Quantum credit loans

TL;DR

This work presents a quantum-like generalization of credit loans by encoding debt $D$, amortization $A$, interest $Y$, and installments $Q$ as operators on a finite-dimensional vector space of dimension $M$. A generalized Heisenberg algebra is used, and an $SO(M)$ rotation of the eigenbasis yields transformed operators whose mean values reproduce different repayment schedules without altering the total amount paid ($Tr(Q)$) or total amortization. The authors analyze concrete cases with $M=2$ and $M=3$, demonstrating how basis rotations produce superpositions of installment schedules and how to recover or mix French and German style payments; they also extend the framework to indexed loans such as UVA, where payments depend on an inflation-linked unit. The framework offers new degrees of freedom for tailoring borrower-friendly schedules, suggests connections to quantum finance constructs like coherent states and path integrals, and points to applications in entangled loan pools and risk-aware credit design. Overall, the quantum credit loan model provides a flexible, algebraic approach to redesigning repayment structures while preserving lender profitability.

Abstract

Quantum models based on the mathematics of quantum mechanics (QM) have been developed in cognitive sciences, game theory and econophysics. In this work a generalization of credit loans is introduced by using the vector space formalism of QM. Operators for the debt, amortization, interest and periodic installments are defined and its mean values in an arbitrary orthonormal basis of the vectorial space give the corresponding values at each period of the loan. Endowing the vector space of dimension M, where M is the loan duration, with a SO(M) symmetry, it is possible to rotate the eigenbasis to obtain better schedule periodic payments for the borrower, by using the rotation angles of the SO(M) transformation. Given that a rotation preserves the length of the vectors, the total amortization, debt and periodic installments are not changed. For a general description of the formalism introduced, the loan operator relations are given in terms of a generalized Heisenberg algebra, where finite dimensional representations are considered and commutative operators are defined for the specific loan types. The results obtained are an improvement of the usual financial instrument of credit because introduce several degrees of freedom through the rotation angles, which allows to select superposition states of the corresponding commutative operators that enables the borrower to tune the periodic installments in order to obtain better benefits without changing what the lender earns.

Figures (5)

• Figure 1: Non-indexed credit loan with $d_{0}=100$, $M=10$ and $t=0.2$. Left: French system. Right: German system.
• Figure 2: $\mathbf{Q}_{F}-\mathbf{Q}_{G}$ difference, where $\mathbf{Q}_{F}$ is the sum of the periodic installments in the French system and $\mathbf{Q}_{G}$ is the sum of the periodic installments in the German system for different values of $t/d_{0}$ and $M/d_{0}$.
• Figure 3: Indexed credit loan with $d_{0}=100$, $M=10$, $t=0.2\,$ and $u_{n}=a^{n}$ where $a=1.1$. Left: French system. Right: German system.
• Figure 4: Transformed periodic installments $\overline{q}_{1}/q_{1}$ and $\overline{q}_{2}/q_{1}$ as a function of $x=\sin \phi$ with $a=1.1$. Vertical lines indicates the values $x=\pm 1/\sqrt{2}$.
• Figure 5: Regions where $\overline{q}_{1}-q_{1}<0$, $\overline{q}_{2}-q_{2}<0$ and $\overline{q}_{3}-q_{3}>0$ holds, with $a=1.05$. Right: $z=\sin \gamma =0.6$. Left: $z=\sin \gamma =0.7$.