Chris Adcock

Skewed distributions in finance and actuarial science: a review

That the returns on financial assets and insurance claims are not well described by the multivariate normal distribution is generally acknowledged in the literature. This paper presents a review of the use of the skew-normal distribution and its extensions in finance and actuarial science, highlighting known results as well as potential directions for future research. When skewness and kurtosis are present in asset returns, the skew-normal and skew-Student distributions are natural candidates in both theoretical and empirical work. Their parameterization is parsimonious and they are mathematically tractable. In finance, the distributions are interpretable in terms of the efficient markets hypothesis. Furthermore, they lead to theoretical results that are useful for portfolio selection and asset pricing. In actuarial science, the presence of skewness and kurtosis in insurance claims data is the main motivation for using the skew-normal distribution and its extensions. The skew-normal has been used in studies on risk measurement and capital allocation, which are two important research fields in actuarial science. Empirical studies consider the skew-normal distribution because of its flexibility, interpretability, and tractability. This paper comprises four main sections: an overview of skew-normal distributions; a review of skewness in finance, including asset pricing, portfolio selection, time series modeling, and a review of its applications in insurance, in which the use of alternative distribution functions is widespread. The final section summarizes some of the challenges associated with the use of skew-elliptical distributions and points out some directions for future research.

Extensions of Stein's lemma for the skew-normal distribution

When two random variables have a bivariate normal distribution, Steins lemma (Stein, 1973 1981), provides, under certain regularity conditions, an expression for the covariance of the first variable with a function of the second. An extension of the lemma due to Liu (1994) as well as to Stein himself establishes an analogous result for a vector of variables which has a multivariate normal distribution. The extension leads in turn to a generalization of Siegels (1993) formula for the covariance of an arbitrary element of a multivariate normal vector with its minimum element. This article describes extensions to Steins lemma for the case when the vector of random variables has a multivariate skew-normal distribution. The corollaries to the main result include an extension to Siegels formula. This article was motivated originally by the issue of portfolio selection in finance. Under multivariate normality, the implication of Steins lemma is that all rational investors will select a portfolio which lies on Markowitzs mean-variance efficient frontier. A consequence of the extension to Steins lemma is that under multivariate skew-normality, rational investors will select a portfolio which lies on a single mean-variance-skewness efficient hyper-surface.

Exploiting skewness to build an optimal hedge fund with a currency overlay

Abstract This paper documents an investigation into the use of portfolio selection methods to construct a hedge fund with a currency overlay. The fund, which is based on number of international stock and bond market indices and is constructed from the perspective of a Sterling investor, allows the individual exposures in the currency overlay to be optimally determined. As well as using traditional mean variance, the paper constructs the hedge funds using portfolio selection methods that incorporate skewness in the optimisation process. These methods are based on the multivariate skewnormal distribution, which motivates the use of a linear skewness shock. An extension to Steins lemma gives the ability to explore the mean-variance-skewness efficient surface without the necessity to be concerned with the precise form of an individual investors utility function. The results suggest that it is possible to use mean variance optimisation methods to build a hedge fund based on the assets and return forecasts described. The results also suggest that the inclusion of a skewness component in the optimisation is beneficial. In many of the cases reported, the skewness term contributes to an improvement in performance over and above that given by mean variance methods.

Mean–variance–skewness efficient surfaces, Stein’s lemma and the multivariate extended skew-Student distribution

Recent advances in Stein’s lemma imply that under elliptically symmetric distributions all rational investors will select a portfolio which lies on Markowitz’ mean–variance efficient frontier. This paper describes extensions to Stein’s lemma for the case when a random vector has the multivariate extended skew-Student distribution. Under this distribution, rational investors will select a portfolio which lies on a single mean–variance–skewness efficient hyper-surface. The same hyper-surface arises under a broad class of models in which returns are defined by the convolution of a multivariate elliptically symmetric distribution and a multivariate distribution of non-negative random variables. Efficient portfolios on the efficient surface may be computed using quadratic programming.

Are Chinese stock and property markets integrated or segmented

This paper explores the empirical question of whether Chinese stock and property markets are integrated or segmented. We find that, at the national level, investment returns in property and the A-share markets were co-integrated in the long run. In the short run, property price Granger caused A-share prices, but not vice versa. However, the B-share prices were negatively correlated with property prices. Furthermore, the linkage between city-level property prices and stock prices showed significant variations across the country. These findings reveal that property and stock markets were integrated at the national level but the property markets were reasonably segmented among cities. They suggest that investment portfolios pursuing risk diversification should include both A and B shares and properties from different cities.

