Analysis of Apple Stock Returns and Market Returns

This study analyzes the stock returns of Apple and their relationship with market returns. Descriptive statistics, regression analysis, and ARIMA modeling are used to examine the data. The results show that Apple stock returns are riskier but have higher returns compared to the market. The regression analysis reveals that market returns significantly predict Apple stock returns. The ARIMA model suggests that lagged returns are significant contributors to the model. The study concludes with suggestions for further research.

MG3002-Assignment -2019
Student Name:
Instructor Name:
Course Number:
15th May 2019

Part A
Calculate the returns using the following formula:
Returns=ln (Stock Price)t −ln( Stock Price)t−1
Calculate also the square returns and the absolute returns.
Do a full descriptive statistics analysis for the returns, square returns and absolute
returns including the following:
a. Graphs (for example, t-plots, histograms).
Answer
Figure 1: Time series plot for the returns on APPLE
Figure 2: Time series plot for the square returns of the APPLE

Figure 3: Time series plot for the absolute returns of APPLE
Figure 4: Histogram for the returns on APPLE
Figure 5: Histogram for the square returns of the APPLE

Figure 6: Histogram for the absolute returns of APPLE
b. Arithmetic measures (for example, mean, median maximum, minimum,
standard deviation). Provide brief comments on your findings.
Answer
Table 1: Descriptive (Summary) Statistics
APPLE
Returns
Square
returns
APPLE
Absolute
returns
APPLE
S & P
Returns
Square
returns S &
P
Absolute S
& P
Mean 4.96E-06 0.000366 0.013023 0.000164 9.13E-05 0.006444
Standard
Error
0.001209 6.14E-05 0.000885 0.000604 1.43E-05 0.000446
Median 0.001326 7.86E-05 0.008867 0.000676 1.64E-05 0.004052
Standard
Deviation
0.019159 0.000973 0.014028 0.009573 0.000226 0.00707
Sample
Variance
0.000367 9.46E-07 0.000197 9.16E-05 5.12E-08 5E-05
Kurtosis 5.179957 61.23986 9.680335 4.240517 44.03966 7.076903
Skewness -0.68957 6.718747 2.598877 -0.16071 5.686191 2.296106
Range 0.172977 0.011009 0.104878 0.08182 0.002343 0.048396
Minimum -0.10492 2.17E-09 4.65E-05 -0.03342 4.72E-11 6.87E-06
Maximum 0.068053 0.011009 0.104924 0.048403 0.002343 0.048403
Sum 0.001245 0.091764 3.268687 0.041281 0.022919 1.61756
Count 251 251 251 251 251 251
Table 1 above presents the descriptive statistics for both the stock (Apple) and the
market (S & P 500). From the above table, we can clearly see that the distribution of
the apple returns is slightly negative (left) skewed as the skewness value is given as -
0.69 (a value less than -0.5). However, the distribution of the market returns (S & P
500 index) is seen to follow a normal distribution (skewness value closer to zero). The
standard deviation for the Apple stock returns is much larger than that of the market

returns (S & P 500 returns) which shows that the Apple stock returns are much riskier
as compared to the market returns.
Part B
Consider the following model:
yt =c +∑
i=1
k
yt −i +∑
i=0
j
f t −i+ut
Where,
yt= a company’s returns
f t=market’s returns (a general index like S&P 500)
Using Linear Regression Analysis estimate the model and do the appropriate
misspecification test to determine the number of the lags ensuring that the model is
statistically adequate.
Discuss the results fully explaining in detail the relevant hypothesis testing.
Answer
In this section, we present the results of the regression analysis where we have the
model summary given in table 2 below.
Table 2: Model Summary
Model R R Square Adjusted R
Square
Std. Error of the
Estimate
1 .746a .556 .552 .01284
a. Predictors: (Constant), S & P 500 returns, Lag
From table 2 above, we can see that the value of R-Squared is given as 0.556; this
implies that 55.6% of the variation in the stock returns (Apple stock returns) is

