Mortgage calculator

Last updated August 04, 2024

Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. Mortgage calculators are used by consumers to determine monthly repayments, and by mortgage providers to determine the financial suitability of a home loan applicant.^[2] Mortgage calculators are frequently on for-profit websites, though the Consumer Financial Protection Bureau has launched its own public mortgage calculator.^[3]^{: 1267, 1281–83}

The major variables in a mortgage calculation include loan principal, balance, periodic compound interest rate, number of payments per year, total number of payments and the regular payment amount. More complex calculators can take into account other costs associated with a mortgage, such as local and state taxes, and insurance.

Mortgage calculation capabilities can be found on financial handheld calculators such as the HP-12C or Texas Instruments TI BA II Plus. There are also multiple free online free mortgage calculators, and software programs offering financial and mortgage calculations.

Uses

When purchasing a new home, most buyers choose to finance a portion of the purchase price via the use of a mortgage. Prior to the wide availability of mortgage calculators, those wishing to understand the financial implications of changes to the five main variables in a mortgage transaction were forced to use compound interest rate tables. These tables generally required a working understanding of compound interest mathematics for proper use. In contrast, mortgage calculators make answers to questions regarding the impact of changes in mortgage variables available to everyone.

Mortgage calculators can be used to answer such questions as:

If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? The answer is $2,142.42.

A potential borrower can use an online mortgage calculator to see how much property he or she can afford. A lender will compare the person's total monthly income and total monthly debt load. A mortgage calculator can help to add up all income sources and compare this to all monthly debt payments.^{[ citation needed ]} It can also factor in a potential mortgage payment and other associated housing costs (property taxes, homeownership dues, etc.). One can test different loan sizes and interest rates. Generally speaking, lenders do not like to see all of a borrower's debt payments (including property expenses) exceed around 40% of total monthly pretax income. Some mortgage lenders are known to allow as high as 55%.

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. The monthly payment formula is based on the annuity formula. The monthly payment c depends upon:

r - the monthly interest rate. Since the quoted yearly percentage rate is not a compounded rate, the monthly percentage rate is simply the yearly percentage rate divided by 12. For example, if the yearly percentage rate was 6% (i.e. 0.06), then r would be $0.06/12$ or 0.5% (i.e. 0.005).
N - the number of monthly payments, called the loan's term, and
P - the amount borrowed, known as the loan's principal .

In the standardized calculations used in the United States, c is given by the formula:^[4]

c={\begin{cases}{\frac {rP}{1-(1+r)^{-N}}}={\frac {rP(1+r)^{N}}{(1+r)^{N}-1}},&r\neq 0;\\{\frac {P}{N}},&r=0.\end{cases}}

For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is $P=200000$ , the monthly interest rate is $r=0.065/12$ , the number of monthly payments is $N=30\cdot 12=360$ , the fixed monthly payment equals $1,264.14. 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 these formulas:

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

The following derivation of this formula 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. This fact results in the debt schedule:

Amount owed ...	Formula
at initiation	$P$
after 1 month	$(1+r)P-c$
after 2 months	$(1+r)((1+r)P-c)-c=(1+r)^{2}P-(1+(1+r))c$
⋮	⋮
after N months	$(1+r)^{N}P-(1+(1+r)+(1+r)^{2}+\cdots +(1+r)^{N-1})c$

The polynomial $p_{N}(x)=1+x+x^{2}+\cdots +x^{N-1}$ appearing before the fixed monthly payment c (with $x=1+r$ ) is a geometric series, which has a simple closed-form expression obtained from observing that $xp_{N}(x)-p_{N}(x)=x^{N}-1$ because all but the first and last terms in this difference cancel each other out. Therefore, solving for $p_{N}(x)$ yields the much simpler closed-form expression

p_{N}(x)=1+x+x^{2}+\cdots +x^{N-1}={\frac {x^{N}-1}{x-1}}

.

