FRG: The Mathematics of Financial Risk Management

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

The goal of this project is to develop rational and implementable methods to measure and control risk in the finance industry. The topic is timely because there have been some spectacular failures in the finance industry, because regulations are under discussion which would institutionalize quantitative risk measures, and because the leading risk measure, "value- at-risk," has undesirable properties. Chief among these is that in certain situations, situations that often arise in credit-risky markets, meeting regulatory value-at-risk requirements encourages concentration of risk rather than diversification. The proposal builds on the idea of coherent risk measures developed by a team, which includes one of the principal investigators on this proposal. This team developed a set of axioms deemed to be desirable, and then characterized those "coherent" risk measures which satisfy these axioms. Their work showed that every coherent risk measure can be characterized by a set of probability measures over possible future scenarios. A three- part project is proposed. The first part is to develop a decentralized implementation of risk control based on coherent risk measurement. This requires development of duality theory and equilibrium pricing within a firm. The second is to elicit scenarios and estimate model parameters in real settings. This mandates refining the mathematical models to the point that a manageable number of measures over future scenarios present themselves, and appealing to statistical theory to make those choices. Finally, it is proposed to extend the existing theory of coherent risk measures. This project is being supported under the Focused Research Groups in the Mathematical Sciences activity.