A Markov Process Model of Joint Liability and Loan Repayment for Sustainable Microfinance

We develop a discrete-time Markov process model for group lending with joint liability to analyze repayment dynamics, income sustainability, and optimal contract design in microfinance. Borrower groups transition between application, beneficiary, delay, and re-application states, with transition probabilities determined by repayment success, delay persistence, partial repayments, and re-entry mechanisms. Using the stationary distribution of the resulting Markov chain, we derive closed-form expressions for expected discounted group income under both full and partial repayment regimes. The analysis shows that expected income increases with repayment success, re-entry probability, and partial repayment effectiveness, while higher delay persistence and default risk substantially reduce long-run wealth. We derive an endogenous optimal interest rate satisfying a lender break-even condition with operational costs and joint-liability spillovers. The resulting pricing rule generalizes zero-net-present-value loan rates to environments with delay risk and burden sharing. A sensitivity analysis indicates that larger borrower pools reduce equilibrium interest rates through risk sharing, whereas higher default burdens and costs necessitate higher rates. The modelling approach provides a tractable analytical basis for pricing and sustainability in joint-liability microfinance lending.