International Journal of Science and Research (IJSR), India Online ISSN: 2319-7064 Volume 2 Issue 7, July 2013 www.ijsr.net Risk Measurement Strategy; An Alternative to Merging that Offers A Capital Relief in Risk Management Godswill. U. Achi 1 , Ogwo Obiageri 2 , Solomon Okechukwu 3 1 Department of Mathematics, Abia State Polytechnic, Aba, PMB 7166 Aba ,Abia State,Nigeria 2, 3 Department of Mathematics, Abia state University, Uturu, Abia State, Nigeria Abstract: In this paper, we suggest the authenticity of a distortion risk measurement strategy that can be used instead of the risk management slogan ‘Avoiding merging increases shortfall’ which justifies the well known advice ‘ don’t put all your eggs in one basket’. There are lots of distortion risk measures like conditional value at risk (expected shortfall) or the Wang transform risk measure, in spite of being coherent they do not always provide incentive for risk management because of lack of giving a capital relief in some simple two scenarios situation of reduced risk. To prevent the existence of such pathological counter examples, we introduce a Weibull distortion measure that preserves the higher degree stop loss order and offer a capital relief. Keywords: Coherent risk measure, tail free distortion risk measures, distortion risk measures, merging, Weibull distortion measure. 1. Introduction The use of distortion risk measures to determine capital requirements of a risky business can be found in many papers (e.g. . To provide incentive for active risk management, it is argued that some coherent distortion risk measure should preserve some higher degree stop loss order. Such risk measures are called tail free risk measures. The axiomatic approach to risk measures is an important and very active subject, which applies to different topics of actuarial and financial interest like premium calculation and capital requirements. Besides the coherent risk measures by , one is interested in the distortion risk measures by . Under certain circumstances, distortion risk measures are coherent risk measures (e.g. ). Consequently, they can be used to determine the capital requirements of a risky business, as suggested by several authors including . However, in spite of being coherent, a lot of distortion risk measures, like conditional value-at-risk (identical to expected shortfall) or the very Wang transform risk measure, do not always provide incentive for risk management because of its inability to giving a capital relief in some simple two scenarios situations of reduced risk (see Examples 1 and 2). In this paper, we consider the known risk management slogan ‘Avoiding merging increases shortfall’ which justifies the well known advice ‘ don’t put all your eggs in one basket’ which if not adhered to, may lead to higher loss in capital of a risky portfolio. This is a desirable property because increased risk should be penalized with an increased risk measure .To prevent the existence of such pathological counterexamples, we are interested in a weibull distortion risk measures ( that preserve the higher degree stop- loss orders (e.g. ) and offers a capital relief which can serve as an alternative to the known slogan. 2. Formulation of Problems 2.1 Diversification and Sub-additive Axioms On Capital Requirement Given two portfolios with respective losses and . Assume that the solvency capital requirement imposed by the regulator is given by the risk measure P, if each portfolio is not liable for the shortfall of the other one, the capital requirement for each portfolio is given by P( ), if the two portfolios are (together) both liable for the eventual shortfall of the aggregate loss , we will say that the portfolios are merged. Therefore, the solvency capital requirement imposed by the supervisory authorities will in this case be equal to ). Merging the two portfolios will lead to a decrease in shortfall given by (1) where the capital requirement for each portfolio = P( ) = solvency capital for the risk exposure imposed by the regulators. Eventual shortfall or aggregate loss. The solvency capital requirement for the aggregate risk = + . The following inequality holds with probability 1 + (2) (See ) This inequality states that, the shortfall of the merged portfolio is always smaller than the sum of the shortfall of the separate portfolios, when adding capitals. It expresses, that from point of view of the regulatory authorities that a merger adding the capital is to be preferred in the sense that the shortfall decreases. The underlying reason is that, within the merged portfolios, the shortfall of one of the entities can be compensated by the gain of the other one. This 27