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The purpose of this paper is to present the method for efficient computation of risk measures using Fourier transform technique. Another objective is to demonstrate that this technique enables an efficient computation of risk measures beyond value-at-risk and expected shortfall. Finally, this paper highlights the importance of validating assumptions behind the risk model and describes its application in the affine model framework.

The method proposed is based on Fourier transform methods for computing risk measures. The authors obtain the loss distribution by fitting a cubic spline through the points where Fourier inversion of the characteristic function is applied. From the loss distribution, the authors calculate value-at-risk and expected shortfall. As for the calculation of the entropic value-at-risk, it involves the moment generating function which is closely related to the characteristic function. The expectile risk measure is calculated based on call and put option prices which are available in a semi-closed form by Fourier inversion of the characteristic function. We also consider mean loss, standard deviation and semivariance which are calculated in a similar manner.

The study offers practical insights into the efficient computation of risk measures as well as validation of the risk models. It also provides a detailed description of algorithms to compute each of the risk measures considered. While the main focus of the paper is on portfolio-level risk metrics, all algorithms are also applicable to single instruments.

The algorithms presented in this paper require little computational effort which makes them very suitable for real-world applications. In addition, the mathematical setup adopted in this paper provides a natural framework for risk model validation which makes the approach presented in this paper particularly appealing in practice.

This is the first study to consider the computation of entropic value-at-risk, semivariance as well as expectile risk measure using Fourier transform method.

This paper aims to propose two portfolio selection models with hesitant value-at-risk (HVaR) – HVaR fuzzy portfolio selection model (HVaR-FPSM) and HVaR-score fuzzy portfolio selection model (HVaR-S-FPSM) – to help investors solve the problem that how bad a portfolio can be under probabilistic hesitant fuzzy environment.

It is strictly proved that the higher the probability threshold, the higher the HVaR in HVaR-S-FPSM. Numerical examples and a case study are used to illustrate the steps of building the proposed models and the importance of the HVaR and score constraint. In case study, the authors conduct a sensitivity analysis and compare the proposed models with decision-making models and hesitant fuzzy portfolio models.

The score constraint can make sure that the portfolio selected is profitable, but will not cause the HVaR to decrease dramatically. The investment proportions of stocks are mainly affected by their HVaRs, which is consistent with the fact that the stock having good performance is usually desirable in portfolio selection. The HVaR-S-FPSM can find portfolios with higher HVaR than each single stock and has little sacrifice of extreme returns.

This paper fulfills a need to construct portfolio selection models with HVaR under probabilistic hesitant fuzzy environment. As a downside risk, the HVaR is more consistent with investors’ intuitions about risks. Moreover, the score constraint makes sure that undesirable portfolios will not be selected.

This paper aims to characterize and identify the existing research on risk management in the export development of high-tech products.

The authors conducted a systematic mapping study to identify and analyze related literature. The authors identified 96 primary studies, dated from 2000 to 2018 and classified them with respect to research focus, types of research and contribution.

A total of 32 studies were identified and mapped, synthesizing the available evidence on risk management in the export development of high-tech products. “Currency risk” with 13 articles is the dominant research focus. Regarding the research type, “Solution proposal” is the most frequently used research type. “Case study”, “Regression analysis” and “Survey”, respectively, were the most used research methods. However, “FANNIS”, “FAHP” and “Discussion paper” were used less often. “Solution proposal” was the most common research type between 2000 and 2018. Further, the number of publications has declined between 2010 and 2012.

This mapping study provides the first systematic exploration of the state-of-art on risk management research in the export development of high-tech products. The existing body of knowledge is limited to a few high-quality studies.

The purpose of this paper is to create a universal (asset-class-independent) portfolio risk index for a global private investor.

The authors first discuss existing risk measures and desirable properties of a risk index. Then, they construct a universal (asset-class-independent) portfolio risk measure by modifying Financial Turbulence of Kritzman and Li (2010). Finally, the average portfolio of a representative global private investor is determined, and, by applying the new portfolio risk measure, they derive the Private investor Risk IndeX.

The authors show that this index exhibits commonly expected properties of risk indices, such as proper reaction to well-known historical market events, persistence in time and forecasting power for both risk and returns to risk.

A dynamic asset allocation example illustrates one potential practical application for global private investors.

As of now, a risk index reflecting the overall risk of a typical multi-asset-class portfolio of global private investors does not seem to exist.

The idea of using stochastic control methods for theoretical studies of portfolio management has long been standard, with maximum expected utility criteria commonly being used. But in recent years a new kind of criterion, the risk sensitive criterion, has emerged from the control theory literature and been applied to portfolio management. This paper studies various economic properties of this criterion for portfolio management, thereby providing justification for its theoretical and practical use. In particular, it is shown that the risk sensitive criterion amounts to maximizing a portfolio's risk adjusted growth rate. In other words, it is essentially the same as what is commonly done in practice: find the best trade‐off between a portfolio's average return and its average volatility.

Knowledge of the dependence structure between financial assets is crucial to improve the performance in financial risk management. It is known that the copula completely summarizes the dependence structure among multiple variables. We propose a multivariate exponential series estimator (ESE) to estimate copula densities nonparametrically. The ESE has an appealing information-theoretic interpretation and attains the optimal rate of convergence for nonparametric density estimations in Stone (1982). More importantly, it overcomes the boundary bias of conventional nonparametric copula estimators. Our extensive Monte Carlo studies show the proposed estimator outperforms the kernel and the log-spline estimators in copula estimation. It also demonstrates that two-step density estimation through an ESE copula often outperforms direct estimation of joint densities. Finally, the ESE copula provides superior estimates of tail dependence compared to the empirical tail index coefficient. An empirical examination of the Asian financial markets using the proposed method is provided.

It has long been challenged that the distributions of empirical returns do not follow the log-normal distribution upon which many celebrated results of finance are based including the Black–Scholes Option-Pricing model. Borland (2002) succeeds in obtaining alternate closed form solutions for European options based on Tsallis distribution, which allow for statistical feedback as a model of the underlying stock returns. Motivated by this, we simulate two distinct time series based on initial data from NIFTY daily close values, one based on the Gaussian return distribution and the other on non-Gaussian distribution. Using techniques of non-linear dynamics, we examine the underlying dynamic characteristics of both the simulated time series and compare them with the characteristics of actual data. Our findings give a definite edge to the non-Gaussian model over the Gaussian one.

The purpose of this paper is to identify how project portfolio selection (PPS) methods have evolved and which approaches are more suitable for radical innovation projects. This paper addressed the following research question: how have the selection approaches evolved to better fit within radical innovation conditions? The current literature offers a number of selection approaches with different and, in some cases, conflicting nature. Therefore, there is a lack of understanding regarding when and how to use these approaches in order to select a specific type of innovation projects (from incremental to more radical ones).

Given the nature of the research question, the authors perform a systematic literature review method and analyze 48 portfolio selection approaches. The authors then classified and characterized these articles in order to identify techniques, tools, required data and types of examined projects, among other aspects.

The authors identify four key features related to the selection of radical innovation projects: dynamism, interdependency management, uncertainty treatment and required input data. Based on the content analysis, the authors identified that approaches based on different sources and nature of data are more appropriated for uncertain conditions, such as behavioral methods, information gap theory, real options and integrated approaches.

The research provides a comprehensive framework about PPS methods and how they have been evolving over time. This portfolio selection framework considers the particular aspects of incremental and radical innovation projects. The authors hope that the framework contributes to reinvigorating the literature on selection approaches for innovation projects.