Portfolio Management of Copula-Dependent Assets Based on P(Y < X) Reliability Models: Revisiting Frank Copula and Dagum Distributions

Modern portfolio theory indicates that portfolio optimization can be carried out based on the mean-variance model, where returns and risk are represented as the average and variance of the historical data of the stock’s returns, respectively. Several studies have been carried out to find better risk proxies, as variance was not that accurate. On the other hand, fewer papers are devoted to better model/characterize returns. In the present paper, we explore the use of the reliability measure P(Y<X) to choose between portfolios with returns given by the distributions X and Y. Thus, instead of comparing the expected values of X and Y, we will explore the metric P(Y<X) as a proxy parameter for return. The dependence between such distributions shall be modelled by copulas. At first, we derive some general results which allows us to split the value of P(Y<X) as the sum of independent and dependent parts, in general, for copula-dependent assets. Then, to further develop our mathematical framework, we chose Frank copula to model the dependency between assets. In the process, we derive a new polynomial representation for Frank copulas. To perform a study case, we considered assets whose returns’ distributions follow Dagum distributions or their transformations. We carried out a parametric analysis, indicating the relative effect of the dependency of return distributions over the reliability index P(Y<X). Finally, we illustrate our methodology by performing a comparison between stock returns, which could be used to build portfolios based on the value of the the reliability index P(Y<X).

Implementing quantile selection models in Stata

In this article, we describe qregsel, a community-contributed command that implements a copula-based sample-selection correction for quantile regression recently proposed by Arellano and Bonhomme (2017, Econometrica 85: 1–28). The command allows the user to model selection in quantile regressions by using either a Gaussian or a one-dimensional Frank copula. We illustrate the use of qregsel with two examples. First, we apply the method to the fictional dataset used in the Stata Base Reference Manual for the heckman command. Second, we replicate part of the empirical application of the original article using data for the United Kingdom that cover the period 1978–2000 to compare wages of males and females at different quantiles.

Optimization of Electric Vehicles Based on Frank-Copula-GlueCVaR Combined Wind and Photovoltaic Output Scheduling Research

Improving the efficiency of renewable energy and electricity utilization is an urgent problem for China under the objectives of carbon peaking and carbon neutralization. This paper proposes an optimization scheduling method of electric vehicles (EV) combined with wind and photovoltaic power based on the Frank-Copula-GlueCVaR. First, a joint output model based on copula theory was built to describe the correlation between wind and photovoltaic power output. Second, the Frank-Copula-GlueCVaR index was introduced in a novel way. Operators can now predetermine the future wind–photovoltaic joint output range based on this index and according to their risk preferences. Third, an optimal scheduling model aimed at reducing the group charging cost of EVs was proposed, thereby encouraging EV owners to participate in the demand response. Fourth, this paper: proposes the application of a Variant Roth–Serve algorithm; regards the EV group as a multi-intelligent group; and finds the Pareto optimal strategy of the EV group through continuous learning. Finally, case study results are shown to effectively absorb more renewable energy, reduce the consumption cost of the EV group, and suppress the load fluctuation of the whole EV group, which has a practical significance and theoretical value.

Introducing Copula as a Novel Statistical Method in Psychological Analysis

During the past decades, the relationship between various psychological parameters had been studied in detail. However, the dependency structure of correlated parameters was rarely investigated. Knowing the dependence structure helps in finding the probability matrix of the interaction between the parameters. In this research, a novel approach was introduced in psychological analysis using copula functions. For this purpose, the self-esteem and anxiety of 141 university students in Iran were extracted using the Coopersmith Self-esteem Inventory and the Zang Anxiety Scale. Then the dependence structure of self-esteem and anxiety were established using copula functions. The Frank copula achieved the best fit for the joint variables of self-esteem and anxiety. Finally, the probability matrix of different classes of anxiety, taking into account self-esteem classes, was extracted. The results indicated that poor self-esteem leads to severe or very severe anxiety, with more than 98% probability, while strong self-esteem may lead to normal and mild anxiety, with about 80% probability. It can be concluded that the method was promising, and that copula functions can open a window to the dependence structure analysis of psychological parameters.

Copula approach for simulated damages caused by landfalling US hurricanes

Hurricanes are destructive forces of nature that have the ability to cause vast devastation both economically and socially. Estimating the potential damage caused by hurricanes aids local, state and federal governments as well as insurance and reinsurance companies to plan for future hurricanes. Direct damages caused by hurricanes are difficult to estimate. There are multiple factors that could contribute to the damages caused by a hurricane. Wind is typically considered the most important factor to account for when estimating potential damage. Aside from the complex physical processes, the difficultly in estimating hurricane damages is further compounded by limited data and a changing climate. Fitting models with limited data presents a series of challenges. These challenges include outliers that could form a large proportion of the data, overfitting, missing data and it becomes difficult to leave out a portion of the data for external validation. This study found a significant positive correlation using the Kendall rank correlation coefficient between hurricane damages, measured by the area of total destruction and the maximum landfalling wind speed (τ = 0.451). A copula-based approach was used to model their dependency. Both bivariate Archimedean and elliptical copulae families were assessed as potential models. A bivariate Frank copula with Weibull marginals was found to be the most appropriate fitting model based on a visual inspection of the contour plots of the fitted copulae. Simulation from the fitted copula was qualitatively similar to observation. This study demonstrated a potential method to overcome the limitation of small data facing models to estimate hurricane damages.

