For example during the world war with all the sunken merchant ships which I'm assuming where owned by private companies were shot down who assumed the cost of the lost property? Was it the government or the owning company. If it was the company why would companies be willing to lose that much money (I know patriotic spirit and all that, but still.)
Well, both, sort of... Merchant ships would be insured, and in the event of a loss, the insurance company would be covering the losses. Obviously, however, in war time, premiums are going to rise considerably, which means that the owner is paying higher operating costs. And likewise, an insurance company faces risks that could be potentially crippling in a wartime situation, so often will decline to insure for certain situations, or else raised rates by 400 percent or more, which requires the government to step in and be the insurer. I don't have anything that talks about World War II, but in World War I, many Merchant Marine ships were thus insured through the American Bureau of War Risk Insurance, constituted in late 1914:
Whereas the foreign commerce of the United States is now greatly impeded and endangered through the absence of adequate facilities for the insurance of American vessels and their cargoes against the risks of war; and whereas it is deemed necessary and expedient that the United States shall temporarily provide for the export shipping trade of the United States adequate facilities for the insurance of its commerce against the risks of war; therefore be it enacted, etc.
It would be later amended to also provide life insurance policies for the sailors on board. After the war, the Bureau brought many indemnity claims against Germany in the Mixed Claims Commission, which dealt with various financial obligations of Germany, to recover the costs.
Sovereignty at Sea: US Merchant Ships and American Entry into World War I by Rodney Carlisle
Purpose and Scope of War Risk Insurance by Samuel McCune Lindsay - Very in depth article on the War Risk Insurance