Mortgage calculator

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Mortgage calculators are used to help a current or potential real estate owner determine how much they can afford to borrow to purchase a piece of real estate. Mortgage calculators can also be used to compare the costs or real [[interest rates between several different loans, determine the impact on the length of the mortgage loan of making added principal payments or bi-weekly instead of monthly payments. A mortgage calculator is an automated tool that enables the user to quickly determine the financial implications of changes in one or more variables in a mortgage financing arrangement. The major variables include loan principal balance, periodic interest rate compound interest, number of payments per year, total number of payments and the regular payment amount.

Mortgage calculator capability can be found on most financial calculators such as the HP-12C, in most desktop spreadsheet programs such as Microsoft Excel and on the Web.

Contents

Uses

When purchasing a new home most buyers choose to finance a portion of the purchase price via the use of 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 I borrow $250,000 at a 7% annual interest rate and pay the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and .5% annual private mortgage insurance payment, what will my monthly payment be? The answer is $2,142.42.

You can use an online mortgage calculator to see how much property you can afford. A lender will compare your total monthly income and your total monthly debt load. A mortgage calculator can help you add up all your income sources and compare this to all your monthly debt payments. It can also factor in a potential mortgage payment and other associated housing costs (property taxes, homeownership dues, etc.). You can test different loan sizes and interest rates. Generally speaking, lenders do not like to see all of your debt payments (including your property expense) exceed around 40% of your 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. This monthly payment c depends upon the monthly interest rate r (expressed as a fraction, not a percentage, i.e., divide the quoted yearly 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 known as the loan's principal; c is given by the formula:

<math>c = (r/(1-(1+r)^{-N}))P</math>

For example, for a home loan for $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is <math>P=200000</math>, the monthly interest rate is <math>r=6.5/100/12</math>, the number of monthly payments is <math>N=30\cdot 12=360</math>, 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
<math>{}=1264.14</math>

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. This fact results in the debt schedule:

Amount owed at month 0: <math>P</math>
Amount owed at month 1: <math>(1+r)P-c</math>
Amount owed at month 2: <math>(1+r)((1+r)P-c)-c = (1+r)^2P - (1+(1+r))c</math>
Amount owed at month 3: <math>(1+r)((1+r)((1+r)P-c)-c)-c = (1+r)^3P - (1+(1+r)+(1+r)^2)c</math>
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Amount owed at month N: <math>(1+r)^NP - (1+(1+r)+(1+r)^2+ \cdots +(1+r)^{N-1})c</math>

The polynomial <math>p_N(x)=1+x+x^2+ \cdots +x^{N-1}</math> appearing before the fixed monthly payment c (with <math>x=1+r</math>) is called a cyclotomic polynomial; it has a simple closed-form expression obtained from observing that <math>xp_N(x)-p_N(x)=x^N-1</math> because all but the first and last terms in this difference cancel each other out. Therefore, solving for <math>p_N(x)</math> yields the much simpler closed-form expression

<math>p_N(x)=1+x+x^2+ \cdots +x^{N-1} = \frac{x^N-1}{x-1}.</math>

Applying this fact about cyclotomic polynomials to the amount owed at the Nth month,

Amount owed at month N
<math>

\begin{align} & {} = (1+r)^NP - p_N(1+r)c \\ & {} = (1+r)^NP - \frac{(1+r)^N-1}{1+r-1} c \\ & {} = (1+r)^NP - \frac{(1+r)^N-1}{r} c. \end{align} </math>

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:

<math>

\begin{align} c & {} = \frac{r(1+r)^N}{(1+r)^N-1}P \\ & {} = \frac{r}{1-(1+r)^{-N}}P \end{align} </math>

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.

Total Interest Paid Formula

The total amount of interest <math>I</math> that will be paid over the lifetime of the loan is given by:

<math>I = cN + (rP-c)\frac{(1+r)^N-1}{r}

</math> where <math>c</math> is the fixed monthly payment, <math>N</math> is the number of payments that will be made, <math>r = (interestrate/100)/12</math>, and <math>P</math> is the principle balance left on the loan.

The derivation of the total interest paid formula is as follows.

The total interest paid will be the sum of the interest paid on each of the payments. For instance, the total interest paid on the first three months of payment would be <math>rP_0 + rP_1 + rP_2</math> . The notation <math>P_i</math> is taken to mean the principle balance after <math>i</math> payments have been made. Therefore <math>P_0</math> is the initial amount of the loan and <math>P_N = 0</math> because after N payments the loan is paid off.

<math>

\begin{align} I & {} = rP_0 + rP_1 + \cdots + rP_{N-1} \\ & {} = r(P_0 + P_1 + \cdots + P_{N-1}) \\ & {} = r\sum_{i=0}^{N-1}P_i \end{align} </math>

Recall from above that the principle after <math>i</math> months of payment is:

<math>P_i = (1+r)^iP_0 - \frac{(1+r)^i-1}{r} c</math> note that <math>i</math> is being used in the place of <math>N</math> and <math>P_0</math> is being used in place of <math>P</math> as the starting balance (the balance after 0 payments)

Now, substituting for <math>P_i</math>

<math>

\begin{align} I & {} = r\sum_{i=0}^{N-1}[(1+r)^iP_0 - \frac{(1+r)^i-1}{r}c] \\ & {} = r\sum_{i=0}^{N-1}(1+r)^iP_0 - r\sum_{i=0}^{N-1}[\frac{(1+r)^i-1}{r}c] \\ & {} = rP_0\sum_{i=0}^{N-1}(1+r)^i-c[\sum_{i=0}^{N-1}(1+r)^i-\sum_{i=0}^{N-1}1] \\ & {} = rP_0\sum_{i=0}^{N-1}(1+r)^i-c\sum_{i=0}^{N-1}(1+r)^i+cN \\ & {} = cN + (rP_0-c)\sum_{i=0}^{N-1}(1+r)^i \\ & {} = cN + (rP_0-c)\frac{(1+r)^N-1}{(1+r)-1} \\ & {} = cN + (rP_0-c)\frac{(1+r)^N-1}{r} \end{align} </math>


the summation in the third to last line above turns out to be a cyclotomic polynomial and is simplified in the same way as earlier in the article.

See also

External links

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