Project Methodologies

This study focuses on project methodologies, specifically mean variance analysis and covariance matrix. It evaluates the stationary factor and provides regression statistics for ASTRAZENECA, BAE SYSTEMS, COCA COLA HOLDING AG, and BRITISH AMERICAN TOBAE SYSTEMSCO PLC. The study also discusses the concept of efficient frontier and its application in portfolio optimization.

PROJECT METHODOLOGIES

TABLE OF CONTENTS
INTRODUCTION...........................................................................................................................1
(A) Evaluation of stationary factor..................................................................................................1
B.......................................................................................................................................................6
(1) Mean variance analysis..........................................................................................................6
(2) Covariance matrix.................................................................................................................7
3 Efficient frontier.......................................................................................................................7
........................................................................................................................................................10
(C) GARCH results........................................................................................................................11
........................................................................................................................................................13
(D) ARMA.....................................................................................................................................20
CONCLUSION..............................................................................................................................22
REFERENCE................................................................................................................................................24
Table 1ASTRAZENECA PACF.....................................................................................................1
Table 2BAE SYSTEMS PACF.......................................................................................................2
Table 3COCA COLA HOLDING AG PACF.................................................................................3
Table 4BRITISH AMERICAN TOBAE SYSTEMSCO PLC PACF.............................................4
Table 5Covariance matrix................................................................................................................7
Table 6Descriptive statistics............................................................................................................7
Table 7Covariance matrix................................................................................................................8
Table 8Target return and average return.........................................................................................9
Table 9Input for efficient frontier....................................................................................................9
Table 10Efficient frontier..............................................................................................................10
Table 15GARCH for ASTRAZENECA 10 months......................................................................11
Table 11ASTRAZENECA regression...........................................................................................11
Table 16Statistics for ASTRAZENECA 10 months.....................................................................12
Table 17GARCH for 2 months......................................................................................................13
Table 13ASTRAZENECA 2 months regression...........................................................................13

Table 18ASTRAZENECA for two months...................................................................................15
Table 19BAE SYSTEMS 10 months............................................................................................16
Table 12BAE SYSTEMSL Regression.........................................................................................16
Table 20Statistics for BAE SYSTEMS 10 months.......................................................................17
Table 21BAE SYSTEMS 2 months..............................................................................................18
Table 14BAE SYSTEMS 2 months regression.............................................................................18
Table 22Statistics for BAE SYSTEMS 2 months.........................................................................20
Table 23ASTRAZENECA regression...........................................................................................20
Table 24BAE SYSTEMS Regression...........................................................................................21

INTRODUCTION
Investment is the one of the sources of income for corporates and small investors. Risk on equity is high and return is also high. It
became always difficult for the investor to make prudent decisions. Hence, efficient frontier approach must be used by the investors in
order to make prudent investment decisions. In the current study, GARCH and ARCH models are prepared and along with them
efficient frontier are also prepared. ACF and PACF calculation is also done and in this way entire research work is carried out.
(A) Evaluation of stationary factor
Table 1ASTRAZENECA PACF
Regression Statistics
Multiple R 0.024623
R Square 0.000606
Adjusted R
Square -0.01158
Standard Error 0.014324
Observations 250
ANOVA
df SS MS F
Significance
F
Regression 3 3.06E-05 1.02E- 0.04974 0.985299
1

05 4
Residual 246 0.050471
0.00020
5
Total 249 0.050502
Coefficient
s
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept 0.000596 0.000909
0.65583
8
0.51254
1 -0.00119 0.002386 -0.00119 0.002386
Lag 2 0.023965 0.063764 0.37583
0.70736
7 -0.10163 0.149559 -0.10163 0.149559
Lag 3 -0.00472 0.063757 -0.0741
0.94099
3 -0.1303 0.120855 -0.1303 0.120855
Lag 4 -0.0036 0.062989 -0.05708
0.95453
1 -0.12766 0.120471 -0.12766 0.120471
Table 2BAE SYSTEMS PACF
Regression Statistics
Multiple R 0.115003739
R Square 0.01322586
Adjusted R
Square 0.001192029
2

