I think the formula is correct as it stands You can check it out on some test data eg loan of  million for  years   monthly Formula gives half life as  years near as d to  months and using Excels PV function for the remainder  months we are a tad short of half way at  With the minus sign in we get  years  oh dear Derivation is from a continuous time mortgage model where the balance on a mortgage Pt is given by the equation     rinterest rate Ttime period of loan and P is the original loan amount Once again this is something I have derived for myself but no doubt also once again there are more authoritative sources