Asymmetric and leptokurtic distribution for heteroscedastic asset returns: The SU-normal distribution

This paper proposes the S_U-normal distribution to describe non-normality features embedded in financial time series, such as: asymmetry and fat tails. Applying the S_U-normal distribution to the estimation of univariate and multivariate GARCH models, we test its validity in capturing asymmetry and excess kurtosis of heteroscedastic asset returns. We find that the S_U-normal distribution outperforms the normal and Student-t distributions for describing both the entire shape of the conditional distribution and the extreme tail shape of daily exchange rates and stock returns. The goodness-of-fit (GoF) results indicate that the skewness and excess kurtosis are better captured by the S_U-normal distribution. The exceeding ratio (ER) test results indicate that the S_U-normal is superior to the normal and Student-t distributions, which consistently underestimate both the lower and upper extreme tails, and tend to overestimate the lower tail in general.     

Alternative statistical distributions for estimating value-at-risk: theory and evidence

A number of applications presume that asset returns are normally distributed, even though they are widely known to be skewed leptokurtic and fat-tailed and excess kurtosis. This leads to the underestimation or overestimation of the true value-at-risk (VaR). This study utilizes a composite trapezoid rule, a numerical integral method, for estimating quantiles on the skewed generalized t distribution (SGT) which permits returns innovation to flexibly treat skewness, leptokurtosis and fat tails. Daily spot prices of the thirteen stock indices in North America, Europe and Asia provide data for examining the one-day-ahead VaR forecasting performance of the GARCH model with normal, student??s t and SGT distributions. Empirical results indicate that the SGT provides a good fit to the empirical distribution of the log-returns followed by student??s t and normal distributions. Moreover, for all confidence levels, all models tend to underestimate real market risk. Furthermore, the GARCH-based model, with SGT distributional setting, generates the most conservative VaR forecasts followed by student??s t and normal distributions for a long position. Consequently, it appears reasonable to conclude that, from the viewpoint of accuracy, the influence of both skewness and fat-tails effects (SGT) is more important than only the effect of fat-tails (student??s t) on VaR estimates in stock markets for a long position.     

Credit contagion in the presence of non-normal shocks

We generalize existing structural credit risk models that account for contagion effects across economic sectors, to capture the impact of neglected skewness and excess kurtosis in the asset return process, on the shape of the credit loss distribution. We specify Skew-Normal and Skew-Student t densities for the underlying asset return process and estimate the derived credit loss density using sector default rates based on proprietary data from the Central Bank of Mexico for six firm sectors. We show that, out of the six sectors analyzed, there is a significant contagion effect in ‘Commerce’, ‘Services’ and ‘Transport’. Moreover, we show that the non-Gaussian modelling of the common factor provides a better characterization than its Gaussian counterpart for the ‘Services’ sector. This result has a significant impact on the shape and the corresponding Value-at-Risk levels of the ‘Services’ credit loss distribution. In this context, traditional Basel and vendor-based credit risk models are inadequate as these do not consider the individual or the joint impact of contagion and non-Gaussian asset returns.     

Approximation methods for multiple period Value at Risk and Expected Shortfall prediction

In this paper, we are interested in predicting multiple period Value at Risk and Expected Shortfall based on the so-called iterating approach. In general, the properties of the conditional distribution of multiple period returns do not follow easily from the one-period data generating process, rendering this a non-trivial task. We outline a framework that forms the basis for setting approximations and study four different approaches. Their performance is evaluated by means of extensive Monte Carlo simulations based on an asymmetric GARCH model, implying conditional skewness and excess kurtosis in the multiple period returns. This simulation-based approach was the best one, closely followed by that of assuming a skewed t-distribution for the multiple period returns. The approach based on a Gram–Charlier expansion was not able to cope with the implied non-normality, while the so-called Root-k approach performed poorly. In addition, we outline how the delta-method may be used to quantify the estimation error in the predictors and in the Monte Carlo study we found that it performed well. In an empirical illustration, we computed 10-day Value at Risk’s and Expected Shortfall for Brent Crude Oil, the EUR/USD exchange rate and the S&P 500 index. The Root-k approach clearly performed the worst and the other approaches performed quite similarly, with the simulation based approach and the one based on the skewed t-distribution somewhat better than the one based on the Gram–Charlier expansion.     

