# An optimization approach to adaptive multi-dimensional capital management

## TL;DR

This paper introduces a Bayesian calibration method for modeling dependencies in multi-line insurance firms, enabling optimal capital allocation considering correlated risks and environmental factors.

## Contribution

It presents a novel Bayesian approach to calibrate environmental states and determine optimal capital levels in multi-dimensional insurance models with dependent risks.

## Key findings

- Calibration converges to true environmental state
- Method effectively models dependence between business lines
- Enables optimal capital allocation under ruin probability constraints

## 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 Cram\'{e}r-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.