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 excel's 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 p t is given by the equation r interest rate t time 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