An Emerging Credit Risk Framework

The main result of this paper is the establishment of an analytic formula for the estimation capital requirements of an individual loan. The derived formula, may be considered as a direct analogue to the Basel risk-weight functions for credit risk, first presented in Basel II framework, with the additional advantage of utilizing a lifetime horizon, thus being suitable for IFRS9/GAAP purposes. Essentially, it bridges the gap between Basel and IFRS9 frameworks as:  The 1-year horizon incorporated in the Basel PD is extended up to maturity, following the IFRS9 rationale;  The notion of unexpected losses, already supplied by Basel Framework, is added to the IFRS9 logic in an analytic fashion. Going one step further, the generalization of the risk variables used in the formula, as Kumaraswamy identically distributed variables, allows for the benchmarking of the total loss of a credit portfolio, with the single knowledge of its current non-performing loans percentage. This conclusion is successfully verified against EBA stress test results for the period 2018-2020.

The necessary input is = Loss given default = Obligor risk curve parameter = Loan maturity This result will be gradually developed in section 0

Benchmarking Bank Stressed Capital Requirements
In Addition, if = The percentage of non-performing loans, of a financial institution, then equation calculated on monthly basis, offers an adequate upper limit benchmark for the stressed losses per 1€ of the Bank's loan portfolio. The accompanying lower limit for the loss estimation is The portfolio level estimation will be built and tested in section 0.

Definitions & Notation
In the current segment we refer to some basic terminology. What follows is the gradual presentation of the risk parameters that will be included in our credit formula, as presented in sections 0 and 0. 0 = Time index indicating current end of month = Time distance in months from point 0 .
We assume that an obligor has exposures in two main loan categories (segments), business loans and consumer/mortgage/personal loans, indicated by ∈ * , + = The situation in which the obligor makes loans payments according to schedule, or not 100% according to schedule, where any minor delays do not indicate that he will not be able to fulfill the terms and conditions of his contract(s).
= The situation in which reduced or delayed payments are observed at least in one of the obligor's loans, indicating that he will not be able to fulfill the terms and conditions of his contract(s).
= The situation that usually follows in which the obligor does not make any payments, at least in one of his loans, and the financial institution has initiated a recovery process, accompanied by legal procedures.
= The expected credit loss (ECL) at time , the loan amount that under the existing (or expected) economic conditions we expect not to be returned to the financial institution.
= Exposure at default, is the on-balance loan amount plus any approved limits that the obligor may use without further notification to the financial institution, if default happens at time .

Theoretical Loss Assessment
The total expected loss of a client at current time 0 , is comprised of the loss in each major loan category.
For simplicity reasons during the presentation of the analysis, we will initially focus on the process ignoring separate loan categories ( subscript). So, for a loan in accruing state at present time 0 , the expected losses throughout its remaining life until maturity, calculated at present time ( 0 ), should be: (5) ( , + -= the probability of the obligor to default during the month interval following month = The maturity of the loan expressed in months; = The expected annual market rate for equivalent risk loans.
Month adjustments of annual rates: Assuming loan initiation at 0 , the monthly installment is calculated as The amount at risk, at each point in time is the remaining number of installments that is approximately ( − ) • , which includes both capital and interest.
The present value of the series of the remaining installments at each point in time ( ) , is given by a geometric progression: The proper amount at risk is given by the combination of (6), (7) Where ( ) = A decay function which indicates the percentage of the original loan 0 , at each point in time , that is at risk, given the agreed interest rate and the prevailing market rate .
The decay function is approached next.

Time Decay Function Approximation ( )
The decay function ( ) could be adequately approximated by a 3-parameter continuous function = The remainder percentage at the last installment which is set at = 0.001 (10) = The curvature parameter, which represents in our definition, the percent of initial loan considered as amount at risk at the loan half life With the valid business assumption that the loan rate will always converge close enough to prevailing market rates, the following detailed approximation may be applied Thus, with the use of relations (11), (12) and (13) we are able to approximate the percentage of a current ( 0 ) exposure at any point of the remaining exposure life.

