Quantitative Business Analysis

This article discusses statistical analysis of expected returns, correlation matrix and portfolio optimization in quantitative business analysis. It also includes a non-parametric test to check the hypothesis of same returns for different assets and suggests an optimal portfolio with weights for two assets. The assumption of independence is also discussed.

Quantitative business analysis
Name:
Institution:
6th June 2018

Q4: Are the expected returns statistically different from zero for each asset?
To answer the above question, the following hypothesis were to be tested;
For the AAPL returns
H0 : μAAPL=0
H A : μAAPL ≠ 0
Tested at α =0.05
For the HPQ returns
H0 : μHPQ =0
H A : μHPQ ≠ 0
Tested at α =0.05
For the INTC returns
H0 : μINTC =0
H A : μINTC ≠ 0
Tested at α =0.05
For the MSFT returns
H0 : μMSFT =0
H A : μMSFT ≠ 0
Tested at α =0.05
One-Sample Statistics
N Mean Std. Deviation Std. Error Mean
AAPL_returns 3417 .0499 4.08332 .06985
HPQ_returns 3417 .0040 2.32050 .03970
INTC_returns 3417 .0199 1.77920 .03044
MSFT_returns 3417 .0348 1.63408 .02795

One-Sample Test
Test Value = 0
t df Sig. (2-tailed) Mean Difference 95% Confidence Interval of the
Difference
Lower Upper
AAPL_returns .715 3416 .475 .04994 -.0870 .1869
HPQ_returns .102 3416 .919 .00405 -.0738 .0819
INTC_returns .654 3416 .513 .01990 -.0398 .0796
MSFT_returns 1.246 3416 .213 .03483 -.0200 .0896
A one-sample t-test was performed to check whether the expected returns statistically different
from zero for each asset. As can be seen, the p-value for all the returns is greater than α = 0.05.
This means that the null hypothesis (H0) is not rejected in all the four asset returns. This leads to
a conclusion that the expected returns are not statistically different from zero for each of the
assets.
Q5: Are the mean returns statistically different from each other?
To answer the above question, the following hypothesis were to be tested;
H0 : μAAPL=μHPQ=μINTC=μMSFT
H A : At least one of the asset returns is different
Tested at α =0.05
To test this, analysis of variance (ANOVA) test was performed. The results are given below;
Anova: Single Factor
SUMMARY
Groups
Cou
nt Sum
Averag
e
Varian
ce
AAPL_returns
341
5
170.66
13
0.0499
74
16.683
26
HPQ_returns 341 13.824 0.0040 5.3847

7 54 46 34
INTC_returns
341
5
67.982
77
0.0199
07
3.1674
04
MSFT_returns
341
7
119.02
39
0.0348
33
2.6702
07
ANOVA
Source of
Variation SS df MS F
P-
value F crit
Between
Groups
3.98382
6 3
1.32794
2
0.19037
1
0.9030
1
2.60555
9
Within Groups
95285.8
6 13660
6.97553
8
Total
95289.8
4 13663
From the ANOVA table above, the p-value is 0.903 (a value greater than 5% level of
significance), the null hypothesis (H0) is rejected implying that there is no significant statistical
evidence to conclude that the mean returns statistically different from each other.
Q6: Correlation matrix of the returns
AAPL_r
eturns
HPQ_r
eturns
INTC_r
eturns
MSFT_r
eturns
AAPL_re
turns 1
HPQ_ret
urns
0.15901
7 1
INTC_re
turns
0.23292
5
0.4296
85 1
MSFT_r
eturns
0.21948
3
0.3879
43
0.5822
71 1
As can be seen from the above table, a strong positive relationship was observed between the
INTC returns and the MSFT return (r = 0.5823) while weak positive returns was observed
between AAPL and HPQ returns (r = 0.1590).

Q7: Is the assumption of independence realistic?
No the assumption is not realistic since we can see there are significant correlation between the
returns. We applied a non-parametric (Kruskal-Wallis) test to test the hypothesis that the returns
for the assets are the same. The results are given below;
Kruskal-Wallis Test
Ranks
Asset N Mean Rank
Returns
AAPL 3417 6995.81
HPQ 3417 6808.58
INTC 3417 6776.47
MSFT 3417 6757.14
Total 13668
Test Statisticsa,b
Returns
Chi-Square 7.911
df 3
Asymp. Sig. .048
a. Kruskal Wallis Test
b. Grouping Variable: Asset
A Kruskal-Wallis H test showed that there was a statistically significant difference in expected
returns between the different stock assets, χ2(3) = 7.911, p = 0.048, with a mean rank returns
score of 6995.81 for asset AAPL, 6808.58 for HPQ, 6776.47 for INTC and 6757.14 for MSFT.
Q8: If you can only choose maximum of two assets into a portfolio, which will you choose?

If I was to choose maximum of two assets then I will choose MSFT and INTC. The optimal
weights for this assets is given below;
Asset Weight Expected returns
AAPL returns 0.040 0.049974
HPQ returns 0.150 0.004046
INTC returns 0.301 0.019907
MSFT returns 0.510 0.034833
As can be seen, MSFT has the highest weight followed by INTC.
The objective function for the portfolio is to minimize the risk of the asset.
Q9.Bonus question: Why is it not realistic to assume these rates follow a normal distribution?
It is not generally realistic to assume that the rates follow a normal distribution since the rates
keep on changing every day and it is very flexible.