Sustainability in Portfolio Optimization

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Sustainability in Portfolio Optimization

Author: Chamika Porage (880926)

Autumn 2020

Statistics, Second Cycle and 15.0hp Subject: Independent Project I, ST433A Örebro University School of Business Supervisor: Olha Bodnar

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Abstract

Sustainability has become a global trend due to the remarkable growth of the demand for sustainable practices when controlling the risks associated with ESG (Environmental, Social and Governance) aspects worldwide. Hence, the socially responsible investors (or ESG investors) are curious to know whether investing in portfolios containing sustainable assets would create better investment opportunities compared to the portfolios consisted of both sustainable and non-sustainable assets. In this study, mean-variance spanning tests that are based on classical optimal portfolio theory, are performed to examine whether there is a statistically significant difference between the minimum variance frontier of sustainable stocks and the minimum variance frontier comprising both the sustainable and non-sustainable stocks in OMXC25 index. For the analysis, weekly returns of fifteen stocks of OMXC25 index are used over a three year period from 2017 to 2019. Statistically significant results of the performed mean-variance spanning tests suggest that inclusion of unsustainable stocks will statistically significantly improve the performance of the portfolio compared with a portfolio of sustainable stocks solely in the OMXC25 index of the Danish capital market.

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Preface

This is a master‟s degree thesis worth 15 credits (ECTS) at Örebro University. I am extremely grateful to my supervisor Olha Bodnar for her excellent guidance, advices, and support during this process and for introducing this project to me.

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Abbreviations

CML Capital Market Line

ESG Environmental, Social and Governance ETF Exchange Traded Fund

LM Lagrange Multiplier LR Likelihood Ratio

MVP Minimum-Variance Portfolio OMXC25 OMX Copenhagen 25

OMXS30 OMX Stockholm 30 SR Socially Responsible

SRI Socially Responsible Investment TP Tangency Portfolio

UK United Kingdom

US United States

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List of Tables

Table 1 : Company, sector, ticker symbol, total ESG Score and total ESG risk rating of twenty five stocks of OMCX25 index. ... 12 Table 2 : Number of test and benchmark assets under four screening criteria ... 15 Table 3 : The test statistics and their corresponding p-values of Likelihood Ratio test (LR-test), Wald test (W-test) and Lagrange Multiplier test (LM-test) on the weekly returns in OMCX25 index for three year period from 2017 to 2019. ... 15 Table 4 : The p-values of asymptotic and exact mean-variance spanning tests on the weekly returns for different values of T, N, K.. ... 17 Table 5 : Test statistics and p-values of the generalized Shapiro-Wilk‟s test and Mardia‟s test for

normality of weekly returns. ... 18 Table 6 : The probabilities of rejecting null hypothesis when H0 is true for the asymptotic mean-variance

tests, for normally distributed and t-distributed residuals using 1000 simulations for T=157 weekly returns. ... 20

List of Figures

Figure 1 : The plot (a) indicates the distribution of total ESG score of fifteen stocks of OMXC25 index and the plot (b) shows the number of assets falling under five categories of the total ESG risk ratings. ... 14 Figure 2 : Chi-squared QQ-plots of residuals from regression models for checking normality ... 19 Figure 3 : Power curves for three spanning tests for rejecting null hypothesis when H0 is true for

significance level α = 0.05 for N=2, K=13 and T=157. The left power plot is from the simulation of multivariate normal distribution and the right power plot depicts the simulation from multivariate t-distribution ... 21 Figure 4 : Power curves for three spanning tests for rejecting null hypothesis when H1 is true for

significance level α = 0.05 for N=2, K=13 and T=157. The left power plot is from the simulation of multivariate normal distribution and the right power plot depicts the simulation from multivariate t-distribution ... 21

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1 Introduction

1.1 Background

Harry Markowitz, the pioneer of the modern portfolio theory, theorized the construction of optimal portfolio on risk-averse investors. Portfolio optimization is defined as the maximization of the expected return for particular level of market risk or minimization of the risk for particular level of market return. “Do not put all your eggs in one basket” is one of the famous phrases that financial advisers give their investors as the first advice on portfolio diversification. Diversification strategy minimizes the risk associated with portfolio construction.

Sustainability has become an increasingly significant aspect among global investors due to increasing pressure and growing community concerns on issues such as climate conditions, labor condition and human rights, energy use and conservation, waste management, cultural heritage, natural resources etc. As a result, asset managers and asset owners are making allowances for use of Environmental, Social and Governance (ESG) information or so called three dimensions of Socially Responsible (SR) criteria when making their management and investment decisions in portfolio construction.

The global trend towards “Responsible” and “Sustainable” finance has changed the attitude of investors to balance the risk and return of their investment by “doing the right thing”. As stated by KPMG in 2019, the estimated value of $30tn sustainable investing assets are managed globally and it is 34% increase in two years. Further, it revealed $78bn of net inflow in worldwide ESG strategies in 2018 and $400bn growth in ESG ETFs was forecasted over the next decade. Moreover, it was indicated that Europe had 48.8% of sustainable investment relative to total managed assets in 2018.

“Every dark cloud has the silver lining” is the best quote to describe the global financial crisis in 2008 and its impact on the rapid growth of ESG related sustainable investment around the world during the last decade. Before 2008, many financial markets were not interested with ESG principles. However, those financial institutions were accused after the crisis and they were strongly requested to revise the capital allocation. As a result, investors were attracted to the

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responsible and sustainable investment as reasonable number of long term investors desired to mitigate risk associated with standard ESG performance.

As financial experts predicted, the current COVID-19 crisis would be a milestone of sustainable investing due to the unstoppable interest of investors towards the ESG investing. Therefore it is vital for investors to examine whether there are opportunities to gain positive returns on sustainable investment.

1.2 Thesis objective and project outline

The primary objective of this study is to investigate whether the returns are statistically significantly different between more sustainable stocks and all other stocks in OMXC25 index. The findings of the study would provide contribution not only for the asset managers and asset owners but also for the academic literature on financial markets.

The outline of the thesis is structured as follows. Analysis and summary of previous research on this subject is discussed under the literature review in chapter 2.

Chapter 3 explains the theoretical framework and the methodology of the thesis. The total ESG scores and total risk ratings from Sustainalytics are used to identify the sustainable stocks and the Minimum Variance Portfolio (MVP) is used to reach the optimal portfolio.

Chapter 4 consists of description of data, the source of the data and variable definitions. For this task, weekly stock returns in OMXC25 index are collected from Yahoo finance over three year period from 2017 to 2019.

Lastly, the readers may find interesting analysis and results in Chapter 5 and the final chapter of the thesis is a general discussion and conclusions. Thesis findings are summarized with the limitations of the study and suggested improvements for future research.

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2 Literature Review

Socially responsible investment (SRI) has become one of the very fashionable industries not only among the investors but also the academics all over the world due to its growth sensitivity to ESG issues. Accordingly, many researchers on this subject have set their main objective to compare the financial performance between ethical and conventional funds.