Standard Error 0.014060995
Observations 250
ANOVA
df SS MS F
Significance
F
Regression 3 0.000652 0.000217 1.099056 0.350169
Residual 246 0.048637 0.000198
Total 249 0.049289
Coefficients
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept 0.000380909 0.00089 0.427991 0.669032 -0.00137 0.002134 -0.00137 0.002134
Lag 2 0.113199299 0.063782 1.774779 0.077171 -0.01243 0.238828 -0.01243 0.238828
Lag 3 0.007945121 0.064332 0.123502 0.901811 -0.11877 0.134657 -0.11877 0.134657
Lag 4
-
0.017024478 0.064132 -0.26546 0.790876 -0.14334 0.109293 -0.14334 0.109293
Table 3COCA COLA HOLDING AG PACF
Regression Statistics
Multiple R 0.183819
3

R Square 0.03379
Adjusted R
Square 0.022006
Standard Error 0.015573
Observations 250
ANOVA
df SS MS F
Significance
F
Regression 3 0.002086 0.000695 2.867637 0.037184
Residual 246 0.05966 0.000243
Total 249 0.061746
Coefficients
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept -1.5E-05 0.000985 -0.01552 0.987632 -0.00196 0.001925 -0.00196 0.001925
Lag 2 -0.14893 0.063662 -2.33935 0.020118 -0.27432 -0.02354 -0.27432 -0.02354
Lag 3 -0.13129 0.063893 -2.05479 0.040956 -0.25713 -0.00544 -0.25713 -0.00544
Lag 4 -0.05105 0.062434 -0.81772 0.414306 -0.17403 0.07192 -0.17403 0.07192
4

Table 4BRITISH AMERICAN TOBAE SYSTEMSCO PLC PACF
Regression Statistics
Multiple R 0.068414
R Square 0.00468
Adjusted R
Square -0.00746
Standard
Error 0.018061
Observations 250
ANOVA
df SS MS F
Significance
F
Regression 3 0.000377 0.000126 0.385602 0.76347
Residual 246 0.080245 0.000326
Total 249 0.080623
Coefficients
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept -0.00011 0.001145 -0.09206 0.926725 -0.00236 0.002149
-
0.00236 0.002149
Lag 2 0.063239 0.059399 1.064646 0.28808 -0.05376 0.180235 - 0.180235
5

0.05376
Lag 3 -0.0077 0.059623 -0.12912 0.897368 -0.12514 0.109739
-
0.12514 0.109739
Lag 4 -0.00894 0.059364 -0.15059 0.880426 -0.12587 0.107988
-
0.12587 0.107988
In case of ASTRAZENECA correlation value for time series with lag is 0.02. In case of BAE SYSTEMS correlation coefficient
value is 0.11. On other hand, for BRITISH AMERICAN TOBAE SYSTEMSCO PLC correlation coefficient value is 0.08 and in case
of COCA COLA HOLDING AG correlation coefficient value is -0.12. All these values are indicating that there is no or less auto
correlation between multiple time series of each company. Hence, it can be said that ACF is weak across firms and there is high
volatility due to this reason there is no autocorrelation between time series of specific company. Apart from ACF PACF is another
approach that is used to measure correlation between time series. It can be seen from the regression results of ASTRAZENECA that
multiple lag time series are not significantly different from main time series data and it can be said that there is no significant
difference between main time series and lagged time series as p values are greater then 0.05. Minor relation is observed between main
time series and lagged versions as reflect by the coefficients. It may be observed from the results of BAE SYSTEMS that varied lag
time series are not significantly different from main time series data and it may be assumed that there is no significant difference
between main time series and lagged time series as p values are greater then 0.05. Minor relation is observed between main time series
and lagged versions as reflect by the coefficients. It may be observed from the results of COCA COLA HOLDING AG that varied lag
time series are not significantly different from main time series data and it may be assumed that there is no significant difference
between main time series and lagged time series as p values are greater than 0.05. Minor relation is observed between main time series
and lagged versions as reflect by the coefficients. It may be observed from the results of BRITISH AMERICAN TOBAE
SYSTEMSCO PLC that varied lag time series are not significantly different from main time series data and it may be assumed that
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there is no significant difference between main time series and lagged time series as p values are greater than 0.05. Minor relation is
observed between main time series and lagged versions as reflect by the coefficients.
B
(1) Mean variance analysis
Mean variance analysis is the part of the modern portfolio theory. It is assumed that investor is able to make rational investment
decisions and can make accurate decisions if complete set of information will be available to them. Under this theory it is assumed
that investor want low risk and maximum profit. Hence, there are factors on which due importance is given in mean variance analysis
and they are variance and return. Variance indicate extent to which return deviate from its mean value and expected return reflect
percentage that can be gained on security if investment is made on it. This theory believed that if variance and expected return of an
asset is identified then in that case by making change in weight of securities portfolio return can be maximized by the business firm.
Due to this reason concept of efficient frontier is used by the business firms so that optimum portfolio can be identified where risk can
be reduced and profit can be maximized.
(2) Covariance matrix
Table 5Covariance matrix
ASTRAZENECA
BAE
SYSTEMS.L
BRITISH
AMERICAN
TOBAE
SYSTEMSCO
PLC.L
COCA
COLA
HOLDING
AG.L GSK PRU PSN TSCO
ASTRAZENEC
A 0.02% 0.00% 0.00% 0.00% 0.01% 0.00% 0.00% 0.00%
BAE SYSTEMS. 0.00% 0.02% 0.01% 0.00% 0.00% 0.01% 0.01% 0.00%
7

