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Espacios. Vol. 37 (Nº 14) Año 2016. Pág. 9

Economic Capital and Rating: Analysis and Minimum Requirements

Capital Econômico e Classificação de Risco: Análise e Requisitos Mínimos

Flavio BARBOZA 1; Herbert KIMURA 2; Leonardo F. C. BASSO 3; Vinicius A. SOBREIRO 4

Recibido: 02/02/16 • Aprobado: 03/03/2016

Contenido

1. Introduction

2. Literature review

3. Measurement model and simulation procedures

4. Analysis and results

5. Final considerations

References

1. Introduction

Risk is a basic business element because the profitability of any economic activity involves risks. However, the possibility of risk does not mean that it can be ignored and diversified investments can be made different degrees of risk, where it is expected that variable results will be obtained. Thus, specific controls may be required to enforce different types of risk by an organization. Previous studies have stressed the importance of risk management for companies in the financial and non-financial sectors (Stoian and Balan, 2012).

During the 2008 financial crisis, which affected the financial sector as well as other companies, the area of risk was the focus of careful analysis in order to obtain more accurate quantifications of risk, particularly credit risk. Institutions became aware that good risk estimates provide greater control over its effects and various strategies can be employed in certain circumstances (Bag and Jacobs Jr, 2011). In particular, according to Bade et al. (2010), the crisis showed that the credit risk models used for portfolio management had a poor degree of transparency, i.e., the estimators produced by these models were not clear (or even consistent, as mentioned by many authors, such as Jokivuolle and Peura (2010)).

Due to the need of covering losses, financial institutions are developing more robust and accurate models. Imprecise models would imply an unnecessary portion of capital requirements, which leads to an inefficiency of the business. More than this, to remain protected against counterparty defaults as successfully as possible and to comply with agreements, banks also need to reliably determine the specific minimum requirements of capital in their transactions. This commitment can avoid credit constraints, bad reputation or even failure in extreme cases. However, the economic capital can generate positive incentives in asset allocation by banks with potential impacts on the total volume of credit granted by the financial system.

Research in the area of credit risk is comparatively recent (Ericsson and Renault, 2006). However, in the late 1990s, Gordy (2000) suggested that important advances were made in credit risk modelling at the portfolio level. More recently, Giese (2005) indicated that a portfolio credit risk model is extremely important because credit risk makes the greatest contribution to increased economic capital in banks.

Economic capital, as a research theme, is not so common when comparing with credit risk in academic studies. However, Das (2007) discusses how Basel II complemented Basel I accord. The author emphasized the importance of the third pillar which presents the capital requirement regulation and purposes for banks. In another study based on empirical evidences, Jokivuolle and Peura (2010) suggested a model for measuring economic capital considering annual loss distribution and ratings as assumptions. Looking at capital requirements from another point of view, Kopecky and Van Hoose (2012) analyze a model considering the government objectives and observe that this regulatory component is a useful tool for controlling social-private relationship.

After the Basel accords determined the minimum capital requirements as one of the most important tools for measuring credit risk, financial institutions attempted to develop efficient models for their parameters and aimed to minimize the capital required and/or maximize profits, i.e., complying with the agreed directives while still satisfying their shareholders. These regulations helped companies to determine their own models as well as intensifying scientific research in more than one study area in the financial sector. The possibility to create their own internal models implies greater interest by banks in economic capital modelling.

It is important to stress that the increased focus on credit risk studies is justified, because there is a lack of a consistent approach that is accepted theoretically and in the market as a whole. This is a very interesting research problem that affects society greatly and its study has gained ground in recent years, mainly due to regulations by financial institutions.

Ratings agencies also perform comprehensive and relevant studies (Wengner et al., 2015) each year in order to organize companies under a specific methodology according to their sole exposure to credit risk. Thus, these agencies consider the default events that occurred over the period and they conduct an analysis for each risk group, where each group is associated with a certain acceptable risk level (Sundmacher and Ellis, 2011).

On the other hand, financial institutions have to do their analyses because they have large historical data that can provide a better understanding about their own clients, giving more precise measures of probability by default.

Economic capital is a relevant topic, mainly for banks. Jorion and Zhang (2009) suggested that financial institutions can choose to whom they lend. Therewith, using large estimates for economic capital, these companies would have protection from collapses in the credit market. So, the search for better strategies is an eternal task in this context. Modelling economics is an open problem as well, since the regulation encourages the development of reliable models.

This paper, in essence, tackles the modelling issue and what strategies banks could apply to avoid a large amount of economic capital charge. Specifically, we try to know how a financial institution would analyse the credit portfolio and select the best possible combination of borrowers, which could provide lower economic capital when borrowers' creditworthiness is also employed in the measurement of economic capital charge.

