  In this paper, we investigate the optimization of mutual proportional
reinsurance --- a mutual reserve system that is intended for the collective
reinsurance needs of homogeneous mutual members, such as P&I Clubs in marine
mutual insurance and reserve banks in the U.S. Federal Reserve. Compared to
general (non-mutual) insurance models, which involve one-sided impulse control
(i.e., either downside or upside impulse) of the underlying insurance reserve
process that is required to be positive, a mutual insurance differs in allowing
two-sided impulse control (i.e., both downside and upside impulse), coupled
with the classical proportional control of reinsurance. We prove that a special
band-type impulse control $(a,A,B,b)$ with $a=0$ and $a<A<B<b$, coupled with a
proportional reinsurance policy (classical control), is optimal when the
objective is to minimize the total maintenance cost. That is, when the reserve
position reaches a lower boundary of $a=0$, the reserve should immediately be
raised to level $A$; when the reserve reaches an upper boundary of $b$, it
should immediately be reduced to a level $B$. An interesting finding produced
by the study reported in this paper is that there exists a situation such that
if the upside fixed cost is relatively large in comparison to a finite
threshold, then the optimal band control is reduced to a downside only (i.e.,
dividend payment only) control in the form of $(0,0;B,b)$ with $a=A=0$. In this
case, it is optimal for the mutual insurance firm to go bankrupt as soon as its
reserve level reaches zero, rather than to jump restart by calling for
additional contingent funds. This finding partially explains why many mutual
insurance companies, that were once quite popular in the financial markets, are
either disappeared or converted to non-mutual ones.

Optimization and Control