L
BRITISH
AMERICAN
TOBAE
SYSTEMSCO
PLC.L 0.00% 0.01% 0.04% 0.01% 0.00% 0.01% 0.00% 0.00%
COCA COLA
HOLDING AG.L 0.00% 0.00% 0.01% 0.03% 0.00% 0.01% 0.00% 0.00%
GSK 0.01% 0.00% 0.00% 0.00% 0.01% 0.00% 0.00% 0.00%
PRU 0.00% 0.01% 0.01% 0.01% 0.00% 0.03% 0.01% 0.01%
PSN 0.00% 0.01% 0.00% 0.00% 0.00% 0.01% 0.04% 0.01%
TSCO 0.00% 0.00% 0.00% 0.00% 0.00% 0.01% 0.01% 0.02%
Covariance reflect change that come in one variable with change in another one. It can be observed from the above table that
covariance percentage is quite low and remain in range of 0.00% to 0.004%. Hence, it can be said that if firm prepare portfolio of
above given firms then in that case any big change will not come in one security with rise or fall in price of another security. Hence,
managers can not follow pair trading strategy in the business. Under this strategy usually association between varied firm’s equity is
identified and with decline in share price of one investment is made on other security. By doing so loss faced in equity of one
company is offset by gain made on another firm share.
8

3 Efficient frontier
Table 6Descriptive statistics
ASTRAZENECA
BAE
SYSTEMS.
L
BRITISH
AMERICAN
TOBAE
SYSTEMSCO
PLC.L
COCA
COLA
HOLDING
AG.L GSK PRU PSN TSCO
Mean 0.1% 0.0% -0.1% 0.0% 0.0% 0.0% 0.0% 0.1%
Variance 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
STDEV 0.01 0.01 0.02 0.02 0.01 0.02 0.02 0.01
Annualized return 22% 12% -13% 8% 12% -9% 4% 13%
Table 7Covariance matrix
Portfolio 0.56 0.00 0.00 0.00 0.61 0.00 0.05 0.00
Expected return 22% 12% -13% 8% 12% -9% 4% 13%
Variance covariance
matrix
ASTRAZENECA BAE BRITISH COCA GSK PRU PSN TSCO
9

SYSTEMS.
L
AMERICAN
TOBAE
SYSTEMSCO
PLC.L
COLA
HOLDING
AG.L
ASTRAZENECA 5.21% -0.05% -0.35% 0.09% 1.81% -0.32% -0.29% -0.24%
BAE SYSTEMS. L -0.05% 4.82% 2.02% 1.01% 0.00% 2.33% 1.45% 0.95%
BRITISH AMERICAN
TOBAE SYSTEMSCO
PLC.L -0.35% 2.02% 9.22% 2.00% -0.02% 2.16% 0.76% 0.74%
COCA COLA
HOLDING AG.L 0.09% 1.01% 2.00% 6.26% -0.01% 1.94% 0.72% 0.55%
GSK 1.81% 0.00% -0.02% -0.01% 2.78% -0.28% 0.42% 0.45%
PRU -0.32% 2.33% 2.16% 1.94% -0.28% 7.92% 3.65% 1.89%
PSN -0.29% 1.45% 0.76% 0.72% 0.42% 3.65% 10.81% 3.18%
TSCO -0.24% 0.95% 0.74% 0.55% 0.45% 1.89% 3.18% 4.79%
Table 8Target return and average return
Portfolio variance 0.00041231
STDEV 0.020305523
Average return 20%
Target return 20%
10

