Title: An optimization approach to adaptive multi-dimensional capital management,
Abstract: Firms should keep capital to offer sufficient protection against the risks
they are facing. In the insurance context methods have been developed to
determine the minimum capital level required, but less so in the context of
firms with multiple business lines including allocation. The individual capital
reserve of each line can be represented by means of classical models, such as
the conventional Cramér-Lundberg model, but the challenge lies in soundly
modelling the correlations between the business lines. We propose a simple yet
versatile approach that allows for dependence by introducing a common
environmental factor. We present a novel Bayesian approach to calibrate the
latent environmental state distribution based on observations concerning the
claim processes. The calibration approach is adjusted for an environmental
factor that changes over time. The convergence of the calibration procedure
towards the true environmental state is deduced. We then point out how to
determine the optimal initial capital of the different business lines under
specific constraints on the ruin probability of subsets of business lines. Upon
combining the above findings, we have developed an easy-to-implement approach
to capital risk management in a multi-dimensional insurance risk model.