Understanding the math behind mortgages, APR, and more is critical for anybody who hopes to eventually buy a home.

Case Study: The Basics of Buying a Home

Case Study: The Basics of Buying a Home

Many people consider buying a home to be a step in fulfilling the “American Dream.” People often see it as an indication of achieving success. But in order to secure this part of the “American Dream,” most people have to borrow a large share of the money required to pay for it. The buyer borrows the proceeds from a mortgage loan lender. This is generally a savings and loan, bank, credit union, mortgage banker, or other authorized lending institution.


There has been an ongoing debate as to whether or not the purchase of a home is an investment. Some suggest that there are much better ways to invest your money and receive a greater return on your initial investment (i.e. your down payment). They argue that the monthly payments – which include interest over the life of the loan – along with periodic payout for maintenance, insurance, and property taxes, generally lessen the investment potential when compared with alternative investments.


Others argue that people who are willing to hold on to the property “for the long term” will generally realize gains. This, they argue, is because of the growing equity (or net worth) that results from the pay-down on the original mortgage balance combined with the potential for appreciation (or increase) of the property’s value.


Prospective homeowners should do their homework and learn about the various types of loans that may be available to them.Click To Tweet


They should also know the amount of their down payment, as well as their creditworthiness, employment history, location of the desired property (urban, suburban, rural), the type of residence (see below), appraised value and sales prices, and the general eligibility requirements for the mortgage loan.


Types of Mortgage Loans

Conventional Mortgage

The most common type of mortgage loan is a conventional mortgage. These make up nearly two-thirds of all mortgages in the U.S.


Generally, any mortgage loan that isn’t government-insured or guaranteed and secured from an established lender is considered to be a conventional mortgage loan. The term of the loan will usually last for either 15 or 30 years. The lender generally requires a down payment of at least 20 percent and loan you 80 percent or less of the loan-to-value (LTV) ratio. For example, if the value of the home is $100,000, the maximum amount to borrow would be $80,000.


While many lenders will make loans in excess of the 80 percent LTV, the borrower will most likely need to pay an additional amount each month known as private mortgage insurance (PMI). This protects the lender in the event that the borrower defaults. The interest rate will also be higher if you have a higher LTV and lower down payment. In either instance, the mortgage rate is fixed over the term of the loan.


Adjustable-Rate Mortgage

For borrowers willing to take on a bit more risk, there is the adjustable-rate mortgage (ARM). There are various terms available, but three-, five-, and seven-year terms are the most common. The borrower enters the loan agreement with a slightly lower interest rate and the understanding that the rate could go up (or down) when reviewed and potentially adjusted in line with the prevailing interest rate at the time of the review (three, five, or seven years in the future).


FHA Loans

The Federal Housing Administration (FHA) generally gives home loans at a slightly lower-than-average interest rate. These loans generally go to borrowers who may not typically qualify for a conventional mortgage. Those approved for FHA loans have to pay mortgage insurance for the life of the loan.


One advantage of an FHA mortgage is that the loan is assumable.


In other words, a future buyer can take on the same loan and interest rate as the original owner.


This is particularly appealing should interest rates rise between the time of the original purchase and that of the later sale.


For example, let’s say that you purchased a home in 2010 and the interest rate was three percent. You decide to sell the home in 2020, and the interest rate is six percent. The new buyer may assume the three percent interest rate. Generally, an assumable mortgage makes the home more marketable.


VA Home Loans

The Veterans Administration (VA) may need to provide a veteran with appropriate discharge papers (DD 214) so that he or she can obtain a Certificate of Eligibility. A vet can then use that certificate to obtain a home loan. In some instances (depending on the house’s appraised value and the veteran’s income and creditworthiness), a veteran be may be eligible for 100 percent financing – no down payment required.


USDA Home Loans

Finally, the United States Department of Agriculture (USDA) provides mortgage loan in certain designated rural areas throughout the country. Prospective borrowers should consult the USDA website for the approved areas. Those interested in a USDA home loan may be eligible for a loan with a low down payment and a high LTV. That said, income restrictions apply according to family size and area.


The Math Behind the Mortgages

The following table includes the various types of mortgages that we discussed and their potential interest rates. The annual percentage rates (APR) – or the true cost of the money borrowed – are also included here, as well as the number of “points,” or additional fees paid at the loan closing.


Loan Type Interest Rate APR (approx.) Points
Conventional 30-Year Fixed

80% LTV

3.25 3.55 1-2
Conventional 15-Year Fixed 

80% LTV

2.50 2.95 1-2
Conventional 15-Year Fixed

50% LTV

2.25 2.45 1
90% LTV 30 Year Fixed


3.50 3.95 2-3
5-Year ARM 2.375 3.675 1-2
FHA 30-Year Fixed 3.00 3.85 2-3
FHA 15-Year Fixed 2.75 3.65 1-2
VA 30-Year Fixed 3.00 3.35 2-3
USDA 30-Year Fixed 3.25 3.75 2-3


Types of Residence

Single-Family (detached): a free-standing property.


Townhouse: typically a row house that shares at least one of its walls with another dwelling that may be owned and occupied by someone else.


Condominium: part of a building or buildings that at are owned and governed by a homeowners association.


Planned Unit Development (PUD): typically detached homes within a homeowners’ association.


Multifamily: a building or complex with distinct separate units. The residence is meant to be occupied by different residents (or families).


Modular Housing: housing that is built in a factory. The materials are delivered to and assembled at the site. The house is then permanently affixed to a foundation and looks very similar to a site-built house.


Buying a Home: Scenarios

The current median price of a single-family house in the United States is approximately $200,000. All of the following prospective borrowers plan to purchase a $200,000 home:


1. Milagros and Dakari will make a down payment of 25 percent and opt for a five-year ARM.


2. Holly and Jose are both veterans and decide to use their certificates of eligibility in order to get a VA home loan. They could qualify for 100-percent financing. But in order to lower their monthly payment, they plan to make a down payment of $10,000.


3. Marcos is doing quite well in his IT career. He managed to save a considerable sum of money and now plans to make a 50-percent down payment. He also plans to take out a 15-year fixed-rate conventional mortgage.


4. Liam and Aisha have decided to move from Denver with their four children. They want like to finally purchase a home as a gift to each other for their 10th wedding anniversary. They will be relocating to jobs in Cheyenne, Wyoming and have decided to commute from the countryside.


5. Luis and Terry will put $25,000 down for their condo in Atlanta. They believe that a 30-year fixed is the best deal for them.


Use the information from the table above and an online mortgage calculator to determine what their monthly payment will be.


Check List

Key Terms

Types of Mortgage:

  • Conventional Mortgage
  • Adjustable-Rate Mortgage
  • FHA Loan
  • VA Home Loan
  • USDA Home Loan


Types of Residence:

  • Single-Family
  • Townhouse
  • Condominium
  • Planned Unit Development (PUD)
  • Multifamily
  • Modular Housing