Probability of Default
As we saw earlier, in 0, ( , +represents the probability that the obligor will transfer from Accruing State into a Default State during the semi-open interval ( , + -. We may generalize for not to stand for 1 month, but a time fraction expressed in years.
For an obligor with a loan granted at time 0 the cumulative probability function is considered exponential, given by If we consider that 0 = 0 months the expression simplifies to The exponential function described in ( _10) is suitable for the depiction of cumulative PD as it exhibits the following properties (17) Expresses the initial condition of non-default.
(18) Is the boundary condition (19) Shows that the cumulative probability increases with time (20) Declares that the cumulative probability increases in a decreasing rate, as we approach loan expiration date.
The one period discrete marginal probability applied at specific month points in time are given by = The exponential function risk parameter that defines the obligor PD curve, estimated at each point in time The continuous marginal probability is the cumulative function derivative

Modelling the Alpha Risk Parameter with the Use of Existing PDs
In this paper we will not argue for the creation of another PD model. We logically adopt the point of view, that there is at least one PD model in a financial institution for every obligor / product type. The existing model scores will be recalibrated as close as possible to the cumulative probability of the form depicted in (16) Again, providing some necessary definitions:   Equal sample fragments are used in each case, using either increasing or decreasing model score in order to have increasing default frequencies  A bounded curve in ,0,1is constructed with the use of average points, which is either monotonous or has one change in monotonicity. In case no more than 1 intervals can be created, all obligors are gathered in one interval Each obligor is evaluated by one or more models in each segment, and receives multiple scores  In each Segment, every model receives a weight, according to its Accuracy Ratio ( )  Each obligor, through his assigned scores, receives the AR-weighted for every combination of −  If the obligor is not a part of the specific segment and has no calculated scores, he receives ̅̅̅̅ ( , ) Using least square minimization criterion in segmentobligor level  Two alpha type ( ) parameters are evaluated for each obligor for every segment So, every obligor receives 4 alpha parameters, 2 in each portfolio , ∈ * , The following conditions are examined: , > , ⟹ The long-term historical default risk is less than the most recent estimated one. Economy is possibly entering a recession phase and so the most conservative estimation is required, , .
, < , ⟹ The long-term historical probability of default is enhanced compared to the most recent estimated PD. Economy is possibly exiting a recession phase. The most conservative estimation in this case, , will result in reduced and more normalized credit expansion. In that case it would seem more appropriate to use , parameter in the process of unexpected credit losses (extra required capital) and not for provision purposes.
Concluding the previous observations, we end up with: The effect of the macroeconomic environment is captured through the historical evolution and the current level of percentage. , = The amount of non-performing loan amounts at point in time (month) for segment ∈ * , + , = The total amount loans at point in time (month) for segment , = The percentage of non-performing loan amounts, defined as: The time series of , contains the measurable effect of all macroeconomic factors and their respective lags, on the Bank's segment portfolios. The adversity of macroeconomic conditions' cumulative effect is correctly portrayed through the increase of , . If: ( +1, ) = the expected value to prevail next month + 1 , = the observed value at month The maximum expected gross change, is: ̂= The maximum expected change of , through the next month period The maximum change may be analyzed as follows: = The set of all macro and market factors, with their respective lags, that have an influence on current percent ( ). We assume ∈ (−∞, +∞) ∈ (0,1) may be expressed as a sigmoid function of The marginal change of , , in real terms the change over a month, is given by From ( may be estimated. We assume that in the long term ( ) = 0 and maintain a conservative value of = 3 • for use in equation (30). The worst expected value for +1, is given by a rearrangement of (30) The worst expected default rate: ̂, = The default rate, including considering the maximum expected deterioration of , with the assumption that , remains stable during the period , , + 1 - With the help of ̂, we end with an incremental alpha risk parameter estimation, for the specific credit

Loss Given Default
As established in 0 ( , + -| = The final percentage loss, measured until recovery time , that the bank will incur if obligor enters into ijef.ccsenet.org International Journal of Economics and Finance Vol. 12, No. 5; 2020 a (assumed to be later followed by the ) during the semi-open interval ( , + -= A Bank policy variable, the maximum allowed time interval, measured in months, during which all recovery is accumulated. An indicative value set is = 5 .
Though we do not propose a new LGD model, we consider it logical to adopt the assumption that after entering into , obligors cannot be efficiently categorized with the use of a classical rating system, since the explanatory power of all atomic "pre-default" characteristics has collapsed.
The only thing that guarantees defaulted loan partial repayments, and perhaps acts as an additional motivation for the obligor to repay, is the level and quality of collateral.
For the downturn LGD value, we will assume that impaired collateral values are applied, in any LGD model calculations, representing the recession (downturn) period of the economic circle.
, 0 = worst period LGD as defined at time snapshot 0 This essentially is the same definition implied in the Basel formulas.

