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Concept# Value at risk

Summary

Value at risk (VaR) is a measure of the risk of loss of investment/Capital. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses.
For a given portfolio, time horizon, and probability p, the p VaR can be defined informally as the maximum possible loss during that time after excluding all worse outcomes whose combined probability is at most p. This assumes mark-to-market pricing, and no trading in the portfolio.
For example, if a portfolio of stocks has a one-day 95% VaR of $1 million, that means that there is a 0.05 probability that the portfolio will fall in value by more than$1 million over a one-day period if there is no trading. Informally, a loss of $1 million or more on this portfolio is expected on 1 day out of 20 days (because of 5% probability).

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Since the 2008 Global Financial Crisis, the financial market has become more unpredictable than ever before, and it seems set to remain so in the forseeable future. This means an investor faces unprecedented risks, hence the increasing need for robust portfolio optimization to protect them against uncertainty, which is potentially devastating if unattended yet ignored in the classical Markowitz model, whose another deficiency is the absence of higher moments in its assumption of the distribution of asset returns. We establish an equivalence between the Markowitz model and the portfolio return value-at-risk optimization problem under multivariate normality of asset returns, so that we can add these excluded features into the former implicitly by incorporating them into the latter. We also provide a probabilistic smoothing spline approximation method and a deterministic model within the location-scale framework under elliptical distribution of the asset returns to solve the robust portfolio return value-at-risk optimization problem. In particular for the deterministic model, we introduce a novel eigendecomposition uncertainty set which lives in the positive definite space for the scale matrix without compromising on the computational complexity and conservativeness of the optimization problem, invent a method to determine the size of the involved uncertainty sets, test it out on real data, and explore its diversification properties. Although the value-at-risk has been the standard risk measure adopted by the banking and insurance industry since the early nineties, it has since attracted many criticisms, in particular from McNeil et al. (2005) and the Basel Committee on Banking Supervision in 2012, also known as Basel 3.5. Basel 4 even suggests a move away from the

`what" value-at-risk to the `

what-if" conditional value-at-risk' measure. We shall see that the former may be replaced with the latter or even other risk measures in our formulations easily.This thesis considers technological risks and their constitution as public problems. It supports that the main difference among risks is not to see between proved risks and uncertain risks, on the basis of the objective knowledge, but rather separates risks which have already reached compromises at both levels of evaluation and responsibility and those which are still controversial. The starting observation is that social acceptability of risk is out of the scope of the objective approach. Actors use categories of "objective risk" and "perception of risk" within struggles to define what is at stake. It shows how risks, as soon as their evaluation starts, are negotiated within controversies, mobilizing experts, stakeholders, and actors of society. With many environmental risk, and with the globalization of hazards such as climate change, the preventive paradigm is overflowed; and with the emergence of new uncertainties, such as the introduction of genetically modified organisms in agriculture and food, expertise as the only method to deal with risk is contested in society. The precautionary approach, which asks to act against uncertainty before the completion of scientific evidence of risk is declining in the United-States while its integration is increasing in public policy in Europe. Have we entered the risk society? Or, is the society of the developed countries safer than it has ever been? Are risk increasing or decreasing? These apparent paradoxes shows there is different social theories of risk with different possible balance between objective and social factors to explain them. The role of technology is ambivalent, generating new risk and in the same time privileged as a solution to uncertainty. The thesis puts forward "science, technology and society" approach in risk studies, and it aims at a sociology where experts are not considered neutral actors. In this perspective, robustness of expertise does not rely directly on objective knowledge, but on the relation of this knowledge with its context of application. After studying the role of risk in modernization and rationalization processes, the thesis focuses the mobilization of actors in risk controversies and the role of some new institutions in the public sphere such as the ethical committees and the technology assessment agencies. Two case studies are investigated which take place in Switzerland end of the 1990s. The first relates to the introduction of genetically modified organisms in agriculture and environment. A controversy occurred about field test, i.e. scientific experiments refused by the authorities in 1999 and 2001. The role of research in biotechnology is considered as well as the different role of expertise in the public sphere. Expertise in biosafety and social acceptability of biotechnology are negotiated simultaneously. The second case study is the climate change issue and the negotiation of measures to control greenhouse gas. The role of research policy in the governance of climate risk is studied. Both case studies are interested with the role of experts in the public sphere, with the coordination of expertise and decision making, and with the degree of democracy achieved in sociotechnical controversies. The main result of this study is that a linear model of risk dominates approaches and constitutes an obstacle to the democratization of technological change. Instead, the thesis proposes a model where rationality is not defined beforehand, but appears as the proper result of controversies. In this view, risks are socially negotiated, already during the production of expertise, and all along the decision making and the implementation processes.

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