Applying this formula to the amount owed at the end of the Nth month gives (using $p_{N}$ to succinctly denote the function value $p_{N}(x)$ at argument value $x=(1+r)$ ):

Amount owed at end of month N

{\begin{aligned}&{}=(1+r)^{N}P-p_{N}c\\&{}=(1+r)^{N}P-{\frac {(1+r)^{N}-1}{(1+r)-1}}c\\&{}=(1+r)^{N}P-{\frac {(1+r)^{N}-1}{r}}c.\end{aligned}}

The amount of the monthly payment at the end of month N that is applied to principal paydown equals the amount c of payment minus the amount of interest currently paid on the pre-existing unpaid principal. The latter amount, the interest component of the current payment, is the interest rate r times the amount unpaid at the end of month N–1. Since in the early years of the mortgage the unpaid principal is still large, so are the interest payments on it; so the portion of the monthly payment going toward paying down the principal is very small and equity in the property accumulates very slowly (in the absence of changes in the market value of the property). But in the later years of the mortgage, when the principal has already been substantially paid down and not much monthly interest needs to be paid, most of the monthly payment goes toward repayment of the principal, and the remaining principal declines rapidly.

The borrower's equity in the property equals the current market value of the property minus the amount owed according to the above formula.

With a fixed rate mortgage, the borrower agrees to pay off the loan completely at the end of the loan's term, so the amount owed at month N must be zero. For this to happen, the monthly payment c can be obtained from the previous equation to obtain:

{\begin{aligned}c&{}={\frac {r(1+r)^{N}}{(1+r)^{N}-1}}P\\&{}={\frac {r}{1-(1+r)^{-N}}}P\end{aligned}}

which is the formula originally provided. 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.

Adjustable interest rates

While adjustable-rate mortgages have been around for decades,^[5] from 2002 through 2005 adjustable-rate mortgages became more complicated as did the calculations involved.^[6] Lending became much more creative which complicated the calculations. Subprime lending and creative loans such as the “pick a payment”,^[7] “pay option”,^[8] and “hybrid” loans brought on a new era of mortgage calculations. The more creative adjustable mortgages meant some changes in the calculations to specifically handle these complicated loans. To calculate the annual percentage rates (APR) many more variables needed to be added, including: the starting interest rate; the length of time at that rate; the recast; the payment change; the index; the margins; the periodic interest change cap; the payment cap; lifetime cap; the negative amortization cap; and others.^[9] Many lenders created their own software programs, and World Savings even had contracted special calculators to be made by Calculated Industries specifically for their “pick a payment” program.^[10] However, by the late 2000s the Great Recession brought an end to many of the creative “pick-a-payment” type of loans which left many borrowers with higher loan balances over time, and owing more than their houses were worth.^[11] This also helped reduce the more complicated calculations that went along with these mortgages.

Total interest paid formula

The total amount of interest $I$ that will be paid over the lifetime of the loan is the difference of the total payment amount ( $cN$ ) and the loan principal ( $P$ ):

I=cN-P

where $c$ is the fixed monthly payment, $N$ is the number of payments that will be made, and $P$ is the initial principal balance on the loan.

The cumulative interest paid at the end of any period N can be calculated by:

(Pr-c){\frac {((1+r)^{N}-1)}{r}}+cN

Outside the U.S.

In the United Kingdom, the FCA - Financial Conduct Authority (formerly the FSA - Financial Services Authority) regulates loans secured on residential property. It does not prescribe any specific calculation method. However, it does prescribe that, for comparative purposes, lenders must display an Annual Percentage Rate as prominently as they display other rates.

In Spain, the regulatory authority (Banco de España) has issued and enforced some good practices, such as clearly advertising the Annual Percentage Rate and stating how and when payments change in variable rate mortgages.^[12]

Related Research Articles

In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be equal or more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater than tomorrow. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of borrowed funds is less than the total amount of money paid to the lender.

In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum, at a particular rate. It is distinct from a fee which the borrower may pay to the lender or some third party. It is also distinct from dividend which is paid by a company to its shareholders (owners) from its profit or reserve, but not at a particular rate decided beforehand, rather on a pro rata basis as a share in the reward gained by risk taking entrepreneurs when the revenue earned exceeds the total costs.

The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference.

In finance, a loan is the transfer of money by one party to another with an agreement to pay it back. The recipient, or borrower, incurs a debt and is usually required to pay interest for the use of the money.

Compound interest is interest accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower.