Development of a prognostic prediction model to estimate the risk of multiple chronic diseases

IntroductionThe ability to estimate risk of multimorbidity will provide valuable information to patients and primary care practitioners in their preventative efforts. Current methods for prognostic prediction modelling are insufficient for the estimation of risk for multiple outcomes, as they do not properly capture the dependence that exists between outcomes. ObjectivesWe developed a multivariate prognostic prediction model for the 5-year risk of diabetes, hypertension, and osteoarthritis that quantifies and accounts for the dependence between each disease using a copula-based model. MethodsWe used data from the Canadian Primary Care Sentinel Surveillance Network (CPCSSN) from 2009 onwards, a collection of electronic medical records submitted by participating primary care practitioners across Canada. We identified patients 18 years and older without all three outcome diseases and observed any incident diabetes, osteoarthritis, or hypertension within 5-years, resulting in a large retrospective cohort for model development and internal validation (n=425228). First, we quantified the dependence between outcomes using unadjusted and adjusted ϕ coefficients. We then estimated a copula-based model to quantify the non-linear dependence between outcomes that can be used to derive risk estimates for each outcome, accounting for the observed dependence. Copula-based models are defined by univariate models for each outcome and a dependence function, specified by the parameter θ. Logistic regression was used for the univariate models and the Frank copula was selected as the dependence function. ResultsAll outcome pairs demonstrated statistically significant dependence that was reduced after adjusting for covariates. The copula-based model yielded statistically significant θ parameters in agreement with the adjusted and unadjusted ϕ coefficients. Our copula-based model can effectively be used to estimate trivariate probabilities. DiscussionQuantitative estimates of multimorbidity risk inform discussions between patients and their primary care practitioners around prevention in an effort to reduce the incidence of multimorbidity.

Based on M-Copula Reliability Analysis of Random Load Correlation

Load is one of the main causes of structural failure, and the correlation among loads would affect the evaluation results of structural performance. The purpose of this paper is to analyze the influence of the correlation among multiple loads on the structural reliability. In this paper, the nonparametric kernel density estimation (NKDE) method is used to estimate the probability density function (PDF) of related loads. In addition, the mixed copula (M-Copula) model is proposed, which combines Gumbel copula, Frank copula, Clayton copula, and weight coefficient, and the model parameters are fitted by MATLAB software to get the correlation of related loads. The reliability based on the related load combination is calculated according to the constructed model. After analyzing three numerical cases, the results show that the probability characteristics of NKDE estimation are very close to the actual conditions, and the reliability calculated by the M-Copula model is larger than those calculated by JCSS, Turkstra, and Gong methods. Using the M-Copula model for load correlation would avoid underestimating the reliability of the structure, which is conducive to structural economic development.

Parametric Vine Copula Construction for Flood Analysis for Kelantan River Basin in Malaysia

The multivariate approach of flood characteristics such as flood peak flow (P), volume (V), and duration (D) is much beneficial in recognizing the critical behaviour of flood episodes at a river basin scale. The incorporation of 2-dimensional copulas for establishing bivariate flood dependency frequently appears, but it could be more comprehensive if we focus all the three flood characteristic simultaneously. In such circumstances, incorporation of vine or Pair-Copula Construction (PCC) could produce a better approximation of joint probability density and much practical approach in the uncertainty analysis, in comparison with conventional trivariate copula distribution. This study demonstrated the efficacy of parametric vine copula in the modelling of trivariate flood characteristics for the Kelantan River basin in Malaysia. The D-vine tree structure is selected where the Gaussian and Frank copula is recognized for bivariate flood pairs (P-V) and (P-D) pairs in the first stage, using the maximum-pseudo-likelihood (MPL) estimation procedure. Similarly, the Gumbel copula is selected in the modelling of conditioned data obtained through the conditional distribution function of bivariate copulas selected in the previous stage based on the partial differentiation, also called h-function. Finally, the full density function of the 3-dimension structure is derived and compared with the observed flood characteristics. Furthermore, tail dependence properties and behaviour of D-vine copula are also investigated, which reveals for well capturing the general behaviour of Gaussian and Frank copula fitted to flood pair (P-V) and (V-D) and reproduces the overall flood correlation structure fairely well. Both the primary ‘OR’ and ‘AND’ joint return periods for trivariate flood events are estimated which pointing that ‘AND’ joint case produces higher return value than ‘OR’ case.