Compare and contrast the approaches of Basel II and Solvency II in assessing and managing the financial risks faced by banks and insurance companies, respectively. Specifically:

Describe Pillar $1$ established by Basel II for determining the minimum capital requirements for banks with respect to credit, market, and operational risks. Elaborate on the methods banks can use under each type of risk to calculate their capital requirements. $(5$ marks$)$

Explain how Solvency II regulates the capital requirements of insurance companies, focusing on the levels of capital requirements and the types of risks that must be accounted for. $(5$ marks$)$

Highlight the key differences in the risk categories and the calculation methodologies of capital requirements between Basel II for banks and Solvency II for insurance companies. Discuss the reasons behind these differences in regulatory approaches and the implications they have on the stability and risk management of each sector. $(5$ marks$)$