Table 9Input for efficient frontier
SD Return
P1 1.98% 5%
P2 1.90% 7%
P3 1.89% 9%
P4 1.89% 11%
P5 1.91% 14%
P6 1.93% 16%
P7 1.98% 18%
P8 2.03% 20%
11

Figure 1Efficient frontier
Efficient frontier is the approach where by investor identify portfolio that must be taken in to account so that risk can be reduced and
profit can be maximized. If one will select portfolio that is on top then in that case risk and return both will be high. Opposite to this, if
one pick a portfolio that is at start of cure then risk will be moderate and return will be low ( Modern portfolio theory efficient and
optimal portfolios., 2019). Portfolio that is on the lowest standard deviation is considered as one of the best portfolios. In the above
chart two portfolio are indicated with green colour that are one of the best option to be picked because standard deviation is less then
1.90% which is lowest in above given chart and return percentage of 10% which to some extent can be considered good and moderate
return in the above chart. Hence, it can be said that efficient frontier assists one to choose best one from available options.
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(C) GARCH results
Figure 2GARCH for ASTRAZENECA 10 months
Table 10ASTRAZENECA regression
ASTRAZENECA
(10 months)
SUMMARY
OUTPUT
Regression Statistics
Multiple R 0.409025156
13

R Square 0.167301578
Adjusted R
Square 0.163584174
Standard Error 2.418216429
Observations 226
ANOVA
df SS MS F
Significance
F
Regression 1 263.1786 263.1786 45.00495 1.59E-10
Residual 224 1309.901 5.847771
Total 225 1573.079
Coefficients
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept 7.224394936 4.961806 1.456001 0.146793 -2.5534 17.00219 -2.5534 17.00219
FTSE 100 0.004617626 0.000688 6.708573 1.59E-10 0.003261 0.005974 0.003261 0.005974
Table 11Statistics for ASTRAZENECA 10 months
Overall MLE -205.4008547
Persistence 7.229012562
14

ω 6.29
α 7.224394936
β 0.004617626
Figure 3GARCH for 2 months
Table 12ASTRAZENECA 2 months regression
ASTRAZENECA
2 months
SUMMARY
15

OUTPUT
Regression Statistics
Multiple R 0.854636533
R Square 0.730403603
Adjusted R
Square 0.720034511
Standard Error 1.149655381
Observations 28
ANOVA
df SS MS F
Significance
F
Regression 1 93.10168 93.10168 70.44046 7.12E-09
Residual 26 34.36439 1.321707
Total 27 127.4661
Coefficients
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept
-
96.59056107 16.9486 -5.69903 5.37E-06 -131.429 -61.7522 -131.429 -61.7522
FTSE 100 0.019620779 0.002338 8.392881 7.12E-09 0.014815 0.024426 0.014815 0.024426
16

Table 13ASTRAZENECA for two months
Overall MLE 4.233831614
Persistence -96.57094029
ω -1.16
α -96.59056107
β 0.019620779
Interpretation
Volatility forecasting power of the GARCH model is not good as it can be observed that in both charts reflect different trends.
In the first chart 10 months data is taken and by running regression alpha and beta values are estimated. As part of omega ratio Sharpe
ratio is computed which reflect return that is gained on each unit of risk. In second chart two months’ time period is taken in to
account. It can be observed that both time period overall observes different trends. Hence, alpha, beta and Sharpe ratio values are
completely different and due to this reason GARCH chart reflect different trends.
17

Figure 4BAE SYSTEMS 10 months
Table 14BAE SYSTEMSL Regression
BAE
SYSTEMSL (10
months)
SUMMARY
OUTPUT
Regression Statistics
18

