAPL 
MSFT 

Mean 
0.085724153 
0.057207522 
Variance 
3.53180602 
2.956383348 
Observations 
2518 
2518 
Pearson Correlation 
0.477990101 

Hypothesized Mean Difference 
0 

df 
2517 

t Stat 
0.776144947 

P(T<=t) onetail 
0.218868165 

t Critical onetail 
1.645459242 

P(T<=t) twotail 
0.437736331 

t Critical twotail 
1.96090693 
The tstatistic is 0.776 which is less than critical t(1.96), implying that the mean difference between the APPL and MSFT is 0
tTest: Paired Two Sample for Means 

HQP 
INTC 

Mean 
0.006166509 
0.03131784 
Variance 
4.530370335 
3.199130054 
Observations 
2518 
2518 
Pearson Correlation 
0.526587893 

Hypothesized Mean Difference 
0 

df 
2517 

t Stat 
0.654355571 

P(T<=t) onetail 
0.256471267 

t Critical onetail 
1.645459242 

P(T<=t) twotail 
0.512942535 

t Critical twotail 
1.96090693 
The tstatistic is 0.654, which is less than the critical t(1.96), implying that the mean difference between HQP and INTC.
tTest: Paired Two Sample for Means 

HQP 
MSFT 

Mean 
0.006166509 
0.057207522 

Variance 
4.530370335 
2.956383348 

Observations 
2518 
2518 

Pearson Correlation 
0.474940557 

Hypothesized Mean Difference 
0 

df 
2517 

t Stat 
1.278939683 

P(T<=t) onetail 
0.100518098 

t Critical onetail 
1.645459242 

P(T<=t) twotail 
0.201036196 

t Critical twotail 
1.96090693 
The tstatistics is 1.279, which is absolutely less than the critical t( 1.96), suggesting that the mean difference between the HQP and MSFT is 0.
tTest: Paired Two Sample for Means 

INTC 
MSFT 

Mean 
0.03131784 
0.057207522 

Variance 
3.199130054 
2.956383348 

Observations 
2518 
2518 

Pearson Correlation 
0.631248664 

Hypothesized Mean Difference 
0 

df 
2517 

t Stat 
0.861721004 

P(T<=t) onetail 
0.194461575 

t Critical onetail 
1.645459242 

P(T<=t) twotail 
0.38892315 

t Critical twotail 
1.96090693 
The tstatistics is 0.862, which is less than the critical t (1.96), implying the mean difference between INTC and MSFT is 0.
Since the tstatistics in all the six pairs is less than the critical t, the average returns in among the four assets are not t significantly different from each other.
The result, in this case, concurs with the results in Task 5 above that is no significant difference between the mean returns of the four assets.
Optimization in the portfolio
The assets that will be chosen are APPL and MSFT. The two assets are optimal weights are 0.78 and 0.22 respectively. The optimal portfolio returns are 0.08.
The average returns of the four assets were computation
Asset 
Mean 
Variance 
Std Dev 
AAPL 
0.09 
3.53 
1.879 
HQP 
0.01 
4.53 
2.128 
INTC 
0.03 
3.20 
1.789 
MSFT 
0.06 
2.96 
1.719 
The covariance matrix of the returns of the four assets
Covariance matrix 

AAPL 
HQP 
INTC 
MSFT 

AAPL 
3.530403397 
1.62698538 
1.68819326 
1.5439214 
HQP 
1.626985381 
4.52857114 
2.0039236 
1.73745663 
INTC 
1.688193258 
2.0039236 
3.19785955 
1.94054753 
MSFT 
1.543921401 
1.73745663 
1.94054753 
2.95520925 
Inference of working weight: choose the equal weight of 0.25
Computation the portfolio means from the average returns of individual assets and inferred weight. Returns matrix is multiplied by the weight vector.
Computation of the portfolio standard deviation from the covariance matrix and the inferred weights
Determination of the objection function using the Sharp ratio
Sharpe ratio is given by
These results were factored in the excel solver to obtain optimal values as shown in the figure below
Notes
Constraint sum of Weights of the portfolio assets
Changing variables were the weights
Sharpe ratio was the objective function
The following are the optimal results obtained
weight 

AAPL 
0.78 
HQP 
0.00 
INTC 
0.00 
MSFT 
0.22 
sum 
1 
Mean of portfolio 
0.08 
Variance 
2.829 
std dev 
1.682 
Rf=1.50% 

Sharpe ratio 
0.0032 
Test for Normality of the four price series.
The test was done using a kurtosis level (3) of a normally distributed variable.
AAPL 
HQP 
INTC 
MSFT 

Kurtosis 
0.644 
0.825 
0.623 
0.881 
Skewness 
0.395 
0.006 
0.889 
1.252 
From the table, it's clear that no kurtosis value that is equal to 3 or approximately 3, therefore, four prices series are not normally distributed.
References
Petters, Arlie O., and Xiaoying Dong. An Introduction to Mathematical Finance with Applications. Springer New York:, 2016.
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