Building Society initially keeps bad accounts (5)
I believe the answer is:
bills
'accounts' is the definition.
(billing is a kind of accounting)
'building society initially keeps bad' is the wordplay.
'initially' suggests taking the first letters.
'keeps' means one lot of letters goes inside another.
'bad' becomes 'ill' ('ill' can be similar in meaning to 'bad'**).
The first letters of 'building society' is 'bs'.
'bs' enclosing 'ill' is 'BILLS'.
(Other definitions for bills that I've seen before include "Draft laws" , "Invoices" , "Beaks" .)