Bauer, Koedijk, and Otten (2002) applied a multi-factor model of Carhart (1997) to identify slightly noticeable significant differences in risk-adjusted returns between ethical and conventional funds for the period 1990-2001 for a database consisting 103 German, UK and US ethical mutual funds. Additionally, they found that these differences were insignificant after controlling for common factors such as size, book to market and momentum. When explaining the fund performance, they illustrated that the performance of ethical indices were worse than the standard indices and “German and US funds under-perform while UK ethical funds out-perform their conventional peers.”

Ortas, Moneva, and Salvador (2012) conducted their study in order to investigate whether social and environmental processes influence the performance of SRI indexes compared to their official benchmarks in terms of risk-adjusted returns and systematic risk levels. They found that differences obtained by the SRI indexes and benchmarks are statistically insignificant. Moreover, compared to the benchmarks, the results depicts that higher systematic risk were associated with SRI indexes and that higher systematic risk is influenced by higher screening intensity.

As demonstrated in Pena and Cue Corte‟s (2017) study, there is no financial return sacrifice with ethical investment policies. In their study, they considered 330 US and European SRI mutual funds over the period 2003-2014 to identify the relationship between risk-adjusted performance and screening strategies. Furthermore, the results depict a curvilinear relationship between screening intensity and performance in US and Scandinavian mutual funds. According to the study, it is evident that types of screening activities significantly influence the risk-adjusted return. Finally, they concluded that positive screening impacts lower performance of the global sample and for US funds since the funds are financially affected by meeting superior ESG standards.

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Anane (2019) has applied Markowitz mean-variance framework for a dataset consisting of 30 constituents of Dow Jones Industrial Average stocks over the period 2008-2018 and identified that portfolio returns are higher for the selected stocks based on environmental and governance ratings.

However, Wong‟s (2020) study is the most related and recent literature for this thesis. The researcher has used mean-variance spanning tests for data consisting of weekly and monthly returns of 21 constituents in OMX Stockholm 30 (OMXS30) index over the period 2008-2019. The aim of the study was to investigate “..whether the differences between the efficient frontiers of considered more sustainable assets and the efficient frontier of all 21 assets are statistically significant.” After applying the mean-variance spanning tests with four screenings, statistically non -significant differences between the efficient frontiers was obtained and the study concludes that there is no need to include unsustainable assets into portfolio. A similar performance can be reached when the portfolio consists of sustainable assets only.

3 Theoretical framework and methodology

This chapter comprises two sections. In the first section, theoretical background and some concepts of the thesis are discussed and the second section contains the method that is used in the analysis.

3.1 Optimal portfolio theory

This thesis is mainly based on optimal portfolio theory. An investor may always look into a portfolio, a combination of financial investments containing both risky and risk-free assets such as stocks, bonds, mutual funds etc that maximize expected return with minimum risk level. Hence, the process of analyzing and assessing the assets in portfolio is known as “the portfolio analysis” and a collection of risky or risk-free financial assets that produce the maximum expected return for a given level of acceptable risk is referred to as “the optimal portfolio”. The “efficient frontier” is the combination of optimal portfolios which are expected to offer the highest returns for the given level of risk. Generally, the portfolio risk is measured by the volatility of returns. Therefore, investors are interested to examine the optimal portfolio through mean-variance analysis.

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3.1.1 Minimum Variance portfolio

Minimum Variance Portfolio (MVP) is the portfolio of risky assets with the lowest variance for the rate of expected return among all feasible portfolios.

Let us consider the portfolio of N risky assets.

Thus the vector of portfolio weights can be written as

w = (w1, w2, w3,…….wN)T , where wi indicates the weight of ith risky asset.

Since the sum of the weights of N risky assets is 1, wT1N=1, where 1N is a size N vector of ones.

The vector of expected returns of N risky asset is µ = (µ1, µ2, µ3, …….µN)T

and the covariance matrix of the N risky assets

V=(

)

where Vij indicates the covariance between the returns of the ith and jth risky assets for i=1,2,3,…,N and j=1,2,3…,N and V is symmetric and positive semi-definite matrix.

To find the MVP, the minimization problem is constructed as subject to the constraint wT

1N =1

The minimization problem is solved using the Lagrange multiplier method and the weights of the MVP is obtained as

3.1.2 Tangency portfolio

The tangency portfolio (TP) is important with the presence of the risk free assets in the portfolio. Generally, combinations of the tangency portfolio with risk free assets are selected by investors that use mean-variance analysis. The capital market line (CML) graphically reflects the portfolios that optimally combine risk and return and the slope of the CML is referred to as the

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Sharp ratio. The portfolio of risky assets that maximizes the Sharp ratio is called the tangency portfolio or in other words the most efficient portfolio, TP is created as a result of the intercept (0, Rf) of CML and the efficient frontier. The expected return of the risk free assets is

symbolized as Rf and its standard deviation is zero.

Let us assume the feasible portfolio P of N risky assets and and are the expected return and the standard deviation of portfolio P respectively. And also µ, w and V are the vector of expected return, the vector of the weights and the covariance matrix respectively, as mentioned in the 3.1.1 section. Then the Sharp ratio is,

=

√

To find the weights of TP, the maximization problem is constructed as

√ subject to the constraint w

T

1N =1

After solving the maximization problem, the weights of the TP is obtained as

3.1.3 Two fund-Theorem

Let the weights of the two different portfolios constructed by N risky assets that lie on the minimum- variance frontier be two vectors of size N and denoted by w1 and w2. The expected return of two portfolios are denoted µ1 and µ2, where µ1 ≠ µ2. Then w3, the weights of the third portfolio that lies on the minimum frontier with expected return µ3 can be obtained by the

weights (w1 and w2) and the expected returns (µ1 and µ2) of two known portfolios by the

following formula

where ⁄ 3.2 Sustainability

The concept of sustainability is mainly comprised of Environmental, Social and Governance (ESG) factors. Socially responsible investors use the ESG concept to screen their investment.

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 Environmental (E): The environmental criteria emphasize the protection of environment. Furthermore, it discusses the company‟s attitude and actions towards environmental risks associated with pollution and waste, climatic changes, natural resources, energy efficiency etc.

 Social (S): The social criteria concentrate on people and relationships. Civil rights, labour standards, Community affairs, Gender and diversity are among the social characteristics of ESG factors

 Governance (G): The Governance criterion consists of standards such as board composition, audit committee, Executive compensation, bribery and corruption etc for managing a company.

Socially responsible investors, or in other words ESG investors consider that company‟s ESG information is important in portfolio construction.

3.3 Mean-variance spanning tests

Kan and Zhou‟s (2012) research paper on mean-variance spanning tests is the main reference for this section and the notations and the technique mentioned by Kan and Zhou‟s in their paper are closely followed in this thesis.

