On Partial Hedging and Counter-Monotonic Sums

In this article, we show, in the context of partial hedging, that some important relationships about comonotonicity and convex order cannot be translated to counter-monotonicity in general because of the possibility of over-hedging. We propose a new notion called proper hedge that can e effectively avoid over-hedging. Diff erent characterizations of a proper hedge are given, and we show that this notion is useful in translating relationship between comonotonicity and convex order to the case of counter-comonotonicity.As an application, we apply our results to identify desirable structural properties of insurance indemnities that make an insurance contract appealing to both the policyholder and the insurer.