HW 3 Brief Solution：
1. Abstract
In this case study, we take weekly data on six European equity indices and achieve several findings:
1. We compare the performance of 6 indices. We find IBEX is the best performer while AEX is
the poorest performer. 2. We generate return table of the 6 indices, covariance and correlation
matrix of weekly returns of the indices. 3. We do the PCA analysis and briefly discuss the major
components.
2. Introduction (It is a general part and I skip it for the solution, you need to generally introduce
what we want to do, where we get the data and other necessary background descriptions)
e.g.: We take weekly data on six indices, i.e. AEX 25, CAC 40, DAX 30, FTSE 100, IBEX 35 and
MIB 30, from 2 Jan 2001 until 17 July 2006.
3. Results and Analyses (I write simply here. I find some groups did very good analyses and
did extra work more than I show here. However, most groups did not write well.)
To compare the performance of different indices, first we standardized the index by applying the
formula as follows,
indext = 100 ∗ /0
Here 0 represents the first date (2001-01-02), and t represents the following dates
Figure 1. Six European Equity Indices (standardized)

From Figure 1, we can see indices decreased from 2001 to the beginning of 2003 and then started
to increase afterwards. All indices show very similar patterns. In later correlation test of index
returns can show the high positive correlations between indices. Among these six indices, we
observe the best performing index is IBEX 35 and the worst performing index is AEX 25. For
detailed information, please refer to the attached Python code and corresponding results.
Correspondingly, we also generate the weekly rate of return (Table 1). The return is calculated
by the formula: weekl rate of return =
indext+1
indext
.
Table 1. Return Table (As a sample, I listed the first 10 weeks)
Date AEX_new CAC_new DAX_new FTSE_new IBEX_new MIB_new
2001-01-02
2001-01-08 0.006386 0.013255 0.016878 -0.00526 0.030079 0.030517
2001-01-15 -0.01716 0.001952 0.024884 0.007104 0.006602 0.003724
2001-01-22 0.005868 0.013666 0.006565 0.013689 0.01352 0.009715
2001-01-29 -0.00251 -0.01675 -0.00851 -0.00602 -0.02394 -0.01898
2001-02-05 -0.0132 -0.01957 -0.02126 -0.01472 0.008915 -0.03355
2001-02-12 -0.00764 -0.02052 -0.0089 -0.01233 -0.01477 -0.02126
2001-02-19 -0.05276 -0.04867 -0.05652 -0.02375 -0.04543 -0.04207
2001-02-26 0.008509 -0.00581 0.013774 -0.01432 0.024195 -0.02107

Furthermore, we calculate the correlation and covariance tables (Table 2 and Table 3) for the index
returns. From the tables (especially Table 2 for correlations), we are aware of high correlations
between different indices, ranging from 0.8229 to 0.936586. It may imply that a portfolio which
invests only in these 6 markets is not well diversified.
Table 2. Correlation table
AEX_new CAC_new DAX_new FTSE_new IBEX_new MIB_new
AEX_new 1 0.936586 0.884453 0.864977 0.822904 0.854891
CAC_new 0.936586 1 0.911694 0.893084 0.847069 0.89628
DAX_new 0.884453 0.911694 1 0.83672 0.830307 0.862673
FTSE_new 0.864977 0.893084 0.83672 1 0.789698 0.855689
IBEX_new 0.822904 0.847069 0.830307 0.789698 1 0.833185
MIB_new 0.854891 0.89628 0.862673 0.855689 0.833185 1

Table 3. Covariance Matrix
AEX_new CAC_new DAX_new FTSE_new IBEX_new MIB_new
AEX_new 0.001075 0.000864 0.000996 0.000604 0.00071 0.000752
CAC_new 0.000864 0.000792 0.000881 0.000535 0.000627 0.000677
DAX_new 0.000996 0.000881 0.00118 0.000612 0.000751 0.000796
FTSE_new 0.000604 0.000535 0.000612 0.000454 0.000443 0.000489
IBEX_new 0.00071 0.000627 0.000751 0.000443 0.000693 0.000589
MIB_new 0.000752 0.000677 0.000796 0.000489 0.000589 0.00072
Then, we perform a PCA on the equally weighted correlation matrix of the weekly returns to these
indices. The PCA is derived from Eigen values and Eigen vectors and the results are shown in
Table 4. We can see the first principal component explains 88.5% of the variation, the second
explains 3.7% and so on. The cumulative variation show that the first three components would
explain 95% of the variation of the whole system over the period.
Table 4. Eigen Value and Eigen Vectors
PC1 PC2 PC3 PC4 PC5 PC6
eigen
values 5.309113 0.222699 0.164578 0.141118 0.109087 0.053406
explained 88.485% 3.712% 2.743% 2.352% 1.818% 0.890%
cumulative 88.485% 92.197% 94.940% 97.292% 99.110% 100.000%
Eigen
Vectors 1 2 3 4 5 6
-0.41279 -0.22049 -0.38922 -0.31086 -0.5607 -0.46742
-0.42211 -0.16492 -0.17504 -0.04412 -0.1889 0.852266
-0.40974 -0.00074 -0.54717 0.28586 0.653225 -0.15588
-0.403 -0.45789 0.579695 -0.37715 0.371975 -0.10621
-0.39347 0.844687 0.162974 -0.32275 0.029987 -0.00801
-0.40782 0.032124 0.394804 0.757165 -0.28976 -0.13971

By using the first three components, we can set up a three-component representation. If we take
AEX 25 as an example,
RAEX = −0.412791 − 0.22049P2 − 0.389223.

Further analysis on the first component, we can find all the returns have a similar coefficient on
the first component, P1. If we only change the first component, then all the returns will change by
approximately the same amount. In other words, the first component captures the common trend
that is shared by these indices.

4. Conclusion (I skip here)

5. Appendix
Please see attached Python code.