My Cayman company only holds investments. Does it still file with the DITC?

Yes. Exempted companies file an economic substance notification with the Department for International Tax Cooperation each year, even when the answer is that they carry on no relevant activity. A company that does carry on one, such as holding, headquarters or financing business, also files an economic substance return showing its activity in Cayman (confirm the filing dates for your year end with your specialist).

Source: Economic substance, Department for International Tax Cooperation, Cayman Islands