3.3.1 The multivariate regression model

Let us assume a dataset of returns of Y, of K+N assets over T time points. Then the regression model

Y= XB + E

where, Y=[ ] , X=[

] , B= [ ] , E=[ ]

The matrix Y is a matrix of the size TxN of the returns of N test assets over time points t=1,2,….,T. The matrix X is a matrix of size Tx(K+1), the first column consists of ones and Rt

denotes the other columns that contain the returns of K benchmark assets, for t =1,2,….,T. Also, B= [α,β]T is the parameter matrix of size (K+1) x N where α and β are constants over time, α is a

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vector containing N intercepts and is a KxN matrix. Moreover, E is a TxN matrix of disturbances, where is a vector of disturbances at time point t for t =1,2,….,T.

3.3.2 Model assumptions of the multivariate regression model

1. T ≥ N+K+1, this implies the existence of an inverse of the matrix XTX

2. The disturbances for t =1,2,….,T, conditioned on the returns of K benchmark assets Rt for t =1,2,….,T are independent and identically distributed (i.i.d) and follow

N(0N, ) that is multivariate normal distribution with a mean vector comprising only

zeros, of size N and a covariance matrix . Thus, E is distributed as N(0TxN,IT ⊗ ),

where 0TxN is a zero matrix of size TxN and ⊗ indicates the Kronecker product.

3.3.3 The maximum likelihood estimators of B and

The unconstrained maximum likelihood estimators of B and matrices can be obtained by: ̂ [ ̂ ̂] = [ ]

̂ ( ̂) ( ̂)

The distribution of the maximum likelihood estimator of B conditioned on the K benchmark assets is obtained as follows, assuming that the disturbances, are i.i.d and follow N(0N, )

̂|X ~ N [B, _{⊗ ]}

and the vectorization of ̂T

conditioned on X is also normally distributed as vec( ̂T

)|X ~ N [vec (BT), ⊗ ]

Additionally, under the normality assumption, T ̂ follows N-dimentional central Wishart distribution with T-K-1 degrees of freedom and covariance matrix ,

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3.3.4 Hypothesis testing

H0: There is no difference between the minimum-variance frontier of the K+N assets and the

minimum variance frontier of K assets, H0:

where [ ] and and = 0 imply testing that TP and MVP of the K+N assets have zero weights in N assets, respectively.

Let us write , where [

] is a matrix of size 2x(K+1) and [ ] is a matrix of size 2xN.

The estimator ̂= A ̂ + C = [ ̂ ̂ ] produces the maximum likelihood estimator of , and the vectorization of ̂T

is normally distributed as vec ( ̂T_{)| X ~ N [vec (}̂),( ̂ ⊗ ] with ̂ = TA _AT Moreover, the matrix H is defined as

̂ ̂ ̂ _̂

Let be the two eigen values of the matrix ̂ ̂ _where _{≥ 0.}

3.3.5 Asymptotic mean-variance spanning tests

The following tests are taken from Kan and Zhou‟s (2012) study. The asymptotic likelihood ratio test (LR- test) can be expressed as

LR = T [ ] = T ∑

Secondly, the asymptotic Wald test (W- test) can be expressed as

W = T

Thirdly, the asymptotic Lagrange multiplier test (LM-test) can be obtained from

LM = T ∑

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3.3.6 Spanning test for small samples

If the sample is not sufficiently large, the results of asymptotic tests will be misleading. If so, a monotone transformation of the likelihood ratio test is proposed.

Hence, when N 2 and T > K+N, the exact distribution of the test statistic under H0 is F-

distribution with the degrees of freedom 2N and 2(T-K-N). (√ ) ( ) ~F 2N, 2(T-K-N )

3.3.7 The goodness-of-fit tests for multivariate normality

The normality assumption of the multivariate regression model is examined using the generalized Shapiro-Wilk‟s test, Mardia‟s test and graphical chi-squared QQ-plots.

The null- hypothesis of the generalized Shapiro-Wilk‟s test is that the tested data are normally distributed and Mardia‟s test checks for both multivariate skewness and multivariate kurtosis. In R, the two packages referred to as „mvShapiro test‟ and „MVN‟ perform the generalized Shapiro-Wilk‟s test and Mardia‟s test, respectively. The chi-squared QQ-plot is a graphical approach to check multivariate normality and the plot should reflect approximately a straight line, if the data are multivariate normal.

3.4 Simulation study for robustness analysis

In this thesis, a simulation study is designed to examine the robustness of non-normality of the asymptotic mean-variance spanning tests. The residuals are simulated from a multivariate normal distribution and a multivariate t distribution in order to study the deviation of the performance of spanning tests for i.i.d but non-normally distributed residuals.

A simulated sample of E and ϵt is defined as E* and ϵt*, respectively for t=1,2…..T and the steps

of the simulation procedure are as follows. It should be noted that the steps 2 to 6 are same for both the simulations from multivariate normal and multivariate t-distribution.

1. Firstly, each row of ϵt* for t=1,2…..T, is simulated from N (0N, and from the

multivariate t- distribution with five degrees of freedom and covariance matrix to produce the matrix E* for both multivariate normal and multivariate t-distribution, respectively. can be any Nx N positive semi-definite matrix and in this study Σ is

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defined as a matrix with same correlation ( and standard deviations drawn from the Uniform (0.1,0.5) distribution.

2. Using the formula Y*= XB+E*, the new returns of the test assets (Y*) are obtained. In this study, the matrix X is generated from the normal/t-distribution in each simulation run and the matrix B of size Nx(K+1) is drawn from Uniform distribution and then it is modified to the null hypothesis, that is the first row should be zero and the column sum should be the vector of ones.

3. As mentioned in section 3.3.3, the maximum likelihood estimators of B and are computed using X and Y*. The new eigen values are obtained as in section 3.3.4.

4. The asymptotic spanning tests are performed using the obtained eigen values (refer section 3.3.5)

5. Steps 1 to 4 are repeated to have 1000 simulations.

6. For each spanning test, the number of times that the null hypothesis is rejected at 5% significance level is counted. Then the each count is divided by 1000 to get the actual probabilities of rejecting the null hypothesis when H0 is true.

4 Data description

In this thesis, primarily, a dataset of weekly returns of fifteen stocks in OMXC25 index is used for three year period from January 2017 to December 2019. The OMXC25 index is a market-weighted price index, which contains the largest and most traded stocks on Nasdaq Nordic Exchange Copenhagen.

Firstly, a dataset containing total ESG scores and the total ESG risk ratings for twenty five companies of OMXC25 index is obtained from Yahoo Finance.

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Table 1 : Company, sector, ticker symbol, total ESG Score and total ESG risk rating of twenty five stocks of OMCX25 index.