Credit Losses
With the combination of the results provided in 0, 0, 0, 0 we will end up in the specification of equation (1), for expected and unexpected losses, into the final equations (38), (39) and (40).

Expected Credit Losses
The theoretical measurement of expected loss for a single exposure is provided by (5) Replacing into (5) the relations (9), (22), we construct the continuous time analogue of equation (5) The amount of 0 is the amount to be held in provisions, for one exposure of level 0 at observation time 0 .

Unexpected Credit Losses
For the calculation of losses under stressed conditions, we will apply the result of (35) into (38), and simplify the notation of functions Unexpected loss, that is excess capital reserve over provisions, for the one exposure, is estimated as the difference: The amount of 0 is the amount to be held in capital for the absorbance of extra-ordinary losses.

Portfolio Level Losses with the Use of Aggregate Data
If loan and obligor data are available, the application of (38), (39), (40) will provide an accurate estimation of lifetime losses for 1 exposure which is afterwards summed for the total portfolio. ijef.ccsenet.org International Journal of Economics and Finance Vol. 12, No. 5;2020 However, when only aggregate figures are published for a credit portfolio, we will use a generalized version of equation ( Assuming perfectly correlated and identically distributed risk factors, the two risk variables above, may be represented by ∈ ,0,1-, leading to the next equation = / • 2 (43)

The Kumaraswamy Distribution Assumption
For a variable ∈ ,0,1-, > 0 the probability density and probability cumulative functions are Imposing on (43) ⟶ ( , ) We arrive at the loss of the total portfolio as Where the expected loss for every 1€ of the performing part of the portfolio is: Is the average loss of the performing, 1 − portfolio segment. If we consider that = 1 for obligors that belong in the segment of the portfolio, being the non-performing loan percentage, then the expected loss for every 1€ of the non-performing part of the portfolio is: The total percentage loss (per 1€ of portfolio value) is The solution of which produces

Numerical Approximation of Loss Curves
For the numerical calibration of ( , , ) loss function, we perform the following minimization procedure , that is the average non-performing percent observed in the portfolio is subject to the same risk distribution imposed on the other correlated risk factors of (43)  the pairs of { , } are approximated by the cdf function Results are portrayed on the Table 1, along with the relevant graph and the expected loss function (51) inside the table.
For any level of of a credit portfolio, (51) yields the percent of the portfolio value that should be kept as provisions.  Vol. 12, No. 5;2020 4.2

.1 Unexpected Losses Function
If is the current level of percentage non-performing loans of the portfolio, then the value of (51) should be kept as provisions. However, a higher amount should be set aside in order to cover for unexpected worsening of economic conditions. We assume that the potential of in any time snapshot may be described by a sigmoid function The worsening of over the next monthly snapshot is given by Repeating the procedure described in 0 but with the use of +1 value of (53) we estimate the "total loss curve" as specified in Table 2. Total loss based on worse expected npl level, where the relevant function (54) and data are included in the table.

Benchmarking with EBA 2018 Stress Test Results
In 2018 EBA performed stress tests on 48 selected European Banks ,9-. Based on data of 31/12/2017, 2 scenarios were applied, baseline and adverse, for the following 3-year period up to 2020.
For the purpose of comparison, we will estimate the resulting provisions for each scenario as a function of the restated NPL percent ( ) of 31/12/2017. For each Bank, we calculate the following: Non-performing loan percentage The percentage loss is Where:  From Figure 2. Percentage loss for end 2017, it is established that the application of our benchmarking policy equations , provide full coverage for the current year (2017 in our example). The two curves of , may serve as lower and upper limit respectively for the percentage level of losses.
Nevertheless, the value of , goes beyond the current year. The next 2 graphs, Figure 3 and Figure 4, illustrate that for a specific time snapshot (month), if the necessary capital to cover for credit risk losses is calculated according to curve, then the portfolio ison averageadequately covered against extreme macroeconomic fluctuation effects (EBA scenarios) for the next 3-year period.

Conclusion
Throughout this paper, we have demonstrated that analytic solutions can be derived for the estimation of lifetime credit losses, based on a set of business and statistical assumptions. We have approached this issue both form the point of view of the financial institution, developing analytic solution for individual loans, and from the perspective of financial authorities or even investors, providing benchmarks for the level of losses in terms of large credit portfolios.