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The Math Behind Purchasing a Home

Updated: Apr 9, 2021

Hey Everyone!

Growing up in the United States we are told that buying a home is the American dream. Many people pursue this goal not out of necessity or the desire to build wealth, but because we are told that being a homeowner is the most American thing one can do. But is it really a good idea financially? This post will discuss the Home Buying Calculator, found on the Downloads page, and how purchasing a home impacts your financial position.

As usual, this post and the Home Buying calculator are not necessarily meant to be taken as advice, but is meant to provide you with tools that will help you make decisions that are the best for you and your financial position.

The Mortgage

Before going into the intricacies of a mortgage, we should know what it is. A mortgage is a loan provided by a lender to allow a borrower (the purchaser or owner) to purchase or refinance a home. By getting a mortgage, the borrower can put a smaller down payment down on the house than if they had to purchase the home in cash. In return, the bank charges a few percentage points of annual interest throughout the term of the loan, called the rate. The most common benchmark for an interest rate is a 10-year or 30-year Treasury. The bank will then add percentage points to the treasury rate as a mechanism to hedge riskier loans and to earn a profit. Reasons why a bank may add on percentage points are the borrowers credit history, risky location, mortgage amount, borrower's financial condition (a lot of debt, few assets), and downpayment amount. A borrower with poor credit history that is buying a house in an economically stagnant town with a 10% downpayment will have a higher interest rate than a borrower with a 750+ credit score purchasing a home in a metropolis and putting a 50% downpayment on the home.

The bank also has a lien on the property until the loan is paid off. This allows the bank to take the property and the borrower's personal assets if the mortgage goes unpaid. This is called recourse.

Another mortgage term is amortization - this is how many months or years it will take a mortgage balance to reach $0. For residential mortgages, the amortization of the mortgage will match the term of the mortgage 95% of the time, this is called a fully-amortizing loan.

The mortgage payment is a calculation that takes amortization, rate, and the loan balance into account to calculate the amount the borrower will need to pay on a monthly basis in order to pay off the mortgage when the mortgage reaches amortization. The payment is made up of two parts; interest and principle. Interest is the portion of the payment that went towards paying off the mortgage interest, and principle is the remainder of the payment - what actually went towards decreasing the loan balance. For every loan, the borrower will be paying a significant amount of interest early on in the mortgage term, this is because the loan balance is higher than in latter parts of the loan, so the interest paid will naturally decrease as the balance decreases.

Home Owning Costs

While home-shopping, one ought to consider more factors than just the mortgage, as there are many hidden costs when owning a home including HOA fees, taxes, homeowners insurance, and mortgage insurance.

HOA fees differ from community to community and are typically based on how new the property is and how well the property is kept, new well-kept properties will have higher HOA fees for their homeowners than an older one that is dingy and has no amenities. Taxes is dependent on the city, county, and state that the home is in, this expense should be researched by identifying the home's taxing authority and visiting the authority's Assessor page. Homeowner's insurance is dependent on the carrier, which I cannot speculate on. Mortgage insurance is required by lenders if the borrower's down payment is less than 20% of the home's value or purchase price and mortgage insurance is required until the borrower has 20% equity in the home.

Mortgage Inputs Tab

The Maximum Affordable Housing Payment assumes that the borrower does not want to pay more than 30% of their gross income for housing. This is discussed in the first blog post and is a common metric for financial planners to use.

In order to use the calculator, input the required data in the gray boxes, the yellow and blue boxes are calculated by formulas. A few key items for you to do some research on are the Annual Taxes, Homeowner's Insurance, and even the Annual Mortgage Insurance Fee. The calculator uses Georgia's tax rate, the median tax rate of states the nation. This is a general approach and should just be used as a benchmark, taxes are highly variable. Homeowner's Insurance is also variable carrier-by-carrier and will depend on bundle packages and discounts. Please do your research for this line item. lastly, the Annual Mortgage Insurance Fee is calculated automatically, but based on the average of what is charged on an annual basis (0,5%-2.55%) again, this isn't the best approach, but good enough to use for getting a ballpark number. The example above uses a $200,000 home purchase, 20% down payment, and a 30-year mortgage with a 3.00% interest rate. The calculator is good for all fixed rate mortgages that are fully-amortizing.

