Mathematics of Home Buying

Summary:Interest rates on mortgages are set using sophisticated mathematical techniques.

Homes are the largest single purchase most people make in their lifetimes. Buyers usually take out a home loan, called a “mortgage,” rather than pay cash. When the desired property is identified and funds for a down payment (typically 20%) are acquired, home buyers work with a bank or other lender to finance the purchase. When determining what a person can afford to borrow, lenders consider several variables. In the past, these judgments were often highly subjective decisions made by individual lender agents, but the increased popularity and availability of credit cards in the latter half of the twentieth century, as well as federal legislation designed to combat discrimination in lending, required more objective methods of assessment.

For example, FICO scores mathematically measure risk of nonpayment. Created by Fair Isaac Company (FICO), founded by engineer Bill Fair and mathematician Earl Isaac, an individual’s FICO score is a weighted combination of variables such as previous credit performance, current debt, and length of credit history. Also, lenders may use debt-to-income ratios to indicate the size of payment a borrower can afford. Comparing home loans can be challenging because different lenders may use this information differently. Also, interest rates, closing costs, and additions to the base payments need to be considered in the comparison. The process of buying a house may involve additional expenditures beyond the mortgage. Buyers routinely hire a home inspector to independently assess the condition of the home, and many such inspectors charge a fee based on the square footage. In some areas, radon tests or soil analyses might be required. If problems are found, either the buyer or the seller may have to hire a structural engineer or other professional to rectify the problem before an agreement is reached or the loan approved. Property taxes, based on the assessed value of the home and land, and homeowners insurance, which is also a function of the assessed value and replacement cost of the home and its contents, are also part of almost all home-buying transactions. Home buying tax credits or reduced interest rates may offer the buyer additional options and are designed to stimulate the economy.

FICO Scores and Credit Ratings

Each of the three major credit bureaus (Experian, TransUnion, and Equifax) calculate a credit score based upon advice from the Fair Isaac Company, an independent company that specializes in business analysis, including risk assessment. However, not all three companies use identical inputs, and each may yield a different result. A FICO score is between 300 and 850, with higher scores indicating better risks. The exact formula used for FICO score calculation is proprietary and changes periodically, but the personal data incorporated in the FICO formula include, in order of importance, payment history; amounts owed; length of credit history; new credit applied for; and types of credit used. The FICO score influences requests for credit, and many banks charge higher interest rates to people with lower scores.

Debt-to-Income Ratio

Most standard loan applications request information on both income (annual income, bank account balances, investments such as stocks and bonds) and debt (amount owed on loans, credit cards, and standard monthly bills like car payments, utilities, insurance). This information can be used to determine the percentage of income already committed to be spent each year: debt-to-income ratio = total debt ÷ total income.

If a prospective borrower’s expected debt-to-income ratio is higher than the bank’s cutoff (most banks have a limit between 32% and 40%), a loan may be denied, particularly if the prospective borrower’s FICO score is low. A high FICO score might result in the prospective borrower being granted the loan even with a higher debt-to-income ratio.

Calculating a Mortgage Payment

The principal (amount to be borrowed), the annual interest rate, the payment schedule (for example, monthly or bimonthly), and the length of the loan (10, 15, 30 years, for instance) all factor into the payment. The formula for the payment (R) is given below, where P is the principal, r is the adjusted interest rate (the annual rate divided by the number of payments in one year), and n is the number of payments to be made over the life of the loan

Additions to this base payment include the following:

• • PMI: Borrowers paying less than 20% down may be required to purchase private mortgage insurance (PMI) to protect the lender’s investment. The cost of this insurance is added to the loan payment.
• • Escrow: Many banks require that payments be made into an escrow account to accrue funds to pay property tax and insurance.

Finalizing a Mortgage Loan

Transaction fees for processing a mortgage are more commonly called “closing costs”; hence finalizing a home loan is called “closing.” Charges to be paid when closing on a loan typically include an origination fee for the lender, appraisal fee for the appraiser, title search and recording fees for the attorney, and points. Points are up-front interest fees charged by the lender, with one point costing 1% of the principal. Banks often give borrowers the option of purchasing additional points at closing to decrease the interest rate on a mortgage.

The reduction of interest for purchasing a point can vary from bank to bank; one point may reduce the rate by as little as 0.1% or as much as 0.25%. The cost of the points purchased at closing is tax deductible, as is any interest paid over the life of the mortgage.

