  We consider the optimal risk transfer from an insurance company to a
reinsurer. The problem formulation considered in this paper is closely
connected to the optimal portfolio problem in finance, with some crucial
distinctions. In particular, the insurance company's surplus is here (as is
routinely the case) approximated by a Brownian motion, as opposed to the
geometric Brownian motion used to model assets in finance. Furthermore, risk
exposure is dialled "down" via reinsurance, rather than "up" via risky
investments. This leads to interesting qualitative differences in the optimal
designs.
  In this paper, using the martingale method, we derive the optimal design as a
function of proportional, non-cheap reinsurance design that maximises the
quadratic utility of the terminal value of the insurance surplus. We also
consider several realistic constraints on the terminal value: a strict lower
boundary, the probability (Value at Risk) constraint, and the expected
shortfall (conditional Value at Risk) constraints under the $\mathbb{P}$ and
$\mathbb{Q}$ measures, respectively. In all cases, the optimal reinsurance
designs boil down to a combination of proportional protection and option-like
protection (stop-loss) of the residual proportion with various deductibles.
Proportions and deductibles are set such that the initial capital is fully
allocated. Comparison of the optimal designs with the optimal portfolios in
finance is particularly interesting. Results are illustrated.
