Portfolio selection based on asymmetric Laplace distribution, coherent risk measure, and expectation-maximization estimation

Variational Bayesian inference for interval regression with an asymmetric Laplace distribution

Mixture of linear regression models provide a popular treatment for modeling nonlinear regression relationship. The traditional estimation of mixture of regression models is based on Gaussian error assumption. It is well known that such assumption is sensitive to outliers and extreme values. To overcome this issue, a new class of finite mixture of quantile regressions (FMQR) is proposed in this article. Compared with the existing Gaussian mixture regression models, the proposed FMQR model can provide a complete specification on the conditional distribution of response variable for each component. From the likelihood point of view, the FMQR model is equivalent to the finite mixture of regression models based on errors following asymmetric Laplace distribution (ALD), which can be regarded as an extension to the traditional mixture of regression models with normal error terms. An EM algorithm is proposed to obtain the parameter estimates of the FMQR model by combining a hierarchical representation of the ALD. Finally, the iterated weighted least square estimation for each mixture component of the FMQR model is derived. Simulation studies are conducted to illustrate the finite sample performance of the estimation procedure. Analysis of an aphid data set is used to illustrate our methodologies.

Linear mixed-effects models are powerful tools for analyzing complex datasets with longitudinal data, but any violation of model assumptions, such as heteroscedasticity, implies invalidity of the model. We develop a new estimator based on the asymmetric Laplace distribution, which explains the heteroscedasticity between different groups of data, establishing its asymptotic properties. Our model focuses on the determination of the data dispersion. Using the permissive formalism of the asymmetric Laplace distribution, we demonstrate the asymptotic properties of a class of estimators defined by a generalized optimization problem inspired by maximum likelihood. More generally, this article highlights the theoretical opportunities offered by the asymmetric Laplace distribution in the framework of linear mixed models and longitudinal data.

We revisit mean‐risk portfolio selection in a one‐period financial market where risk is quantified by a positively homogeneous risk measure . We first show that under mild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact. However, unlike in classical mean‐variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation ‐arbitrage, and prove that it cannot be excluded—unless is as conservative as the worst‐case risk measure. After providing a primal characterization of ‐arbitrage, we focus our attention on coherent risk measures that admit a dual representation and give a necessary and sufficient dual characterization of ‐arbitrage. We show that the absence of ‐arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual representation of . A special case of our result shows that the market does not admit ‐arbitrage for Expected Shortfall at level if and only if there exists an EMM such that .

This paper proposes a new approach to analyze stock return asymmetry and quantiles. We also present a new scale mixture of uniform (SMU) representation for the asymmetric Laplace distribution (ALD). The use of the SMU for a probability distribution is a data augmentation technique that simplifies the Gibbs sampler of the Bayesian Markov chain Monte Carlo algorithms. We consider a stochastic volatility (SV) model with an ALD error distribution. With the SMU representation, the full conditional distribution for some parameters is shown to have closed form. It is also known that the ALD can be used to obtain the coefficients of quantile regression models. This paper also considers a quantile SV model by fixing the skew parameter of the ALD at specific quantile level. Simulation study shows that the proposed methodology works well in both SV and quantile SV models using Bayesian approach. In the empirical study, we analyze index returns of the stock markets in Australia, Japan, Hong Kong, Thailand, and the UK and study the effect of S&P 500 on these returns. The results show the significant return asymmetry in some markets and the influence by S&P 500 in all markets at all quantile levels. Copyright © 2014 John Wiley & Sons, Ltd.

This paper concentrates on solving the portfolio selection problem. It starts with an extension of the well-known optimization framework for Conditional Value-at-Risk (CVaR)-based portfolio selection problems [1, 2] to optimization over a more general class of risk measure - known as the class of Coherent Distortion Risk Measure (CDRM). The CDRM class of risk measures is the intersection of Coherent Risk Measure (CRM) and Distortion Risk Measure (DRM). It concludes with showing that many of the well-known risk measures are of special cases of the CDRM class what may facilitate to deal with the portfolio optimization problem.

This chapter looks at several methods for quantile and expectile regression that fall within the VGLM/VGAM framework. The following main categories are described: LMS-type quantile regression methods, the classical method (based on a loss or check function) and its connection with the asymmetric Laplace distributions (ALD), and expectile regression. A parallelism assumption for the ALD and ER allows for one solution to the quantile-crossing problem. The location parameter of the ALD can be modelled using link functions, therefore responses such as counts can be potentially handled. A second solution to the quantile-crossing problem is called the ‘onion’ method, which is likened to estimating successive layers of an onion. The VGAM package is used to illustrate the models.

ABSTRACTValue‐at‐risk (VaR) is a standard measure of market risk in financial markets. This paper proposes a novel, adaptive and efficient method to forecast both volatility and VaR. Extending existing exponential smoothing as well as GARCH formulations, the method is motivated from an asymmetric Laplace distribution, where skewness and heavy tails in return distributions, and their potentially time‐varying nature, are taken into account. The proposed volatility equation also involves novel time‐varying dynamics. Back‐testing results illustrate that the proposed method offers a viable, and more accurate, though conservative, improvement in forecasting VaR compared to a range of popular alternatives. Copyright © 2013 John Wiley & Sons, Ltd.

Bayesian analysis of a Tobit quantile regression model

A Dual Characterisation of Regulatory Arbitrage for Coherent Risk Measures

We establish the first axiomatic theory for diversification indices using six intuitive axioms: nonnegativity, location invariance, scale invariance, rationality, normalization, and continuity. The unique class of indices satisfying these axioms, called the diversification quotients (DQs), are defined based on a parametric family of risk measures. A further axiom of portfolio convexity pins down DQs based on coherent risk measures. The DQ has many attractive properties, and it can address several theoretical and practical limitations of existing indices. In particular, for the popular risk measures value at risk and expected shortfall, the corresponding DQ admits simple formulas, and it is efficient to optimize in portfolio selection. Moreover, it can properly capture tail heaviness and common shocks, which are neglected by traditional diversification indices. When illustrated with financial data, the DQ is intuitive to interpret, and its performance is competitive against other diversification indices. This paper was accepted by Manel Baucells, behavioral economics and decision analysis. Funding: X. Han is supported by the National Natural Science Foundation of China [Grants 12301604, 12371471, and 12471449). L. Lin is supported by the Hickman Scholarship from the Society of Actuaries. R. Wang is supported by the Natural Sciences and Engineering Research Council of Canada [Grants CRC-2022-00141 and RGPIN-2024-03728] and the Sun Life Research Fellowship. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.00513 .

Multifractality attributed to dual central limit-like convergence effects

Chapter 1 - Bayesian quantile regression with the asymmetric Laplace distribution

In this paper, we take a new perspective to describe the model uncertainty, and thus propose two new classes of risk measures under model uncertainty. To be precise, we use an auxiliary random variable to describe model uncertainty. By proposing new sets of axioms under model uncertainty, we axiomatically introduce and characterize conditional coherent and convex risk measures under a random environment, respectively. As examples, we also discuss the connections of the introduced conditional coherent risk measures under random environments with two existing risk measures. This paper mainly gives some theoretical results, and it is expected to make meaningful complement to the study of coherent and convex risk measures under model uncertainty.

Regulatory Arbitrage of Risk Measures