Amortization of Loans and Fixed Rates

Loan payments include a portion that reduces the principal balance and a portion that the lender keeps—the interest. The amount of interest included in a payment varies over the life of the loan, but can be determined by remembering that each payment includes “simple interest payable on the balance.” Calculating the schedule of payments, including the split between principal and interest for each payment, is referred to as “amortizing.” The Latin roots of the term mean “death pledge,” indicating linguistically the willingness to forfeit something of great value if the debt is not paid. In this case, failure to pay the debt results in foreclosure by the bank and the loss of the property.

To illustrate, suppose a home buyer borrows $100,000 at 6% fixed annual interest (the interest rate does not change over the life of the loan) payable monthly for 30 years (360 payments). Using the loan formula, the monthly payment is $599.55, assuming no PMI or escrow. Since the homebuyer is paying 6% annual interest and 12 payments a year, the adjusted (monthly) interest rate is 0.5% for all payments.

Thus, the homebuyer owes the lender 0.5% of $100,000 in the very first payment; $500 will be kept by the lender as interest and $99.55 will be used to reduce the principal. For the next payment the balance is $99,900.45; the interest will be 0.5% of that balance or $499.50. These calculations can be summarized in an amortization table, which is usually provided to the buyer as part of the mortgage agreement. The first and last several rows for this example are presented in Table 1. As the loan progresses, the interest portion decreases and the remaining amount applied to the principal increases.

Variable Rates

The example uses a “fixed” interest rate, but many lenders also offer variable rate loans, meaning that the interest rate may be changed according to some economic indicator (called the “index”), such as the prime rate. There are legal restrictions on this practice: the lender must inform the borrower of the size and frequency of such changes (called the “interval”), and the maximum (called the “cap”) for the rate. For example, a two-year adjustable rate mortgage (ARM) payable monthly over 30 years might have the following particulars: 6% to start indexed on the 6-month U.S. Treasury Bill, and adjustments of at most 2% are allowed every two years with a cap of 10%.

The initial calculations, such as down payments or the payment amount, do not change in this case. For this two-year ARM, the first two years of the amortization table above does not change. One important question to ask the lender when considering an adjustable rate loan is what happens to the payment when the interest changes? Most redo the amortization calculations starting at the next payment, so a new rate would mean a new payment amount.

The housing crisis of 2007–2009 resulted in many homeowners finding themselves with homes whose values had decreased to the point that they were worth less than the amount owners still owed on their mortgages. There were many factors that influenced this outcome. Prior to the Great Depression, home ownership was much more rare than in the early twenty-first century when homes were often financed with balloon-payment mortgages in which a loan is amortized over only part of its lifetime, leaving a large principal payment due at the end. The federal push to open the housing market using fully amortized, fixed-interest mortgages required lenders to assume much greater financial risk, which can be mathematically modeled but not perfectly predicted. To manage that risk, mortgages became financial commodities in the larger financial marketplace. Housing prices, interest rates, and other aspects of financial markets are highly variable, and some people blamed the housing crisis on too much reliance on sophisticated mathematics.

In general, it was probably not the models themselves but the sometimes-incorrect ways in which the models were often used. In addition, many lenders ignored reliable risk predictors, such as FICO scores and debt-to-income ratios, resulting in more people taking on higher loan payments than they could afford. Home prices rose from demand to the point where properties were extremely overvalued. They later decreased in value, so the property was worth much less than the balance on the loan, leading to a large increase in foreclosures. Homeowners defaulted on loans, ruining their credit ratings, and banks paid large foreclosure fees. There were also more short sales, where banks agreed to accept less than the mortgage balances when homes sold to avoid foreclosure charges and poor credit ratings for the homeowners.

Bibliography

Johnson, Tim. “Paying the Price.” Plus Magazine (July 14, 2009). http://plus.maths.org/content/paying-price.

Perry, Timothy, and Daniel Prouty. The Book of Home Purchase: Make Quick, Simple On-Site Room-to-Room Calculations of Repair and Replacement Costs. Bergamo, Italy: Bergamo Publications, 1998.

Shestopaloff, Yuri. Mortgages and Annuities: Mathematical Foundations and Computational Algorithms. Toronto: AKVY Press, 2009.