Company Sector Ticker Symbol Total ESG

score

Total ESG risk rating

1 AMBU Health care AMBU-B NA NA

2 Bavarian Nordic Health care BAVA NA NA

3 Carlsberg Group Food, beverage, tobacco CARL-B 18 Low

4 Chr. Hansen Health care CHR 19 Low

5 Coloplast Health care COLO-B 13 Low

6 Danske Bank Banks DANSKE 29 Medium

7 Demant Health care DEMANT NA NA

8 DSV Panalpina Industrial goods and services DSV 16 Low

9 FLSmidth & Co. Construction and material FLS NA NA

10 Genmab Health care GMAB 27 Medium

11 GN Store Nord Health care GN NA NA

12 ISS Industrial goods and services ISS 15 Low

13 Jyske Bank A/S Banks JYSK NA NA

14 Lundbeck Health care LUN 23 Medium

15 Maersk (class A) Industrial goods and services MAERSK-A NA NA 16 Maersk (class B) Industrial goods and services MAERSK-B 22 Medium

17 Novo Nordisk Health care NOVO-B 22 Medium

18 Novozymes Health care NZYM-B 21 Medium

19 Orsted Utilities ORSTED 21 Medium

20 Pandora A/S Consumer products and services PNDORA 12 Low

21 Royal Unibrew Food, beverage, tobacco RBREW NA NA

22 Rockwool International Construction and material ROCK-B NA NA

23 SimCorp Technology SIM NA NA

24 Tryg Insurance TRYG 21 Medium

25 Vestas Wind Systems A/S Energy VWS 16 Low

According to table 1, it is observed that twenty five stocks of OMXC25 belong to twenty four distinct companies since the company Maersk has a Class A share and a Class B share. And also the majority, 40% of the stocks of OMXC25 belongs to the health care sector.

The Yahoo finance provides company‟s total ESG score and total ESG ratings which is powered by Sustainalytics, the prominent independent global provider of ESG research and ratings. Company‟s total ESG score is an overall score and it measures the unmanaged risk level of the

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company. The range of total ESG score is from 0 to 100, with 0 indicating no unmanaged ESG risks and 100 indicating the highest unmanaged risk level driven by ESG factors. Therefore, it is obvious that a company with lower ESG score attracts investors who are interested in sustainable investing.

Moreover, the five categories of the total ESG risk ratings is created using the total ESG score as a company with ESG score; 0-10 denotes negligible risk, 10-20 low risk, 20-30 medium risk, 30-40 high risk and the final category 40-100 represents severe risk. However, ten stocks are excluded after some data cleaning due to missing values of company‟s ESG information in Yahoo Finance and the final dataset of weekly return is consists of 15 stocks out of 25 stocks of OMXC25 index.

Secondly, the weekly closing prices for fifteen stocks from 2017 to 2019 are obtained from Yahoo finance to calculate weekly returns. Each stock is consisted of 157 observations for given three year time period.

Furthermore, the asset based screening procedure and mean-variance spanning tests are used to analyze if there exist a statistically significant difference between the efficient frontier of the sustainable stocks and the efficient frontier of all fifteen stocks. The screening method contains four categories as 10%, 25%, 50% and one other category refer to as “All”. The stocks are screened based on the 90th, 75th and 50th percentiles of total ESG scores to obtain 10%, 25% and 50% screening respectively. The test assets of the 10% screening are observed by screening all the stocks with greater than the 90th percentile of the total ESG score while the remaining stocks are considered as benchmark assets. Therefore, it is obvious that those benchmark assets are more sustainable compared to the test assets. Likewise all the stocks with higher than 75th and 50th percentiles of total ESG score are screened as the test assets and the rest of the stocks are set as benchmark assets of 25% and 50% screening respectively. Moreover, the stocks are screened on the total ESG risk rating to obtain the forth screening category which is referring to “All”. The purpose of introducing “All” category is to perform screening on all fifteen stocks with the stocks from medium to severe total ESG risk rating. In this category, all the stocks with a total ESG risk rating from medium to severe are screened as test assets and the remaining assets (assets with negligible and low total ESG risk rating) are set as benchmark assets.

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Additionally, some notations are used throughout the thesis. The number of test assets and the number of benchmark assets are indicated as N and K respectively, and T denotes the length of the time series.

The statistical software R (version 3.6.1) is used for data analysis in this study (R codes used in the thesis are available in Appendices).

5 Analysis and results

5.1 Screening

In this thesis, the screening is based on the total ESG score and the total ESG risk rating.

a.) Total ESG score

b.) Total ESG risk rating

Figure 1 : The plot (a) indicates the distribution of total ESG score of fifteen stocks of OMXC25 index and the plot (b) shows the number of assets falling under five categories of the total ESG risk ratings. 0 1 2 3 4 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 Nu m b er o f ass ets

Total ESG score

0 2 4 6 8 10

Negligible Low Medium High Severe

Nu m b er o f ass ets

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The plot (a) illustrates the range of the distribution of total ESG scores, which is from 12 to 29. From plot (b), it can be seen that the total ESG risk ratings are from low to medium for all fifteen stocks and there is no stock belongs to negligible, high or severe ESG risk ratings.

The 10% screening is based on the assets higher than the 90th percentile of the distribution of total ESG scores. Hence, there are two test assets for 10% screening as the 90th percentile of the distribution of total ESG score is 25.4. Likewise, 75th, 50th and 25th percentile of the distribution of total ESG score are 22, 21 and 16 respectively. Table 2 reveals the number of test and benchmark assets falling under four screening criteria used throughout this thesis.

Table 2 : Number of test and benchmark assets under four screening criteria

Screening Number of assets

N (Test ) K ( Benchmark)

10% 2 13

25% 3 12

50% 7 8

All 8 7

The results of three asymptotic mean-variance spanning tests on the weekly returns in OMXC25 index are summarized in table 3.

5.2 Results of mean-variance spanning tests

Table 3 : The test statistics and their corresponding p-values of Likelihood Ratio test (LR-test), Wald test (W-test) and Lagrange Multiplier test (LM-test) on the weekly returns in OMCX25 index for three year period from 2017 to 2019.

Asymptotic tests

LR-test W-test LM- test

Screening N K T LR p-value W p-value LM p-value

10% 2 13 157 12.66 0.0131 13.11 0.0108 12.24 0.0157

25% 3 12 157 13.48 0.0360 13.99 0.0298 13.01 0.0429

50% 7 8 157 26.55 0.0220 28.13 0.0137 25.10 0.0335

All 8 7 157 49.76 2.50E-05 55.89 2.53E-06 44.58 0.0002

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According to table 3, it is interesting to note that all three asymptotic tests on 10%, 25% and 50% screening are statistically significant at 5% level or in other words statistically significant difference between two efficient frontiers are discovered. Additionally, when all unsustainable stocks (stocks with medium, high or severe total ESG risk rating) are screened, the null hypothesis is rejected at 1% level of significance indicating that the seven sustainable stocks do not span the fifteen assets at any significance level. Thus, it is evident that the difference between the minimum-variance frontier of all the fifteen stocks and each of the minimum-variance frontiers of the smaller number of stocks which are considered as more sustainable are highly statistically significant in OMXC25 index. The rejection of the null hypothesis means that it is not enough to consider only sustainable assets in the Danish capital market. The inclusion of unsustainable stocks will statistically significantly improve the performance of the portfolio. This is an interesting result, since the opposite was observed in the Swedish capital market where it is enough to invest into the sustainable assets only (Wong 2020).