Models of asset returns: changes of pattern from high to low event frequency

In this paper we have analysed asset returns of the New York Stock Exchange and the Helsinki Stock Exchange using various time-independent models such as normal, general stable, truncated Lévy, mixed diffusion-jump, compound normal, Student t distribution and power exponential distribution and the time-dependent GARCH(1, 1) model with Gaussian and Student t distributed innovations. In order to study changes of pattern at different event horizons, as well as changes of pattern over time for a given event horizon, we have analysed high-frequency or short-horizon intraday returns up from 20 s intervals to a full trading day, medium-frequency or medium-horizon daily returns and low-frequency or long-horizon returns with holding period up to 30 days. As for changes of pattern over time, we found that for medium-frequency returns there are relatively long periods of business-as-usual when the return-generating process is well-described by a normal distribution. We also found periods of ferment, when the volatility grows and complex time dependences tend to emerge, but the known time dependences cannot explain the variability of the distribution. Such changes of pattern are also observed for high-frequency or short-horizon returns, with the exception that the return-generating process never becomes normal. It also turned out that the time dependence of the distribution shape is far more prominent at high frequencies or short horizons than the time dependence of the variance. For long-horizon or low-frequency returns, the distribution is found to converge towards a normal distribution with the time dependences vanishing after a few days.     

Zur Risikoanalyse von Hedgefonds: Ein datenanalytischer Ansatz

Hedge fund returns have a number of specific features compared to traditional investments which result in problems when applying traditional methods of risk analysis (Markowitz portfolio selection theory, Sharpe Ratio, value at risk calculation based on normal returns). These problems have to be considered adequately by insurance companies when constructing internal risk models and performing risk management for hedge funds in their investment.The present paper has its focus on the departure of hedge fund returns from the normality hypothesis, especially with respect to the statistical quantities skewness and kurtosis (fat tail problem). A statistical analysis of hedge fund index returns gives evidence that the majority of hedge fund returns show substantial departures from normality. In addition, the analysis shows that hedge fund returns are adequately represented by the family of GH-distributions developed in exploratory data analysis. Following this result a risk analysis of hedge fund strategies is performed on the basis of the GH-value at risk.     

Testing for persistence in stock returns with GARCH-stable shocks

We investigate persistence in CRSP monthly excess stock returns, using a state space model with stable disturbances. The non-Gaussian state space model with volatility persistence is estimated by maximum likelihood, using the optimal filtering algorithm given by Sorenson and Alspach (1971 Automatica 7 465–79). The conditional distribution has a stable α of 1.89, and normality is strongly rejected even after accounting for GARCH. However, stock returns do not contain a significant mean-reverting component. The optimal predictor is the unconditional expectation of the series, which we estimate to be 9.8% per annum.     

SKEWNESS AND KURTOSIS IN JAPANESE EQUITY RETURNS: EMPIRICAL EVIDENCE

In this paper, the distribution of equity returns on the Tokyo Stock Exchange is examined from 1965 to 1984, and significant and persistent skewness and kurtosis are found. The deviation of security returns from normality declines with increasing portfolio size and appears to be greater than the non-normality evidenced in U.S. security returns. Further, these deviations from normality persist even after controlling for January and firm size effects.     

The shifting dependence dynamics between the G7 stock markets

The growing interdependence between financial markets has attracted special attention from academic researchers and finance practitioners for the purpose of optimal portfolio design and contagion analysis. This article develops a tractable regime-switching version of the copula functions to model the intermarkets linkages during turmoil and normal periods, while taking into account structural changes. More precisely, Markov regime-switching C-vine and D-vine decompositions of the Student’s t copula are proposed and applied to returns on diversified portfolios of stocks, represented by the G7 stock market indices. The empirical results show evidence of regime shifts in the dependence structure with high contagion risk during crisis periods. Moreover, both the C- and D-vines highly outperform the multivariate Student’s t copula, which suggests that the shock transmission path is as important as the dependence itself, and is better detected with a vine copula decomposition.     

