Notes on the Simple Mathematics of Mortgages

To explore the mathematics of mortgages, let's use the following notation:

\(M\) - Original mortgage balance
\(N\) - Mortgage maturity, number of periods for which payments are due
\(R\) - Mortgage interest rate
\(INT_t\) - Mortgage interest payments in period t
\(PAY_t\) - Mortgage payment due in period t
\(B\) - Outstanding mortgage balance at the end of period t, \(B_0\) = M

The variable t refers to a particular period and t = 1 to N. For level payment mortgages \(PAY_t\) will be me same for all periods and we can drop the subscript t. The examples below measure interest rates on an annual basis. For a twenty five year 9% mortgage, R=0.09 and N=25. To use the formulas for mortgages with monthly payments one needs to replace the Variable R with R/12 and the variable N with 12N. For example, a twenty five year 9% mortgage would have R=0.0075 and N=300.

Level Payment Mortgages

Mortgage payments, PAY, are determined such that the present value of the N mortgage payments, when discounted at the interest rate R, is equal to the original mortgage balance M. The basic formula for a fully amortized, level payment loan where payments are made at the end of each period is:

$$M = \sum_{t=1}^N\frac{PAY}{(1 + R)^t}$$

A fully amortized loan is one where the periodic payments cover interest and principal. There is no outstanding loan balance following the last payment.

One can manipulate the basic equation to derive the expression for PAY shown below. This expression involves four variables, PAY, M, R and N. Given any three variables it can be solved for the fourth. For example, one can use the formula to determine the payment, PAY, necessary to service a loan of M for N periods at interest rate R. Alternatively, one can use me equation to find the maximum loan, M, that can be supported with payments of PAY, based on an interest rate of R for N periods. The equation can also be solved for R given PAY, M and N or it can be solved for N given PAY, M and R. It is not possible to write a simple equation for R. The expression for N involves logarithms.

$$PAY = R{\cdot}M\left[\frac{(1 + R)^N}{(1 + R)^N - 1}\right]$$

A fully amortized loan covers interest and principal. For a level payment mortgage the expression above shows that PAY is greater than interest on the original loan balance as the term in brackets is greater than 1.0.

One can also derive equations for the following:

The outstanding loan balance at the end of any period.

$$B_t = M\left[\frac{(1 + R)^N - (1 + R)^t}{(1 + R)^N - 1}\right]_{t=1, 2 ... N}$$

Note that at higher interest rates the outstanding loan balance declines more slowly in the early years of a mortgage.

The interest payment for any period, \(INT_t\):

$$INT_t = R{\cdot}B_{t-1} = PAY\left[1 - \frac{(1 + R)^{t-1}}{(1 + R)^N}\right]_{t=1, 2 ... N}$$

Interest payments in period t are proportional to the outstanding loan balance at the beginning of the period which is the same as the outstanding loan balance at end of the previous period. The decline in the outstanding balance over time leads to a decline in interest payments over time.

The principal payment for any period, \(PRIN_t\):

$$PRIN_t = PAY - INT_t = PAY\left[\frac{(1 + R)^{t-1}}{(1 + R)^N}\right]_{t=1, 2 ... N}$$

As principal payments plus interest payments equal the level payment each period, we know that the principal portion of mortgage payments will increase as the interest portion decreases.

If the interest rate is higher, mortgage payments must also be higher. It turns out that as interest rates increase the dollars devoted to initial principal repayment decrease. On net the increase in the mortgage payment is less than the increase in interest dollars that must be paid.

Points and Closing Costs. At the time of mortgage origination there are typically additional costs that borrowers must pay. These include charges for specific services such as attorney fees, property appraisal, and filing fees. In addition, borrowers may be asked to pay points, a charge that is computed as a percentage of the mortgage loan. One point would be equal to 1% of the mortgage loan, two points 2%. Points are a form of prepaid interest. Different mortgage interest rates from different lenders are often offset by differences in points. A lender quoting a low interest rate will typically charge more points and vice versa.

The impact of points on the effective mortgage rate depends upon when a mortgage is paid off. If a mortgage is prepaid after one year, one point is equivalent to raising the mortgage rate by one percentage point. As a mortgage is held longer the impact of points is less. The exact impact can be calculated by investigating an amended form of the first equation that includes the payment of points at the time of origination, periodic mortgage payments for the estimated holding period, and the payment of the outstanding balance at the end of the holding period. Remembering that PAY is fixed by the contract interest rate, one then solves for the interest rate that equates the present value of points, periodic payments, and the outstanding balance at the end of the holding period to the original loan balance, M.

