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HW 3 Brief Solution
Creation date：2024-05-17 17:52:49
Belonging category：金融Finance
Tag HW 3
equity indices

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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.
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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