  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".
