Some Approaches to the Calibration of Internal Rating Models

This article covers the peculiarities of calibration of internal rating models which are the most popular approach to assessing credit risks. The authors address the most common approaches and methods used for rating models calibration, as well as propose their own algorithm for calibration, the main feature of which is taking into account the forecasted probability of default on the portfolio. Research methods include regression analysis, time series analysis (ARIMA models development). Reliability of the proposed approach has been verified on the basis of the portfolio, based on the debt obligations of Russia financial institutions. The advantage of the proposed approach towards determination of the average default probability of credit portfolio is that one gets a tool that allows you to construct a rating model that is “forward looking”, respectively, appear to more quickly adapt to the changing patterns of rating environment.


Introduction
Recent global crisis indicated high interrelation between both financial institutions and even regional financial systems with the global systematic risk.Though fiscal policy of regional regulators can reduce the impact of negative trends in the local market, absence of such regulating authority on the global scale stipulates that reduction of "regional systematic" risk does not mean proper mitigation of the "world systematic" risk.Accordingly, a crisis in one region or country can cause the "domino effect" and consume all other regions and countries, despite the futile efforts of local regulators.This effect is especially material for developing countries.The impact could be observed via the so-called "flight to quality" effect, e.g. the size of the foreign investment in a particular country.Being the economic growth driver of any country, credit institutions are at the same time are the most susceptible to systematic risk: hence only proper risk mitigation policies can be viewed as pillars for survival and successful doing business for credit institutions.Traditional banks are by default exposed to credit risks (which is the nature of their business) and hence proper evaluation of those risks is the cornerstone of their profitability and survival.
It should be noted that the main study of statistical approaches towards rating model development are associated with the work of Altman E., Englmanna B., Erlenmeyer W., Hayden E., Tasha D. and others.For the validation of rating models should be noted works Kohavi R., Cook D., Picard R., Rauhmaer R. and others.The Russian researchers on these issues should be noted works Ayvazian S. A., Bukhtin M. A., Halavan S. V., Karminsky A. M., Lobanov A. A., Peresetsky A. A, Pomazanov M. V., Putilovsky V. A. et al.
However, the works of these authors largely contain general description of the problem, or are considering the individual stages of the development of rating models.Little relates to the issues of development of rating models for sub-portfolios with low or no defaults.(Burakov, 2014b;2014c) Most of existing research relates also to the allocation of economic capital and implication of the risk weighting calculation as proposed by the Basel Committee.
To remain a going concern a credit institution must have adequate methods and tools to differentiate borrowers by level of credit risk.In accordance with Basel II internal rating models can be viewed as such instrument, as providing for the best way to estimate the level of capital adequacy to cover credit risks because rating models providing representative result.

1) Development of the scoring model,
2) Calibration of the rating model.
The 1 st step is to setup the following equation: Where Score is defined as value assigned to the borrower and reflecting its relative creditworthiness (Nurlybayeva & Balakayeva, 2013); x 1 ,.., x n -borrowers quantitative factors (for example, leverage, return on assets, current liquidity, etc.); y 1 ,…, y n -borrowers qualitative factors (for example, quality of management, market position, credit history, etc.); f (.)-the functional relationship that translates quantitative and qualitative factors in scoring points (usually a function of the logistic or normal distribution).
The scoring itself can only tell whether 1 st Borrower is better than 2 nd Borrower (for instance, scoring of the 1 st Borrower is higher than that of the 2 nd ).This approach, however, does not give an answer to the question of the probability of default of the 1 st Borrower relative to the 2 nd Borrower.Another issue is lack of comparative basis of the scoring results of regular borrowers with borrowers having external ratings (assigned by global rating agencies-Moody's, Standard & Poor's, Fitch).

