Catch the Heterogeneity: The New Bank Tailored Integrated Rating

We develop a banks’ specific integrated rating approach, tailored incorporating the various heterogeneity dimensions characterizing financial institutions (see Mantovani et al., 2013 and 2014 regarding the heterogeneity risk analysis in corporate firms), named “bank tailored integrated rating” (BTIR) and able to catch the financial cycle. The approach is inherently coherent with the challenging frontier of forecasting tail risk in financial markets (De Nicolò and Lucchetta, 2017) since it considers the downside risk in the theoretical framework. The innovation consists in using the integrated rating (IR) with the pre-selection of the variables through a statistical procedure that takes into account the characteristics of risk and greater heterogeneity of the banks.


I. INTRODUCTION
The capital regulatory policies imposed on banking institutions, increasingly reveal the need to consider the heterogeneity of regulated entities and, at the same time, to avoid obvious errors above or under assessment of the risks inherent in the various business models of modern banks. A recent paper of Ongena et al (2017) suggests that the traditional ratings, based on risk rating agencies, are biased by the "too big to fail" phenomenon and are therefore not reliable. The corporate performance literature introduces the Lintner's model (1965) as an alternative approach to appraise firms and their performance, through the companies' asset-side capability of the management in the long term. The analysis is useful to understand whether there is an appropriate allocation of financial resources, in line with the goodness of the performance and it is important to assess company pay-out and managerial rents as in Lambrecht and Stewart (2012). The theoretical framework replaces the estimation of market risk premia for discounting rates with the use of the certainty equivalent approach. This substitution is possible, because the certainty equivalent of expected cash flows is discounted (at a free-risk rate) instead than the volatile expectations of cash flows (at a risky market rate). Furthermore, it requires a total risk-aversion input to estimate the confident equivalent, instead of the adoption of the systematic-risk-aversion, exactly as applied to a CAL-portfolio, used as benchmark. However, Leibowitz and Heriksson (1989) noted that it is important to consider a shortfall approach that looks more on a "confident equivalent", rather that the Lintner's "certainty equivalent", which is a minimum threshold that may be overpassed, according to a certain confidence percentage. Determining either the threshold and the confidence is up to the investor, even before choosing the investment. Indeed, in banking analysis the downside risk is particularly important since "tail risk" is considered an important component in financial market analysis as underlined in De Nicolò and Lucchetta (2017). Economic cycle may matter. The first to underline the importance of the cycle is Löffler (2004). This author proposes the Kalman filter procedure to distinguish between cyclical component and firm-specific component. The cited current literature on risk assessment concentrates on corporate firms and the "tail risk" analysis is mainly oriented to macroeconomic risk measures. This paper fills these gaps and contributes to the identification of a synthetic indicator of company performance and long-term creditworthiness, which is also able to take into consideration the investor's risk aversion and the downside risk component: the "bank tailored integrated rating" (BTIR). This need arises from studies on rating modelling in order to make easier the implementation and use of the results within banking organizations. Indeed, it must be ensured that the indicator has three characteristics: (i) scientifically reliable and (ii) comprehensible to customers, finally (iii) consistent with the credit policies adopted. The Integrated Rating (IR, Mantovani et Al., 2015) indicator provides a comparison between the permanent ROI of a given company indexed by "i" (used as a proxy for company performance) and a threshold value, called threshold ROI, for the same company. The ROI threshold is calculated through panel regressions, which consider 25 indicators calculated on company balance sheets, which include the income, risk, economic performance, financial management and technological status of the company. The difference between permanent ROI and threshold ROI shows how the company performs better (worse) than its target. The target considers the company indicators and the weights of the reference market, that is, the coefficients of the regressions. The problem encountered in the development of the integrated rating methodology is that, despite the underlying logic is rather easy (the higher the rating, the better performing the company), there are no extreme limits, neither lower nor higher than the numerical result; this involves some problems in the communication and understanding of the rating to third parties. We hypothesized to apply a mathematical treatment appropriate to the raw results in order to obtain a transformation to the Integrated Rating indicator, which allows a simple, clear and linear reading. To transform the indicator, through a logistic transformation, deriving from the logistic function is found to be the better fitting methodology into the whole model. The logistic transformation allows us to have an indicator included in a range between 1 and -1 and a unit standardize and concave curvature. However, it is possible to investigate to detect a multiplicative constant in the exponential component, which changes the degree of curvature of the function, going to change the degree of discrimination of the data set, compared to more extreme values. Therefore, it is precisely on this last point on which the analysis and research that underlie the paper are concerned. In finding an optimal method that determines the degree of curvature of the function, one would discover an optimal methodology for identifying the degree of risk aversion of the investor. The degree of curvature, therefore, would represent the degree of risk aversion of the investor. The logistics transformation makes it possible to discriminate companies in the set of observations, but making the observations rather similar with anomalous behavior (so-called outliers). This allows us not to overestimate companies that have a better performance, than expectations and do not underestimate companies that are in line with expectations. This effect can be regulated by a multiplicative constant in the exponential component and allows to determine a degree of convexity / concavity that can adapt to the needs. The goal is to investigate an optimal method for determining the correct degree of convexity and concavity. This finally, proxies, the differences in risks' attitude of the institutions. In conclusion, the bank specific integrate rating project, here detailed, focus our research on the development of a mathematical / econometric method that allows us to identify the best algorithm, to determine a correct degree of convexity and concavity (and therefore, consequently, the correct degree of risk aversion of the investor), which can be dynamic and adaptable, consequently to heterogeneous banks. To take into account the characteristics of risk and greater heterogeneity of the banks, we propose a challenge procedure that cluster banks by risk. This further step allows us to design our "bank tailored integrated rating" (BTIR). The approach is inherently coherent with the challenging frontier of forecasting tail risk in financial markets.
The rest of the paper introduces at section II the basic model and the statistical procedure adopted. Section III extends the IR model to the challenging definition of BTIR that better fits risk and heterogeneity of the banking sector. Section IV discusses further developments 4 to better catch the financial cycle. In our example, we show how heterogeneity is important in banks' rating, but the systemic risk is considered implicitly. Therefore, section IV proposes a methodology to deal explicitly with the common factors (for instance using as explanatory variables the Factors of a VAR). Finally, section V concludes.