Other Mortgages
Adjustable Rate Mortgages

Under ARMs the interest rate may change. Most ARMs in the United States link changes in the mortgage rate to changes in a specified market interest rate. Different ARMs index the mortgage rate to different market interest rates. When the index interest rate changes, mortgage payments are recalculated using the equation for PAY. M would be replaced by the outstanding loan balance and N would be replaced by the remaining number of periods. Typically ARMs have limits on how much the interest rate can increase per adjustment period, so-called brakes, along with an overall upper limit or cap. At times ARMs are offered with a very low initial interest rate that is adjusted after six months or a year regardless of changes in market interest rates. In these cases the initial interest rate is called a teaser rate.

Graduated Payment Mortgages

GPMs call for escalating mortgage payments. Payments start low and end high. Most real world GPMs call for a period of constant lower mortgage payments, perhaps three to five years and then an adjustment to constant higher payments for the remainder of the life of the loan. Under a GPM with low payments for 5 years and higher payments for the remainder of the loan, payments would have to satisfy the following expression where PAY is replace by \(P_L\) and \(P_H\). This equation involves five unknowns, M, R, N, \(P_L\), and \(P_H\).

$$M = \sum_{t=1}^5\frac{P_L}{(1 + R)^t} + \sum_{t=6}^N\frac{P_H}{(1 + R)^t}$$

To solve for initial payments in terms of M, R and N, one needs to specify the relation between \(P_L\) and \(P_H\). For example, if mortgage payments will increase by 20% after five years, then \(P_H = {1.2\cdot}P_L\)

A simple form of GPMs for analytic manipulation is one that calls for mortgage payments to grow at rate g each period, that is \(PAY_t = P(1+g)^t\). In this case our basic formula can be written as

$$M = \sum_{t=1}^N\frac{PAY_t}{(1 + R)^t} = \sum_{t=1}^N\frac{P(1+g)^t}{(1 + R)^t}$$

and the initial mortgage payment, \(PAY_1 = P(1+g)\), can be written as

$$PAY_1 = M(R-g)\left[\frac{(1 + R)^N}{(1 + R)^N - (1 + g)^N}\right]$$

If g = 0 this expression reverts to the earlier equation for level payment mortgages. Note that the loan balance increases initially. The hump pattern reflects the fact that for this example initial loan payments are not sufficient to cover interest due. The initial unpaid interest gets added to the loan balance.

GPMs were developed as a possible alternative during periods of high inflation. The inflation premium built into the mortgage rate is in anticipation of future increases in prices. Under fixed payment mortgages, the inflation premium increases mortgage payments from the very beginning of the loan before much inflation has occurred. GPMs postpone increases in mortgage payments until later in the life of the loan when it was expected that inflation will have increased wages and salaries making the increase in mortgage payments easier to handle. If the graduation factor, g, matches the rate of inflation and this inflation applies equally to income and house prices, then income will increase in step with mortgage payments and increases in house prices will more than match the increase in the outstanding loan balance during the early years of the loan. If inflation is less than the rate of graduation or if it does not apply equally to income and house prices, homeowners could find themselves in a difficult position.

  Mortgage Calculators

What is a mortgage? In a nutshell, a mortgage is a loan you take out to help you pay for a house. In exchange for the loan, you commit to making monthly payments to your lender until the loan is paid in full or you sell the property and repay the loan. Each monthly payment includes an interest payment (the money you give the bank that makes the loan worth their while) and a principal payment (money that decreases the amount you owe the bank). Over the life of your loan, you keep making payments to decrease the amount you owe the bank and once you have paid the loan to the bank your mortgage has become "Amortized".

Important note about mortgages. Once you have a mortgage, it's important to remember that you do not fully own your house until you have repaid the loan. When you buy with a mortgage, the lender gets a "lien" against your house, which is another way of saying that the lender can take possession of your home if you fail to make your payments.

What is a mortgage calculator? A mortgage calculator is a tool that helps the borrower determine how much money is required and whether the borrower can afford the property to be purchased. This calculator can be used to compare costs, rates, schedule of payments or determine the loan term by adding principal payments. The Mortgage Calculator allows you to automatically calculate the financial implications of mortgage payments for numerous variables such as the loan amount, interest rate, frequency, amount of regular payments and several others. The calculation result depends on mortgage loan type and can be presented in the form of tables, graphs, charts and drawings.

Our Mortgage Calculator Toolkit App provides users with a wide collection of mortgage calculators packed in one app. The Mortgage Calculator Toolkit consist of Fixed Rate Loan Payments, Mortgage Terms Comparison, Mortgage Qualifier, Mortgage APR, Adjustable Rate Loan Payments, Balloon Mortgage

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. For user convenience each calculator has the following sections: Input page, Charts page, Amortization table page, Summary page. Built-in on-screen keyboard enables user to modify values easily. Touch any mortgage value on Input page to open keyboard. Each calculator automatically generates report that includes all available charts and amortization schedule. This report can be shared via email with friends.

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