Furthermore, three asymptotic mean-variance spanning tests and an exact mean-variance spanning LR test are performed on fifteen stocks of different dimensions and different shorter series of observations. In table 4, three series of observations of 50, 100 and 157 are used and exact mean- variance spanning LR test would be more appropriate in order to obtain accurate results for fairly small samples as well. To construct the first row of table 4, initially, two stocks with the highest ESG score are excluded as test assets. Secondly, the first two stocks from the remaining thirteen stocks are obtained as benchmark assets (K=2). Then four spanning tests are performed over three series of observations of 50,100 and 157.

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Table 4 : The p-values of asymptotic and exact mean-variance spanning tests on weekly returns for different values of T, N, K.

Asymptotic tests Exact test

N K T LR W LM LR

2

50 0.001 0.000 0.003 0.002

100 4.96E-05 1.02E-05 0.000 7.45E-05 157 9.19E-09 4.08E-10 1.21E-07 1.46E-08 5 50 0.011 0.006 0.018 0.023 100 0.019 0.014 0.024 0.026 157 8.47E-05 3.52E-05 0.000 0.000 10 50 0.011 0.006 0.020 0.041 100 0.021 0.016 0.026 0.036 157 0.002 0.002 0.003 0.004 13 50 0.072 0.054 0.092 0.192 100 0.037 0.032 0.043 0.069 157 0.013 0.011 0.016 0.022 5 2 50 0.001 1.46E-05 0.008 0.002 100 0.001 0.000 0.004 0.002

157 8.48E-08 1.46E-09 2.04E-06 1.74E-07 5 50 0.001 0.000 0.009 0.007 100 0.019 0.010 0.034 0.034 157 0.000 0.000 0.001 0.001 10 50 0.229 0.176 0.287 0.482 100 0.253 0.225 0.281 0.367 157 0.184 0.160 0.208 0.245 10 2

50 3.57E-09 2.92E-26 0.001 4.52E-07 100 1.25E-12 4.70E-23 2.55E-07 2.95E-11 157 2.39E-22 6.31E-40 3.16E-13 6.41E-21 5

50 3.70E-05 4.09E-11 0.013 0.002

100 8.63E-08 3.77E-12 3.78E-05 1.51E-06 157 9.75E-15 8.70E-23 8.26E-10 2.33E-13

As shown in table 4, when considering N=2, all four spanning tests for all dimensions over three series of observations are statistically significant at 5% level except two situations. When K=13 and T=100 the exact LR test is significant at 10% level and when K=13 and T=50 all the asymptotic mean-variance spanning tests are statistically significant at 10% level while the exact

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mean-variance spanning test is insignificant. However, the results of asymptotic tests may be misleading when the sample size is small.

Moreover, when N=5 and K= 10, it is interesting to note that all four spanning tests over three sample sizes are statistically insignificant. Furthermore, it can be seen that all the tests are highly statistically significant for all the dimensions over three sample sizes when the number of test stocks is ten (N=10).

5.3 Checking for normality

When working with models, it is important to verify that the normality assumption of the residuals in the multivariate regression model since the mean-variance spanning tests are based on the multivariate regression model. Therefore, the generalized Shapiro-Wilk‟s test, Mardia‟s test and graphical chi-squared QQ-plots are used to verify the normality assumption. Apart from the normality, it should also be noted that the residuals are assumed to be independent and identically distributed, homoscedastic and mean vector comprising only zeros in multivariate regression model.

Table 5: Test statistics and p-values of the generalized Shapiro-Wilk‟s test and Mardia‟s test for normality of weekly returns.

Generalized Shapiro-Wilk's

test Mardia's test

Screening Test

statistic p-value

Test statistic

for skewness p-value

Test statistic

for kurtosis p-value

10% 0.9424 < 2.2e-16 999.48 7.24E-43 20.03 0.000

25% 0.9481 < 2.2e-16 726.64 2.29E-26 18.12 0.000

50% 0.9412 < 2.2e-16 324.00 1.14E-20 20.97 0.000

All 0.9318 < 2.2e-16 276.97 1.90E-22 24.04 0.000

As indicated in table 5, both the generalized Shapiro-Wilk‟s test for normality and Mardia test for both skeweness and kurtosis are highly significant. Hence, the null hypothesis of the normality is rejected and it can be concluded the presence of non-normal residuals for the used regression model.

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(a) 10% screening (b) 25% screening

(c) 50% screening (d) All screening

Figure 2 : Chi-squared QQ-plots of residuals from regression models for checking normality Furthermore, as per figure 2, it is obvious that the squared Mahalanobis distances of all the Chi-squared QQ plots are considerably deviated from the straight line and it also confirms the violation of normality assumption.

Therefore, the simulation study is designed in order to investigate the robustness of the non-normality of the asymptotic mean-variance spanning tests. In this study, new weekly returns of the test assets are simulated by simulating residuals from both the multivariate normal and the multivariate t-distribution.

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5.4 Simulation study for robustness analysis

Table 6 : The probabilities of rejecting null hypothesis when H0 is true for the asymptotic

mean-variance tests, for normally distributed and t-distributed residuals using 1000 simulations for T=157 weekly returns. Multivariate Normal Distribution Multivariate t-Distribution Screening N K LR W LM LR W LM 10% 2 13 0.074 0.082 0.067 0.075 0.081 0.067 25% 3 12 0.075 0.078 0.070 0.074 0.089 0.067 50% 7 8 0.061 0.091 0.047 0.073 0.093 0.061 All 8 7 0.09 0.129 0.061 0.093 0.119 0.064

As shown in table 6, it is obvious that the probabilities of rejecting null hypothesis under the multivariate normal distribution are quite similar to corresponding probabilities from multivariate t-distribution. Also, the probabilities from Wald test are higher compared to the corresponding Likelihood Ratio test and Lagrange Multiplier test. However, for each asymptotic spanning test, the probabilities of rejecting null hypothesis when H0 is true are slightly higher

than the expected 5% level of significance, but less than the 10% level of significance except for the Wald test with all screening. Therefore, it can be concluded that getting the type 1 error is a bit higher than the expected 5% level of significance level, if the residuals are i.i.d and non-normal. However, these probabilities of rejecting null hypothesis when H0 is true will be lower if

we increase the number of simulations.

Furthermore, the power curves for three spanning tests are plotted for two scenarios of rejecting null hypothesis when H0 is true and rejecting null hypothesis when H1 is true at significance

level, α = 0.05. The following plots depict the comparison between the simulations from multivariate normal and multivariate t-distribution for newly simulated 157 weekly observations for 10% screening when the number of test asset is 2 and the number of benchmark assets is 13.

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(a) Multivariate normal distribution (b) Multivariate t-distribution Figure 3 : Power curves for three spanning tests for rejecting null hypothesis when H0 is true for

significance level α = 0.05 for N=2, K=13 and T=157. The left power plot is from the simulation of multivariate normal distribution and the right power plot depicts the simulation from multivariate t-distribution.

According to figure 3, it is evident that the power curves of three spanning tests are coincided. The power curve from multivariate normal distribution is a bit smoother than the multivariate t- distribution.

