Definitions

# Fixed rate mortgage

[fikst-reyt]
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, adjustable rate mortgage, 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 $P_0$ known as the loan's principal; rearranging the formula for the present value of an ordinary annuity we get the formula for $c$:

$c = \left(r/\left(1-\left(1+r\right)^\left\{-N\right\}\right)\right)P_0$
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 $P_0=200000$, 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
$\left\{\right\}=1264.14$

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:
$P_0$
Amount owed at month 1:
$P_1 = P_0+P_0*r-c$ (principle + interest - payment)
$P_1 = P_0\left(1+r\right)-c$ (equation 1)
Amount owed at month 2:
$P_2 = P_1\left(1+r\right)-c$
Using equation 1 for $P_1$
$P_2 = \left(P_0\left(1+r\right)-c\right)\left(1+r\right)-c$
$P_2 = P_0\left(1+r\right)^2- c\left(1+r\right)- c$ (equation 2)
Amount owed at month 3:
$P_3 = P_2\left(1+r\right) - c$
Using equation 2 for $P_2$
$P_3 = \left(P_0\left(1+r\right)^2- c\left(1+r\right)- c\right)\left(1+r\right) - c$
$P_3 = P_0\left(1+r\right)^3- c\left(1+r\right)^2- c\left(1+r\right) - c$
Amount owed at month N:
$P_N = P_\left\{N-1\right\}\left(1+r\right) - c$
$P_N = P_0\left(1+r\right)^N - c\left(1+r\right)^\left\{N-1\right\} - c\left(1+r\right)^\left\{N-2\right\} .... - c$
$P_N = P_0\left(1+r\right)^N - c \left(\left(1+r\right)^\left\{N-1\right\} + c\left(1+r\right)^\left\{N-2\right\} .... + 1\right)$
$P_N = P_0\left(1+r\right)^N - c \left(S\right)$ (equation 3)
Where $S = \left(1+r\right)^\left\{N-1\right\} + c\left(1+r\right)^\left\{N-2\right\} .... + 1$ (equation 4) (see geometric progression)
$S\left(1+r\right) = \left(1+r\right)^N + c\left(1+r\right)^\left\{N-1\right\} .... + \left(1+r\right)$ (equation 5)
With the exception of two terms the $S$ and $S\left(1+r\right)$ series are the same so when you subtract all but two terms cancel:
Using equation 4 and 5
$S\left(1+r\right)-S = \left(1+r\right)^N - 1$
$S\left(\left(1+r\right)-1\right) = \left(1+r\right)^N - 1$
$S\left(r\right) = \left(1+r\right)^N - 1$
$S = \left(\left(1+r\right)^N - 1\right)/r$ (equation 6)
Putting equation 6 back into 3:
$P_N = P_0\left(1+r\right)^N - c \left(\left(\left(1+r\right)^N - 1\right)/r\right)$
$P_N$ will be zero because we have paid the loan off.
$0 = P_0\left(1+r\right)^N - c \left(\left(\left(1+r\right)^N - 1\right)/r\right)$
We want to know $c$
$c = \left(r\left(1+r\right)^N/\left(\left(1+r\right)^N-1\right)\right)P_0$
Divide top and bottom with $\left(1+r\right)^N$
$c= \left(r/\left(1-\left(1+r\right)^\left\{-N\right\}\right)\right)P_0$

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.

## Characteristics

### Index

Unlike adjustable rate mortgages, fixed rate mortgages are not tied to an index. Instead, the interest rate is set (or "fixed") in advance to an advertised rate, usually in increments of 1/4 or 1/8 percent.

## Terminology

• Fully Indexed Rate—The price of the FRM as calculated by adding Index + Margin = Fully Indexed Rate. This is the interest rate for the life of the loan.
• Term—The length of time of the loan. The number of payments is independent of this term, so a 30-year term would have 30 payments for a yearly payment plan, but 360 payments for a common monthly plan.

## Popularity

Fixed rate mortgages are the most classic form of loan for home and product purchasing in the United States. The most common terms are 15-year and 30-year mortgages, but shorter terms are available, and 40-year and 50-year mortgages are now available (common in areas with high priced housing, where even a 30-year term leaves the mortgage amount out of reach of the average family).

Outside the United States, fixed-rate mortgages are less popular, and in some countries, true fixed-rate mortgages are not available except for shorter-term loans. For example, in Canada the longest term for which a mortgage rate can be fixed is typically no more than ten years, while mortgage maturities are commonly 25 years. In Australia banks are unable to offer fixed rates for terms longer than 15 years due to funding constraints.

## Pricing

Fixed rate mortgages are usually more expensive than adjustable rate mortgages. Due to the inherent interest rate risk, long-term fixed rate loans will tend to be at a higher interest rate than short-term loans. The difference in interest rates between short and long-term loans is known as the yield curve, which generally slopes upward (longer terms are more expensive). The opposite circumstance is known as an inverted yield curve and is relatively infrequent.

The fact that a fixed rate mortgage has a higher starting interest rate does not indicate that this is a worse form of borrowing compared to the adjustable rate mortgages. If interest rates rise, the ARM cost will be higher while the FRM will remain the same. In effect, the lender has agreed to take the interest rate risk on a fixed rate loan. Some studies have shown that the majority of borrowers with adjustable rate mortgages save money in the long term, but that some borrowers pay more. The price of potentially saving money, in other words, is balanced by the risk of potentially higher costs. In each case, a choice would need to be made based upon the loan term, the current interest rate, and the likelihood that the rate will increase or decrease during the life of the loan.

## Prepayment

In the United States, fixed rate mortgages, like other types of mortgage, may offer the ability to prepay principal (or capital) early without penalty. Early payments of part of the principal will reduce the total cost of the loan (total interest paid), and will shorten the amount of time needed to pay off the loan. Early payoff of the entire loan amount through refinancing is sometimes done when interest rates drop significantly.

Some mortgages may offer a lower interest rate in exchange for the borrower accepting a prepayment penalty.

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