On distributional and asymptotic results for exponential functional of renewal -- reward processes describing risk models

TL;DR

This paper models lender cash flow using exponential functionals of renewal-reward processes, analyzing the asymptotic behavior of the probability of full loan repayment and providing explicit calculations in specific scenarios.

Contribution

It introduces a novel approach to model risk in loan repayment using renewal-reward processes and derives asymptotic and exact results for repayment probabilities.

Findings

Finite-time repayment probability converges exponentially fast to the infinite-time probability.

Explicit formulas for infinite-time repayment probabilities in certain scenarios.

Analysis applicable to risk models inspired by real-world mortgage problems.

Abstract

Inspired by the double-debt problem in Japan where the mortgagor has to pay the remaining loan even if their house was destroyed by a catastrophic event, we model the lender's cash flow, by an exponential functional of a renewal-reward process. We propose an insurance add-on to the loan repayments and analyse the asymptotic behavior of the distribution of the first hitting time, which represents the probability of full repayment. We show that the finite-time probability of full loan repayment converges exponentially fast to the infinite-time one. In a few concrete scenarios, we calculate the exact form of the infinite-time probability and the corresponding premiums.