Accurate minimum capital risk requirements: A comparison of several approaches

In this paper we estimate, for several investment horizons, minimum capital risk requirements for short and long positions, using the unconditional distribution of three daily indexes futures returns and a set of short and long memory stochastic volatility and GARCH-type models. We consider the possibility that errors follow a t-Student distribution in order to capture the kurtosis of the returns’ series. The results suggest that accurate modelling of extreme observations obtained for long and short trading investment positions is possible with an autoregressive stochastic volatility model. Moreover, modelling futures returns with a long memory stochastic volatility model produces, in general, excessive volatility persistence, and consequently, leads to large minimum capital risk requirement estimates. Finally, the models’ predictive ability is assessed with the help of out-of-sample conditional tests.     

Tests of the multiperiod two-parameter model

Although investors face multiperiod decision problems, there are conditions under which the results of the one-period two-parameter model apply period by period. In addition to the assumptions made in the development of the two-parameter model itself (a perfect capital market, investor risk aversion, and normal distributions of one-period portfolio returns), the critical assumption in a multiperiod context is that, for any t, returns on portfolio assets from t?1 to t are independent of stochastic elements of the state-of-the-world at time t that affect investor tastes for given levels of wealth to be obtained at t.One such element of the state-of-the-world is the nature of investment opportunities to be available at t. For example, if the level of expected returns on investment portfolios to be available at time t is uncertain at time t?1, and if the returns from t?1 to t on some investment assets are more strongly related to the level of expected returns at t than returns on other assets, then the former assets are better vehicles for hedging against the level of expected returns at t. This can affect the demands for assets and their prices in such a way that the simple results of the one-period two-parameter model do not hold.The empirical tests of this paper reveal no evidence of measurable relationships between the returns on portfolio assets from t?1 to t and the level of expected returns to be available at t. Indeed, in our opinion there is no reliable evidence that the level of expected returns changed during the 1953–1972 period.     

Elliptical tempered stable distribution

Elliptical distributions are useful for modelling multivariate data, multivariate normal and Student t distributions being two special classes. In this paper, we provide a definition for the elliptical tempered stable (ETS) distribution based on its characteristic function, which involves a unique spectral measure. This definition provides a framework for creating a connection between the infinite divisible distribution (in particular the ETS distribution) with fractional calculus. In addition, a definition for the ETS copula is discussed. A simulation study shows the accuracy of this definition, in comparison to the normal copula for measuring the dependency of data. An empirical study of stock market index returns for 20 countries shows the usefulness of the theoretical results.     

Forecasting VaR using analytic higher moments for GARCH processes

It is widely accepted that some of the most accurate Value-at-Risk (VaR) estimates are based on an appropriately specified GARCH process. But when the forecast horizon is greater than the frequency of the GARCH model, such predictions have typically required time-consuming simulations of the aggregated returns distributions. This paper shows that fast, quasi-analytic GARCH VaR calculations can be based on new formulae for the first four moments of aggregated GARCH returns. Our extensive empirical study compares the Cornish–Fisher expansion with the Johnson SU distribution for fitting distributions to analytic moments of normal and Student t, symmetric and asymmetric (GJR) GARCH processes to returns data on different financial assets, for the purpose of deriving accurate GARCH VaR forecasts over multiple horizons and significance levels.     

The role of autoregressive conditional skewness and kurtosis in the estimation of conditional VaR

This paper investigates the role of high-order moments in the estimation of conditional value at risk (VaR). We use the skewed generalized t distribution (SGT) with time-varying parameters to provide an accurate characterization of the tails of the standardized return distribution. We allow the high-order moments of the SGT density to depend on the past information set, and hence relax the conventional assumption in conditional VaR calculation that the distribution of standardized returns is iid. The maximum likelihood estimates show that the time-varying conditional volatility, skewness, tail-thickness, and peakedness parameters of the SGT density are statistically significant. The in-sample and out-of-sample performance results indicate that the conditional SGT-GARCH approach with autoregressive conditional skewness and kurtosis provides very accurate and robust estimates of the actual VaR thresholds.     