II. THE BASIC MODEL
In Integrated Rating (Mantovani et Al. 2015), determining either the threshold and the confidence, is up to the investor, even before choosing the investment. Equation (1) explains the relationship between the expected return for a specific investment (i-th) and the confident equivalent return (Rce), supposing a 10% confidence for the overall market: (1)

Where
Equation (1) indicates that the investor's risk aversion makes her accept an ex-post return below Rce only once each 10 cases for the entire investment's holding period. This equation is indeed a Shortfall Line according to Leibowitz and Hericksson (1989) The Capital Allocation Line can be seen as an original case of a Shortfall Line if: a. The confidence is higher than 50%, since the Sharpe ratio is positive to back investors' risk aversion; b. A bottom threshold determined at the risk-free rate.
If a risk-free investment is not found, Fisher Black's zero-beta model must be considered to identify the market portfolio through the CAL (Black, 1972). With this methodology, the zero-beta return is seen as a confident equivalent return identified between the market portfolios lying on the efficient frontier; consequently, the slope of the CAL is linked directly to a probability. In Black's model, in order to determine the downside threshold-return without leading an analytical estimation of the investors' risk aversion, it is adopted a market replicability of zero-beta return through the efficient frontier. Indeed, as with Linter's certainty equivalent of a specific investment which moves towards the equilibrium as proposed in classic CAPM, similarly the explained confident equivalent for a specific investor directs to the equilibrium as shown in Black's Zero-beta model. The strong points of this methodology are: 1. Risk aversion is not a punctual data; 2. The estimation of the confident equivalent for a specific investment is sufficient with an ex-ante probability; These points lead to the possibility of using this method also in incomplete markets, in particular for unlisted and private firms. Indeed, the methodology is useful to rank private/unlisted Firm/investments by taking into account their expected long-term return rate (e.g. ) and standard deviation ( ), given a market driven z-number: (2) Consequently, a comparison with the (unique) market confident equivalent allows to rank the best performing firms, as suggested in the original Lintner's model: (3) Alternatively, a threshold ROI (T-ROI) can be computed, given the and the standard deviation of ROI of the specific business as in Equation (4): (4) If the permanent (or sustainable) expected ROI is higher than T-ROI, i.e. , than the company is compliant with the investor's risk tolerance and, accordingly, it is creating sustainable value based on the 'Integrated Rating approach. Hence, the gap allows to rank the firms in the long run, since they have a given confidence to trespass the threshold in the next time horizon.
The case of private firms (i.e. SMEs) is quite more difficult to deal because of the lack of sufficient data to estimate mainly due the higher endogeneity that characterize the corporate risk vs. the market risk (Mantovani et al., 2015). Consequently, the market benchmarking might be difficult as well and investments' comparison gets more complex. But the assumptions under the application of the confident equivalent model to the case of firms, let us find a measure of based on the components of the overall firm's endogenous risk. The risk component cannot be estimated considering time series data because the volatility of firms' performance is not caused by past return only, but mainly by all (i.e. present and forthcoming) the strategic decisions considered by managers, together with governance and management quality. This leads to the explanation of why the corporate-risk is more stable and mean reverting than market-risks (Mantovani et al, 2015). Therefore, measures of firm strategy, governance and management decisions for unlisted companies can be proxied by balance sheet indexes, and melted into an integrated approach which relates them to firm's as in equation (5).
where, is the Return on Investments, is the constant component, is the matrix of independent variables each measuring specific components of the corporate risk, is the vector of single risk relations for independent variables, and a random component. If the relation found in equation (5) is robust, we are also able to find, the standard deviation of , by integrating formulas with the covariance matrix of risk indexes, for a sample of firms. , is the matrix of variance and covariance of the risk independent variables and is the transposed vector of risk relations.
The same computation is possible for a single firm if a specific corporate risk analysis is conducted in order to find out sound relations between the different elements composing the overall corporate risk (i.e. the corporate variance-covariance matrix). In any case, even with sufficient long-time series, the connections estimated by equation (6) are worthless in the case of a single firm. Indeed, a firm is by definition an entity in continuous evolution because of the changes in strategies, economic context, technologies, competitors, etc. Also returnrisk relations change in time. However, return to risk relations as estimated through equation (5) and equation (6) over a sample of similar firms, can be very useful in case of assessment of the value of a portfolio of financial credits or for the benchmarking of a specific firm with the strategies adopted by its peer companies. If we find out trustable evidence of the dependence over on such relations, we can re-arrange the above equations' term as follows: The level of that we find in equation (7) is the same as that one used in the confident equivalent method, and can be implemented for single firm rating estimation as in Eq.8.
In banking practices you can identify whether a firm that: 1) presents an average relation of or , 2) raises more or less financial resources, proxied by Debt on Operating Revenue (DEBT/OPRE) than the sample average, or 3) pays more or less interest rates for them, proxied by the ratio Interests on Debt (INTE/DEBT) we are able to determine a measure of the banks' ability to detect risk components in firms' performance. In order to get a more comprehensible reading of the results, a transformation is needed for the difference between the Permanent and the Threshold to make it usable even to nonacademic individuals. We state the following proposition.

