Fixed rate mortgage

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A fixed-rate mortgage (FRM) is a
mortgage loan A mortgage loan or simply mortgage (), in civil law (legal system), civil law jurisdicions known also as a hypothec loan, is a loan used either by purchasers of real property to raise funds to buy real estate, or by existing property owners ...
where the
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
on the
note Note, notes, or NOTE may refer to: Music and entertainment * Musical note In music, a note is the representation of a musical sound. Notes can represent the Pitch (music), pitch and Duration (music), duration of a sound in musical notation. A ...
remains the same through the term of the loan, as opposed to loans where the interest rate may adjust or "float". As a result, payment amounts and the duration of the loan are fixed and the person who is responsible for paying back the loan benefits from a consistent, single payment and the ability to plan a budget based on this fixed cost. Other forms of mortgage loans include interest only mortgage, graduated payment mortgage, variable rate mortgage (including
adjustable-rate mortgage A variable-rate mortgage, adjustable-rate mortgage (ARM), or tracker mortgage is a mortgage loan with the interest rate on the mortgage note, note periodically adjusted based on an index which reflects the cost to the lender of borrowing on the cre ...
s and
tracker mortgage A variable-rate mortgage, adjustable-rate mortgage (ARM), or tracker mortgage is a mortgage loan with the interest rate on the mortgage note, note periodically adjusted based on an index which reflects the cost to the lender of borrowing on the cre ...
s), negative amortization mortgage, and
balloon payment mortgage A balloon payment mortgage is a mortgage loan, mortgage which does not fully amortization (business), amortize over the term of the mortgage note, note, thus leaving a balance due at Maturity (finance), maturity.Wiedemer, John P, ''Real Estate Fin ...
. Unlike many other loan types, FRM interest payments and loan duration is fixed from beginning to end. Fixed-rate mortgages are characterized by amount of loan,
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
, compounding frequency, and duration. With these values, the monthly repayments can be calculated. Fixed-rate mortgages are vulnerable to inflation risk, which means that borrowers with such mortgages are better off under unexpectedly high inflation (as the inflation lowers the real present value of their loan repayments), while they are worse off if there is a drop in inflation that lowers interest rates. Fixed-rate mortgages usually charge higher interest rates than those with adjustable rates. According to scholars, "borrowers should generally prefer adjustable-rate over fixed-rate mortgages, unless interest rates are low."

# Overview

Unlike
adjustable-rate mortgage A variable-rate mortgage, adjustable-rate mortgage (ARM), or tracker mortgage is a mortgage loan with the interest rate on the mortgage note, note periodically adjusted based on an index which reflects the cost to the lender of borrowing on the cre ...
s (ARM), 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. 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.

# Usage throughout the world

The availability of fixed-rate mortgages varies between countries. In the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...
,
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...
Federal Housing Administration The Federal Housing Administration (FHA), also known as the Office of Housing within the Department of Housing and Urban Development The United States Department of Housing and Urban Development (HUD) is one of the executive departments o ...
(FHA) helped develop and standardize the fixed rate mortgage as an alternative to the
balloon payment mortgage A balloon payment mortgage is a mortgage loan, mortgage which does not fully amortization (business), amortize over the term of the mortgage note, note, thus leaving a balance due at Maturity (finance), maturity.Wiedemer, John P, ''Real Estate Fin ...
by insuring them and by doing so helped the mortgage design garner usage. Because of the large payment at the end of the older, balloon-payment loan,
refinancing risk Refinancing risk, in banking and finance, is the possibility that a borrower cannot refinance by borrowing to repay existing debt Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to anoth ...
resulted in widespread foreclosures. The fixed-rate mortgage was the first mortgage loan that was fully amortized (fully paid at the end of the loan) precluding successive loans, and had fixed interest rates and payments. Fixed-rate mortgages are the most classic form of loan for home and product purchasing in the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...
. 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). 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 Denmark, fixed-rate 30-year mortgages are the standard form of home loan. A fixed rate mortgage in Singapore has the interest rate fixed for only the first three to five years of the loan, and it then becomes variable. In Australia, "honeymoon" mortgages with introductory rates are common, but can last as short as a year, and may instead offer a fixed reduction in interest rate rather than a fixed rate itself. Furthermore, they are often combined with properties of flexible mortgages to create what is known as an Australian mortgage, which often allow borrowers to overpay to reduce interest charges and then draw on these overpayments in the future. The mortgage industry of the United Kingdom has traditionally been dominated by
building societies A building society is a financial institution owned by its members as a mutual organization. Building societies offer banking and related financial services Financial services are the Service (economics), economic services provided by the ...
, whose raised funds must be at least 50% deposits, so lenders prefer variable-rate mortgages to fixed-rate mortgages to reduce asset–liability mismatch due to
interest rate risk In finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of Production (economics), production, Distribution (economics), distribution, and Consumpt ...
. Lenders, in turn, influence consumer decisions which already prefer lower initial monthly payments. Nationwide Commercial recently issued a 30-year fixed rate mortgage as bridging finance.

