On the Behavior of the Payoff Amounts in Simple Interest Loans in Arbitrage-Free Markets

TL;DR

This paper analyzes the behavior of payoff amounts in simple interest loans within arbitrage-free markets, deriving a new formula, examining its unique properties, and introducing a polynomial notation for better modeling.

Contribution

It derives a new payoff formula for simple interest loans in arbitrage-free markets and explores its distinct behavior compared to compound interest, including a novel polynomial notation.

Findings

Payoff amounts increase then decrease after a critical time when interest exceeds a threshold.

The critical interest rate depends on the number of installments and decreases as installments increase.

For two installments, the critical rate equals the golden section.

Abstract

The Consumer Financial Protection Bureau defines the notion of payoff amount as the amount that has to be payed at a particular time in order to completely pay off the debt, in case the lender intends to pay off the loan early, way before the last installment is due (CFPB 2020). This amount is well-understood for loans at compound interest, but much less so when simple interest is used. Recently, Aretusi and Mari (2018) have proposed a formula for the payoff amount for loans at simple interest. We assume that the payoff amounts are established contractually at time zero, whence the requirement that no arbitrage may arise this way The first goal of this paper is to study this new formula and derive it within a model of a loan market in which loans are bought and sold at simple interest, interest rates change over time, and no arbitrage opportunities exist. The second goal is to show that this formula exhibits a behaviour rather different from the one which occurs when compound interest is used. Indeed, we show that, if the installments are constant and if the interest rate is greater than a critical value (which depends on the number of installments), then the sequence of the payoff amounts is increasing before a certain critical time, and will start decreasing only thereafter. We also show that the critical value is decreasing as a function of the number of installments. For two installments, the critical value is equal to the golden section. The third goal is to introduce a more efficient polynomial notation, which encodes a basic tenet of the subject: Each amount of money is embedded in a time position (to wit: The time when it is due). The model of a loan market we propose is naturally linked to this new notation.