Rafael Schmidt

Tail dependence for elliptically contoured distributions

Abstract. The relationship between the theory of elliptically contoured distributions and the concept of tail dependence is investigated. We show that bivariate elliptical distributions possess the so-called tail dependence property if the tail of their generating random variable is regularly varying, and we give a necessary condition for tail dependence which is somewhat weaker than regular variation of the latter tail. In addition, we discuss the tail dependence property for some well-known examples of elliptical distributions, such as the multivariate normal, t, logistic, and Bessel distributions.

Copula-Based Measures of Multivariate Association

This chapter constitutes a survey on copula-based measures of multivariate association - i.e. association in a d-dimensional random vector \(X = (X_1 , \ldots ,X_d )\) where \(d \ge 2\). Some of the measures discussed are multivariate extensions of wellknown bivariate measures such as Spearman’s rho, Kendall’s tau, Blomqvist’s beta or Gini’s gamma. Others rely on information theory or are based on L p-distances of copulas. Various measures of multivariate tail dependence are derived by extending the coefficient of bivariate tail dependence. Nonparametric estimation of these measures based on the empirical copula is further addressed.

Credit Risk Modelling and Estimation via Elliptical Copulae

Dependence modelling plays a crucial role within internal risk models. The theory of copulae, which describes the dependence structure between a multi-dimensional distribution function and the corresponding marginal distributions, provides useful tools for dependence modelling. The difficulty in employing copulae for internal credit risk models arises from the appropriate choice of a copula function.

Future life expectancy in Australia, Europe, Japan and North America

Human life expectancy has risen in most developed countries over the last century, causing the observed demographic shifts. Babel, Bomsdorf and Schmidt (forthcoming) introduce a stochastic mortality model using panel data procedures which distinguishes between a common time effect and a common age effect of mortality evolvement. Using this mortality model, the present paper provides forecasts of future life expectancy for 17 countries divided into 12 regions: Australia, Alps, Bene, Canada, England and Wales, France, Germany, Italy, Japan, Spain, Scandinavia and the United States of America. We consider (traditional) period life expectancies as well as cohort life expectancies, the latter being a more realistic approach but less common. It turns out that a continuing increase of life expectancy is expected in all considered countries. Further, we show that the probabilistic uncertainty of forecast life expectancies is different if either period life expectancies or cohort life expectancies are considered and, moreover, the uncertainty increases substantially if the error of parameter estimation is included.

Statistical Inference for Sharpe Ratio

Sharpe ratios (Sharpe 1966) are the most popular risk-adjusted performance measure for investment portfolios and investment funds. Given a riskless security as a benchmark, its Sharpe ratio is defined by

Non-parametric Estimation of Tail Dependence

SR = \frac{{\mu - z}}{{\sqrt {{\sigma ^2}} }}

Multivariate conditional versions of Spearman's rho and related measures of tail dependence

where μ and σ2 denote the portfolio’s mean return and return volatility, respectively, and z represents the riskless return of the benchmark security. From an investor’s point of view, a Sharpe ratio describes how well the return of an investment portfolio compensates the investor for the risk he takes. Financial information systems, for example, publish lists where investment funds are ranked by their Sharpe ratios. Investors are then advised to invest into funds with a high Sharpe ratio. The rationale behind this is that, if the historical returns of two funds are compared to the same benchmark, the fund with the higher Sharpe ratio yields a higher return for the same amount of risk. Though (ex post) Sharpe ratios are computed using historical returns, it is assumed that they have a predictive ability (ex ante). We refer to Sharpe (1994) for related discussions and further references.

Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence

A general, copula-based framework for measuring the dependence among financial time series is presented. Particular emphasis is placed on multivariate conditional Spearmans rho (MCS), a new measure of multivariate conditional dependence that describes the association between large or extreme negative returns—so-called tail dependence. We demonstrate that MCS has a number of advantages over conventional measures of tail dependence, both in theory and in practical applications. In the analysis of univariate financial series, data are filtered to remove temporal dependence as a matter of routine. We show that standard filtering procedures may strongly influence the conclusions drawn concerning tail dependence. We give empirical applications to two large data sets of high-frequency asset returns. Our results have immediate implications for portfolio risk management, derivative pricing and portfolio selection. In this context we address portfolio tail diversification and tail hedging. Amongst other aspects, it is shown that the proposed modeling framework improves the estimation of portfolio risk measures such as the value at risk.