Papers by Zinoviy Landsman
Multivariate Families with Mixture Dependence
Social Science Research Network, 2017
This paper introduces a multivariate tail covariance (MTCov) measure, which is a matrix-valued ri... more This paper introduces a multivariate tail covariance (MTCov) measure, which is a matrix-valued risk measure designed to explore the tail dispersion of multivariate loss distributions. The MTCov is the second multivariate tail conditional moment around the MTCE, the multivariate tail conditional expectation (MTCE) risk measure. Although MTCE was recently introduced in (Landsman et al., 2016), in this paper we essentially generalize it, allowing for quantile levels to obtain the different values corresponded to each risk. The MTCov measure, which is also defined for the set of different quantile levels, allows us to investigate more deeply the tail of multivariate distributions, since it focuses on the variance-covariance dependence structure of a system of dependent risks. As a natural extension, we also introduced the multivariate tail correlation matrix (MTCorr). The properties of this risk measure are explored and its explicit closed-form expression is derived for the elliptical family of distributions. As a special case, we consider the normal, Student-t and Laplace distributions, prevalent in actuarial science and finance. The results are illustrated numerically with data of some stock returns.
Random volatility and option prices with the generalized Student-t distribution
Advances and applications in statistics, 2006
ABSTRACT
A New Class of Generalised Hyper-Elliptical Distributions and Their Applications in Computing Conditional Tail Risk Measures
Social Science Research Network, 2020
This paper introduces a new family of Generalized Hyper-Elliptical (GHE) distributions providing ... more This paper introduces a new family of Generalized Hyper-Elliptical (GHE) distributions providing further generalization of the generalized hyperbolic (GH) family of distributions, considered in Ignatieva and Landsman. The GHE family is constructed by mixing a Generalized Inverse Gaussian (GIG) distribution with an elliptical distribution. We present an innovative theoretical framework where a closed form expression for the tail conditional expectation (TCE) is derived for this new family of distributions. We demonstrate that the GHE family is especially suitable for a heavy - tailed insurance losses data. Our theoretical TCE results are verified for two special cases, Laplace - GIG and Student-t - GIG mixtures. Both mixtures are shown to outperform the GH distribution providing excellent fit to univariate and multivariate insurance losses data. The TCE risk measure computed for the GHE family of distributions provides a more conservative estimator of risk in the extreme tail, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from extreme losses. Our multivariate analysis allows to quantify correlated risks by means of the GHE family: the TCE of the portfolio is decomposed into individual components, representing individual risks in the aggregate loss.
Communications in Statistics, Oct 23, 2019
This paper generalizes Stein's Lemma recently obtained for elliptical class distributions to the ... more This paper generalizes Stein's Lemma recently obtained for elliptical class distributions to the generalized skew-elliptical family of distributions. Stein's Lemma provides a useful tool for deriving covariances between functions of component random variables. This Lemma has applications in finance, notably for portfolio selection and hence for the capital asset pricing model (CAPM), as well as technical applications such as the computation of moments. It also leads to important propositions concerning the mean and variance of generalized skew-elliptical variables.
Statistical meaning of Carlen's superadditivity of the Fisher information
Statistics & Probability Letters, Mar 1, 1997
In Carlen (1991) a property of the Fisher information called “superadditivity”, was proved via an... more In Carlen (1991) a property of the Fisher information called “superadditivity”, was proved via analytic means.We show that the superadditivity is a corollary of the following simple statistical principle which is of an independent interest. The Fisher information about a parameter θ contained in an observation X = (Y,Z) with a density f(y − θ,z) is never less than the
Asymptotic behavior of the fisher information contained in additive statistics
Springer eBooks, 1976
Asymptotic tests for mean location on manifolds
Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 1996
ABSTRACT
Relation between the covariance and Fisher information matrices
Statistics & Probability Letters, Mar 1, 1999
It is proved that for any two positive definite Hermitian m×m matrices I and V subject to I⩾V−1th... more It is proved that for any two positive definite Hermitian m×m matrices I and V subject to I⩾V−1there exists an m-variate random vector X with V as its covariance matrix and I its matrix of Fisher information.
Social Science Research Network, 2015
This paper addresses one of the main challenges faced by insurance companies and risk management ... more This paper addresses one of the main challenges faced by insurance companies and risk management departments, namely, how to develop standardised framework for measuring risks of underlying portfolios and in particular, how to most reliably estimate loss severity distribution from historical data. This paper investigates tail conditional expectation (TCE) and tail variance premium (TVP) risk measures for the family of symmetric generalised hyperbolic (SGH) distributions. In contrast to a widely used Value-at-Risk (VaR) measure, TCE satisfies the requirement of the ''coherent'' risk measure taking into account the expected loss in the tail of the distribution while TVP incorporates variability in the tail, providing the most conservative estimator of risk. We examine various distributions from the class of SGH distributions, which turn out to fit well financial data returns and allow for explicit formulas for TCE and TVP risk measures. In parallel, we obtain asymptotic behaviour for TCE and TVP risk measures for large quantile levels. Furthermore, we extend our analysis to the multivariate framework, allowing multivariate distributions to model combinations of correlated risks, and demonstrate how TCE can be decomposed into individual components, representing contribution of individual risks to the aggregate portfolio risk.
