Most factor-based forecasting models for the term structure of interest rates depend on a fixed number of factor loading functions that have to be specified in advance. In this study, we relax this assumption by building a yield curve forecasting model that learns new factor decompositions directly from data for an arbitrary number of factors, combining a Gaussian linear state-space model with a neural network that generates smooth yield curve factor loadings. In order to control the model complexity, we define prior distributions with a shrinkage effect over the model parameters, and we present how to obtain computationally efficient maximum a posteriori numerical estimates using the Kalman filter and automatic differentiation. An evaluation of the model’s performance on 14 years of historical data of the Brazilian yield curve shows that the proposed technique was able to obtain better overall out-of-sample forecasts than traditional approaches, such as the dynamic Nelson and Siegel model and its extensions.

Primary energy consumption is one of the key drivers of global CO₂ emissions that, in turn, heavily depends on the efficiency of involved technologies. Either improvement in technology efficiency or the expansion of non-fossil fuel consumption requires large investments. The planning and financing of such investments by global policy makers or global energy firms require, in turn, reliable measures of associated global spread and their evolution in time, at least from the point of view of the principles for responsible investment (PRI). In this paper, our main contribution is the introduction of index measures for accessing global spread (that is, measures of inequality or inhomogeneity in the statistical distribution of a related quantity of interest) of technology efficiency and CO₂ emission in primary energy consumption. These indexes are based on the Gini index, as used in economical sciences, and generalized entropy measures. Regarding primary energy sources, we consider petroleum, coal, natural gas, and non-fossil fuels. Between our findings, we attest some stable relations in the evolution of global spreads of technology efficiency and CO₂ emission and a positive relation between changes in global spread of technology efficiency and use of non-fossil fuel.

Conditional value at risk (CVaR), or expected shortfall, is a risk measure for investments according to Rockafellar and Uryasev. Yamai and Yoshiba define CVaR as the conditional expectation of loss given that the loss is beyond the value at risk (VaR) level. The VaR is a risk measure that represents how much an investment might lose during usual market conditions with a given probability in a time interval. In particular, Rockafellar and Uryasev show that CVaR is superior to VaR in applications related to investment portfolio optimization. On the other hand, the Shannon entropy has been used as an uncertainty measure in investments and, in particular, to forecast the Bitcoin’s daily VaR. In this paper, we estimate the entropy of intraday distribution of Bitcoin’s logreturns through the symbolic time series analysis (STSA) and we forecast Bitcoin’s daily CVaR using the estimated entropy. We find that the entropy is positively correlated to the likelihood of extreme values of Bitcoin’s daily logreturns using a logistic regression model based on CVaR and the use of entropy to forecast the Bitcoin’s daily CVaR of the next day performs better than the naive use of the historical CVaR.

Primary energy consumption is one of the key drivers of global CO2 emissions that, in turn, heavily depend on the efficiency of involved technologies. Either the improvement in technology efficiency or the expansion of non-fossil fuel consumption require large investments. The planning and financing of such investments, by policy makers or global energy firms, require, in turn, reliable measures of associated global spreads and their evolution in time. In this paper, our main contribution is the introduction of index measures for accessing global spreads (that is, measures of inequality or inhomogeneity in the statistical distribution of a related quantity of interest) of technology efficiency and CO2 emission in primary energy consumption. These indexes are based on the Gini index, as used in economical sciences, and generalised entropy measures. Regarding primary energy sources, we consider petroleum, coal, natural gas and non-fossil fuels. Between our findings, we attest some stable relations in the evolution of global spreads of technology efficiency and CO2 emission, and a positive relation between changes in global spreads of technology efficiency and use of non-fossil fuel.

The first portfolio risk diversification strategy was put into practice by the All Weather fund in 1996. The idea of risk diversification is related to the risk contribution of each available asset class or investment factor to the total portfolio risk. The maximum diversification or the risk parity allocation is achieved when the set of risk contributions is given by a uniform distribution. Meucci (2009) introduced the maximization of the Rényi entropy as part of a leverage constrained optimization problem to achieve such diversified risk contributions when dealing with uncorrelated investment factors. A generalization of the risk parity is the risk budgeting when there is a prior for the distribution of the risk contributions. Our contribution is the generalization of the existent optimization frameworks to be able to solve the risk budgeting problem. In addition, our framework does not possess any leverage constraint.