The table above shows the increase in costs as one inputs the annual or monthly fees associated with owning a home. Even without paying homeowner's insurance (about 1.5% of the mortgage balance annually), a monthly payment of $675 increases to $1,243, an 84% increase over the mortgage payment. Over the term of the 30-year loan, this means that the homeowner will have paid $205,000 of additional expenses associated with homeownership.

As we'll see in the next section, the amortization schedule reveals more detail about a mortgage payment and where the money is going.

Amortization Schedule

The Amortization Schedule shows the total payment of a loan, including columns for principle and interest paid. This allows the borrower, the banks, and investors to see how the loan is being paid in detail, and is important for bond remittances, mortgage payoffs, and other financial metrics that investors have to monitor. Further, it is very informational to see how the ratio of interest to principle amount changes over the course of a loan. At first, interest makes up $400 of the $675 mortgage payment in our example. But, by the twelfth year, 143rd payment, the interest portion of the loan payment has decreased to $282. Overall, using the example mortgage, the borrower would pay about $82,000 in interest over the course of the 30-year loan.

The Numbers

So after looking at the $205,000 of homeownership expenses and $82,000 of interest expenses, the math shows that a homeowner would pay about $287,000 of additional expenses over the 30-year mortgage term. Adding back the $160,000 mortgage and the $40,000 down payment, the homeowner put about $487,000 into owning a $200,000 home, 245% greater than the home's purchase price. The important thing to find out is how much the home would sell for. Let's look to some data to find out.

The above chart shows that renting a home is cheaper than buying for the first thirteen years, after that point, owning the home becomes a better deal, but cost is not everything. Let's see how the renter would fare when putting the would-be down payment into an S&P Index ETF.

This chart, found on the Return tab, shows the Home Price Index and the S&P 500 index from YE1991 to YE2019. Both lines are indexed at 100 beginning at YE1991, this means that, for each period, the index will be relative the initial 100 index. This is a simple way to compare numbers of vastly different scales, like in our example.

So let's dig in. If the homeowner were to buy their house in 1991 for Index 100 and sell in 2019 (almost 30-years), the house would be worth an Index of 277. This equates to a 177% increase, meaning the home would be worth about $354,000 after purchasing the home for $200,000. Not counting for broker fees and other third-party fees, this nets the homeowner a $133,000 loss after sale when including the total cost of purchasing the home.

How does the return on investment from selling your home compare to investing in an S&P 500 index fund? Much worse. The S&P Index reached an Index of 817 at YE2019. If a renter in the same financial condition was to put $40,000 (equivalent to the homeowner's down payment) in the S&P Index in 1991, it would be worth about $287,000 in 2019, a $247,000 profit. But, the renter had to continue renting an apartment for those thirty years. How much would that cost?

In the Cost Comparison tab we assume that the renter paid the same initial rent as the homeowner's all-in monthly mortgage payment and that rent will increase 3.00% annually. In total, the rent cost would be $709,000 throughout 30 years of rent payments. Despite the $247,000 profit from the sale of S&P Index funds, the renter would lose $462,000 due to the prohibitive cost of renting long-term.


Financially, the best way to purchase a home is to put as much downpayment on the home as you can afford. The larger the down payment means that the mortgage balance will be lower, and so will the mortgage rate. Those factors make your mortgage as cheap as possible. If you are not in a financial position to put down 20%+ into a downpayment, then it may be best to continue renting in order to save up and build credit. Remember, debt is not a financial tool that should be used casually, but one that should be used sparingly and when it can help increase one's net worth. Buying a house with a small downpayment and large mortgage can lead one to spend too much money on housing, not keep up with mortgage payments, or even find oneself with an underwater mortgage; when the mortgage balance is greater than the value of the house.


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