# Comparisons

Fixed-rate mortgages are usually more expensive than adjustable rate mortgages. The inherent
interest rate risk In finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of Production (economics), production, Distribution (economics), distribution, and Consumpt ...
makes long-term fixed rate loans tend to have a higher interest rate than short-term loans. The relationship between interest rates for short and long-term loans is represented by the
yield curve In finance, the yield curve is a graph which depicts how the Yield to maturity, yields on debt instruments - such as bonds - vary as a function of their years remaining to Maturity (finance), maturity. Typically, the graph's horizontal or ...
, which generally slopes upward (longer terms are more expensive). The opposite circumstance is known as an
inverted yield curve In finance, an inverted yield curve happens when a yield curve graph of typically government bonds inverts in the opposite direction and the shorter term United States Treasury security, US Treasury bonds are offering a higher Yield (economi ...
and occurs less often. The fact that a fixed-rate mortgage has a higher starting interest rate does not indicate that it is a worse type of borrowing than an adjustable-rate mortgage. If interest rates rise, the ARM will cost more, but the FRM will cost the same. In effect, the lender has agreed to take the
interest rate risk In finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of Production (economics), production, Distribution (economics), distribution, and Consumpt ...
on a fixed-rate loan. Some studieshttp://www.ifid.ca/pdf_workingpapers/WP2001A.pdf have shown that the majority of borrowers with adjustable rate mortgages save money in the long term but also 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.

# Pricing

*''Note'': Fixed-rate mortgage interest may be compounded differently in other countries, such as in Canada, where it is compounded every 6 months. 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 An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
$r$ (expressed as a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, 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 $c$: ::$c = P_0$ 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 $N=30\cdot 12=360$, the fixed monthly payment $c=\1264.14$. This formula is provided using the financial function in a
spreadsheet A spreadsheet is a computer application for computation, organization, analysis and Data storage, storage of data in table (information), tabular form. Spreadsheets were developed as computerized analogs of paper accounting Worksheet#Accounting, ...
such as
Excel ExCeL London (an abbreviation for Exhibition Centre London) is an exhibition centre, international convention centre and former hospital in the Custom House A custom house or customs house was traditionally a building housing the offices ...
. In the example, the monthly payment is obtained by entering either of these formulas: :: :: ::$=1264.14$ 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: :::$P_1 = P_0+P_0*r-c$ ( principal + 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: :::$P_N = P_\left(1+r\right) - c$ :::$P_N = P_0\left(1+r\right)^N - c\left(1+r\right)^ - c\left(1+r\right)^ .... - c$ :::$P_N = P_0\left(1+r\right)^N - c \left(\left(1+r\right)^ + \left(1+r\right)^ .... + 1\right)$ :::$P_N = P_0\left(1+r\right)^N - c \left(S\right)$ (equation 3) :::Where $S = \left(1+r\right)^ + \left(1+r\right)^ .... + 1$ (equation 4) (see
geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
) :::$S\left(1+r\right) = \left(1+r\right)^N + \left(1+r\right)^ .... + \left(1+r\right)$ (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 :::$S\left(1+r\right)-S = \left(1+r\right)^N - 1$ :::$S\left(\left(1+r\right)-1\right) = \left(1+r\right)^N - 1$ :::$S\left(r\right) = \left(1+r\right)^N - 1$ :::$S =$ (equation 6) ::Putting equation 6 back into 3: :::$P_N = P_0\left(1+r\right)^N - c$ ::$P_N$ will be zero because we have paid the loan off. :::$0 = P_0\left(1+r\right)^N - c$ ::We want to know $c$ :::$c = P_0$ ::Divide top and bottom with $\left(1+r\right)^N$ :::$c = P_0$ 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. The vendor may sell off the fix element as a "fixed to floating" derivative. Look up "Black-Scholes" to see how this is done.

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Adjustable-rate mortgage A variable-rate mortgage, adjustable-rate mortgage (ARM), or tracker mortgage is a mortgage loan with the interest rate on the mortgage note, note periodically adjusted based on an index which reflects the cost to the lender of borrowing on the cre ...
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Singapore Swap Offer Rate (SOR) Singapore Dollar Swap Offer Rate (SOR) is an implied interest rate, determined by examining the spot and forward foreign exchange rate between the US dollar (USD) and Singapore dollar (SGD) and the appropriate US dollar interest rate for the ...
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SIBOR SIBOR stands for Singapore Interbank Offered Rate
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VA loan *
FHA loan An FHA insured loan is a US Federal Housing Administration The Federal Housing Administration (FHA), also known as the Office of Housing within the Department of Housing and Urban Development The United States Department of Housing and ...