The Distribution of Stock Returns When the Market Is Up

Abstract

For some investments, the relation between stock returns and the market proxy is conventionally described by a linear regression model with the normality assumption. This paper derives the distribution of stock returns for a security in an upgrade (or downgrade) market with the assumption that the log stock returns of the market proxy follow a mixture of normal distributions. We discuss MLE and the method of moment estimation for parameters involved in the model. An analysis of stock data in Johannesburg Stock Exchange is included to illustrate the model. This note explains the phenomenon in financial analysis regarding the shape of the distribution of long-run stock returns limited on an upgrade or downgrade market index.     

Influence diagnostics in the capital asset pricing model under elliptical distributions

In this paper we consider the Capital Asset Pricing Model under Elliptical (symmetric) Distributions. This class of distributions, which contains the normal distribution, t, contaminated normal and power exponential, among others, offers a more flexible framework for modelling asset prices or returns. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators, the local influence method was implemented. The results are illustrated by using a set of shares from companies who trade in the Chilean Stock Market. Our main conclusion is that symmetric distributions having heavier tails than those of the normal distribution, especially the t distribution with small degrees of freedom, show a better fit and allow the reduction of the influence of atypical returns in the maximum likelihood estimators.     

A semi-parametric method for estimating the beta coefficients of the hidden two-sided asset return jumps

ABSTRACT

We introduce a new methodology for estimating the parameters of a two-sided jump model, which aims at decomposing the daily stock return evolution into (unobservable) positive and negative jumps as well as Brownian noise. The parameters of interest are the jump beta coefficients which measure the influence of the market jumps on the stock returns, and are latent components. For this purpose, at first we use the Variance Gamma (VG) distribution which is frequently used in modeling financial time series and leads to the revelation of the hidden market jumps' distributions. Then, our method is based on the central moments of the stock returns for estimating the parameters of the model. It is proved that the proposed method provides always a solution in terms of the jump beta coefficients. We thus achieve a semi-parametric fit to the empirical data. The methodology itself serves as a criterion to test the fit of any sets of parameters to the empirical returns. The analysis is applied to NASDAQ and Google returns during the 2006–2008 period.     

Modeling stock markets’ volatility using GARCH models with Normal,Student’s <Emphasis Type="Italic">t</Emphasis> and stable Paretian distributions

As GARCH models and stable Paretian distributions have been revisited in the recent past with the papers of Hansen and Lunde (J Appl Econom 20: 873–889, 2005) and Bidarkota and McCulloch (Quant Finance 4: 256–265, 2004), respectively, in this paper we discuss alternative conditional distributional models for the daily returns of the US, German and Portuguese main stock market indexes, considering ARMA-GARCH models driven by Normal, Student’s t and stable Paretian distributed innovations. We find that a GARCH model with stable Paretian innovations fits returns clearly better than the more popular Normal distribution and slightly better than the Student’s t distribution. However, the Student’s t outperforms the Normal and stable Paretian distributions when the out-of-sample density forecasts are considered.     

Influential observations in GARCH models

This paper examines local influence assessment in generalized autoregressive conditional heteroscesdasticity models with Gaussian and Student-t errors, where influence is examined via the likelihood displacement. The analysis of local influence is discussed under three perturbation schemes: data perturbation, innovative model perturbation and additive model perturbation. For each case, expressions for slope and curvature diagnostics are derived. Monte Carlo experiments are presented to determine the threshold values for locating influential observations. The empirical study of daily returns of the New York Stock Exchange composite index shows that local influence analysis is a useful technique for detecting influential observations; most of the observations detected as influential are associated with historical shocks in the market. Finally, based on this empirical study and the analysis of simulated data, some advice is given on how to use the discussed methodology.     

上海银行间同业拆放利率ES风险度量研究

随着基准利率地位的不断变化,上海银行间同行业拆放利率（SHIBOR）市场风险管理对金融机构将会越来越重要。然而同正态分布相比而言,SHIBOR收益率变量具有偏态等特征。提出采用广义双曲线分布来拟合收益率序列。为了解决参数估计难的问题,提出利用强有力的EM算法对于解决像包含Bessel函数这样复杂、具有大量局部最优解的优化问题,具有很现实的意义,同时利用蒙特卡罗模拟方法来计算广义双曲线分布下的VaR值、ES值,最后讨论广义双曲线分布在SHIBOR市场风险度量中的应用。     

Estimation and diagnostic for skew-normal partially linear models

Partially linear models (PLMs) are an important tool in modelling economic and biometric data and are considered as a flexible generalization of the linear model by including a nonparametric component of some covariate into the linear predictor. Usually, the error component is assumed to follow a normal distribution. However, the theory and application (through simulation or experimentation) often generate a great amount of data sets that are skewed. The objective of this paper is to extend the PLMs allowing the errors to follow a skew-normal distribution [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178], increasing the flexibility of the model. In particular, we develop the expectation-maximization (EM) algorithm for linear regression models and diagnostic analysis via local influence as well as generalized leverage, following [H. Zhu and S. Lee, Local influence for incomplete-data models, J. R. Stat. Soc. Ser. B 63 (2001), pp. 111–126]. A simulation study is also conducted to evaluate the efficiency of the EM algorithm. Finally, a suitable transformation is applied in a data set on ragweed pollen concentration in order to fit PLMs under asymmetric distributions. An illustrative comparison is performed between normal and skew-normal errors.     