The Tail Stein's Identity with Applications to Risk Measures
The North American Actuarial Journal, Oct 1, 2016
In this article, we examine a generalized version of an identity made famous by Stein, who constr... more In this article, we examine a generalized version of an identity made famous by Stein, who constructed the so-called Stein's Lemma in the special case of a normal distribution. Other works followed to extend the lemma to the larger class of elliptical distributions. The lemma has had many applications in statistics, finance, insurance, and actuarial science. In an attempt to broaden the application of this generalized identity, we consider the version in the case where we investigate only the tail portion of the distribution of a random variable. Understanding the tails of a distribution is very important in actuarial science and insurance. Our article therefore introduces the concept of the “tail Stein's identity” to the case of any random variable defined on an appropriate probability space with a Lebesgue density function satisfying certain regularity conditions. We also examine this “tail Stein's identity” to the class of discrete distributions. This extended identity allows us to develop recursive formulas for generating tail conditional moments. As examples and illustrations, we consider several classes of distributions including the exponential family, and we apply this result to demonstrate how to generate tail conditional moments. This holds a large promise for applications in risk management.
The location of a minimum variance squared distance functional
Insurance Mathematics & Economics, Jul 1, 2022
The Tail Stein's Identity with Actuarial Applications
Social Science Research Network, 2015
In this article, we examine a generalized version of an identity made famous by Stein (1981) who ... more In this article, we examine a generalized version of an identity made famous by Stein (1981) who constructed the so-called Stein's Lemma in the special case of a normal distribution. Other works later followed to extend the lemma to the larger class of elliptical distributions, e.g. Landsman (2006) and Landsman and Neslehova (2008). The lemma has had many applications in statistics, finance, insurance and actuarial science. In an attempt to broaden the application of this generalized identity, we consider the version in the case where we investigate only the tail portion of the distribution of a random variable. Understanding the tails of a distribution is widely important in actuarial science and insurance. Our paper therefore introduces the concept of the "tail Stein's identity" to the case of any random variable defined on an appropriate probability space with a Lebesque density function satisfying certain regularity conditions. We also examined this "tail Stein's identity" to the class of discrete distributions. This extended identity allowed us to develop recursive formulas for generating tail conditional moments. As examples and illustrations, we consider several classes of distributions including the exponential family, and we apply this result to demonstrate how to generate tail conditional moments. This has a large promise of applications in risk management.
The classes of distribution families with the lower bound of fisher information and its meaning in the statistical estimation
Springer eBooks, 1983
Journal of Soviet mathematics, Feb 1, 1988
For the density V2(x) we obtain the expression 1 o~r , (log X+ log y) B'(g) dg.
Social Science Research Network, 2017
This paper deals with the estimation of loss severity distributions arising from historical data ... more This paper deals with the estimation of loss severity distributions arising from historical data on univariate and multivariate losses. We present an innovative theoretical framework where a closed-form expression for the tail conditional expectation (TCE) is derived for the skewed generalised hyperbolic (GH) family of distributions. The skewed GH family is especially suitable for equity losses because it allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to the widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its special cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. The GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on equity losses. The TCE risk measure computed for the skewed family of GH distributions provides a conservative estimator of risk, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolios of losses, allowing the multivariate GH distribution to capture the combination of correlated risks and demonstrate how the TCE of the portfolio can be decomposed into individual components, representing individual risks in the aggregate (portfolio) loss.
Social Science Research Network, 2012
Systematic improvements in mortality results in dependence in the survival distributions of insur... more Systematic improvements in mortality results in dependence in the survival distributions of insured lives. This is not allowed for in standard life tables and actuarial models used for annuity pricing and reserving. Systematic longevity risk also undermines the law of large numbers; a law that is relied on in the risk management of life insurance and annuity portfolios. This paper applies a multivariate gamma distribution to incorporate dependence. Lifetimes are modelled using a truncated multivariate gamma distribution that induces dependence through a shared gamma distributed component. Model parameter estimation is developed based on the method of moments and generalized to allow for truncated observations. The impact of dependence on the valuation of a portfolio, or cohort, of annuitants with similar risk characteristics is demonstrated by applying the model to annuity valuation. The dependence is shown to have a significant impact on the risk of the annuity portfolio as compared with traditional actuarial methods that implicitly assume independent lifetimes.
A class of generalised hyper-elliptical distributions and their applications in computing conditional tail risk measures
Insurance Mathematics & Economics, Nov 1, 2021
Abstract This paper introduces a new family of Generalised Hyper-Elliptical (GHE) distributions p... more Abstract This paper introduces a new family of Generalised Hyper-Elliptical (GHE) distributions providing further generalisation of the generalised hyperbolic (GH) family of distributions, considered in Ignatieva and Landsman (2019) . The GHE family is constructed by mixing a Generalised Inverse Gaussian (GIG) distribution with an elliptical distribution. We present an innovative theoretical framework where a closed form expression for the tail conditional expectation (TCE) is derived for this new family of distributions. We demonstrate that the GHE family is especially suitable for heavy-tailed insurance losses data. Our theoretical TCE results are verified for two special cases, Laplace - GIG and Student-t - GIG mixtures. Both mixtures are shown to outperform the GH distribution, providing excellent fit to univariate and multivariate insurance losses data. The TCE risk measure computed for the GHE family of distributions provides a more conservative estimator of risk in the extreme tail, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from extreme losses. Our multivariate analysis allows to quantify correlated risks by means of the GHE family: the TCE of the portfolio is decomposed into individual components, representing individual risks in the aggregate loss.
A New Approach to Multivariate Archimedean Copula Generation
Uploads
Papers by Zinoviy Landsman