An effective interest rate cap: a clarification

The national legislation of many countries imposes restrictions on lending rates known as interest rate caps. In most cases, the effective (rather than nominal) interest rate is restricted, which includes all commissions and fees associated with a loan. Typically, the generic wording of this restriction is ambiguous in three respects. First, the literature provides several nonequivalent concepts of internal rate of return (IRR). Since the effective interest rate is the IRR of the cash flow stream of a loan, the wording should specify which concept of IRR is used. Second, most definitions of IRR are partial in the sense that there are cash flow streams that have no IRR. Thus, the wording is vague for loans that have no IRR. Third, when loan advances and repayments alternate in time, the respective roles of the borrower and lender can be blurred. In this case, it is unclear to which of the parties of the loan contract the regulatory measure should be addressed and, if it treats the contract as illegal, which party should be brought to justice. This paper aims to resolve these ambiguities. We start by clarifying the concept of IRR. We axiomatize the conventional definition of IRR (as a unique root of the IRR polynomial) and show that any extension to a larger domain necessarily violates a natural axiom. Given this result, we show how to derive an effective interest rate cap. We prove that there is a unique solution consistent with a set of reasonable conditions. Finally, we suggest a way to avoid the problem of identification of the roles of the loan contract parties by imposing both a floor and a ceiling on lending rates.