Exponentially Smoothing the Skewed Laplace Distribution for Value‐at‐Risk Forecasting

In this paper, portfolio selection problem is studied under Asymmetric Laplace Distribution(ALD) framework. Asymmetric Laplace distribution is able to capture tail-heaviness, skewness, andleptokurtosis observed in empirical financial data that cannot be explained by traditional Gaussiandistribution. Under Asymmetric Laplace distribution framework, portfolio selection methods basedon di erent risk measures are discussed. Moreover, we derived the Expectation-Maximization (EM)procedure for parameter estimation of Asymmetric Laplace distribution. Performance of the proposedmethod is illustrated via extensive simulation studies. Two real data examples are complemented toconfirm that the Asymmetric Laplace distribution based portfolio selection models are effcient.

Recently, researchers have been paying increasingly more attention to risk-adjusted performance of investment funds. Nevertheless, the asymmetry, the smoothness or the peakedness of returns have been poorly explored. In addition, most researches have been carried out in well developed, liquid and efficient markets.In this study, risk-adjusted performance of investment funds traded in the stock exchange of the Baltic countries was assessed using Sharpe and Sortino ratios as well as skewness and kurtosis. There are 38 investment funds traded in the stock exchange of the Baltic countries. However, most of these funds have been offered to public only recently. Thus, in order to have a research period as long as possible, only 14 funds were selected: 4 equity funds, 4 bond funds and 6 funds of funds. The period of investigation is 2008-2016.The research results revealed that only bond funds and one equity fund (ZPR Global Equity) were efficient in 2008-2016. This equity fund mostly invested in companies from the USA, where the economy, hit by the financial crisis, began to grow at a fast pace. All funds of funds were inefficient. This research was conducted in a relatively illiquid and inefficient market, but the investors’ behaviour during the economic downturn and slow growth period was the same as in efficient markets: they shifted their investments from equities to bonds. Consequently, additional demand increased bond market prices and returns.Despite the fact that only bond funds were efficient, their skewness and kurtosis were negative and greater in comparison with equity funds and funds of funds – the return distribution in bond had a greater downside. Moreover, bond funds were: 1) more overestimated by Sharpe ratio than equity funds and funds of funds; 2) their return distribution had a greater gain/loss than could be expected from a normal distribution; 3) their kurtosis were greatest in comparison with the kurtosis of equity funds and funds of funds. It should be emphasised that if the investor considered large negative skewness and very large positive excess kurtosis, he would come to a different conclusion. This is particularly important for intermediately and highly risk-averse investors.The analysis showed that almost all investment funds (with exception of one) had negative skewness (greater downside events), making the use of standard deviation and reliance upon normal distributions much less relevant for understanding the true risk of investment funds. 13 investment funds had leptokurtic return distributions, and 1 fund – platykurtic. Compared to a normal distribution, the latter tails are shorter and thinner. An asymmetrical return distribution and fat tails lead to the danger of underestimating risk and overestimating risk-adjusted performance with standard risk and performance measures. The correlation between the Sharpe and Sortino ratios is almost linear, which allows for concluding that investment funds can be equally ranked by both indicators. Contrarily, if some investment funds had negative and the others – positive skewness, the correlation would be lower (Cibulskiene et al., 2017) and these funds would be differently ranked by the Sharpe and Sortino ratios. This statement contradicts the results of Eling (2008), who found that investment funds could be equally ranked by different risk-adjusted performance indicators.

Variational Bayesian inference for interval regression with an asymmetric Laplace distribution

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.

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 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.

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.

Tail risk of hedge funds: an extreme value application

Quantile regression is a regression method that modelling a relationship between quantile of variable response and one or more variable predictors. Quantile regression has advantages that linear regression does not have; it is robust against outliers and can model heteroscedasticity data. The parameters of quantile regression can be estimated using the Bayesian method. The Bayesian method is a data analysis tool derived based on the Bayesian inference principle. Bayesian inference is the process of studying data analysis inductively with the Bayes theorem. To estimate regression parameters with Bayesian inference, it is necessary to find the posterior distribution of the regression parameters where the posterior distribution is proportional to the product of the prior distribution and its likelihood function. Since the calculation of the posterior distribution analytically is difficult to do if more parameters are estimated, the Markov Chain Monte Carlo (MCMC) method is proposed. The use of the Bayesian method in quantile regression has advantages, namely the use of MCMC has the advantages of obtaining sample parameter values from an unknown posterior distribution, using computationally efficient, and easy to implement. Yu and Moyeed (2001) introduced Bayesian quantile regression using the likelihood function of errors with an Asymmetric Laplace Distribution (ALD) and found that minimizing parameter estimates in quantile regression is the same as maximizing the likelihood function of errors with an Asymmetric Laplace Distribution (ALD). The method used to estimate quantile regression parameters is Gibbs sampling from the ALD, which is a combination of the exponential and normal distributions. To find the parameters of the regression model by sampling the posterior distribution found in this thesis. The results obtained from Gibbs sampling are a sample sequence of estimated parameters. After obtaining the sample sequences, the sample lines are averaged to obtain an estimated regression parameter. The case study in this thesis discusses the effect of risk factors from motor vehicle insurance customers on the size of claims submitted by customers.