Power-law scaling behavior analysis of financial time series model by voter interacting dynamic system

We investigate the power-law scaling behaviors of returns for a financial price process which is developed by the voter interacting dynamic system in comparison with the real financial market index (Shanghai Composite Index). The voter system is a continuous time Markov process, which originally represents a voter's attitude on a particular topic, that is, voters reconsider their opinions at times distributed according to independent exponential random variables. In this paper, the detrended fluctuation analysis method is employed to explore the long range power-law correlations of return time series for different values of parameters in the financial model. The findings show no indication or very weak long-range power-law correlations for the simulated returns but strong long-range dependence for the absolute returns. The multiplier distribution is studied to demonstrate directly the existence of scale invariance in the actual data of the Shanghai Stock Exchange and the simulation data of the model by comparison. Moreover, the Zipf analysis is applied to investigate the statistical behaviors of frequency functions and the distributions of the returns. By a comparative study, the simulation data for our constructed price model exhibits very similar behaviors to the real stock index, this indicates somewhat rationality of our model to the market application.     

Bayesian modeling of autoregressive partial linear models with scale mixture of normal errors

Normality and independence of error terms are typical assumptions for partial linear models. However, these assumptions may be unrealistic in many fields, such as economics, finance and biostatistics. In this paper, a Bayesian analysis for partial linear model with first-order autoregressive errors belonging to the class of the scale mixtures of normal distributions is studied in detail. The proposed model provides a useful generalization of the symmetrical linear regression model with independent errors, since the distribution of the error term covers both correlated and thick-tailed distributions, and has a convenient hierarchical representation allowing easy implementation of a Markov chain Monte Carlo scheme. In order to examine the robustness of the model against outlying and influential observations, a Bayesian case deletion influence diagnostics based on the Kullback–Leibler (K–L) divergence is presented. The proposed method is applied to monthly and daily returns of two Chilean companies.     

Quantile Plots,Partial Orders,and Financial Risk

Quantile-quantile plots are most commonly used to compare the shapes of distributions, but they may also be used in conjunction with partial orders on distributions to compare the level and dispersion of distributions that have different shapes. We discuss several easily recognized patterns in quantile-quantile plots that suffice to demonstrate that one distribution is smaller than another in terms of each of several partial orders. We illustrate with financial applications, proposing a quantile plot for comparing the risks and returns of portfolios of investments. As competing portfolios have distributions that differ in level, dispersion, and shape, it is not sufficient to compare portfolios using measures of location and dispersion, such as expected returns and variances; however, quantile plots, with suitable scaling, do aid in such comparisons. In two plots, we compare specific portfolios to the stock market as a whole, finding these portfolios to have higher returns, greater risks or dispersion, thicker tails than their greater dispersion alone would justify. Nonetheless, investors in these risky portfolios are more than adequately compensated for the risks undertaken.     

Box–Cox realized asymmetric stochastic volatility models with generalized Student's t-error distributions

This study proposes a class of non-linear realized stochastic volatility (SV) model by applying the Box–Cox (BC) transformation, instead of the logarithmic transformation, to the realized estimator. The non-Gaussian distributions such as Student's t, non-central Student's t, and generalized hyperbolic skew Student's t-distributions are applied to accommodate heavy-tailedness and skewness in returns. The proposed models are fitted to daily returns and realized kernel of six stocks: SP500, FTSE100, Nikkei225, Nasdaq100, DAX, and DJIA using an Markov chain Monte Carlo Bayesian method, in which the Hamiltonian Monte Carlo (HMC) algorithm updates BC parameter and the Riemann manifold HMC algorithm updates latent variables and other parameters that are unable to be sampled directly. Empirical studies provide evidence against both the logarithmic transformation and raw versions of realized SV model.     

Bayesian inference for sinh-normal/independent nonlinear regression models

Sinh-normal/independent distributions are a class of symmetric heavy-tailed distributions that include the sinh-normal distribution as a special case, which has been used extensively in Birnbaum–Saunders regression models. Here, we explore the use of Markov Chain Monte Carlo methods to develop a Bayesian analysis in nonlinear regression models when Sinh-normal/independent distributions are assumed for the random errors term, and it provides a robust alternative to the sinh-normal nonlinear regression model. Bayesian mechanisms for parameter estimation, residual analysis and influence diagnostics are then developed, which extend the results of Farias and Lemonte [Bayesian inference for the Birnbaum-Saunders nonlinear regression model, Stat. Methods Appl. 20 (2011), pp. 423-438] who used the Sinh-normal/independent distributions with known scale parameter. Some special cases, based on the sinh-Student-t (sinh-St), sinh-slash (sinh-SL) and sinh-contaminated normal (sinh-CN) distributions are discussed in detail. Two real datasets are finally analyzed to illustrate the developed procedures.     

