  We develop quantile regression models in order to derive risk margin and to
evaluate capital in non-life insurance applications. By utilizing the entire
range of conditional quantile functions, especially higher quantile levels, we
detail how quantile regression is capable of providing an accurate estimation
of risk margin and an overview of implied capital based on the historical
volatility of a general insurers loss portfolio. Two modelling frameworks are
considered based around parametric and nonparametric quantile regression models
which we develop specifically in this insurance setting.
  In the parametric quantile regression framework, several models including the
flexible generalized beta distribution family, asymmetric Laplace (AL)
distribution and power Pareto distribution are considered under a Bayesian
regression framework. The Bayesian posterior quantile regression models in each
case are studied via Markov chain Monte Carlo (MCMC) sampling strategies.
  In the nonparametric quantile regression framework, that we contrast to the
parametric Bayesian models, we adopted an AL distribution as a proxy and
together with the parametric AL model, we expressed the solution as a scale
mixture of uniform distributions to facilitate implementation. The models are
extended to adopt dynamic mean, variance and skewness and applied to analyze
two real loss reserve data sets to perform inference and discuss interesting
features of quantile regression for risk margin calculations.
