Multi-Period Merton-Vasicek-Pykhtin Model

Abstract

We propose a dynamic structural model of a loan portfolio, secured by collaterals. Contrary to existing dynamic models, our model takes into account the time-dependence of the debtors' wealth and the fact that, due to defaults, the financial health within the portfolio improves in comparison to the population. As such, the model replicates the empirically observed decrease of loans' default rates in time. In the model, the debtors' resources, insufficiency of which is assumed to cause defaults, and the prices of the collaterals, determining the losses given default (LGD), depend on common and individual factors. The individual factors follow an AR(1) vector process with general residuals, the common factors are general, possibly dependent on exogenous variables. We show that the mapping transforming the common factors into the conditional probabilities of default (PD) and the LGDs is one-to-one monotonous continuously differentiable. As this transformation is not analytically tractable, we propose an approximation technique which is convergent with polynomial complexity. To demonstrate a possible application of our model, we formulate a decision problem for optimal choice of loan candidates. Further, we discuss estimation of the model's parameters, we suggest ways of modelling heterogenous portfolios, and we give arguments supporting empirical validity of the model.