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A Non-Absorbing Migration Rate with Renewal Approach to the Dynamic

Estimation of Credit Risk Economic Capital

Casellina S.a, Landini S.b, Uberti M.c

aBanca d’Italia, Rome, Italy.

bIRES Piemonte, Turin, Italy.

cDepartment of Management, Universit`a degli studi di Torino, Turin, Italy.

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.

Keywords: Macro prudential and regulatory criteria; Rating classes; Credit risk migration models;

Lenders’ economic capital at risk; Markov processes generators.

Submitted to the 8th MDEF Workshop, Urbino, Italy August 11, 2014