Proposition 1. Consider the bank (firm)"i", it is possible to design its integrated rating IR based on a robust model ranking firms' performance.
Proof. The logistic function spread in Mathematics and Statistics is a useful instrument, in order to apply the transformation. The function is defined as: For all the real values of with codomain , with inflection point in and with slope . The simplest case is the logistic function defined as in the codomain with inflection in : Graphically, it is represented as: In the case of integrated rating, the variable is the difference between Permanent and Threshold , in a way that becomes an indicator in a close and limited interval, with a clear interpretation. The transformation belongs to the interval [-1,1] and the preserving of the algebraic difference. A useful modification of the logistic function might be the following: In this case, is in the interval [-1,1] with a unitary curve. A multiplying constant in the exponential component (e.g., ) change the degree of the function slope, changing the way in which the data might be discriminated, comparing them to more extreme values. Then, it is straightforward to hypothesise the following distribution for the rating indicator: • the firm has performed very bad, and the rating leads to a negative merit of credit valuation • the firm has performed bad, but it is not the only one in the sample. • the firm has performed in a good way, but it has not overpassed the expectations • the firm has over-performed, compared to the benchmark. Q.E.D.

III. EXTENDING RI TO THE BANK TAILORED INTEGRATED RATING BY CLUSTER OF BANKS: BTIR
This section shows how to cluster banks and illustrates a first simple procedure to deal with banks' heterogeneity. We would like to point out that systemic risk is implicitly considered and clustering concerns the effective diversity of banks. Next, we explain how to insert, for example through the factors of a VAR, the systemic component. Actually, our purpose is to demonstrate that the subset procedures are needed to consider the inevitable difference between banks.
We analyze a sample of 227 Italian banks, extracted by ORBIS database (edited by Bureau Van Dijk). This basic example is the origin of what the Bank Tailored Integrated Rating is. The selected banks have as last year of accounts 2016; the banks do show details of their balance sheets for 6 years, in particular, 2016, 2015, 2014, 2013, 2012 and 2011. The estimations are performing using a panel regression, whose coefficients will be used to compute the measure of the rating.
Here, we list the explanatory variables. The explanatory variable that has been chosen to conduct the analysis is a different computation of the Return on Equity. Indeed, ROE is the classic measurement of performance for banks. However, to compute the panel regression, we use a decomposed ROE measure, which considers four different drivers (as suggest by the ECB, 2010): non-recurring assets, leverage, recurring revenues and cost efficiency through the formula: The decomposed ROE considers difference characteristics of performance of the banks. For this reason, it is more suitable for the specific purposes of the paper. The independent variables are made up of indexes that inform about different characteristics of the bank. In particular, there are 26 indexes concerning: asset quality, capital, operations, liquidity and structure.
ASSET QUALITY In order to measure the asset quality of banks, we consider: The following table shows the main estimation results. are not significant and, consequently, their coefficients will not be used in the computation of the "Threshold ROE". The Integrated Rating for this sample has been computed following the original idea, but changing the comparing variables in order to have ratios more suitable for the banking system. In this case, in order to investigate the goodness of the banks, we have considered two measures and we have created four different clusters of banks for each indicator: (i) Non-Performing Loans/Gross Loans, where the four clusters are: "good", for the interval [0 to 7%); "not so good", for [8% to 15%); "bad", for [16% to 23%); "very bad", for [24% to 30%]; (ii) Non-Performing Loans/Tot. Assets where the four clusters are: "good", for the interval [0 to 6%); "not so good", for [7% to 13%); "bad", for [14% to 20%); "very bad", for [21% to 25%]. The intervals expressed, in order to compute the four clusters for each indicator, have been calculated considering the maximum level for each sample, (30% for NPLs/GLs and 25% for NPLs/Tot. Ass.) and then dividing the difference respectively from 0 to 30% and from 0 to 25%, by four. Then, each bank has been observed and inserted inside the interval and it has been assigned to the cluster.
The following charts show the results for the two indicators taken as benchmark. The two matrixes give a very small result of the first version of the Integrated Rating on banks. With the original version proposed in the paper the methodology used is quite different.