Using parametric classification trees for model selection with applications to financial risk management

We describe two parametric classification tree methods, which allow formal selection of a member of a class of generalised distributions. In the paper we consider generalised Beta distributions for non-negative random variables and the generalised skew-Student distribution for random variables distributed on the real line. We introduce a class of symmetric generalised multivariate Student distributions, members of which may also be selected using the classification trees. We present two versions of the parametric classification tree: specific to general and general to specific. We apply the classification methods to daily returns on stocks from a selection of 15 major, mid-cap and emerging markets. The results show that the majority of return distributions follow Students t, but that a non-negligible minority follow a symmetric generalised Student distribution. We confirm a well-known stylised fact about skewness: it tends not to be persistent. By contrast, kurtosis is persistent. Using the symmetric generalised multivariate Student distribution, we present a risk management study based on efficient portfolios constructed from UKFTSE250 stocks and specifically concerned with the computation of value at risk. The case study demonstrates that the model selection procedures based on the classification trees lead to more accurate computation of VaR than those based on the normal distribution or on non-parametric approaches. The study also shows that the normal distribution may be used for VaR computations for larger portfolios when the holding period is longer.

Value-at-Risk Forecasting Ability of Filtered Historical Simulation for Non-Normal GARCH Returns

As a hybrid methodology to estimate VaR, that combines the use of parametric modelling with the use of bootstrapping techniques, filtered historical simulation (FHS) should not be sensitive to the use of alternative distributions assumed in the filtering stage. However, recent studies (Kuester et al. 2006) have found that the distribution used in the filtering stage can influence the VaR estimates obtained in the context of this methodology. Using Extreme Value Theory (EVT) this paper explains that the VaR estimates for lower probabilities should not be sensitive to the distribution assumed in the filtering stage of the FHS method. However, for higher probabilities, the EVT results do not hold and therefore the use of alternative distributions might impact the VaR estimates. These theoretical results are tested using both simulated and real data. Three different realistic data generating processes were considered to generate several series of simulated returns. Additionally, three competing models, differing in the innovations assumption, were tested: a normal-GARCH, a t-GARCH and a skew-t-GARCH. Our backtesting results indicate that FHS can forecast VaR with accuracy for data which exhibits a high incidence of zeros, time-varying skewness, asymmetric effects to return shocks on volatility, as well as other stylized facts. Importantly, our results for the simulated data demonstrate that, for lower probabilities, the choice of the distribution assumed in the filtering stage has no impact on the performance of FHS as an accurate method to forecasting VaR. Additionally, 40 years of daily data on six well known active stock indices are used to empirically evaluate the FHS VaR estimates. Four competing GARCH-type specifications, combined with three different innovation assumptions (normal, Student-t and skew-Student t), are used to capture time series dynamics. Based on a sample of several VaR probabilities, the results of the dynamic quantile (DQ) tests clearly indicate that the use of asymmetric GARCH models (specifically GJR and GJR in Mean) generally improve the VaR forecasting performance of FHS. In addition, the choice of a skew-Student t distribution for the innovation process slightly improves the performance results of the GJR in Mean model. When different VaR probabilities are used, the choice of an appropriate model specification seems to be more important than the choice of a suitable distribution assumption. With respect to the lower VaR probability tested (1%), the results show that, as expected, the VaR estimate is very similar regardless of the GARCH model and distribution assumed.

Derivative activities and Chinese banks'exposures to exchange rate and interest rate movements

This study investigates the impact of Chinese banks’ derivative activities on their exposure to exchange rate and interest rate changes. The standard Jorion [1990. “The Exchange-Rate Exposure of U.S. Multinationals.” Journal of Business 63 (3): 331–345] model provides weak evidence of Chinese banks’ exposure to these risks. However, the exposure increases substantially when time-varying exposure regressions with orthogonalised market returns are used. We also show that Chinese banks exhibit linear and nonlinear exposures to the exchange rate and interest rate fluctuations. Further analysis indicates that the use of derivatives reduces banks’ foreign exchange risk, but does not affect their interest rate exposure. Derivative products are more likely to be used as an integrated part of the Chinese banks’ risk management systems, which could thus help to stabilise the banking system.

Asset Pricing under Financial Repression: Evidence from the Chinese Real Estate Boom during 1999–2010

China is well known for having adopted financial repression policies to achieve faster economic growth (Huang, 2010; Lin, Cai and Li, 1995). The term “financial repression” refers to government policies which regulated interest rates, set high reserve requirements on bank deposits and mandatorily allocated financial resources (MaKinnon, 1973). Such policies are common in developing countries. Despite more than 30 years of economic reform, the Chinese economy still exhibits characteristics typical of financial repression: heavily regulated interest rates, state-influenced credit allocation, frequently adjusted reserve requirements and tightly controlled capital account (Huang and Wang, 2011).

Risk seeking and measures of portfolio performance

This paper considers the case when asset returns follow a multivariate elliptically symmetric distribution. A multivariate extension of the Sharpe ratio is the measure of portfolio performance for all expected utility maximisers. Performance measures in common use arise formally in the multi-asset case using utility functions based on lower and upper partial moments. This utility function allows risk-seeking for some values of portfolio return and permit choice of the preferred degree of loss aversion. Performance measures which have expected excess return in their numerator are equivalent to the multivariate Sharpe ratio. The Farinelli-Tibiletti ratio also arises in the multi-asset setting. General portfolio selection is considered. The paper presents two results which give conditions for a utility function based on partial moments to lead to a portfolio on the efficient frontier. The paper contains an example based on the weekly returns of a number of FTSE indices.