Modeling amortization systems with vector spaces

TL;DR

This work reframes amortization theory within a finite-dimensional vector-space formalism by treating debt, amortization, interest, and payments as commuting operators on an $M$-dimensional space with an $SO(M)$ symmetry. It develops a generalized Heisenberg-algebra framework to recover classical recurrence relations as operator eigenvalues and shows that basis rotations can modify payment schedules without affecting total repayment or lender profit. The authors also introduce loan entanglement via tensor products and an entangling gate, enabling correlated repayment schedules across borrowers and quantifying the resulting entanglement with metrics like entanglement entropy. Collectively, the approach provides a novel, flexible toolkit for designing and analyzing amortization schemes and loan portfolios with potential applications to risk management and macroeconomic resilience.

Abstract

Amortization systems are used widely in economy to generate payment schedules to repaid an initial debt with its interest. We present a generalization of these amortization systems by introducing the mathematical formalism of quantum mechanics based on vector spaces. Operators are defined for debt, amortization, interest and periodic payment and their mean values are computed in different orthonormal basis. The vector space of the amortization system will have dimension M, where M is the loan maturity and the vectors will have a SO(M) symmetry, yielding the possibility of rotating the basis of the vector space while preserving the distance among vectors. The results obtained are useful to add degrees of freedom to the usual amortization systems without affecting the interest profits of the lender while also benefitting the borrower who is able to alter the payment schedules. Furthermore, using the tensor product of algebras, we introduce loans entanglement in which two borrowers can correlate the payment schedules without altering the total repaid.

Figures (6)

• Figure 1: Flexible French system with $M=3$, $d_{0}=100$ y $t=0.2$. Different payment schedules are obtained as a function of the initial payment $q_{1}$. When $q_{1}=q_{F}$ all the payment schedules are identical to $q_{F}$.
• Figure 2: The setup of two-borrower entanglement loan.
• Figure 3: Entangled periodic payments as a function of $\gamma$ for $M=2$. Both borrowers have identical initial debts but different interest rate. a) $\theta _{1}=0$ and $\theta _{2}=0$. b) $\theta _{1}=\pi /3$ and $\theta _{2}=\pi /6$.
• Figure 4: The setup of two-borrower entanglement loan.
• Figure 5: Entangled payments for the first borrower as a function of $\gamma$ for $M=3$ and two specific set of angles. The initial payments are $q_{1}^{(1)}=3$, $q_{2}^{(1)}=6$ and $q_{3}^{(1)}=9$. a) $\theta _{1}=0$, $\phi _{1}=0$, $\psi _{1}=0$. b) $\theta _{1}=\pi /3$, $\phi _{1}=\pi$, $\psi _{1}=\pi /4$. c) $\theta _{2}=0$, $\phi _{2}=0$, $\psi _{2}=0$. d) $\theta _{2}=\pi /3$, $\phi _{2}=\pi$, $\psi _{2}=\pi /4$.