If we are given the following quantities


            P0 – loan amount to buy real estate or the “principle”

            r   - monthly interest rate

            n   - number of monthly payments


Then we can calculate the repeated monthly payment which completely repays the loan in the last month


            M – monthly payment


We first note that the principal changes each month.  Each month the principle, Pi for ith month, increases as a result of cumulative interest but is reduced an even a greater amount by the monthly payment, as follows



and so forth until the last month when the principle is zero having been paid off, as follows


If we let  then we can write





In the last month, the principal will be fully paid as follows



And we can solve directly for the monthly payment






When we borrow money to buy a house, lenders charge fees “C” to originate the loan.  These fees increase the loan amount beyond the actual cost of the house “P0” and effectively appear to the buyer as an increase in the interest rate.


This new interest rate is called the Annual Percentage Rate (APR) which we represent as “A”.   For convenience we will change the annual APR interest “A” to a monthly rate of “a”



To calculate the APR, we first compute the monthly payment “Mnew” necessary to repay the combined amount of “” in “n” installments at the original interest rate of “r”.   This is the actual payment we will have to make.



Then for this monthly payment, we compute the effectively increased monthly APR interest rate “a” by assuming the monthly payment, which we just calculated of “Mnew” was intended to repay just original principle, that is the initial cost of the real estate ”. 


We go to this trouble because we are confronted with a monthly payment of “Mnew” and the only thing of lasting value we have for our money is the real estate.  The original interest rate “r” has been corrupted.  Legislation requires the disclosure of the APR so lenders will not be able to hide, much as they would like to, their incredibly over-inflated charges which are occasionally even for non-existent services.   And do not be fooled as these charges are negotiable and frequently reduced when challenged as they are basically insupportable.  Another common fraud is “title insurance” which, believe it or not, insures nothing.


 In any event the relevant equation involving the increased effective interest rate “a”, or APR, is



Unfortunately, there is no closed form solution for this equation so we have to use the Newton-Raphson approximation to solve for “a”.  This equation can be made homogeneous in the variable “a” as follows



Rearranging terms we can define the equation f(a) and calculate its derivative as follows




We then use successive approximations starting from an initial guess “a0” for the APR.  For each approximation “ai“ we use Newton-Raphson steepest descents to get an even better approximations “ai+1“ as follows






We can get an initial guess of “a0” by considering the basic definition of APR



Then if we let   and recall that



so that for


Then rewriting the above equation we have



And we can solve for “” as follows



Or finally, the improved starting value “a0“ for Newton-Raphson is



Note that this is always slightly less (typically on the order of 10%) than the final value.





Please note that the Newton-Raphson method only converges for starting values of “a0” which are reasonably close to the exact answer.   Otherwise successive approximations may diverge to ever greater incorrect values.   In particular, Newton-Raphson requires that the slope of the curve f(a) not change sign between the initial guess and the final exact solution.   In our case the function f(a) has a local maximum where its slope is zero, that is



For even larger values of “a > astable”, the function f(a) uniformly decreases.  A requirement for convergence is thus that the starting value of “a0” must be greater than this stable value; and twice that value would not be an unreasonable starting point. 


The expression for the smallest allowed value is



The question arises as to whether the improved starting value for “a0” is stable, or



This can be further reduced to



And continuing



But we note from the binominal expansion by expanding the first few terms



And that




So that the equality reduces to



which for positive values of P0, C, n, and r is always true.  And so we note the starting value given above is always in the range of stability (“a0 > astable”).