Standard transition matrices among classes of credit risk are grounded on few fundamental assumptions. Among them, two are specifically relevant. The first standard methodological assumption considers migra-tions are ruled by a transition matrix induced by a homogeneous Markov process generator, even though the recent approaches available in literature are revising its feasibility. In fact, the Markovian assumption implies different borrowers in the same rating class share the same probability to migrate to the same rating class in the future, without care of their rating history. As a second simplifying hypothesis, default state is assumed absorbing even if, very often, empirical evidence rejects it. A further relevant but neglected aspect to take care is that the universe of borrowers is never the same: some borrowers exit the credit system while new ones enter it through time; clearly this extends also to the sample data of a given lender. As a consequence, standard transition matrices are suitable to describe transitions of borrowers already and persistently in the credit system, while they should take care of the system renewal and, therefore, how the system dimension evolves. In this paper, the proposed model tackles the dynamic evolution of the borrowers' universe. New entries as well as exits from the system are regarded as pure births and deaths processes: their dynamics explains the renewal of the system through time. The migrations of borrowers already inside the credit system over a set of rating classes are looked as recombinations. That is, by following a demographic analogy, the former describe the natural balance of population while the latter one concerns the migratory balance. By associating a diversified provision rate to each rating class, it is possible evaluating the amount of own funds the lender must prompt as economic capital at risk to obey regulatory norms. This is in order to make their credit activity more sound and sustainable in terms of macro-prudential and systemic financial stability. Moreover, the proposed model allows for linking the borrowers dynamics to the business cycle, or exoge-nous policy forecasts provided by international economic institutions, to predict the dynamics of the number of the borrowers as well as the dynamics of the economic capital at risk for lenders. In this framework, the change of lender's own funds at risk can also be analysed where, in normal conditions, such a change is financed by means of interest margins. When this is not enough, i.e. when the lenders are stressed, the lender resorts to its own capital. Specifically, the model enables to estimate the dynamics of the lender's portfolio configuration in three situations. The first one concerns the unconditioned expected migrations at one year by using the long run estimate of transition matrices: this provides estimates of the unconditional expected losses, the average cost of risk to be financed by revenues of lenders. The second situation concerns the conditional expected migrations at one year under an extreme case hypothesis: it gives the estimate of the expected losses while conditioning on a given policy or exogenous scenario. The third case provides the unexpected losses estimate the lender uses to set the value of economic capital at risk to be provisioned by conditioning the migration matrix on a macroeconomic scenario.

### A Non-Absorbing Migration Rate with Renewal Approach to the Dynamic Estimation of Credit Risk Economic Capital

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*UBERTI, Mariacristina*

##### 2014-01-01

#### Abstract

Standard transition matrices among classes of credit risk are grounded on few fundamental assumptions. Among them, two are specifically relevant. The first standard methodological assumption considers migra-tions are ruled by a transition matrix induced by a homogeneous Markov process generator, even though the recent approaches available in literature are revising its feasibility. In fact, the Markovian assumption implies different borrowers in the same rating class share the same probability to migrate to the same rating class in the future, without care of their rating history. As a second simplifying hypothesis, default state is assumed absorbing even if, very often, empirical evidence rejects it. A further relevant but neglected aspect to take care is that the universe of borrowers is never the same: some borrowers exit the credit system while new ones enter it through time; clearly this extends also to the sample data of a given lender. As a consequence, standard transition matrices are suitable to describe transitions of borrowers already and persistently in the credit system, while they should take care of the system renewal and, therefore, how the system dimension evolves. In this paper, the proposed model tackles the dynamic evolution of the borrowers' universe. New entries as well as exits from the system are regarded as pure births and deaths processes: their dynamics explains the renewal of the system through time. The migrations of borrowers already inside the credit system over a set of rating classes are looked as recombinations. That is, by following a demographic analogy, the former describe the natural balance of population while the latter one concerns the migratory balance. By associating a diversified provision rate to each rating class, it is possible evaluating the amount of own funds the lender must prompt as economic capital at risk to obey regulatory norms. This is in order to make their credit activity more sound and sustainable in terms of macro-prudential and systemic financial stability. Moreover, the proposed model allows for linking the borrowers dynamics to the business cycle, or exoge-nous policy forecasts provided by international economic institutions, to predict the dynamics of the number of the borrowers as well as the dynamics of the economic capital at risk for lenders. In this framework, the change of lender's own funds at risk can also be analysed where, in normal conditions, such a change is financed by means of interest margins. When this is not enough, i.e. when the lenders are stressed, the lender resorts to its own capital. Specifically, the model enables to estimate the dynamics of the lender's portfolio configuration in three situations. The first one concerns the unconditioned expected migrations at one year by using the long run estimate of transition matrices: this provides estimates of the unconditional expected losses, the average cost of risk to be financed by revenues of lenders. The second situation concerns the conditional expected migrations at one year under an extreme case hypothesis: it gives the estimate of the expected losses while conditioning on a given policy or exogenous scenario. The third case provides the unexpected losses estimate the lender uses to set the value of economic capital at risk to be provisioned by conditioning the migration matrix on a macroeconomic scenario.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.