Quantum Bohmian Model for Amortization Systems

Amortization systems are widely used in finance to design loans and future contracts. In this work, we introduce a generalization of amortization systems based on the de Broglie-Bohm pilot-wave formalism. We derive the Lagrangian of the debt, which allows for the description of the typical dynamical evolution of any amortization system with a finite payment schedule, where the potential is a shifted inverted harmonic oscillator. By knowing the Hamiltonian, we proceed with the first quantization in order to obtain a Schrodinger equation for the debt. Using the wave function, it is possible to derive a quantum potential, thereby altering the debt’s dynamics with quantum corrections proportional to a parameter analogous to Planck’s constant. We then analyze the French amortization system, which features constant payments. We find that the quantum correction introduces a transcendental equation for determining the payment. Furthermore, at low interest rates, the quantum dynamical equation simplifies to the classical debt equation, with an effective interest rate and payment. We analyze the new solutions found for the debt and discuss a possible interpretation of the wave function, still under exploration, and its relevance to discounting.