  In this paper we study a class of insurance products where the policy holder
has the option to insure $k$ of its annual Operational Risk losses in a horizon
of $T$ years. This involves a choice of $k$ out of $T$ years in which to apply
the insurance policy coverage by making claims against losses in the given
year. The insurance product structure presented can accommodate any kind of
annual mitigation, but we present three basic generic insurance policy
structures that can be combined to create more complex types of coverage.
Following the Loss Distributional Approach (LDA) with Poisson distributed
annual loss frequencies and Inverse-Gaussian loss severities we are able to
characterize in closed form analytical expressions for the multiple optimal
decision strategy that minimizes the expected Operational Risk loss over the
next $T$ years. For the cases where the combination of insurance policies and
LDA model does not lead to closed form expressions for the multiple optimal
decision rules, we also develop a principled class of closed form
approximations to the optimal decision rule. These approximations are developed
based on a class of orthogonal Askey polynomial series basis expansion
representations of the annual loss compound process distribution and functions
of this annual loss.