(a) Multivariate normal distribution (b) Multivariate t-distribution Figure 4 : Power curves for three spanning tests for rejecting null hypothesis when H1 is true for

significance level α = 0.05 for N=2, K=13 and T=157. The left power plot is from the simulation of multivariate normal distribution and the right power plot depicts the simulation from multivariate t-distribution

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As indicated from figure 3, figure 4 also reveals the overlapping power curves for three asymptotic spanning tests for rejecting null hypothesis when H1 is true. Although, the shapes of

curves are quite similar, the curves from the multivariate normal simulation are smoother than the curves from multivariate t-distribution.

6 General discussion and conclusion

The purpose of this study was to examine whether there is a statistically significant difference, if an investor would prefer invest in a portfolio containing more sustainable assets than the portfolio consisting of both sustainable and unsustainable assets in OMXC25 Index. The total ESG score and the total risk rating are collected from Yahoo finance as company‟s ESG information. However, ten stocks are removed out of twenty five stocks of OMXC25 index due to the inaccessibility of ESG information. Therefore, a dataset of weekly returns of fifteen stocks in OMXC25 index for three year period from 2017 to 2019 is used for the analysis in this thesis. In this study, Likelihood Ratio test, Wald test and Lagrange Multiple test were used as mean-variance spanning tests under the four screening categories referred to as “10%”, “25%”, “50%”, and “All” to investigate possible statistical significant difference between the minimum variance frontier of more sustainable assets and the minimum variance frontier of all assets containing both sustainable and unsustainable assets. It should be noted that these mean-variance spanning tests are based on the multivariate regression model assuming that the residuals are multivariate normal and i.i.d.

Interestingly, it is evident that all three asymptotic mean-variance spanning tests based on four screening criteria results in statistically significance for fifteen stocks of OMXC25 index, or in other words it reveals a statistically significant difference between the minimum variance frontier containing more sustainable stocks and the minimum variance frontier consists of both sustainable and non-sustainable stocks. Hence, it can be assumed that the investor may have opportunity to improve his investment by adding more unsustainable stocks to the portfolio rather than maintaining a portfolio of sustainable stocks solely.

Furthermore, the normality assumption of the residuals is tested using the generalized Shapiro-Wilk‟s test, Mardia‟s test for skewness and kurtosis and graphical representation of Chi-squared

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QQ plot. The results of the normality tests indicate that the normality assumption of the residuals in multivariate regression model is violated. Therefore, the simulation study is designed to examine the robustness of the non-normality assumption in residuals, by simulating new weekly returns of the test assets. The residuals are simulated from multivariate normal and multivariate t-distribution in order to simulate new weekly returns. The results depict that the probabilities of rejecting null hypothesis when H0 is true for asymptotic mean-variance tests are

bit higher than the expected significance level of 0.05 for i.i.d and non-normal residuals. And also the power curves illustrate that the power of the tests become little weaker when the residuals are i.i.d and non-normal, compared to the normal and i.i.d residuals.

As mentioned in the literature review, this study is closely related to Wong‟s (2020) study. In her study, she obtained non-significant difference between the efficient frontier of more sustainable stocks and the efficient frontier containing both sustainable and non-sustainable stocks of twenty one stocks in OMX Stockholm 30 index over the period 2008-2019 using monthly and weekly returns. Therefore, she concluded that there is no opportunity for investor to make higher returns by adding unsustainable stocks into the portfolio containing more sustainable assets. However, in this study significant difference between the corresponding efficient frontiers were found. Hence, investors may have better opportunities, if they would add unsustainable stocks into their constructed portfolios of sustainable stocks in OMX Copenhagen 25 index. And also, comparing the results of these two studies one may assume that the stocks in OMXS 30 are more sustainable than the stocks in OMXC25.

6.1 Limitations of the study

When considering the ESG information, ten out of twenty five stocks in OMXC25 index are removed due to the unavailability of ESG information which is a limitation of the study. Therefore, the results may differ, if this limitation could be avoided.

6.2 Improvements/ suggestions for future research

In this study, weekly data of OMXC25 index is analyzed solely. It might be better to consider other time frequencies such as daily and monthly when performing the spanning tests.

Additionally, this study has found a significant difference between the efficient frontier containing sustainable assets and the efficient frontier consisted of both sustainable and

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sustainable assets in OMXC25 index. Then, it is important to calculate the positive or negative investment returns caused by the constructed sustainable portfolios in OMXC25 index due to that significant difference between the efficient frontiers. Sometimes, one may be interested to know whether this significant difference is caused due to the TP or MVP and it will also be an interesting topic for a future study.

Moreover, socially responsible investors would be interested to compare the Nordic OMX indices to make investment decisions on ESG investments. Therefore, this study can be extended as a comparison of all the Nordic OMX indices to investigate the indices that may have better investment opportunities towards sustainable investment.

Finally, it can be observed that the assumptions of performed spanning tests are violated in most of the cases. In this study, the simulation study is designed only for the robustness analysis to the non-normal assumption of mean-variance spanning tests. Thus, the simulation study can be extended as the robustness analysis for other model violations such as heteroscedasticity and autocorrelation.

7 References

1. Anane, A.K., 2019. Sustainability for Portfolio Optimization. [online] Available at:

<http://mdh.diva-portal.org/smash/get/diva2:1329098/FULLTEXT01.pdf> [Accessed 03

December 2020].

2. Bauer, R., Koedijk, K. and Otten, R., 2002. International evidence on ethical mutual fund performance and investment style. Discussion paper series. [online] Available at:

<www.cepr.org/pubs/dps/DP3452.asp> [Accessed 03 December 2020].

3. Capinski, M. and Zastawniak, T., 2011 Mathematics for Finance: An Introduction to

Financial Engineering [e-book] Springer. Available at:

<http://poincare.matf.bg.ac.rs/~kmiljan/UFM.pdf> [Accessed 03 December 2020].

4. Friede, G., Busch, T. and Bassen A., 2015. ESG and financial performance: aggregated evidence from more than 2000 empirical studies. Journal of Sustainable Finance &

Investment [online] Available at: <https://doi.org/10.1080/20430795.2015.1118917> [Accessed 03 December 2020].

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5. Herzel, S., Nicolosi, M. and Starica, C., 2012. The Cost of Sustainability in Optimal Portfolio Decisions. The European Journal of Finance,18,3-4, pp. 333-349.

6. Kan, R., and Zhou, G., 2012. Tests of Mean-Variance Spanning, Annals of Economics

and Finance, 13(1), pp. 139-187.

7. KPMG, 2019.The numbers that are changing world. Revealing the growing appetite

for responsible investing, [pdf] KPMG, Available at:

<

https://assets.kpmg/content/dam/kpmg/ie/pdf/2019/10/ie-numbers-that-are-changing-the-world.pdf > [Accessed 03 December 2020].

8. Nasdaq, 2020.Index Methodology.OMX Copenhagen 25 Index, [pdf] Nasdaq, Available at: <https://indexes.nasdaqomx.com/docs/Methodology_OMXC25EXP.pdf> [Accessed 03 December 2020].