Calibration of the Rating Model
The above-mentioned issue can be solved via calibration of the model that is the process of determining a calibration function, which is used to adjust the scores by probability of default and internal ratings (see Figure 1).Practically, this would lead the average portfolio probability of default to reflect the actual (observed) value of probability of default.Depending on the availability of default statistics one of the below calibration functions may be used for calculation purposes.(Table 1) (Ipatyev, 2012(Ipatyev, , 2013) )   As observed from the calibration functions in Table 1, each of them is effectively a translation of the linear function α + β Score.The steeper the lien is, the lower is the differentiation model and hence less is the distribution of the ratings.(Figure 2) (Gurny, 2013) Multiplicative regression ratio β indicates the ratio by which the probability of default index increases while the value of scoring points changes by 1 rating notch.The additive ratio α is defined as the difference between scoring points and multiple of multiplicative factor and the probability of default values for a given rating.The ratios α and β of the calibration curve can be determined based on the following equation: Where N is the number of borrowers in a sub-portfolio for which a rating model is constructed; DR is the observed default rate for the portfolio on the 1-year horizon.
There are several ways to determine the DR Index.The two most common approaches are the so-called "through the cycle" (TTC) and the "point in time" (PIT) (Loffler, 2012).
In accordance with the PIT approach DR can be determined on the basis of the number of default-approaching borrowers in the sub-portfolio for the next year.Thus, to determine DR PIT we need to determine the number of non-defaulting borrowers into credit portfolio as of a year ago, and find out how many of them actually defaulted in the past 12 months.
In accordance with TCC approach it is necessary to determine the term of the credit cycle, i.e. the cycle, during which the banks are overoptimistic regarding the extension of credit facilities.During the next stage this "optimism" results in non-payments, non-performing loans and hence economic downturn, during which the banks (credit institutions) become risk averse and concentrate on bad loans write-offs rather than new loans.After some time the banks recover from the initial shock and resume loan extension, which promotes new economic growth that has a powerful impact on the banks' optimism thus leading to the new credit cycle.Therefore a borrower's probability of default determined by the TTC method resembles the long-term estimation of a borrower's financial stability.
Under Basel II criteria a 7-year cycle is proposed.Analysis of the dynamics of Russia's GDP over the last 20 years shows the validity of this approach for determination of the credit cycle length for the the Russian market.(see Figure 3).(3) For calculation of the economic capital (used to cover unexpected losses) and provisions (used to cover expected losses) Basel II clearly identifies the need to consider the level of losses in the credit cycle, i.e one must use the default probability TTC (Basel, 2001).There is no firm criteria regarding the pricing setup.As already mentioned above, the only criteria explicitly required by the Basel II is the need to use the results of the rating systems for capital allocation, and business purposes, including pricing ("use test").(Basel, 2006) In the case of TTC approach is selected the cost of risk component in pricing will be too high in case of economic growth (leading to a bank losing its competitive edge over peers) while being too low in case of downturn, since expected pricing wouldn't cover the actual level of losses, having a negative impact on the actual profit figure.
Thus application of PIT approach deems more reasonable, however it has some demerits.For instance, in case the loan is provided for longer periods, the performance of the loan portfolio may render the price of the 7 year long-term credit too small compared with the market standard for a particular borrower.(Note 3) (Gobeljic, 2012)

