Hidden Equations of Threshold Risk

We consider the problem of sensitivity of threshold risk, defined as the probability of a function of a random variable falling below a specified threshold level $\delta >0.$ We demonstrate that for polynomial and rational functions of that random variable there exist at most finitely many risk critical points. The latter are those special values of the threshold parameter for which rate of change of risk is unbounded as $\delta$ approaches these threshold values. We characterize candidates for risk critical points as zeroes of either the resolvent of a relevant $\delta-$perturbed polynomial, or of its leading coefficient, or both. Thus the equations that need to be solved are themselves polynomial equations in $\delta$ that exploit the algebraic properties of the underlying polynomial or rational functions. We name these important equations as "hidden equations of threshold risk".