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Jared Bass

# Mortgage Mathematics

Disclaimer: If you are either super comfortable with the math behind mortgages or have absolutely no interest in understanding how principal and interest are calculated, feel free to skip this post and keep an eye out for the next one :)

## mortgage mathematics

A standard mortgage payment includes paying back your loan and any interest that the bank requires for giving you the loan. You may also decide to include taxes and insurance with your monthly mortgage rate, but this may be calculated separately.

An amortization schedule shows you a full breakdown of your mortgage payments month by month. It includes your principal amount, which is the amount towards paying off the loan, as well as the interest amount that goes to the bank. The amortization schedule can help you see milestones in paying off your mortgage and how long it will take.

Before 2008, if you had a good job and \$50,000 to plunk down as a down payment, a bank would generally lend you the \$250,000 you needed, and getting a 30-year fixed rate mortgage at 6% wouldn't be a big deal. You'd probably pay about \$1,500/month with total interest over the life of the loan almost equal to the loan itself.

These numbers will help you figure out whether you're in over your head. If the monthly payment is a few times what you pay in rent, you may be scrambling each month to pay. And if your mortgage term is really long, you'll end up paying more for your house overall. And don't forget: no matter how nice and low that monthly amount is, it doesn't include property taxes, maintenance, and new designer furniture for your place.

A long term loan that is used for the purchase of a house is called a mortgage. It is called a mortgage because the lending agency requires that the house be used as collateral for the loan. This means that if the mortgage holder is unable to make the payments the lender can take possession of the house.

Mortgages generally tend to be for longer time periods than an installment loan and the terms of the mortgage will often change over the course of the mortgage. Take for example the purchase of a house with a twenty year mortgage. The purchaser might sign a mortgage agreement for a five year term. The mortgage agreement will include the interest rate, the frequency of payments and additional rules which may allow the mortgage holder to make lump sum payments or change the payment amount. At the end of the five year term a new agreement will be required and the conditions of the mortgage usually change.

A young couple have received an inheritance and they now have enough money for a down payment on their first home. They plan to take out a 25 year mortgage at an interest rate of 3.8%. They are considering a new house for \$750,000 or a smaller older home for \$380,000. If they purchase the larger house they plan to make a 20% down payment. With the less expensive smaller house they can afford a 35% down payment.

A couple has won \$50,000 in the lottery and they decide to put this towards the purchase of a vacation cottage or a house. They plan to make a 10% down payment and are considering a 25 year mortgage at a rate of 2.9%. They are deciding between the purchase of a cottage for \$500,000 or a house for \$880,000.

Find a recurrence relation for the amount of money outstanding on a \\$40,000 mortgage after n years. The interest rate on the mortgage is 10% and the yearly payment is \\$2,000( the yearly payment is paid at the end of each year after the interest has been computed).

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.

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.

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%.

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:

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:

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:

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]

She has over 35 years of financial, banking, accounting and business experience. Since 1994 when she started originating retail residential mortgages she has directly overseen and managed in excess of \$1 billion in closed residential real estate loans, representing thousands of customers.Debra's experience and educational focus has given her an edge in the marketplace and a strong foundation as a passionate trainer for mortgage originators and Realtors offering customized sales and and mortgage-related training courses.

In 2010, Debra co-authored one of the first NMLS approved 20-hour Pre-licensing SAFE Act courses, required for all state-licensed mortgage originators by Title V of the SAFE Act. She also develops for continuing education classes for Realtors.

More often than not, a homeowner who borrowed money to buy a house is making one lump-sum monthly payment to their mortgage lender. But while it may be called the monthly mortgage payment, it includes more than just the cost of repaying their loan and interest.

For many of the millions of American homeowners carrying a mortgage, the monthly payment also includes private mortgage insurance, homeowners insurance, and property taxes. This is known as PITI: principal, interest, taxes, and insurance.

The most common terms for a fixed-rate mortgage are 30 years and 15 years. To get the number of monthly payments you're expected to make, multiply the number of years by 12 (number of months in a year).

Private mortgage insurance (PMI) is required if you put down less than 20% of the purchase price when you get a conventional mortgage, or what you probably think of as a "regular mortgage." Most commonly, your PMI premium will be added to your monthly mortgage payments by the lender.

A monthly mortgage payment will often include property taxes, which are collected by the lender and then put into a specific account, commonly called an escrow or impound account. At the end of the year, the taxes are paid to the government on the homeowners' behalf.

How much you owe in property taxes will depend on local tax rates and the value of the home. Just like income taxes, the amount the lender estimates the homeowner will need to pay could be more or less than the actual amount owed. If the amount you pay into escrow isn't enough to cover your taxes when they come due, you'll have to pay the difference, and your mortgage payment will likely increase going forward.

Once you calculate M (monthly mortgage payment), you can add in the monthly property tax and homeowners insurance premium, if you have them. These are fixed costs that aren't determined by how much you borrow from the bank, so they can easily be added to the monthly cost.

In the aftermath of the 2007-2008 financial crisis, there has been criticism of mathematics and the mathematical models used by the finance industry. We answer these criticisms through a discussion of some of the actuarial models used in the pricing of credit derivatives. As an example, we focus in particular on the Gaussian copula model and its drawbacks. To put this discussion into its proper context, we give a synopsis of the financial crisis and a brief introduction to some of the common credit derivatives and highlight the difficulties in valuing some of them. 041b061a72