Evaluating effects of excess kurtosis on VaR estimates: Evidence for international stock indices

The calculus of VaR involves dealing with the confidence level, the time horizon and the true underlying conditional distribution function of asset returns. In this paper, we shall examine the effects of using a specific distribution function that fits well the low-tail data of the observed distribution of asset returns on the accuracy of VaR estimates. In our analysis, we consider some distributional forms characterized by capturing the excess kurtosis characteristic of stock return distributions and we compare their performance using some international stock indices. JEL Classification C15 · G10     

Effects of skewness and kurtosis on production and hedging decisions: a skewed t distribution approach

This paper assumes that the spot price follows a skewed Student t distribution to analyze the effects of skewness and kurtosis on production and hedging decisions for a competitive firm. Under a negative exponential utility function, the firm will not over-hedge (under-hedge) when the spot price is positively (negatively) skewed. The extent of under-hedge (over-hedge) decreases as the forward price increases. Compared with the mean-variance hedger, the producer will hedge more (less) when negative (positive) skewness prevails. In addition, an increase in the skewness reduces the demand for hedging. The effect of the kurtosis, however, depends on the sign of the skewness. When the spot price is positively (negatively) skewed, an increase in kurtosis leads to a smaller (larger) futures position.     

Increasing correlations or just fat tails?

Increasing correlation during turbulent market conditions implies a reduction in portfolio diversification benefits. We investigate the robustness of recent empirical results that indicate a breakdown in the correlation structure by deriving theoretical truncated and exceedance correlations using alternative distributional assumptions. Analytical results show that the increase in conditional correlation could be a result of assuming conditional normality for the return distribution. When assuming a popular alternative distribution – the bivariate Student-t_r – we find significantly less support for an increase in conditional correlation and conclude that this is due to the presence of fat tails when assuming normality in the return distribution.     

Normality of stock returns with event time clocks

We introduce a different way to measure time using event clocks, with which we can observe a normal distribution of intraday stock returns. Most finance studies employ a ‘default’ time measurement that uses a calendar clock. Cumulative evidence from prior literature shows that returns with a calendar clock follow a distribution with an excess kurtosis and a heavier tail, relative to a normal distribution. We examine the distribution of intraday stock returns using different clocks. We find that returns do not follow a normal distribution with a traditional calendar clock, but do follow a normal distribution when event clocks are applied.     

Implicit expectiles and measures of implied volatility

It is well known that the normal distribution is inadequate in capturing the skewed and heavy-tailed behaviour of exchange rate returns. To this end, various flexible distributions that are capable of modelling the asymmetric and tailed behaviour of returns have been proposed. In this paper, we investigate the performance of the generalized lambda distribution (GLD) to capture the skewed and leptokurtic behaviour of exchange rate returns. We do this by conducting a comprehensive numerical study to compare the performance of the GLD against the performances of the skewed t distribution, the unbounded Johnson family of distributions and the normal inverse Gaussian (NIG) distribution. Our results suggest that in terms of the value-at-risk and expected shortfall, the GLD shows at least similar performance to the skewed t distribution and the NIG distribution. Considering the ease in GLD’s use for random variate generation in Monte Carlo simulations, we conclude that the GLD can be a good alternative in various financial applications where modelling of the heavy tail behaviour is critical.     

Why are excess returns on China’s Treasury bonds so predictable? The role of the monetary system

It is well documented that the time-varying bond excess returns can be explained by predetermined variables such as information in the term structure and macro economic variables. Recent studies suggest that demand and supply of bonds influence bond excess returns. We extend the literature and find that monetary system attributes affect return dynamics in the bond market. By introducing a theoretical model to forecast excess returns on Treasury bonds in the context of China’s unique monetary system, this paper attributes the predicted components of bond excess returns mainly to the inflexible term structures of official interest rates set by China’s central bank.