explained by the two independent variables (lag of stock returns for Apple and the
market returns) in the model.
Table 3: ANOVAa
Model Sum of Squares df Mean Square F Sig.
1
Regression .051 2 .025 154.551 .000b
Residual .041 247 .000
Total .092 249
a. Dependent Variable: Returns
b. Predictors: (Constant), S & P 500 returns, Lag
Table 3 above presents the ANOVA table where we can deduce from the table that the
overall model is significant and fit to predict the stock returns for the Apple [f (2, 247)
= 154.55, p = 0.00].
Table 4: Coefficientsa
Model Unstandardized Coefficients Standardized
Coefficients
t Sig.
B Std. Error Beta
1
(Constant) .000 .001 -.302 .763
Lag .030 .043 .030 .706 .481
S & P 500 returns 1.497 .085 .748 17.557 .000
a. Dependent Variable: Returns
The last table (table 4) above shows the regression coefficients as well as their t-value
and the p-values. We can see that one of the predictor variables is significant while
the other one is insignificant in the model. The significant predictor variable is the S
& P 500 returns (p < 0.05) while the other predictor variable (Lag of the Apple stock
returns) was found to be insignificant in the model (p > 0.05).
The coefficient of the intercept (constant coefficient) was found to be 0.000 and it was
also insignificant implying that the constant coefficient is not important/significant in
predicting the stock returns.
The coefficient of lag of stock returns for the Apple stock was found to be 0.030 with
a p-value of 0.481. This means that there is a positive though insignificant relationship
between the lag of the stock returns and the stock returns (p > 0.05). However the

coefficient of the lag implies that a unit increase in the lag of the stock returns of the
Apple is expected to result in an increase in the stock returns by 0.03. Similarly, a unit
decrease in the lag of the stock returns of the Apple is expected to result in a decrease
in the stock returns by 0.03.
Lastly, the coefficient of the market returns (returns of S & P 500) was found to be
1.497 with a p-value of 0.000. This means that there is a positive and a significant
relationship between the market returns (returns of S & P 500) and the stock returns (p
< 0.05). The coefficient of the lag implies that a unit increase in the market returns
(returns of S & P 500) is expected to result in an increase in the stock returns by 1.497
(Francq & Zakoïan, 2015). Similarly, a unit decrease in the market returns (returns of
S & P 500) is expected to result in a decrease in the stock returns by 1.497.
Part C
Using Linear Regression Analysis estimate an AR(p) Model choosing the appropriate
number of lags. Determine if and how the lags are a significant contributor to our
model explaining in detail the relevant hypothesis testing.
Using the same model make a 5-year forecast. Provide brief comments on your
findings.
Answer
In this section, we present the regression analysis using the Arima Regression (AR)
model. Results shows that the best Arima model is the AR(6). That is, the best model
is one with 6 lags (Bos, et al., 2012). This is based on the fact that the first 6 lags were
found to be significant. The lags are very significant contributors to the model

/sigma .0194805 .0006328 30.79 0.000 .0182402 .0207207
L8. -.098173 .0600179 -1.64 0.102 -.2158058 .0194599
L7. -.1876413 .072483 -2.59 0.010 -.3297054 -.0455772
L6. -.4532446 .0844113 -5.37 0.000 -.6186877 -.2878015
L5. -.6441502 .0860155 -7.49 0.000 -.8127374 -.475563
L4. -.7724717 .0995356 -7.76 0.000 -.9675579 -.5773856
L3. -.8736936 .0979171 -8.92 0.000 -1.065608 -.6817796
L2. -.9395911 .0826401 -11.37 0.000 -1.101563 -.7776194
L1. -.9792604 .0605823 -16.16 0.000 -1.098 -.8605212
ar
ARMA
_cons -.0000626 .0002294 -0.27 0.785 -.0005123 .0003871
returns
D.returns Coef. Std. Err. z P>|z| [95% Conf. Interval]
OPG
Log likelihood = 628.9154 Prob > chi2 = 0.0000
Wald chi2(8) = 320.40
Sample: 3 - 252 Number of obs = 250
ARIMA regression
Iteration 12: log likelihood = 628.91542
Iteration 11: log likelihood = 628.9154
Iteration 10: log likelihood = 628.91524
Iteration 9: log likelihood = 628.91384
Iteration 8: log likelihood = 628.90509
Iteration 7: log likelihood = 628.8599
Iteration 6: log likelihood = 628.75151
Iteration 5: log likelihood = 627.13149
(switching optimization to BFGS)
Iteration 4: log likelihood = 625.06209
Iteration 3: log likelihood = 620.47197
Iteration 2: log likelihood = 612.65534
Iteration 1: log likelihood = 592.88434
Iteration 0: log likelihood = 549.64435
(setting optimization to BHHH)
. arima returns, arima(8,1,0)
As can be seen from the table above, lag 1 to lag 6 were found to be significant (p <
0.05).
The forecast values for the 5-year is given below;
Date AAPL S & P 500 Returns Square returns Absolute returns
5/8/2019 202.9 2879.42 0.000197 3.88588E-08 0.000197126
5/9/2019 200.72 2870.72 -0.0108 0.00011669 0.01080231
5/10/2019 197.18 2881.4 -0.01779 0.000316624 0.017793925
5/13/2019 185.72 2811.87 -0.05988 0.003585233 0.059876813
5/14/2019 188.3844 2845.19 0.014244 0.000202902 0.014244382

