Modeling and Optimization of Risk Measures

点击翻译按钮获取中文摘要

Stunning financial losses have been witnessed in recent years due to risk underestimation. The management of pension funds is one of the many examples, where the failure of measuring the actual risk involved has led to widespread economic instability. This raises a pressing need to design new risk measures that capture risk overlooked by existing measures. Off-the-shelf measures such as Value-at-Risk (VaR) generate numerical risk estimates by assuming that all uncertainty can be represented by a probability distribution, and the rest is just a statistical exercise. The real world however, as evidenced by the 2008 financial crisis, is far more uncertain than what a distribution can describe. This multifaceted uncertainty has now gradually learned by decision makers, but a risk measure that embodies this knowledge is still lacking. More sophisticated measures have been proposed lately. In the area of Robust Optimization (RO), a series of works have been done to design robust measures that allow the use of multiple distributions to describe uncertainty. Meanwhile, in Risk Analysis, axioms have been introduced that define the properties of ideal risk measures. While these works provide a whole new spectrum of measures, they leave the following question open: Which one of them accounts precisely for the actual uncertainty and risk facing decision makers? Modern risk analysis theory does not provide a definite answer to this - largely because there is no single measure that can fit all situations. Decision makers from different sectors or industries can have different perspectives of risk, depending on the nature of the work undertaken. Current models of risk measures provide only narrow views of risk and are not amenable to incorporating decision makers' knowledge of risk. This gap between what current models offer and what decision makers need to measure is one of the greatest impediments toward the widespread deployment of risk measures. The goal of this research program is to close the gap by developing a new interactive scheme that allows decision makers' knowledge of risk to be explored and incorporated into the design of risk measures. This scheme shares the same spirit as Interactive Optimization (IO), where the solution procedure involves user input in seeking preferable solutions. Important research questions to ask while developing this scheme include: - How should we interact with users to best elicit their knowledge of risk? - How can this information be incorporated into the design of risk measures? - How can we provide guarantees on the quality of risk estimates even if only limited resource is available for elicitation? - How can we ensure, given enough resource for elicitation, that the resulting risk measure converges to the one that best represents a user's knowledge of risk? - How can the resulting measures be integrated into a decision optimization process? Our methodology will be developed based on three streams of research: Robust Optimization, axiomatic approaches in Risk Analysis, and Interactive Optimization, which if successfully developed will also provide theoretical evidence of how these approaches can be integrated in the context of risk analytics. The end result of this research program will be a new decision support system that enables decision makers to model their knowledge of risk and to minimize them. The system will be validated using case studies from the sectors of finance, energy, and health care. This will bridge the gap between theory and practice in risk analytics, and encourage practitioners to approach risk management problems analytically, which is critically important in today’s ever-changing environment.