Mathematics of loans

Summary: Determining the terms of a loan so that they are fair but compensate for risk is a challenge of algebra.

Most people have personal experience with one or more types of loans, such as home mortgages, car loans, or home equity loans. In each case, the general format of the loan is the same: the lender provides temporary funds to a borrower, and the borrower repays these funds over a prespecified period of time, according to a prespecified pattern. As it is for any financial asset or liability, mathematics is a critical tool for determining the appropriate parameters of loans, including the periodic payment necessary for the borrower to completely pay off the loan by the end of the loan’s life.

Mathematicians work on many problems related to loans. For example, individuals who take out mortgage loans, are often required to purchase insurance for those loans. Actuaries use mathematical and statistical methods to assess lending risk to decide whether insurance is needed and how much. They also work on more complex problems related to interest rates and credit, such as deciding what constitutes usury (unreasonably high interest rates) for loans whose yield rate is not fixed or determining the reliable predictors of credit risk.

History

Loans appear to have been a part of economic activity ever since economies began to become sophisticated. In response to certain historical unfair lending practices, a number of proscriptions against usury were recorded in ancient sources, such as the Old Testament, and works by Aristotle and Tacitus. More generally, an active lending market is important to an economy, as it facilitates the availability of funds for investment.

Loans, like other financial instruments, are two-sided transactions. There is a lender and there is a borrower, and cash flows are made between them—what one party pays, the other receives. Algebraically, this process is usually reflected by identifying the cash flows as either positive or negative; a positive cash flow for the lender would be a negative cash flow of the same magnitude for the borrower, and vice versa. For the lender, the loan transaction is essentially an investment, and thus an asset. For the borrower, the loan represents a liability and ultimately needs to be paid back.

The most common method in the twenty-first century of paying off a loan is via amortization, in which interest and a portion of the original borrowed principal are paid back in each of the periodic payments. There are a number of parameters associated with the typical amortization loan, including the following:

• • The original amount borrowed (B)
• • The length or term (n) of the loan (for personal loans, such as mortgages and auto loans, the length of the loan is typically measured as the number of monthly payments to be made by the borrower to the lender; theoretically, however, payments can be made according to any schedule, such as weekly, annually, or uneven periods of time)
• • The periodic (for example, monthly) interest rate (i) on the loan, which determines the amount of interest paid by the borrower to the lender
• • The periodic (for example, monthly) payment (R) made by the borrower to the lender

In the most common type of amortized loan, the payment made by the borrower each period is constant over time. Each payment consists of two components: an interest payment and a partial principal repayment. Across the life of the loan, the sum of all of the n partial principal repayments is equal to the total original amount borrowed, B. As each payment is made, the outstanding balance of the loan is lessened by the amount of the partial principal repayment in that payment.

The effect of this approach is that, while each payment R is of the same size, the split between the interest component and the principal component of each payment changes over time. More specifically, as time moves on, the principal component increases and the interest component decreases. This is because the indebtedness (the outstanding balance) of the loan decreases over time, and thus the periodic interest charged on the loan (which is equal to the interest rate multiplied by the loan’s outstanding balance) also decreases over time.

To illustrate, suppose that $1,000 is borrowed, and this four-year loan is to be paid off with four equal annual payments of R, one at the end of each of the four years during the life of the loan. Suppose that the effective annual interest rate i=0.10, or 10%. In this situation, the annual payment R can be determined by the formula

where i=0.10, B=$1,000, and n=4. Thus, R=$315.47.

This value for R can be verified by considering the impact of each annual payment separately. For example, consider the first payment of R. During the first year, the borrower incurs interest charges of 10% of the outstanding balance at the beginning of the year, or $100. Thus, $100 of the $315.47 first payment covers the interest for borrowing the original $1,000 during the first year; the remaining $215.47 of the first payment then serves to partially pay off the loan, leaving an outstanding loan balance, or indebtedness, of $1,000-$215.47=$784.53. During the second year, the borrower incurs interest of 10% of that new outstanding balance, or $78.45. That portion of the second payment of R covers this interest, and the remainder ($315.47-$78.45=$237.02) serves to further pay down the loan. Thus, after the second payment, the borrower has loan indebtedness of $784.53-$237.02=$547.51. Continuing this process through the fourth and final payment will reveal that, after that final payment, the original $1,000 loan has been completely and precisely paid off.

Occasionally, people will pay off installment loans before their final due date by making early payments or paying slightly more than is due at each installment. In this case, they may be entitled to a rebate on some of the originally computed interest. Rebates can be figured using several methods, including variables such as how the interest was originally computed and the way in which the regular and extra payments were divided between principal and interest. The actuarial method of calculation is generally more favorable to the borrower than rebates calculated under other methods, such as the Rule of 78s.

There are other ways of paying off loans; for example, paying the interest regularly and then paying off the entire principal at the end of the loan term. In fact, this process is essentially how a specific type of financial instrument, a bond, works. When corporate or governmental entities issue bonds, they are borrowing money. More precisely, they are borrowing an amount equal to the price of the bond from the investor or investors who purchase the bond. The issuing organization pays periodic interest to the investors (in the form of coupons) and at the expiration date of the bond pays back to the investors a lump sum, known as the “redemption value.”

Bibliography

Broverman, Samuel A. Mathematics of Investment and Credit. Winsted, CT: ACTEX Publications, 2008.

Kellison, Stephen. Theory of Interest. New York: McGraw-Hill, 2008.