The fundamental risk quadrangle in risk management, optimization and statistical estimation

Random variables that stand for cost, loss or damage must be confronted in numerous situations. Dealing with them systematically for purposes in risk management, optimization and statistics is the theme of this presentation, which brings together ideas coming from many different areas.Measures of risk can be used to quantify the hazard in a random variable by a single value which can substitute for the otherwise uncertain outcomes in a formulation of constraints and objectives. Such quantifications of risk can be portrayed on a higher level as generated from penalty-type expressions of “regret” about the mix of potential outcomes. A trade-off between an up-front level of hazard and the uncertain residual hazard underlies that derivation. Regret is the mirror image of utility, a familiar concept for dealing with gains instead of losses, but regret concerns hazards relative to a benchmark. It bridges between risk measures and expected utility, thereby reconciling those two approaches to optimization under uncertainty.Statistical estimation is inevitably a partner with risk management in handling hazards, which may be known only partially through a data base. However, a much deeper connection has come to light with statistical theory itself, in particular regression. Very general measures of error can associate with any hazard variable a “statistic” along with a “deviation” which quantifies the variable’s nonconstancy. Measures of deviation, on the other hand, are paired closely with measures of risk exhibiting “aversity”. A direct correspondence can furthermore be identified between measures of error and measures of regret. The fundamental quadrangle of risk developed here puts all of this together in a unified scheme.