A discrete time Markov chain model for predicting the duration of a retail mortgage in the non-default states.

Author:Hassan, Morsheda


Duration of a retail mortgage, which could provide a snapshot of a portfolio' health status, is defined as the expected number of days before a loan is charged off. Te terms or number of years that are usually specified by the mortgage contract could not be considered as an effective proxy for the expected duration defined above. Ths is because of the extension risk, which is defined as the extension of the mortgage terms because of the past due status of the loan, and the contraction risk, which is defined as the shrinkage of the mortgage terms because of the early payment, or prepayment as most mortgage contracts do not include the prepayment penalty. As a result, an accurate prediction of duration, taking both extension risk and contraction risk into account, could be a major factor in credit risk management.

In this study, the authors propose a discrete Markov chain model to predict the expected duration of the loan in the non-default states. This model could be used to predict the timing of a given credit event for portfolios with different vintages, and thus provide a dynamic comparison between portfolios. For example, a 20-year semi-paid portfolio with expected default at the 21st year is definitely better than a 30-year annually paid portfolio with expected default at the 29th year.

The structure of this study is organized as follows: Section one provides a review of salient studies using Markov chain models to perform credit analysis. Section two gives a theoretical derivation of the portfolio economic assets model based on a Markov chain. Section three provides an empirical application of the model, and section four gives the conclusion and future studies.


There are many quantitative methods in credit asset management. As summarized by White (1993), Markov decision models have been frequently used in 18 areas, including (1) Finance and Investment, (2) Insurance, and (3) Credit Analysis. Of the 98 papers discussed by White, 9 papers relate to finance and investment, 2 to insurance, and 2 to credit analysis. This survey is by no means comprehensive, but it reveals the fact that Markov chains have been used extensively t analyze real world data.

General concepts of Markov processes are presented in Ross (1996). Let [[pi].sub.ij] be the steady state probability or limiting probability of being in state i and adapting policy j, [[pi].sub.ij] = [lim.sub.n[right arrow][infinity]][P.sub.ij](n), where [X.sub.n] n = 1,2,3 ... represent the state of a Markov chain at the nth transition.. As such, the expected benefit is given as

[[summation].sub.i][[summation].sub.i][[pi].sub.ij][R(i,j)-C(i,j)] (1)

where, R(i, j), C(i, j)are defined as the reward function and cost function for being in state / and adopting policy /, respectively. Also, dynamic programming could be used to find an optimal policy / to maximize the expected benefit. To this end, one may maximize

[[summation].sub.i][[summation].sub.i][[pi].sub.ij][R(i,j)-C(i,j)] Subject to [[pi].sub.ij][greater than or equal to]0, and [[summation].sub.i][[summation].sub.i][pi]ij =1 (2)

Consumer credit analysis is used to analyze account receivable, as triggered by credit sales. The model, based on the transition probability between different states, is primarily used by a company to adjust its credit sale and collection policy. Absorbing states could be reached either by collection or bad debt, both of which lead to a decline in the portfolio size.

On the other hand, by defining a past-due period as a different transient state, and default as an absorbing state, Markov models are used to analyze the characteristics of a loan portfolio, namely the estimated duration before an individual default, prediction of economic portfolio balance, and health index. The primary purpose of this study is to develop this type of model for banks and other commercial lending institutes in order to analyze the nature of their products.

Markov Models for Consumer Credit Analysis

Cyert, Davison and Thompson (1962) developed a finite stationary Markov chain model to predict uncollectible amounts (receivables) in each of the past due category. This classic model is referred to as the CDT model. The states of the chain ([S.sub.j], j=0,1,2 ..., J were defined as normal payment, past due, and bad-debt states. The probability [P.sub.ij] of a dollar in state [S.sub.i] at time t transiting to state [S.sub.j] at time t + 1 is given as

[P.sub.ij]=[B.sub.ij]/[[summation].sub.m=0 to J][B.sub.im] (3)


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