In this study, we provided an empirical testability of the FF3FM against the co-moment-augmented FF3FM using the Newey–West GMM methods to account for potential endogeneity. The results showed that value risk is not significantly priced, and systematic skewness and systematic kurtosis play more important roles in explaining asset return than market and size factors. We discovered that there is much to gain in moving from the incumbent FF3FM to the co-moment-augmented FF3FM. By extending this study to Nigeria and using out-of-sample data that span across the recent COVID-19 period, the study has contributed to existing knowledge significantly. The study concludes that the co-moment-augmented FF3FM is superior to the incumbent FF3FM in pricing assets, and the anomalies associated with the return-size risk factor relationship are reduced by accounting for systematic skewness and systematic kurtosis in the FF3FM. Therefore, we recommend that investors should take a long position in size-hedged portfolios and hold systematic skewness and kurtosis as constant, and vice visa.

The contents of Education for Sustainable Development (ESD) should be included in teachers’ initial and advanced training programs. In order to determine the key competences for sustainability, creating a Sustainable Consciousness is one of the main foundations. However, there are not many empirical studies that deal with consciousness from education. In this context, e-portfolio appears as a tool that promotes reflection and critical thinking, key competences for consciousness development. This work intends to propose a categorization system to extract types of consciousness and identify trainee teachers’ levels of consciousness. For this research work, of an eminently qualitative nature, we have selected twenty-five portfolios of students in the last year of the School of Education at the University of Macerata (Italy). The qualitative methodological procedure followed allowed to deduce three bases that shape trainee teachers’ consciousness: thinking, representation of reality and type of consciousness. We concluded that the attainment of a Sustainable Consciousness in teachers requires activating and developing higher levels of thinking, as well as a projective and macrostructural representation of reality.

This study presents an application of fractal mathematics to financial markets through the utilization of the Multifractal Model of Asset Returns (MMAR). The model is employed to analyse a 30-year dataset encompassing three distinct financial markets: the USD/NOK currency pair, the OMXS30 stock index, and the 12-month LIBOR rate. The MMAR model successfully captures essential stylized facts about financial markets, including high kurtosis, non-independent price movements, and clustered volatile days. Simulations generated by the model demonstrate realistic behaviour and multifractal characteristics. However, the model lacks a statistical method for assessing its accuracy and may underestimate kurtosis in high-kurtosis markets. Notably, the model is capable of producing simulations across a wide range of kurtosis values and can be instrumental in stress-testing portfolios. Despite its potential, the model is subject to limitations, such as the need for human judgment, ambiguous predictions, and interpretability issues. Furthermore, the model may become unreliable following a high-kurtosis event.

This study aims to explore methods for automating the replication of correlated portfolios across shares using various algorithms. The goal is to select one algorithm for implementation in a visualization tool that will facilitate portfolio building. The dynamics of the stock market provide a comprehensive view of the global economy, acting as a catalyst for stakeholders to participate and benefit various sectors financially, primarily through investments in companies that yield returns and contribute to the economy. The evaluation and comparison of these algorithms are crucial for making well-informed decisions regarding the core objectives of the study. Ultimately, a determination will be made as to whether the program can effectively analyse data from Yahoo finance to provide a quick overview of the financial market state of companies in the S&P 500 US stock prices.

Portfolio weights solely based on risk avoid estimation error from the sample mean, but they are still affected from the misspecification in the sample covariance matrix. To solve this problem, we shrink the covariance matrix towards the Identity, the Variance Identity, the Single-index model, the Common Covariance, the Constant Correlation and the Exponential Weighted Moving Average target matrices. By an extensive Monte Carlo simulation, we offer a comparative study of these target estimators, testing their ability in reproducing the true portfolio weights. We control for the dataset dimensionality and the shrinkage intensity in the Minimum Variance, Inverse Volatility, Equal-risk-contribution and Maximum Diversification portfolios. We find out that the Identity and Variance Identity have very good statistical properties, being well-conditioned also in high-dimensional dataset. In addition, the these two models are the best target towards to shrink: they minimise the misspecification in risk-based portfolio weights, generating estimates very close to the population values. Overall, shrinking the sample covariance matrix helps reducing weights misspecification, especially in the Minimum Variance and the Maximum Diversification portfolios. The Inverse Volatility and the Equal-Risk-Contribution portfolios are less sensitive to covariance misspecification, hence they benefit less from shrinkage.