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Rewriting: Paraphrase or rewrite your friend's essay with similar meaning at reduced cost   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

• 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 z-test 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

 z-Test: One-Sample 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 z-test 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

 z-Test: One-Sample for Mean AAPL Mean 0.0857 Known Variance 3.53 Observations 2518 Hypothesized Mean Difference 0 z 2.2814 P(Z<=z) one-tail 0.0113 z Critical one-tail 1.6449 P(Z<=z) two-tail 0.0225 z Critical two-tail 1.9600

From table z-statistic (two-tailed) is 2.2814, which is greater than the z-critical, 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 P-value, 0.0225 which is less than 0.05 significant levels, which also allows the rejection of .

HQP Asset

 z-Test: One-Sample for Mean HQP Mean 0.006 Known Variance 4.53 Observations 2518 Hypothesized Mean Difference 0 z 0.145 P(Z<=z) one-tail 0.442 z Critical one-tail 1.645 P(Z<=z) two-tail 0.885 z Critical two-tail 1.960

From table z-statistic (two-tailed) is 0.145, which is less than the z-critical, 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

 z-Test: one Sample for Mean INTC Mean 0.0313 Known Variance 3.2 Observations 2518 Hypothesized Mean Difference 0 z 0.8751 P(Z<=z) one-tail 0.1908 z Critical one-tail 1.6449 P(Z<=z) two-tail 0.3815 z Critical two-tail 1.9600

From the above table z-statistic (two-tailed) is 0.8751, which is less than the z-critical, 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

 z-Test: One-Sample for Mean MSFT Mean 0.0572 Known Variance 2.96 Observations 2518 Hypothesized Mean Difference 0 z 1.6615 P(Z<=z) one-tail 0.0483 z Critical one-tail 1.6449 P(Z<=z) two-tail 0.0966 z Critical two-tail 1.9600

From the above table z-statistic (two-tailed) is 1.662, which is less than the z-critical, 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 P-value 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   F-statistics (0.8271) is less than F-critical (2.6058). Also, the p-value 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.

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.

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 t-test as conducted on different pairs of the four return series. Results are shown in the table below

 t-Test: 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) one-tail 0.034373379 t Critical one-tail 1.645459242 P(T<=t) two-tail 0.068746758 t Critical two-tail 1.96090693

t- Statistic=1.821, which is less than   critical-t(1.96), this implies that the mean difference  0 between  AAPL and HQP

 t-Test: 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) one-tail 0.068057255 t Critical one-tail 1.645459242 P(T<=t) two-tail 0.13611451 t Critical two-tail 1.96090693

The t-statistics is 1.49, which is less than the critical-t (1.96), suggesting that the mean difference between APPL and INTC is 0.

 t-Test: 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) one-tail 0.218868165 t Critical one-tail 1.645459242 P(T<=t) two-tail 0.437736331 t Critical two-tail 1.96090693

The t-statistic is 0.776 which is less than critical t(1.96), implying that  the mean difference between the  APPL and MSFT is 0

 t-Test: 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) one-tail 0.256471267 t Critical one-tail 1.645459242 P(T<=t) two-tail 0.512942535 t Critical two-tail 1.96090693

The t-statistic is 0.654, which is less than the critical t(1.96), implying that the mean difference between HQP and INTC.

 t-Test: 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) one-tail 0.100518098 t Critical one-tail 1.645459242 P(T<=t) two-tail 0.201036196 t Critical two-tail 1.96090693

The t-statistics 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.

 t-Test: 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) one-tail 0.194461575 t Critical one-tail 1.645459242 P(T<=t) two-tail 0.38892315 t Critical two-tail 1.96090693

The t-statistics is 0.862, which is less than the critical t (1.96), implying the mean difference between  INTC and MSFT is 0.

Since the t-statistics 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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