For monitoring network traffic, there is an enormous cost in collecting, storing, and analyzing network traffic datasets. Data mining based network traffic analysis has a growing interest in the cyber security community, but is computationally expensive for finding correlations between attributes in massive network traffic datasets. To lower the cost and reduce computational complexity, it is desirable to perform feasible statistical processing on effective reduced datasets instead of on the original full datasets. Because of the dynamic behavior of network traffic, traffic traces exhibit mixtures of heavy tailed statistical distributions or overdispersion. Heavy tailed network traffic characterization and visualization are important and essential tasks to measure network performance for the Quality of Services. However, heavy tailed distributions are limited in their ability to characterize real-time network traffic due to the difficulty of parameter estimation. The Entropy-Based Heavy Tailed Distribution Transformation (EHTDT) was developed to convert the heavy tailed distribution into a transformed distribution to find the linear approximation. The EHTDT linearization has the advantage of being amenable to characterize and aggregate overdispersion of network traffic in realtime. Results of applying the EHTDT for innovative visual analytics to real network traffic data are presented.

Multifractality attributed to dual central limit-like convergence effects

Chapter 1 - Bayesian quantile regression with the asymmetric Laplace distribution

Energy prices are characterized by distributions that are asymmetric and that have heavier than Gaussian tails. Yet, many researchers continue to employ statistical methods that do not explicitly account for heavy tails and skewness in energy prices. In this chapter, we explicitly account for heavy tails and skewness with an application to electricity price risk using the Generalized Pareto Distribution and the Generalized Extreme Value Distribution. Specifically, we model value-at-risk (VaR) which is a widely-used measure of the maximum potential change in value of a portfolio of financial assets with a given probability over a given time horizon. VaR has become a standard measure of market risk and a common practice is to compute VaR by assuming that changes in value of the portfolio are conditionally normally distributed. However, assets returns usually come from heavy-tailed distributions, so computing VaR under the assumption of conditional normality can be an important source of error. We illustrate in our application to electric power, that VaR estimates based on extreme value theory models — in particular the generalized Pareto distribution — are more accurate than those produced by alternative models such as normality or historical simulation.

Introduction In deriving the Mean-Variance (M-V) rule and the Capital Asset Pricing Model (CAPM), one assumes explicitly or implicitly normal distributions of returns and risk aversion. With these assumptions, it emerges that the variance of the portfolio's return is the measure of the investment's risk. In Chapter 8, we saw that although normality of returns is statistically rejected, the economic loss from assuming normality when the distributions are actually not normal is relatively small. Moreover, in Chapter 4, we saw that even if the distributions of returns are not normal, the M-V rule can be employed as an approximation to expected utility, as long as risk aversion is assumed. Furthermore, it was shown empirically in Chapter 4 that the choices made by the M-V rule are almost fully correlated with the choices made by direct expected utility maximization for a wide set of risk-averse utility function. Considering the relatively small economic loss involved as a result of the normality assumption, in this chapter we assume that the M-V and the CAPM are derived based on this normality assumption, within the expected utility framework, rather than as an approximation to expected utility. However, even in this framework, as we saw in Chapters 9 and 10, the M-V and the CAPM are questionable for the following reasons: By the Prospect Theory (PT), people maximize a value function, which is defined on change of wealth (gains and losses) rather than on total wealth; by expected utility, the M-V rule and the CAPM are defined on total wealth. To derive the M-V rule and the CAPM, risk aversion is needed, whereas by PT, the value function has a risk-seeking segment. The risk- seeking segment of preference is steeper than the risk-aversion segment, implying loss aversion . Adopting the modified version of the PT, called Cumulative Prospect Theory (CPT), people employ cumulative decision weights . Thus, the cumulative distribution of returns is transformed, and in calculating the expected value (utility), the transformed distribution rather than the original distribution is employed. This seems to be a “deathblow” to the M-V rule and the CAPM because even if the original distribution of returns is normal, generally the transformed distribution is not. Moreover, although the normal distribution is symmetric, implying that skewness is equal to zero, the transformed distribution may be skewed, thus relying only on the mean, and variance may be misleading because there is ample evidence that investors like positive skewness and dislike negative skewness. For the preceding reasons, it would seem that the CPT paradigm and the M-V rule plus the CAPM paradigm cannot coexist. Moreover, the fact that Markowitz and Sharpe won the Noble Prize in Economics in 1990 for developing the M-V rule and the CAPM, and Kahneman won the Noble Prize in Economics in 2002 for introducing the PT, emphasizes the seeming contradiction between these two paradigms and makes it even more uncomfortable for researchers. If one adopts PT, it seems the M-V and the CAPM should be abandoned. If one employs the M-V and the CAPM, the important evidence regarding subjects’ choices, as clearly shown in PT experiments, are ignored.

We investigate the relation between various alternative risk measures and future daily returns using a sample of firms over the 1988-2009 time period. Previous research indicates that returns are not normally distributed and that investors seem to care more about downside risk than total risk. Motivated by these findings and mixed empirical evidence supporting theoretical positive risk-return relationship, we model the relation between future returns and risk measures and investigate the following questions: (1) Are investors compensated for total risk and/or asymmetric measures of risk? (2) How does the degree of risk aversion in the lower tail of the return distribution impact the predictability of future returns? (3) Is upside risk or downside risk a better predictor of future returns? We find that, although investors seem to be compensated for total risk, measures of downside risk, such as the lower partial moment, are better at explaining future returns. Further, when comparing downside risk to upside risk, we find that investors are more concerned about downside risk. That is, downside risk is a better predictor of future returns. Our results are robust to the addition of traditional control variables, including size, book-to-market ratio of equity (B/M), leverage, and market risk measures, including beta, downside beta and co-skewness. Our findings are an important contribution to the literature as we document a positive risk-return relationship, using both total and asymmetric measures of risk.