Fixed rate mortgage

A fixed rate mortgage (FRM) is a mortgage loan where the interest rate on the note remains the same through the term of the loan, as opposed to loans where the interest rate may adjust or "float." Other forms of mortgage loan include interest only mortgage, graduated payment mortgage, variable rate (including adjustable rate mortgages and tracker mortgages), negative amortization mortgage, and balloon payment mortgage. Please note that each of the loan types above except for a straight adjustable rate mortgage can have a period of the loan for which a fixed rate may apply. A Balloon Payment mortgage, for example, can have a fixed rate for the term of the loan followed by the ending balloon payment. Terminology may differ from country to country: loans for which the rate is fixed for less than the life of the loan may be called hybrid adjustable rate mortgages (in the United States).
This payment amount is independent of the additional costs on a home sometimes handled in escrow, such as property taxes and property insurance. Consequently, payments made by the borrower may change over time with the changing escrow amount, but the payments handling the principal and interest on the loan will remain the same.
Fixed rate mortgages are characterized by their interest rate (including compounding frequency, amount of loan, and term of the mortgage). With these three values, the calculation of the monthly payment can then be done.

Monthly payment formula

The fixed monthly payment for a fixed rate mortgage is the amount paid by the borrower every month that ensures that the loan is paid off in full with interest at the end of its term. This monthly payment c depends upon the monthly interest rate r (expressed as a fraction, not a percentage, i.e., divide the quoted yearly nominal percentage rate by 100 and by 12 to obtain the monthly interest rate), the number of monthly payments N called the loan's term, and the amount borrowed P0 known as the loan's principal; rearranging the formula for the present value of an ordinary annuity we get the formula for c:

c = (r / (1 − (1 + r) − N))P0

For example, for a home loan for $200,000 with a fixed yearly nominal interest rate of 6.5% for 30 years, the principal is P0 = 200000, the monthly interest rate is r = 6.5 / 100 / 12, the number of monthly payments is N = 30 * 12 = 360, the fixed monthly payment equals $1264.14. This formula is provided using the financial function PMT in a spreadsheet such as Excel. In the example, the monthly payment is obtained by entering either of the these formulas:

=PMT(6.5/100/12,30*12,200000)
=((6.5/100/12)/(1-(1+6.5/100/12)^(-30*12)))*200000
= 1264.14

This monthly payment formula is easy to derive, and the derivation illustrates how fixed-rate mortgage loans work. The amount owed on the loan at the end of every month equals the amount owed from the previous month, plus the interest on this amount, minus the fixed amount paid every month.

Amount owed at month 0:
P0
Amount owed at month 1:
P1 = P0 + P0 * r − c ( principle + interest - payment)
P1 = P0(1 + r) − c (equation 1)
Amount owed at month 2:
P2 = P1(1 + r) − c
Using equation 1 for P1
P2 = (P0(1 + r) − c)(1 + r) − c
P2 = P0(1 + r)2 − c(1 + r) − c (equation 2)
Amount owed at month 3:
P3 = P2(1 + r) − c
Using equation 2 for P2
P3 = (P0(1 + r)2 − c(1 + r) − c)(1 + r) − c
P3 = P0(1 + r)3 − c(1 + r)2 − c(1 + r) − c
Amount owed at month N:
PN = PN − 1(1 + r) − c
PN = P0(1 + r)N − c(1 + r)N − 1 − c(1 + r)N − 2.... − c
PN = P0(1 + r)N − c((1 + r)N − 1 + (1 + r)N − 2.... + 1)
PN = P0(1 + r)N − c(S) (equation 3)
Where S = (1 + r)N − 1 + (1 + r)N − 2.... + 1 (equation 4) (see geometric progression)
S(1 + r) = (1 + r)N + (1 + r)N − 1.... + (1 + r) (equation 5)
With the exception of two terms the S and S(1 + r) series are the same so when you subtract all but two terms cancel:
Using equation 4 and 5
S(1 + r) − S = (1 + r)N − 1
S((1 + r) − 1) = (1 + r)N − 1
S(r) = (1 + r)N − 1
S = ((1 + r)N − 1) / r (equation 6)
Putting equation 6 back into 3:
PN = P0(1 + r)N − c(((1 + r)N − 1) / r)
PN will be zero because we have paid the loan off.
0 = P0(1 + r)N − c(((1 + r)N − 1) / r)
We want to know c
c = (r(1 + r)N / ((1 + r)N − 1))P0
Divide top and bottom with (1 + r)N
c = (r / (1 − (1 + r) − N))P0

This derivation illustrates three key components of fixed-rate loans: (1) the fixed monthly payment depends upon the amount borrowed, the interest rate, and the length of time over which the loan is repaid; (2) the amount owed every month equals the amount owed from the previous month plus interest on that amount, minus the fixed monthly payment; (3) the fixed monthly payment is chosen so that the loan is paid off in full with interest at the end of its term and no more money is owed.