  We investigate an optimal reinsurance problem for an insurance company facing
a constant fixed cost when the reinsurance contract is signed. The insurer
needs to optimally choose both the starting time of the reinsurance contract
and the retention level in order to maximize the expected utility of terminal
wealth. This leads to a mixed optimal control/optimal stopping time problem,
which is solved by a two-step procedure: first considering the pure reinsurance
stochastic control problem and next discussing a time-inhomogeneous optimal
stopping problem with discontinuous reward. Using the classical
Cram\'er-Lundberg approximation risk model, we prove that the optimal strategy
is deterministic and depends on the model parameters. In particular, we show
that there exists a maximum fixed cost that the insurer is willing to pay for
the contract activation. Finally, we provide some economical interpretations
and numerical simulations.
