Yes, central banks cannot go bust

Central banks can, and some do, incur losses without running the risk of going bankrupt. This is already widely known among central bankers.

Still, making this point was worth a story on Bloomberg, referring to an ECB paper on central banks profits and losses:

Central banks cannot run out of money because they are the ones that create the money. And you cannot run out of something you can create yourself.

This is how you basically translate into plain words a footnote included in the ECB's paper, which said - in technical, expert lingo - the following:

Central banks are protected from insolvency due to their ability to create money and can therefore operate with negative equity.

Sure, as it goes high credibility and sound reputation are more valuable to a central bank than an accounting number. As Czech Governor Miroslav Singer once said:

... numerous central banks have negative equity yet have no problems operating if they are credible enough. The real capital of a central bank is its credibility and reputation, not a figure on its balance-sheet. Credibility and reputation are virtues that thrive on consistent and credible policies and communication.