Volume 41, Issue 1-2

The multi-dimensional vector autoregressive (VAR) time series is often used to model the impulse-response functions of macroeconomics variables. However, in some economical applications, the variable of main interest is the product of time series describing market variables, like e.g. the cost, being the product of price and volume. In this paper, we analyze the product of the bi-dimensional VAR(1) model components. For the introduced time series, we derive general formulas for the autocovariance function and study its properties for different cases of cross-dependence between the VAR(1) model components. The theoretical results are then illustrated in the simulation study for two types of bivariate distributions of the residual series, namely the Gaussian and Student’s t. The obtained results are applied for the electricity market case study, in which we show that the additional cost of balancing load prediction errors prior to delivery can be well described by time series being the product of the VAR(1) model components with the bivariate normal inverse Gaussian distribution.

In this paper, we conducted a comprehensive examination of the Extended Gini Shortfall (EGS) as a flexible risk measure for portfolio selection, employing various approaches. The EGS measure possesses desirable properties, such as coherence, risk and variability measurement, and risk aversion. Additionally, we introduced the Reward Risk Ratio induced from EGS and explored its associated properties. Our main focus centered on a convex optimization problem, where the objective was to minimize portfolio risk while adhering to reward and budget constraints. We demonstrated the effectiveness of the obtained theoretical results through a practical application.

We investigate the problem of finding upper and lower bounds for a Choquet risk measure of a nonlinear function of two risk factors, when the marginal distributions of the risk factors are ambiguous and represented by nonadditive measures on the marginal spaces and the joint nonadditive distribution on the product space is unknown. We treat this problem as a generalization of the optimal transport problem to the setting of nonadditive measures. We provide explicit characterizations of the optimal solutions for finite marginal spaces, and we investigate some of their properties. We further discuss the connections with linear programming, showing that the optimal transport problems for capacities are linear programs, and we also characterize their duals explicitly. Finally, we investigate a series of numerical examples, including a comparison with the classical optimal transport problem, and applications to counterparty credit risk.