Predictor distribution and forecast accuracy of threshold models

In the present paper the predictor distribution of a SETAR (Self Exciting Threshold Autoregressive) model (Tong and Lim, 1980) has been investigated when the lead time is greater than the threshold delay.After a brief presentation of the model under study, some relevant aspects of the density forecasts are shown highlighting how they can be used to generate more accurate predictions and to estimate an approximation of the probability density function of the SETAR predictors. The performances of competing predictors have been evaluated through a simulation study and an application to financial market data of the daily Nikkey 300 stock market returns.     

A robust multivariate measurement error model with skew-normal/independent distributions and Bayesian MCMC implementation

Skew-normal/independent distributions are a class of asymmetric thick-tailed distributions that include the skew-normal distribution as a special case. In this paper, we explore the use of Markov Chain Monte Carlo (MCMC) methods to develop a Bayesian analysis in multivariate measurement errors models. We propose the use of skew-normal/independent distributions to model the unobserved value of the covariates (latent variable) and symmetric normal/independent distributions for the random errors term, providing an appealing robust alternative to the usual symmetric process in multivariate measurement errors models. Among the distributions that belong to this class of distributions, we examine univariate and multivariate versions of the skew-normal, skew-t, skew-slash and skew-contaminated normal distributions. The results and methods are applied to a real data set.     

BL-GARCH models with elliptical distributed innovations

In this work, we discuss the class of bilinear GARCH (BL-GARCH) models that are capable of capturing simultaneously two key properties of non-linear time series: volatility clustering and leverage effects. It has often been observed that the marginal distributions of such time series have heavy tails; thus we examine the BL-GARCH model in a general setting under some non-normal distributions. We investigate some probabilistic properties of this model and we conduct a Monte Carlo experiment to evaluate the small-sample performance of the maximum likelihood estimation (MLE) methodology for various models. Finally, within-sample estimation properties were studied using S&P 500 daily returns, when the features of interest manifest as volatility clustering and leverage effects. The main results suggest that the Student-t BL-GARCH seems highly appropriate to describe the S&P 500 daily returns.     

Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

Statistical diagnostics for nonlinear regression models based on scale mixtures of skew-normal distributions

The purpose of this paper is to develop diagnostics analysis for nonlinear regression models (NLMs) under scale mixtures of skew-normal (SMSN) distributions introduced by Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124]. This novel class of models provides a useful generalization of the symmetrical NLM [Vanegas LH, Cysneiros FJA. Assessment of diagnostic procedures in symmetrical nonlinear regression models. Comput. Statist. Data Anal. 2010;54:1002–1016] since the random terms distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as the skew-t, skew-slash, skew-contaminated normal distributions, among others. Motivated by the results given in Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124], we presented a score test for testing the homogeneity of the scale parameter and its properties are investigated through Monte Carlo simulations studies. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. The newly developed procedures are illustrated considering a real data set.     

Analysis of mutual funds’ management styles: a modeling,ranking and visualizing approach

A method to rank mutual funds according to their investment style measured with respect to the returns of a reference portfolio (benchmark) is introduced. It is based on a style analysis model estimating a mutual fund portfolio composition as well as the benchmark one. Starting from such compositions, it computes a proximity measure based on the L ₁ or L ₂ norm to assess the similarity between each mutual fund portfolio returns and the benchmark returns as well as between the returns of each benchmark constituent and that of the corresponding mutual fund constituent. To this purpose the mean integrated absolute error and the mean integrated squared error are computed to derive both a global ranking of mutual fund management styles and partial rankings expressing the over- (under-) weighting of each portfolio constituent. A visual inspection of the results emphasizing main differences in management styles is provided, using a parallel coordinates plot. Since a modeling, a ranking and a visualizing approach are integrated, the method is named MoRaViA. From the practitioners’ point of view, it allows the identification of a specific management style for each mutual fund, discriminating active management funds from passive management ones. To evaluate the effectiveness of MoRaViA, many sets of artificial portfolios are generated and an application on a set of equity funds operating in the European market is presented.     

The Use of Ratios to Compare Volatilities and VaRs

The riskiness of two investments can be compared by looking at the ratio of the respective Value-at-Risk's (VaRs) or the ratio of volatilities. The exact distribution of the ratio of two volatilities calculated from normal observations and an asymptotic confidence interval for the ratio of two VaRs is derived. A simulation study shows good coverage rates for ratios of VaRs calculated from observations from distributions commonly used to model logarithmic returns.     

Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.