9. Ortas, E., Moneva, J.M., and Salvador, M., 2012. Do social and environmental screens influence ethical portfolio performance? Evidence from Europe. BRQ Business Research

Quarterly (2014),17, pp.11-21.

10. Pena, J. and Céu Cortez, M., 2017. Social screening and mutual fund performance: international evidence, [online] Available at: <

https://editorialexpress.com/cgi-bin/conference/download.cgi?db_name=financeforum17&paper_id=96>[Accessed 03

December 2020].

11. Wong, V., 2020, Statistical Aspects of Sustainability in Optimal Portfolio Theory, [online] Available

at:<http://kurser.math.su.se/pluginfile.php/20130/mod_folder/content/0/Master/2020/202

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A Appendix

A.1 R code for screening criteria #Screening

#calculating percentiles

x<-c(12,13,15,16,16,18,19,21,21,21,22,22,23,27,29) quantile(x,probs =c(.25,.5,.75,.9) )

A.2 R code for importing data from excel and convert into data matrix library(readxl)

weeklyret <- read_excel("D:/Orebro University/fall 2020/Master Thesis 1/weeklyret.xlsx") M<-as.matrix(weeklyret)

A.3 R code for test statistics and their corresponding p-values of asymptotic mean-variance spanning tests when T=157, N=2 and K=13 (table 3)

#N=2,K=13 and T=157 #creating Y

AD1 <- M[,-c(1,2,3,4,6,8,9,10,11,12,13,14,15,16)] #creating X

BD1 <- M[,-c(5,7)] #creating beta hat

CD1 <- solve(t(BD1)%*%BD1)%*%t(BD1)%*%AD1 # creating sigma SD1 <- (1/157)*t(AD1-BD1%*%CD1)%*% (AD1-BD1%*%CD1) #creating A XD1 <- matrix(c(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1),nrow =2, ncol = 14, byrow= TRUE) #creating C

YD1 <- matrix(c(0,0,1,1), nrow =2,ncol = 2,byrow = TRUE ) #creating theta OD1 <- (XD1%*%CD1) + YD1 #creating G GD1 <- 157*(XD1%*% solve(t(BD1)%*%BD1)%*% t(XD1)) #creating H HD1<-OD1%*%solve(SD1)%*%t(OD1) # eigen value matrix

ED1<- HD1%*%solve(GD1) eigen(ED1)

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# test statistics are calulated after obtaining the eigen values #LR Test LR1= 157*(log(1.0562141944)+log(1.0001600834)) LR1 #critical value qchisq(0.95,2*2) #pvalue pchisq(LR1,2*2,lower.tail = FALSE) # Asymptotic Wald test

W1= 157*(0.0562141944+0.0001600834) W1 pchisq(W1,2*2,lower.tail = FALSE) # Asymptotic LM test LM1= 157*((.0562141944/1.0562141944)+(.0001600834/1.0001600834)) LM1 pchisq(LM1,2*2,lower.tail = FALSE) A.4 R code for simulation study

A.4.1 R code for calculating the actual probabilities of rejecting null hypothesis under the multivariate normal distribution (table 6)

set.seed(2021)

#portfolios dimension and sample size: N=2 and K=13 and Ts=747 N<-2

K<-13 Ts<-157

#Number of repetitions in the simulation study: B=1000 r<-1000

#Model parameter \Sigma and B

#\Sigma can be theoretically any N\times N positive definite matrix. One can also take it from the empirical study.

#Here, we define Sigma as a matrix with the same correlations \rho=0.6 and standard deviations drawn from

#the uniform distribution on [0.1,0.5] # matrix of standard deviations: D<-diag(runif(N,0.1,0.5)) #correlation matrix rho<-0.6

R<-(1-rho)*diag(rep(1,N))+rho*matrix(1,N,N) #covariance matrix

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Sigma<-D%*%R%*%D

#To generate the matrix of residuals E, we will also need the square root of Sigma. It is obtained as the Cholesky root

sqSigma<-chol(Sigma)

## a - parameter used to model under H1, where the first row in B is obtained by replacing the first row in B by

## the row whose all elements are a. The case a=0 corresponds to H0. a<-seq(0,10,1)*0.05

a_l<-length(a)

#Similarly, we also compute the covariance matrix of X needed in the simulation D_X<-diag(runif(K,0.1,0.5)) #correlation matrix rho_X<-0.5 R_X<-(1-rho_X)*diag(rep(1,K))+rho_X*matrix(1,K,K) #covariance matrix Sigma_X<-D_X%*%R_X%*%D_X sqSigma_X<-chol(Sigma_X)

# we will also have mean vector for X, which we generate from the uniform distribution on [-0.5,0.5] here

mu_X<-runif(K,-0.5,0.5)

#### Two matrices used in the computation of the test statistics #creating A

A <- cbind(c(1,0),rbind(rep(0,K),rep(-1,K))) #creating C

C <- rbind(rep(0,N),rep(1,N))

#p-values of three tests will be saved in a three r\times a_l matrices called "results_LR", "results_W", "results_LM".

results_LR<-matrix(0,r,a_l) results_W<-matrix(0,r,a_l) results_LM<-matrix(0,r,a_l) for (j in 1:a_l) {

#Matrix B: N \times K+1 is drawn here from the uniform distribution (result B0: N \times K+1), and then modified to

#correspond to the alternative hypothesis, that is it first row should have all elements "a" and #the column sum should be the vector of ones.

#We can also take B0 from the empirical study and then modified it to get matrix B. B0<-matrix(runif((K+1)*N),K+1,N)

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B<-rbind(rep(a[j],N),B0[2:(K+1),]/(matrix(1,K,1)%*%colSums(B0[2:(K+1),]))) for (i in 1:r)

{

########Generation of data matrix Y consist of generating X and E

# generation of E: Ts \times N as independent sample from the multivariate normal distribution N(0,Sigma)

E<-matrix(rnorm(Ts*N),Ts,N)%*%sqSigma

# generation of tX: Ts \times K as independent sample from the multivariate normal distribution N(mu_X,Sigma_X)

tX<-matrix(1,Ts,1)%*%mu_X+matrix(rnorm(Ts*K),Ts,K)%*%sqSigma_X # matrix X is obtained from tX by adding the vector of ones at the beginning X<-cbind(rep(1,Ts),tX)

#Computation of Y Y<-X%*%B+E

#######Computation of test statistics #Estimation of B: hB<-solve(t(X)%*%X)%*%t(X)%*%Y #Estimation of Sigma hSigma<-t(Y-X%*%hB)%*%(Y-X%*%hB)/Ts #Computation Theta Theta <- A%*%hB + C #Computation G G <- Ts*A%*%solve(t(X)%*%X)%*% t(A) #Computation H H<-Theta%*%solve(hSigma)%*%t(Theta) # Computation of the eigenvalues of HG^{-1} lambda<- eigen(H%*%solve(G))$values #LR Test LR<- Ts*sum(log(1+lambda)) results_LR[i,j]<-1-pchisq(LR,2*N) #Wald test W<- Ts*sum(lambda) results_W[i,j]<-1-pchisq(W,2*N) # LM test LM<- Ts*sum(lambda/(1+lambda)) results_LM[i,j]<-1-pchisq(LM,2*N) } }

