Mortgage

A mortgage is a device used to create a lien on real estate by contract. The mortgage is an instrument that the borrower (called the mortgagor) uses to pledge real property to the lender (called the mortgagee) as security for a debt, also called hypothecation.

The mortgage instrument contains two parts:

To protect the lender, a mortgage is recorded in the public records creating a lien (when there are multiple liens, order of recording determines priority).

History

At common law, a mortgage was a conveyance that on its face was absolute and conveyed a fee simple estate, but which was in fact conditional, and would be of no effect if certain conditions were met --- usually, but not necessarily, the payment of a debt by the original landowner. Hence the word "mortgage," Law French for "dead pledge;" that is, it was absolute in form and in theory required no further steps to be taken by the creditor.

In many U. S. states, however, a mortgage has been converted by statute to a device for creating a security interest in land. When the landowner fails to perform on the obligation secured by the mortgage, the mortgage holder must file a foreclosure to cause the property to be sold at auction, usually by the sheriff. Since mortgage debt is often the largest debt owed by the debtor, banks and other mortgage lenders run title searches of the real property to make certain that the lien of the mortgage is prior to anyone else's claim.

Mortgage finance industry

Mortgage lending is a major category of the business of finance in the United States of America. Mortgages are commercial paper and can be conveyed and assigned freely to other holders. In the USA the Home Owners Loan Corporation, the Federal Housing Administration administer the programmes colloquially known as "Ginnie Mae" and "Freddie Mac" (aka the GSE's—the government sponsored entities) to foster mortgage lending and thus to encourage home ownership and construction.

Mortgage loan types

There are many types of mortgage loans. The two basic types of amortized loans are the fixed rate mortgage (FRM) and adjustable rate mortgage (ARM).

In a FRM, the interest rate, and hence monthly payment, remains fixed for the life (or term) of the loan. In the US, the term is usually for 10, 15, 20, or 30 years. In the UK the fixed term can be as short as five years, after which the loan reverts to a variable rate.

In an ARM, the interest rate will periodically (annually or even monthly) adjust up or down to some market index. Adjustable rates transfer part of the interest rate risk from the lender to the borrower, and thus are widely used where unpredictable interest rates make fixed rate loans difficult to obtain. Since the risk is transferred, lenders will usually make the initial interest rate of the ARM's note anywhere from 0.5% to 2% lower than the average 30-year fixed rate.

A partial amortization or balloon loan is similar to a FRM, but the balance is due at some point short of the full term.

Other loan types:

Fixed rate mortgage calculations

First the nomenclature.

I - The stated interest rate, for example, 5%/year. This is not the APR (annualized percentage rate).

m - The number of periods in the time frame of I. I is usually based on a year but it could be based on any amount of time.

i - The interest rate for the compounding period which is needed for the calculation. For example, a real property mortgage is usually based on a monthly period. In this case i=I*1/12 where I is based on the normal yearly period. In general i=I/m. Also I needs to be a decimal not a percent thus it also needs to be divided by 100.

n - The total number of periods or payments. Things like mortgages usually cover multiple years.

B - The balance, for example, the balance remaining on the mortgage at any point in time.

Mortgage Calculations:

Let B₀ be the original mortgage.

Let B₁, B₂, B₃ etc. be the balance after the first, second, third period respectively.

Obviously, one can think of B₀ as the balance after the zeroth period namely the beginning balance.

P - The mortgage payment.

Now lets write down the balances. First the initial balance, the amount of the mortgage.

B₀

Now calculate the balance after one period or payment.

During the first period the initial balance has grown by the period interest and has been decreased by the first payment. Similarly

Again

After n periods or payments we have

B_n is set equal to zero. When the mortgage is paid off the balance is zero. Now one can solve for P the payment. Rearranging gives:

The righthand side is a geometric series where each term is equal to the preceding term multiplied by (1 + i) which is known as the common ratio. See geometric sequence for additional details.

Solving for P gives:

The payment can be readily calculated to the penny with a spread sheet or scientific calculator.

Note: B₀ is just a simple multiplier. Therefore one can do the calculation for a unit of currency such as a dollar and then multiply the result by the amount of the loan. In essence B₀ is just a scale factor. For example think of the loan amount as my dollar where my dollar is just a currency whose exchange rate is just the loan amount difference.