Part D
Write a conclusion about your findings making suggestions.
Answer
This study sought to analyse the stock returns of Apple. Results shewed that the Apple
stock returns are much riskier as compared to the market returns. However, they
(apple stock) had more returns as compared to the market. We also performed a
regression analysis to see how market returns affects the Apple stock returns. We
established that 55.6% of the variation in the stock returns (Apple stock returns) is
explained by the two independent variables (lag of stock returns for Apple and the
market returns) in the model (Tofallis, 2012). The overall regression model that was
estimated to predict the stock returns for Apple was found to be significant and fit to
predict the stock returns for the Apple.
The regression coefficients showed that the market returns (S & P 500 returns)
significantly predicts the Apple stock returns (p < 0.05) while the lag returns of the
Apple stock was found to be insignificant in predicting the Apple stock returns
(Aldrich, 2005). The coefficient of the intercept (constant coefficient) was found to be
0.000 and it was also insignificant implying that the constant coefficient is not
important/significant in predicting the stock returns (Long, 2009).
The coefficient of lag of stock returns for the Apple stock was found to be 0.030 with
a p-value of 0.481. This means that there is a positive though insignificant relationship
between the lag of the stock returns and the stock returns (p > 0.05). However the
coefficient of the lag implies that a unit increase in the lag of the stock returns of the
Apple is expected to result in an increase in the stock returns by 0.03. Similarly, a unit
decrease in the lag of the stock returns of the Apple is expected to result in a decrease
in the stock returns by 0.03 (Waegeman, et al., 2008).
Lastly, the coefficient of the market returns (returns of S & P 500) was found to be
1.497 with a p-value of 0.000. This means that there is a positive and a significant

relationship between the market returns (returns of S & P 500) and the stock returns (p
< 0.05). The coefficient of the lag implies that a unit increase in the market returns
(returns of S & P 500) is expected to result in an increase in the stock returns by
1.497. Similarly, a unit decrease in the market returns (returns of S & P 500) is
expected to result in a decrease in the stock returns by 1.497.

References
Aldrich, J., 2005. Fisher and Regression. Statistical Science, 20(4), p. 401–417.
Bos, R., De Waele, S. & Broersen, P. M. T., 2012. Autoregressive spectral estimation
by application of the burg algorithm to irregularly sampled data. IEEE Transactions
on Instrumentation and Measurement, 51(6), p. 1289..
Francq, C. & Zakoïan, J., 2015. Recent results for linear time series models with non
independent innovations. Statistical Modeling and Analysis for Complex Data
Problems, 5(2), p. 241–265.
Long, Y., 2009. Human age estimation by metric learning for regression problems.
International Conference on Computer Analysis of Images and Patterns, 5(2), p. 74–
82.
Tofallis, C., 2012. Least Squares Percentage Regression. Journal of Modern Applied
Statistical Methods, 7(5), p. 526–534.
Waegeman, W., De Baets, B. & Boullart, L., 2008. ROC analysis in ordinal
regression learning. Pattern Recognition Letters, 29(5), p. 1–9.