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######Computation of actual probabilities of rejection the null hypothesis for significance level alp=0.05

alp<-0.05

#Matrix res_dec consist of test decision: 1 -- rejection, 0 -- non-rejection res_dec_LR<-results_LR<alp

res_dec_W<-results_W<alp res_dec_LM<-results_LM<alp ##Actual probability of rejection: prob_LR<-colSums(res_dec_LR)/r prob_W<-colSums(res_dec_W)/r prob_LM<-colSums(res_dec_LM)/r rbind(a,prob_LR,prob_W,prob_LM) #### figure

pdf("H1-alpha-nor.pdf", height=5.5, width=7)

plot(c(a,a,a),c(prob_LR,prob_W,prob_LM), type="n", main="", xlab=expression(a), ylab="Empirical power ", ylim=c(0,1),cex=1.5, lwd = 1.6, axes=T,frame=T)

lines(a, prob_LR, lty=1, col="black", lwd = 1.6) lines(a, prob_W, lty=2, col="red", lwd = 1.6) lines(a, prob_LM, lty=3, col="green", lwd = 1.6)

legend("topleft", lty=c(1,2,3), col=c("black","red","green"), c("LR","W", "LM"), lwd = 1.6) dev.off()

A.4.2 R code for calculating the actual probabilities of rejecting null hypothesis under the multivariate t-distribution (table 6)

set.seed(2021)

#portfolios dimension and sample size: N=2 and K=13 and Ts=747 N<-2

K<-13 Ts<-157

#degrees of freedom fort-distribution d<-5

#Number of repetitions in the simulation study: B=1000 r<-1000

#Model parameter \Sigma and B

#\Sigma can be theoretically any N\times N positive definite matrix. One can also take it from the empirical study.

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#Here, we define Sigma as a matrix with the same correlations \rho=0.6 and standard deviations drawn from

#the uniform distribution on [0.1,0.5] # matrix of standard deviations: D<-diag(runif(N,0.1,0.5)) #correlation matrix rho<-0.6 R<-(1-rho)*diag(rep(1,N))+rho*matrix(1,N,N) #covariance matrix Sigma<-D%*%R%*%D

#To generate the matrix of residuals E, we will also need the square root of Sigma. It is obtained as the Cholesky root

sqSigma<-chol(Sigma)

## a - parameter used to model under H1, where the first row in B is obtaned by replacing the first row in B by

## the row whose all elements are a. The case a=0 corresponds to H0. a<-seq(0,10,1)*0.05

a_l<-length(a)

#Similarly, we also compute the covariance matrix of X needed in the simulation D_X<-diag(runif(K,0.1,0.5)) #correlation matrix rho_X<-0.5 R_X<-(1-rho_X)*diag(rep(1,K))+rho_X*matrix(1,K,K) #covariance matrix Sigma_X<-D_X%*%R_X%*%D_X sqSigma_X<-chol(Sigma_X)

# we will also have mean vector for X, which we generate from the uniform distribution on [-0.5,0.5] here

mu_X<-runif(K,-0.5,0.5)

#### Two matrices used in the computation of the test statistics #creating A

A <- cbind(c(1,0),rbind(rep(0,K),rep(-1,K))) #creating C

C <- rbind(rep(0,N),rep(1,N))

#p-values of three tests will be saved in a three r\times a_l matrices called "results_LR", "results_W", "results_LM".

results_LR<-matrix(0,r,a_l) results_W<-matrix(0,r,a_l)

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results_LM<-matrix(0,r,a_l) for (j in 1:a_l) {

#Matrix B: N \times K+1 is drawn here from the uniform distribution (result B0: N \times K+1), and then modified to

#correspond to the alternative hypothesis, that is it first row should have all elements "a" and #the column sum should be the vector of ones.

#We can also take B0 from the empirical study and then modified it to get matrix B. B0<-matrix(runif((K+1)*N),K+1,N)

B<-rbind(rep(a[j],N),B0[2:(K+1),]/(matrix(1,K,1)%*%colSums(B0[2:(K+1),]))) for (i in 1:r)

{

########Generation of data matrix Y consist of generating X and E. The factor sqrt((d-2)/d) ensures that

########the covariance matrix is the same as in the normal distribution.

# generation of E: Ts \times N as independent sample from the multivariate t-distribution (0,Sigma)

E<-sqrt((d-2)/d)*diag(sqrt(d/rchisq(Ts,d)))%*%matrix(rnorm(Ts*N),Ts,N)%*%sqSigma # generation of tX: Ts \times K as independent sample from the multivariate t-distribution (mu_X,Sigma_X)

tX<-matrix(1,Ts,1)%*%mu_X+sqrt((d-2)/d)*diag(sqrt(d/rchisq(Ts,d)))%*%matrix(rnorm(Ts*K),Ts,K)%*%sqSigma_X # matrix X is obtained from tX by adding the vector of ones at the beginning X<-cbind(rep(1,Ts),tX)

#Computation of Y Y<-X%*%B+E

#######Computation of test statistics #Estimation of B: hB<-solve(t(X)%*%X)%*%t(X)%*%Y #Estimation of Sigma hSigma<-t(Y-X%*%hB)%*%(Y-X%*%hB)/Ts #Computation Theta Theta <- A%*%hB + C #Computation G G <- Ts*A%*%solve(t(X)%*%X)%*% t(A) #Computation H H<-Theta%*%solve(hSigma)%*%t(Theta) # Computation of the eigenvalues of HG^{-1}

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######Computation of actual probabilities of rejection the null hypothesis for significance level alp=0.05

alp<-0.05

#Matrix res_dec consist of test decision: 1 -- rejection, 0 -- non-rejection res_dec_LR<-results_LR<alp

res_dec_W<-results_W<alp res_dec_LM<-results_LM<alp ##Actual probability of rejection: prob_LR<-colSums(res_dec_LR)/r prob_W<-colSums(res_dec_W)/r prob_LM<-colSums(res_dec_LM)/r rbind(a,prob_LR,prob_W,prob_LM) #### figure

pdf("H1-alpha-t5.pdf", height=5.5, width=7)

plot(c(a,a,a),c(prob_LR,prob_W,prob_LM), type="n", main="", xlab=expression(a), ylab="Empirical power ", ylim=c(0,1),cex=1.5, lwd = 1.6, axes=T,frame=T)

lines(a, prob_LR, lty=1, col="black", lwd = 1.6) lines(a, prob_W, lty=2, col="red", lwd = 1.6) lines(a, prob_LM, lty=3, col="green", lwd = 1.6)

legend("topleft", lty=c(1,2,3), col=c("black","red","green"), c("LR","W", "LM"), lwd = 1.6) dev.off()

Sustainability in Portfolio Optimization