IV. FUTURE DEVELOPMENTS TO BETTER CATCH THE FINANCIAL CYCLE
Through this procedure we adapt the Integrated Rating according to the banks' characteristics, which allows us to apply the IR tailored on bank's selected variables. Indeed, as it is possible to observe on the previous example, the adjustment of the original procedure on banks has given good results. In particular, it is possible to see that using two proxies of banks' "health" and crossing them with the rating measure (the difference between the persistent decomposed ROE and the threshold ROE), the results, on average, do fit. In particular, riskiness is a crucial element that needs to be considered; indeed, banks are extremely riskier and more heterogeneous, comparing them to normal enterprises. Past researches already considered these special characteristics belonging to banks and they tried to select banks ex-ante, through a quantile regression, which provides an endogenous risk index (Koenker and Hallock, 2001). Furthermore, more recent researches preselected banks' default risk in two different ways: (i) through a CoVAR to analyse which banks are riskier (Tobias and Brunnermeier, 2016) (ii) through a measure of tail risk, which indicates that banks are different from enterprises, because of this excessive risk, named "tail risk" (De Nicolò and Lucchetta, 2017). The model that we may chose, in order to conduct a further bank pre-selection is the VAR model that allows to extract the common factors, called Factor VAR. The VAR model is a powerful instrument to preselect the important xs and it is not a usual instrument in this field, indeed it is usually applied in researches about financial markets or corporate finance. The BTIR is an important innovation which allows through the VAR model to consider the diversities across banks and the condition of global risk, by selecting the xs through VAR factors. Proposition 2. Consider the bank "i", it is possible to design its BTIR based on the robust model that extends the RI preselecting the variables though a VAR methodology.
Proposition 2 underlines a challenging framework that consist in pre-selecting or clustering the relevant banks' variables though a simple VAR model. This is the further development that we aim to analyze in the next paper.

V. CONCLUSION
The current development of ever-increasing banking regulations requires the study and the development of increasingly precise rating methods that take into account the increasing heterogeneity of banks and the presence of systemic risk, in addition to ongoing contagion relations between financial institutions. Also, the traditional and simple capital regulatory policies imposed on banking institutions, increasingly reveal the need to consider the heterogeneity of regulated entities and, at the same time, to avoid obvious errors above or under assessment of the risks inherent in the various business models of modern banks. Our work considers the extension of the integrated rating (IR) procedure, used primarily for non-financial companies, developing the "bank tailored integrated rating" (BTIR). The approach is inherently coherent with the challenging frontier of forecasting tail risk in financial markets (De Nicolò and Lucchetta, 2017) since it considers the downside risk in the theoretical framework. The innovation consists in using the integrated rating (IR) with the pre-selection of the variables through a statistical procedure that takes into account the characteristics of risk and greater heterogeneity of the banks. In this first proposal, we use a simple panel regression that cluster bank by heterogeneity and considers systemic risk implicitly. However, our innovative procedure may include, in the future, more sophisticated preselection of variables such as Factor VAR (FAVAR) and CoVARs. This work requires testing whether a more sophisticated pre-selection model is better than a traditional VAR. In fact, for simplicity, we believe that starting with a simple methodology is the first step of research. BTIR makes possible to adapt the rating procedures to all banks, even that showing very different characteristics. In fact, the VAR allows to pre-select and to evaluate markets with high systemic risk, avoiding errors due to general market conditions that may differ from country to country. In conclusion, our BTIR opens the door to a new research line to innovative ideas for the development of increasingly accurate ratings for banks embedding the needs of macro-and micro-prudential policies.