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We introduce a new and easy-to-calculate measure for the expected degree of herd behavior or co-movement between stock prices. This forward looking measure is model-independent and based on observed option data. It is baptized the Herd Behavior Index (HIX). The degree of co-movement in a stock market can be determined by comparing the observed market situation with the extreme (theoretical) situation under which the whole system is driven by a single factor. The HIX is then defined as the ratio of an option-based estimate of the risk-neutral variance of the market index and an option-based estimate of the corresponding variance in case of the extreme single factor market situation. The HIX can be determined for any market index provided an appropriate series of vanilla options is traded on this index as well as on its components. As an illustration, we determine historical values of the 30-days HIX for the Dow Jones Industrial Average, covering the period January 2003 to October 2009.

In this paper we examine and summarize properties of several well-known risk measuresthat can be used in the framework of setting solvency capital requirements for a risky business.Special attention is given to the class of (concave) distortion risk measures. We investigatethe relationship between these risk measures and theories of choice under risk. Furthermore weconsider the problem of how to evaluate risk measures for sums of non-independent randomvariables. Approximations for such sums, based on the concept of comonotonicity, are proposed.Several examples are provided to illustrate properties or to prove that certain properties donot hold. Although the paper contains several new results, it is written as an overview andpedagogical introduction to the subject of risk measurement. The paper is an extended versionof Dhaene et al.[11].Keywords Comonotonicity; Distortion; Lognormal; Risk measurer; Theory of choiceunder risk.Mathematics Subject Classification 91B30.

In this paper, the authors consider the present value of a series of fixed cash flows under stochastic interest rates. To model these interest rates, they don’t use the common lognormal model, but stable laws, which better fit in with reality. Their main intention is to derive a result for the distribution function of such a present value. However, due to the dependencies between successive discounted payments, the calculation of an exact analytical distribution is impossible. Therefore, use is made of the methodology of comonotonic random variables and the convex ordering of risks, introduced by the same authors in some previous papers. The present paper starts with a brief overview of properties and features of stable laws, and of the possible application of the concept of convex ordering to sums of risks, which is also the situation for a present value of future payments. Afterwards, the authors show how, for the present value under investigation, an approximation in the form of a convex upper...

n an insurance context, one is often interested in the distribution function of a sum of random variables. Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not realistic. We will determine approximations for sums of random variables, when the distributions of the terms are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. In this paper, the theoretical aspects are considered. Applications of this theory are considered in a subsequent paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented. ispartof: DTEW Research Report 0216 pages:1-50 status: published

In an insurance context, one is often interested in the distribution function of a sum of random variables (rv’s). Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio, at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not a realistic one. In The Concept of Comonotonicity in Actuarial Science and Finance: Theory, we determined approximations for sums of rv’s, when the distributions of the components are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. Practical applications of this theory will be considered in this paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented.

The distribution of the present value of a series of cash flows under stochastic interest rates has been investigated by many researchers. One of the main problems in this context is the fact that the calculation of exact analytical results for this type of distributions turns out to be rather complicated, and is known only for special cases. An interesting solution to this difficulty consists of determining computable upper bounds, as close as possible to the real distribution.In the present contribution, we want to show how it is possible to compute such bounds for the present value of cash flows when not only the interest rates but also volatilities are stochastic. We derive results for the stop loss premium and distribution of these bounds. ispartof: DTEW Research Report 0218 pages:1-24 status: published