# Math

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My monthly mortgage payments are \$4,369.66 for 30 years at 8%, what is the original cost of the house?

• Math -

We need to know how often interest is accrued. It is not always equal to the payment frequency. Some banks compound every 3 months, some six months, and some a year.

For lack of information, we will assume that interest is compounded monthly, which simplifies the calculations.

To do the calculations, we assume:
A=amount of payment per period (month) = \$4369.66
P=principal, amount borrowed
i=interest per period (month)=0.08/12
n=number of periods (month) = 30*12=360

We equate the future value of the amount borrowed and the future value of the payments, as follows:

The first payment is assumed to be made at the end of the first period.

P(1+i)^n
=A(1+i)^(n-1)+A(1+i)^(n-2)...+A(1+i)^1+A(1+i)^0

The last term represents the last payment.

The right hand side factorizes to:
A((1+i)^n -1)/(1+i-1)
=A((1+i)^n -1)/i

So the whole equation becomes:

P(1+i)^n=A((1+i)^n -1) /i

Which means that
P=A((1+i)^n -1)/[i×(1+i)^n]

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