Do you 'own' Bitcoins purchased via an exchange? I have been told by some that bitcoin exchanges do NOT record BTC transactions by a buyer/seller on the blockchain, but rather hold them in their own name and ledger - 
eg, if I buy one BTC via an exchange, I do not own a UTXO on the blockchain, but simply have been 'assigned' a BTC that the exchange owns.  

This of course, would give rise to counterparty risk, even ignoring fraud risk. 

Truth or fiction?
Most exchanges are centralized services, operating in the way you've mentioned. So yes, the exchange owns the bitcoins, so there is significant counterparty risk involved with storing your coins on an exchange.