Analytical Solution For Expected Loss Of A Collateralized Loan With Regime Switching

There are two basic approaches to study the Credit Risk Theory: Reduced Form Approach and Structure Approach. Adopting the Reduced Form Approach, this paper conducts a quantitative analysis of the expected loss of a collateralized loan. To begin with, a continuous-time Markov chain is introduced on the basis of existing models to represent macro-economic state changes and establish a expected loss model with regime-switching. The model mainly involves two stochastic processes: 1. the hazard rate process of the enterprise that controlling events of default; 2. the process in which the transaction counterparty is provided with collateral value in case of default. We presume that collateral value is in geometric Brownian motion, and that the rates of return, fluctuation, and hazard rate are simultaneously subject to economic variants. On that basis, this paper propounds an explicit solution to the expected loss of collateralized loan and works out an explicit solution to the two states of Markov-chain drawing on the notion of "occupation times". Last, the numerical relationship between expected loss, return rate, and time limit is figured out by means of numerical computation.