  Any firm whose business strategy has an exposure constraint that limits its
potential gain naturally considers expansion, as this can increase its
exposure. We model business expansion as an enlargement of the opportunity set
for business policies. However, expansion is irreversible and has an
opportunity cost attached. We use the expected optimization of utility to
formulate this as a novel stochastic control problem combined with an optimal
stopping time, and we derive an explicit solution for exponential utility. We
apply the framework to an investment and a reinsurance scenario. In the
investment problem, the cost and incentives to increase the trading exposure
are analyzed, while the optimal timing for an insurer to launch its reinsurance
business is investigated in the reinsurance problem. Our model predicts that
the additional income gained through business expansion is the key incentive
for a decision to expand. Interestingly, companies may have this incentive but
are likely to wait for a period of time before expanding, although situations
of zero opportunity cost or specific restrictive conditions on the model
parameters are exceptions to waiting. The business policy remains on the
boundary of the opportunity set before expansion during the waiting period. The
length of the waiting period is related to the opportunity cost, return, and
risk of the expanded business.
