This document presents research analyzing the relationship between global stock indices and optimal allocation for a global equity portfolio. It introduces the topic, reviews literature on international diversification, outlines hypotheses about returns and correlations between indices, and describes the methodology including statistical tests and portfolio construction using random weights in Python. Key results include descriptive statistics on the indices, correlation analysis, and statistical tests showing differences in returns, variances, and correlations between index pairs. The document concludes with limitations and implications for optimal portfolios across global markets.

1. RELATIONSHIP BETWEEN GLOBAL STOCK INDICES AND
OPTIMAL ALLOCATION FOR A GLOBAL EQUITY PORTFOLIO
Mathematical Finance Masters
Programming Techniques
Aakash Ahuja | Anne-Sophie Vienne | Miguel Alves | Pedro Matias
52676 53468 52787 47246

2. SUMMARY
Introduction and topic theory
Literature review
Hypothesis
Methodology and statistical model tests
Python code – presentation and analysis
Results, conclusions and limitations

3. INTRODUCTION AND TOPIC THEORY

4. INTRODUCTION AND TOPIC THEORY
Finance theory : Portfolios should be diversified to achieve risk reduction provided the correlation/
covariance between markets is not high
Stock Markets selected:
S&P500 and NASDAQ
FTSE 100
EURONEXT NIKKEI 225
SHANGHAI STOCK

5. INTRODUCTION AND TOPIC THEORY
Main topic : To look at significance of the correlation coefficient and to check differences/senilities between
returns and risks of investing in these stock market indices
Purpose : To study the correlation between Global Stock Market Indices and find an optimal allocation
portfolio
Through the : Examination of the optimal global allocation for investors of the equity markets
Based on : Modern Portfolio theory and mean-variance

6. LITERATURE REVIEW

7. LITERATURE REVIEW
Levy and saranat (1970) and solnik (1974)
Encourage diversification across borders by
citing support based on low cross-country
correlations
Goetzmann, et al. (2011)
The diversification benefits to global investing
are not constant, and they are low compared
to the rest of capital market history
Moldovan (2011)
The links between the stock markets were
more intense during the crisis
Scott, et al. (2019)
Global market capitalization weight serves as
a helpful starting point in determining the
appropriate allocation between domestic
and international equities

8. HYPOTHESIS

9. HYPOTHESIS
Ho: Mean Return of each pair of stock index is equal
H1: Mean Return of each pair of stock index is NOTequal
Ho: Variance of Returns of each pair of stock index is equal
H1: Variance of Returns of each pair of stock index is NOTequal
Ho: Pairwise Correlation between stock Indices is equal to zero
H1: Pairwise Correlation between stock Indices is NOTequal to zero

10. METHODOLOGY AND STATISTICAL MODEL TESTS

11. METHODOLOGY AND STATISTICAL MODEL TESTS
Data
presentation
Data preparation
log 𝑟𝑒𝑡𝑢𝑟𝑛 = ln
𝑤𝑒𝑒𝑘 𝑖
𝑤𝑒𝑒𝑘 𝑖 − 1
Data analysis
𝐶𝑂𝑉 𝑋, 𝑌 =
∑(𝑋 − 𝑋)(𝑌 − 𝑌)
𝑛 − 1
Data modeling
- Statistical model tests
- Portfolio construction
Data collection
- Index price of 6
Stock Exchanges
- Risk Free rate of 5
currencies

12. METHODOLOGY AND STATISTICAL MODEL TESTS
Arbitary portfolios are consutructed with randomly
generated weights over 50.000 simulations.
Data
presentation
T-test Two-sample
T-test
F-test for Equality of
two variances
Data preparation Data analysis Data modeling
- Statistical model tests
- Portfolio construction
All statistical tests are conducted with a 5% significance level
Data collection
Portfolio construction Statistical model tests

13. PYTHON CODE – PRESENTATION AND ANALYSIS

14. PYTHON CODE – PRESENTATION AND ANALYSIS
Libraries imported
Data imported
Data Preparation
- Return
- Correlation
- Covariation

15. PYTHON CODE – PRESENTATION AND ANALYSIS
Returns Plot

16. PYTHON CODE – PRESENTATION AND ANALYSIS
Correlation Heatmap

17. PYTHON CODE – PRESENTATION AND ANALYSIS
Random seed
Matrix with 50.000 different
portfolios
No short positions

18. PYTHON CODE – PRESENTATION AND ANALYSIS
Financial analysis of all portfolio
random generated generated
- Risk
- Return

19. PYTHON CODE – PRESENTATION AND ANALYSIS
Financial analysis of all portfolio
random generated generated
- Sharp Ration

20. PYTHON CODE – PRESENTATION AND ANALYSIS
identify the portfolio with the
highest ratio Sharp Ration

21. PYTHON CODE – PRESENTATION AND ANALYSIS
Plot all the
50.000 portfolios
tested

22. PYTHON CODE – PRESENTATION AND ANALYSIS
Two-Sample t-test for Means
(Assuming Unequal Variances)
Performed for each pair of
Indices with a 95% confidence
level

23. PYTHON CODE – PRESENTATION AND ANALYSIS
F-test for Equality of
Two Variances
Performed for each
pair of Indices with a
95% confidence
level

24. PYTHON CODE – PRESENTATION AND ANALYSIS
Statistical Significance
of Existence of
Correlation Test
Performed for each
pair of Indices with a
95% confidence
level

25. RESULTS, CONCLUSIONS AND LIMITATIONS

26. RESULTS, CONCLUSIONS AND LIMITATIONS
Basic Descriptive Stats:
(not annualized – numbers in percentage)

27. RESULTS, CONCLUSIONS AND LIMITATIONS
Basic Descriptive Stats and Correlation Analysis
(not annualized – numbers in percentage)

28. RESULTS, CONCLUSIONS AND LIMITATIONS
Statistical Tests
Statistical Significance of Existence of Correlation Two-Sample t-test for Means
(Assuming Unequal Variances)

29. RESULTS, CONCLUSIONS AND LIMITATIONS
Statistical Tests
F-test for Equality of Two Variances

30. RESULTS, CONCLUSIONS AND LIMITATIONS
Optimal Portfolio

32. THANK YOU