While technically true, this is misleading. The numbers are still the same whether or not the government runs the bank. The private bank would lose 1 cent, as you mentioned, but no matter what the government is still getting the $2 in taxable revenue resulting in 20 cents to the government at the 10% rate.
Then you also need to factor in that banks have operating costs. In the case where the government gets both the tax revenue and profits (or losses) of the loan, this means that the only way for the government to make this work is if the government can run the bank more efficiently than the private bank would be run. Or, at least equal. Otherwise, the additional operating cost is a net loss to society from the inefficiency.
I don't know about you, but I would never, ever count on the government running anything more efficiently than the private sector. This is a bad idea.