Hybrid Approach towards Calibration of Rating Models
The alternative approach to both methods is use of a hybrid approach, i.e. weight-average between PIT and TTC approach.The longer the loan, the more it should be closer to the approach of TTC.At the same time, short-term loans require repayment of the loan within one year, or for the period during which, as the model assumes, the characteristics of the loan portfolio and external factors change is not significant.In this case, the approach should be close to the PIT.We suggest using the following example of such a function.(Ipatyev, 2012) PD Hybrid = PD PIT (1-(0.1 (D-1))) + PD TTC (0.1 (D-1)) (4) Where in: D-the term of the requested loan in years (from 1 to N years, where N-the maximum number of years over which a bank issues a "standard" loans):  If the loan period is less than a 1 year, then the value is 1;  If the loan period is over N years, then the value is N;  In other cases, if the loan period is not equal to a whole year, it is rounded according to the rules of mathematical rounding.
The obvious disadvantage of the PIT is that calibration is based on sub-portfolio data a year backwards, and not on the current sub-portfolio, making it impossible to determine the percentage of the current portfolio to be allocated to default within the next year.
This disadvantage could be avoided if one knew how many credit institutions currently existing would became defaults during the next 12 months.However, this information is not available.But on the other hand, a tool can be created that would allow estimation of this value and then this estimate could be used instead of the quantity itself.(Burakov, 2014a) In this paper, we propose an approach to solve this problem, which is described by the following algorithm: 1) Correlation between DR and macroeconomic indicators is plotted (RTS / MICEX indices, interbank rates, inflation, etc.): 2) For each of the macroeconomic indicators (x i ) included in the model (5), the time series is constructed (for example, on the methodology of ARIMA); 3) For each of the macroeconomic indicators included in the model ( 5) are built predictive values of the indicators for the year ahead; 4) Predicted macroeconomic indicators are substituted into the model ( 5), so we get forecast DR.
To illustrate the proposed approach, one needs to consider a hypothetical sub-portfolio consisting of all credit institutions in a country (Russia).
To construct the correlation between macroeconomic indicators and defaulted credit organizations we examined the dynamics of shift in the number of banks in the country and the number of banks with licenses recalled (see Figure 4). All macroeconomic indicators are highly correlated with the above indicators.
In the first stage was constructed regression between the DR and the explanatory variables.Selected model, consisting of 5 explanatory indicators (R 2 is 97.57%).More detailed information is shown in Table 4.For each of the explanatory variables were constructed (by using ARIMA method) time series and with 1-year forecast.For example, RTS index RTSog by analyzing the autocorrelation function (ACF) and partial autocorrelation function (PACF) was defined as the time series form ARIMA (2, 0, 0).Statistical indicators and graphical representation of the time series RTSog and correlation analysis are presented in Table 5 and Figure 6, respectively.For example, according to the forecast, the rate RTSog within a year on 01.03.2014 and 01.04.2014 was to retain its value.As seen in Figure 7, despite the fact that the actual value indicator fluctuated around the prediction, the end of the period are substantially aligned.Thus, one can conclude that the constructed time series quite clearly predict the future value of the index.However, due to the other indicators the overall dynamics of the index is less positive, but the observed values of the series does not go beyond the boundaries of the level of confidence (95% level of confidence in the charts-a gray area), which, given the current unstable situation in the markets, is pointing to the sufficient quality of the given time series.As the result we have the value of DR point-in-time on April 4, 2014 and can calibrate rating model for the current portfolio, but not for the portfolio a year earlier.

Conclusion
The advantage of the proposed approach towards determination of the average default probability of credit portfolio is that one gets a tool that allows you to construct a rating model that is "forward looking", respectively, appear to more quickly adapt to the changing patterns of rating environment.
In the example tackled in the article (based on PIT approach) average forecast probability of default on the portfolio will be equal to 2.3%.In case the model calibration is performed on the data a year earlier, the rate would be equal to 4.51%, which would have clearly reduced the competitive advantages of a credit institution, as products offered to them would cost too high.On the other hand, if we observed the opposite picture, and forecast the probability of default would be higher than the observed, the application of the proposed approach would allow preparing for a crisis, while the use of the observed values of DR could cause unexpected losses incurred by the credit institution.

Figure 1 .
Figure 1.Correlation between the scoring figure and probability of default statistics of default, but there is sufficient number of observations with external ratings (ratings assigned by Moody's, S & P and/or Fitch)) (Log) linear regression: ln F (α, β) = α+β Score Absence of sufficient statistics of default and sufficient number of observations with external ratings (ratings assigned by Moody's, S & P and/or Fitch)), but there is available sufficient number observations with the expert rankings assigned by other rating agencies (not Moody's, S & P and/or Fitch) or business experts of the bank Multinomial logistic regression (model of multiple (discrete) choice) (Note 1).

Figure 2 .
Figure 2. View of the calibration portfolio based on the angle of the calibration line

Figure 4 .
Figure 4. Dynamics of changes in the number of credit institutions in the banking system of Russia.The red curve-the dynamics of license recall

Table 3 .
Selected indicators of credit institutions

Table 4 .
Qualitative parameters of a regression model

Table 5 .
Statistical indicators of the time series RTsogIn the third stage were defined parameters of time series and constructed forecast values of the explanatory variables for 12 months ahead.The model was based on data for the period from January 2005 to March 2013 (the same period, we have a time series with DR values).
After the substitution of predicted values in the regression equation we obtained forecast DR on 01.04.2014 equal to 2.3% (i.e., during the period 01.04.2014-01.04.2015), in accordance with our methodology assumes that will be recalled licenses (default) 2.3% of the credit institution of a total of 910 credit institutions, i.e., at 21 banks).Figure 8. Predictive value of DR on March , 01, 2014