A variable-rate mortgage, adjustable-rate mortgage (ARM), or tracker mortgage is a mortgage loan with the interest rate on the note periodically adjusted based on an index which reflects the cost to the lender of borrowing on the credit markets. The loan may be offered at the lender's standard variable rate/base rate. There may be a direct and legally defined link to the underlying index, but where the lender offers no specific link to the underlying market or index, the rate can be changed at the lender's discretion. The term "variable-rate mortgage" is most common outside the United States, whilst in the United States, "adjustable-rate mortgage" is most common, and implies a mortgage regulated by the Federal government, with caps on charges. In many countries, adjustable rate mortgages are the norm, and in such places, may simply be referred to as mortgages.

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". 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.

The term annual percentage rate of charge (APR), corresponding sometimes to a nominal APR and sometimes to an effective APR (EAPR), is the interest rate for a whole year (annualized), rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc. It is a finance charge expressed as an annual rate. Those terms have formal, legal definitions in some countries or legal jurisdictions, but in the United States:

A shared appreciation mortgage often abbreviated as "SAM" is a mortgage in which the purchaser of a home shared a percentage of the appreciation in the home's value with the lender. In return, the lender agrees to charge an interest rate that is lower than the prevailing market interest rate. The lender agrees to receive some or all of the repayment of the loan in the form of a share of the increase in value of the property.

In finance, negative amortization occurs whenever the loan payment for any period is less than the interest charged over that period so that the outstanding balance of the loan increases. As an amortization method the shorted amount is then added to the total amount owed to the lender. Such a practice would have to be agreed upon before shorting the payment so as to avoid default on payment. This method is generally used in an introductory period before loan payments exceed interest and the loan becomes self-amortizing. The term is most often used for mortgage loans; corporate loans with negative amortization are called PIK loans.

Also known as the "Sum of the Digits" method, the Rule of 78s is a term used in lending that refers to a method of yearly interest calculation. The name comes from the total number of months' interest that is being calculated in a year. This is an accurate interest model only based on the assumption that the borrower pays only the amount due each month. The outcome is that more of the interest is apportioned to the first part or early repayments than the later repayments. As such, the borrower pays a larger part of the total interest earlier in the term.

In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, mortgages, medium-term notes, swaps, and forward rate agreements (FRAs). This determines the number of days between two coupon payments, thus calculating the amount transferred on payment dates and also the accrued interest for dates between payments. The day count is also used to quantify periods of time when discounting a cash-flow to its present value. When a security such as a bond is sold between interest payment dates, the seller is eligible to some fraction of the coupon amount.

An amortization calculator is used to determine the periodic payment amount due on a loan, based on the amortization process.

In banking and finance, an amortizing loan is a loan where the principal of the loan is paid down over the life of the loan according to an amortization schedule, typically through equal payments.

A mortgage loan or simply mortgage, in civil law jurisdictions 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 to raise funds for any purpose while putting a lien on the property being mortgaged. The loan is "secured" on the borrower's property through a process known as mortgage origination. This means that a legal mechanism is put into place which allows the lender to take possession and sell the secured property to pay off the loan in the event the borrower defaults on the loan or otherwise fails to abide by its terms. The word mortgage is derived from a Law French term used in Britain in the Middle Ages meaning "death pledge" and refers to the pledge ending (dying) when either the obligation is fulfilled or the property is taken through foreclosure. A mortgage can also be described as "a borrower giving consideration in the form of a collateral for a benefit (loan)".

In finance, the weighted-average life (WAL) of an amortizing loan or amortizing bond, also called average life, is the weighted average of the times of the principal repayments: it's the average time until a dollar of principal is repaid.

In economics, Present value interest factor, also known by the acronym PVIF, is used in finance theory to refer to the output of a calculation, used to determine the monthly payment needed to repay a loan. The calculation involves a number of variables, which are set out in the following description of the calculation:

Analogous to continuous compounding, a continuous annuity is an ordinary annuity in which the payment interval is narrowed indefinitely. A (theoretical) continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity.

An equated monthly installment (EMI) is a fixed payment amount made by a borrower to a lender at a specified date each calendar month. Equated monthly installments are used to pay off both interest and principal each month, so that over a specified number of years, the loan is fully paid off along with interest.