Multiple R 0.277753
R Square 0.077147
Adjusted R
Square 0.073027
Standard Error 34.49748
Observations 226
ANOVA
df SS MS F
Significance
F
Regression 1 22284.67 22284.67 18.72542 2.28E-05
Residual 224 266577 1190.076
Total 225 288861.7
Coefficients
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept 196.7171 70.78349 2.779138 0.005913 57.23038 336.2038 57.23038 336.2038
FTSE 100 0.042491 0.009819 4.327288 2.28E-05 0.023141 0.061841 0.023141 0.061841
19

Table 15Statistics for BAE SYSTEMS 10 months
Overall MLE 0.199849861
Persistence 196.7595873
ω -0.17
α 196.7170964
β 0.042490957
Figure 5BAE SYSTEMS 2 months
20

Table 16BAE SYSTEMS 2 months regression
BAE SYSTEMS
2 months
SUMMARY
OUTPUT
Regression Statistics
Multiple R 0.662362
R Square 0.438723
Adjusted R
Square 0.417135
Standard Error 12.18539
Observations 28
ANOVA
df SS MS F
Significance
F
Regression 1 3017.626 3017.626 20.32293 0.000123
Residual 26 3860.579 148.4838
Total 27 6878.205
21

Coefficients
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept -251.837 179.6411 -1.40189 0.172774 -621.095 117.4205 -621.095 117.4205
FTSE 100 0.111704 0.024779 4.508096 0.000123 0.060771 0.162638 0.060771 0.162638
Table 17Statistics for BAE SYSTEMS 2 months
Overall MLE 0.878116993
Persistence -251.7254192
ω -1.36
α -251.8371235
β 0.111704375
It can be observed in case of BAE SYSTEMS that charts of 10 month and 2 months are identical to some extent in terms of GARCH
value given in the chart. In this case GARCH model forecasting efficiency is high and it can be said that trends almost remain same.
Hence, it can be said that in both cases GARCH model forecasting efficiency is different to large extent.
(D) ARMA
Table 18ASTRAZENECA regression
Regression Statistics
Multiple R 0.390433
22

R Square 0.152438
Adjusted R
Square 0.149074
Standard Error 2.818415
Observations 254
ANOVA
df SS MS F
Significance
F
Regression 1 360.0237 360.0237 45.32325 1.12E-10
Residual 252 2001.753 7.943466
Total 253 2361.777
Coefficients
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept 2.574249 5.719573 0.450077 0.653042 -8.69001 13.8385 -8.69001 13.8385
FTSE 100 0.005338 0.000793 6.732254 1.12E-10 0.003777 0.0069 0.003777 0.0069
Table 19BAE SYSTEMS Regression
Regression Statistics
23

Multiple R 0.282764
R Square 0.079955
Adjusted R
Square 0.076304
Standard Error 36.81953
Observations 254
ANOVA
df SS MS F
Significance
F
Regression 1 29689.03 29689.03 21.89977 4.69E-06
Residual 252 341630.7 1355.678
Total 253 371319.8
Coefficients
Standard
Error t Stat P-value Lower 95%
Upper
95%
Lower
95.0%
Upper
95.0%
Intercept 159.427 74.71999 2.133659 0.033837 12.27175 306.5822 12.27175 306.5822
FTSE 100 0.048476 0.010359 4.679719 4.69E-06 0.028076 0.068877 0.028076 0.068877
Interpretation
24

Sum of square for ASTRAZENECA is 422905.3296 and same in case of BAE SYSTEMS is 59420723. It can be said that
variance is high in case of both firms and this means that shares are fluctuating at rapid pace which means that there is high risk
profile of both firms shares. BAE SYSTEMS shares are highly volatile then ASTRAZENECA shares.
CONCLUSION
On the BAE systemssis of above analysis, it is concluded that regression method is the one of best approach that is used to
compute beta and intercept values. Mentioned tool scope is wide in terms of its use in the business and management. It is also
concluded that modern portfolio theory assists to large extent to investors in terms of making sound investment decisions. By using
efficient frontier model, one can prepare portfolio in such a way which lead to reduced risk and higher return on investment. It is also
concluded that GARCH and ARCH model to large extent assist investor to evaluate trends and making investment decisions.
25

REFERENCE
Books and journals
Modern portfolio theory efficient and optimal portfolios., 2019. [Online]. Available through:<
https://thismatter.com/money/investments/modern-portfolio-theory.htm>.
26