This paper examines the tail conditional expectation risk measure (TCE) in the case of a multivariate gamma portfolio of risks. Explicit formulas for both the TCE and the risk capital allocations based on it are provided in the context of the multivariate model possessing dependent gamma marginals. Some of our results exceed the frameworks of the multivariate gamma distributions and may be appl...

Value at Risk measures the worst loss to be expected of a portfolio over a given time horizon at a given confidence level. Calculation of VaR frequently involves estimating the volatility of return processes and quantiles of standardized returns. In this paper, several semiparametric techniques are introduced to estimate the volatilities . In addition, both parametric and nonparametric techniqu...

In this paper we present a framework for backtesting all currently popular risk measurement methods (including value-at-risk and expected shortfall) using the functional delta method. Estimation risk can be taken explicitly into account. Based on a simulation study we provide evidence that tests for expected shortfall with acceptable low levels have a better performance than tests for value-at-...

In this paper we provide a survey of recent contributions to robust portfolio strategies from operations research and finance to the theory of portfolio selection. Our survey covers results derived not only in terms of the standard mean-variance objective, but also in terms of two of the most popular risk measures, mean-VaR and mean-CVaR developed recently. In addition, we review optimal estima...

Background: Brachytherapy treatment planning in cervix carcinoma patients using two dimensional (2D) orthogonal images provides only point dose estimates while CT-based planning provides volumetric dose assessment helping in understanding the correlation between morbidity and the dose to organs at risk (OARs) and treatment volume. Objective: Aim of present study is to compare Internationa...

We explore generalizations of the pari-mutuel model (PMM), a formalization of an intuitive way of assessing an upper probability from a precise one. We discuss a naive extension of the PMM considered in insurance and generalize the natural extension of the PMM introduced by P. Walley and other related formulae. The results are subsequently given a risk measurement interpretation: in particular ...

This paper addresses the Entropic Value-at-Risk (EV@R), a recently introduced coherent risk measure. It is demonstrated that the norms induced by EV@R induce the same Banach spaces, irrespective of the confidence level. Three spaces, called the primal, dual, and bidual entropic spaces, corresponding with EV@R are fully studied. It is shown that these spaces equipped with the norms induced by EV...

A new one-parameter family of risk measures called Conditional Drawdown (CDD) has been proposed. These measures of risk are functionals of the portfolio drawdown (underwater) curve considered in active portfolio management. For some value of the tolerance parameter α, in the case of a single sample path, drawdown functional is defined as the mean of the worst (1 − α) ∗ 100% drawdowns. The CDD m...

In this paper we analyse nonparametric methods to estimate risk measures in loss distributions. We study kernel estimation for Value-at-Risk and Tail Value-at-Risk based on transformation of the original data. The proposed method consists of a double transformation kernel estimation. We show that a suitable bandwidth selection criterion has a direct expression for the optimal smoothing paramete...

By mid 2004, the Basel Committee on Banking Supervision (BCBS) is expected to launch its final recommendations on minimum capital requirements in the banking industry. Although there is the intention to arrive at capital charges which concur with economic intuition, the risk weight formulas proposed by the committee will lack an adequate treatment of concentration risks in credit portfolios. Th...