Multi-Period Merton-Vasicek-Pykhtin Model

52 Pages Posted: 14 Mar 2018 Last revised: 16 Jun 2018

See all articles by Martin Smid

Martin Smid

Institute of Information Theory and Automation, Prague

Jaroslav Dufek

Institute of Information Theory and Automation, Prague

Jan Vorisek

Academy of Sciences of the Czech Republic

Date Written: June 8, 2018

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.

Keywords: Credit Risk, Structural Factor Models, Loan Portfolio Management

JEL Classification: G32

Suggested Citation

Smid, Martin and Dufek, Jaroslav and Vorisek, Jan, Multi-Period Merton-Vasicek-Pykhtin Model (June 8, 2018). Available at SSRN: https://ssrn.com/abstract=3138545 or http://dx.doi.org/10.2139/ssrn.3138545

Martin Smid (Contact Author)

Institute of Information Theory and Automation, Prague ( email )

Pod vodarenskou vezi 4
Praha, CZ-18208
Czech Republic

HOME PAGE: http://www.klec.cz/martin

Jaroslav Dufek

Institute of Information Theory and Automation, Prague ( email )

Pod vodarenskou vezi 4
Praha, CZ-18208
Czech Republic

Jan Vorisek

Academy of Sciences of the Czech Republic ( email )

Narodni 3, 111 42
Praha 1, 117 20
Czech Republic

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