**MORTGAGES**

**HOW TO COMPUTE A MONTHLY MORTGAGE PAYMENT**

If we are given the following quantities

P_{0} – 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, P_{i} for i^{th} 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

** **

**ANNUAL PERCENTAGE RATE**

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 “P_{0}” 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
“M_{new}” 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 “M_{new}” 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 “M_{new}” 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 “a_{0}” for the APR. For each approximation “a_{i}“ we
use Newton-Raphson steepest descents to get an even better approximations “a_{i+1}“
as follows

**STARTING VALUE**

We can get an initial guess of “a_{0}” 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 “a_{0}“ for
Newton-Raphson is

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

**STABILITY OF THE “APR” CALCULATION**

Please note that the Newton-Raphson method only converges for
starting values of “a_{0}” 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 > a_{stable}”, the
function f(a) uniformly decreases. A requirement for convergence is thus that the
starting value of “a_{0}” 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 “a_{0}” 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 P_{0}, C, n, and r is
always true. And so we note the starting value given above is always in the
range of stability (“a_{0 }> a_{stable}”).