Quantitative risk management is arguably the most fundamental step in every risk management process. It builds the theoretical foundation to quantify and analyze risks in every company. In the present thesis, each chapter contributes to the literature by dealing with aspects of fundamental problems and discussions in quantitative risk management. Special emphasis is put on three major topics: First, on the shortfall probability resp. the risk measure concepts in the context of portfolio allocation problems. Then on the impact of periodic premium payments on pricing, the shortfall probability and the optimal portfolio allocation of an insurance contract. Finally, on time-inconsistent optimal portfolio allocations that stem from uncertainty aspects because of different market states (regimes).