In investment, an annuity is a series of payments made at equal intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Annuities may be calculated by mathematical functions known as "annuity functions".

References

↑ "Mortgage Calculator". Archived from the original on 2022-06-19. Retrieved 2022-06-20.
↑ "How National Australia Bank circumvents rules to stop a property bubble" . Australian Financial Review. Fairfax Media. 2015-04-22. Archived from the original on 2015-04-26. Retrieved 2015-04-28.
↑ Van Loo, Rory (2017-03-01). "Rise of the Digital Regulator". Duke Law Journal. 66 (6): 1267. Archived from the original on 2020-05-30. Retrieved 2020-10-25.
↑ Kohn, Robert. "A capital budgeting model of the supply and demand of loanable funds", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430).
↑ "History of ARMs". Loantech. Archived from the original on 2007-08-27. Retrieved 2011-09-22.
↑ "Nightmare Mortgages". Business Week. Archived from the original on 2010-05-21. Retrieved 2010-05-18.
↑ Block, Sandra (2005-07-19). "'Pick-a-payment' mortgage risks are high". USA TODAY. Archived from the original on 2012-01-04. Retrieved 2011-09-22.
↑ Schoen, John W. (2008-10-12). "'Pay option' loans could swell defaults". NBC News. Archived from the original on 2013-10-12. Retrieved 2011-09-22.
↑ "Pick-a-Payment ARM ~ Using Pay Option ARM Calculator". Mortgage-X.com. Archived from the original on 2011-09-28. Retrieved 2011-09-22.
↑ "Qualifier Plus IIIpx: Pick-A-Payment Calculator". Archived from the original on 2010-03-02.
↑ Boye, Will (2008-06-30). "Wachovia eliminates pick-a-payment mortgage loans - San Francisco Business Times". Bizjournals.com. Archived from the original on 2012-10-25. Retrieved 2011-09-22.
↑ "Criterios de buenas prácticas - Préstamos hipotecarios" [Good Practice Criteria - Mortgage Loans] (in Spanish). Banco de España. Archived from the original on 2014-11-29. Retrieved 2014-11-15.

External links

Mortgage Calculator at NYTimes.com

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[1] "Mortgage Calculator". Archived from the original on 2022-06-19. Retrieved 2022-06-20.

[AFR-2] "How National Australia Bank circumvents rules to stop a property bubble" . Australian Financial Review. Fairfax Media. 2015-04-22. Archived from the original on 2015-04-26. Retrieved 2015-04-28.

[3] Van Loo, Rory (2017-03-01). "Rise of the Digital Regulator". Duke Law Journal. 66 (6): 1267. Archived from the original on 2020-05-30. Retrieved 2020-10-25.

[4] Kohn, Robert. "A capital budgeting model of the supply and demand of loanable funds", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430).

[5] "History of ARMs". Loantech. Archived from the original on 2007-08-27. Retrieved 2011-09-22.

[6] "Nightmare Mortgages". Business Week. Archived from the original on 2010-05-21. Retrieved 2010-05-18.

[7] Block, Sandra (2005-07-19). "'Pick-a-payment' mortgage risks are high". USA TODAY. Archived from the original on 2012-01-04. Retrieved 2011-09-22.

[8] Schoen, John W. (2008-10-12). "'Pay option' loans could swell defaults". NBC News. Archived from the original on 2013-10-12. Retrieved 2011-09-22.

[9] "Pick-a-Payment ARM ~ Using Pay Option ARM Calculator". Mortgage-X.com. Archived from the original on 2011-09-28. Retrieved 2011-09-22.

[10] "Qualifier Plus IIIpx: Pick-A-Payment Calculator". Archived from the original on 2010-03-02.

[11] Boye, Will (2008-06-30). "Wachovia eliminates pick-a-payment mortgage loans - San Francisco Business Times". Bizjournals.com. Archived from the original on 2012-10-25. Retrieved 2011-09-22.

[12] "Criterios de buenas prácticas - Préstamos hipotecarios" [Good Practice Criteria - Mortgage Loans] (in Spanish). Banco de España. Archived from the original on 2014-11-29. Retrieved 2014-11-15.

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