Taking this into account, the aim of the present study is to determine the volume of economic capital made available by a financial institution using an internal risk-based model to estimate credit risk while considering a borrower's credit quality and based on an assessment by a ratings agency regarding the lending institution's credit risk for this volume. We make a model using beta distribution to simulate the losses (in per cent), credit rating for default estimations as inputs. As a result, banks might analyse if they enhance their own credit rating and at the same time minimize the economic capital. This approach allows the lending institution to employ the transaction to improve its own rating, while also minimizing the regulatory capital. We employ the same method applied bySundmacher and Ellis (2011), but using data from other sources as well as PD and LGD from two specific periods (1998 and 2002). We also consider the search for an improved risk rating by financial institutions, where the borrowers have a lower LGD relative to a lower level rating, and thus the minimum capital requirements remain the same.

Bearing this objective in mind, the outline of this paper is as follows: In Section 2, we review previous research in this area, including the risk control guidelines set down by international regulatory bodies, the models used for risk measurement in professional and academic environments, and the importance of risk management in all of these areas. Section 3 explains the methodology employed with the aim of reducing the economic capital required from the lender. In Section 4, we show that the results obtained using methodology have interesting properties, such as much lower economic capital requirements for borrowers where the loss due to default (denoted as the loss given default, or LGD) is above its average rating under several criteria subject to adjustments by the lender. In Section 5, we discuss the results of our simulations and provide suggestions to improve the results obtained.

2. Literature review

The term "credit" is related to confidence and one method for determining this confidence is risk measurement (Das and Teng, 2001). In financial terms, Stoian and Balan (2012) argues that credit may be treated as a sum loaned to somebody who it is expected to return it following a certain period. Thus, credit is related to time and to the trust relationship between the parties involved. Because a sum of money is involved, it is possible for the borrower (who provides credit) to require a minimum compensation in exchange for confidence.

If there is a likelihood of disrupting this confidence, it is clear that there is a risk when giving somebody a certain sum and expecting to receive it in full (or duly compensated) after the agreed period. This unpredictable element is associated with the credit risk. Neal (1996) defined credit risk as the likelihood of a borrower not meeting the commitment of paying a bank's loans or debts. Therefore, measuring this risk may provide information regarding likely losses, thereby allowing a financial agent to better assess its compensation. In addition, Stoian and Balan (2012) defined credit risk as the current or future risk of profits and capital being affected adversely should a debt or contractual obligations fail or undergo damages.

Altman et al. (2005) asserted that credit risk affects most financial agreements; thus, it should be considered during all transaction by attempting to manage it to reduce losses. Therefore, management provides alternatives to reduce risks based on adjustments in the allocation of funds or hedge instruments, with the aim of not providing credit to a company that has unnecessary risks.

In academic research, Ericsson and Renault (2006) noted that credit risk has been under more scrutiny since the seminal study by Merton (1974). Stoian and Balan (2012) confirmed the findings of Ericsson and Renault (2006) and indicated that credit risk analysis and prediction has been an intense research area since the 1970s. However, this task has been shown to be difficult. Lai and Soumaré (2010) added that credit risk is one of the key challenges faced by banks and other creditors.

Moreover, Lai and Soumaré (2010) emphasized risk management is a crucial task for corporations throughout the world. In particular, Balla and Rose (2015) asserted that credit risk management has received much attention from economists, bank supervisors, regulators and financial market professionals. As a possible consequence, according to Schuermann (2004), credit risk management is now common in large financial institutions, where control procedures are required to reduce potential losses due to loan defaults.

To facilitate effective risk management, Stoian and Balan (2012) emphasized that it is important to include credit risk aspects in the overall operating viewpoint as well as the risk associated with a borrower because strategies based on a counterparty's assessment are calculated by considering the probability of default.

During the last decade, banks and insurance companies have made considerable investments in conceiving and implementing risk management systems. Nonetheless, the financial crisis showed clearly that the models produced were insufficient to detect problems, thereby increasing the pressure on financial institutions and regulating bodies. As a result, these issues have produced a challenging environment, which has compelled domestic and international administrators, and other bodies, to reconsider all of the concepts employed, as well as the models. In particular, credit risk models have been subjected to this critical review process due to the dynamics of events (Gordy, 2000).

Neal (1996) noted that numerous methods have been employed for credit risk management. In recent years, an alternative credit risk management approach has focused on the sale of assets with a greater credit risk. According to Castermans et al. (2009), the validation of these methods is a crucial part of credit risk management, but only one approach is often used to assess multiple restrictions on the models employed.

The difficulty of dealing with credit risk is related to its quantitative analysis. Botha and Van Vuuren (2009) showed that credit risk is a complex measure, where the breakdown of institutional credit losses is highly asymmetric, based on mathematics that are sometimes refined (e.g., using copulas to determine the dependence between random variables) to explain the relationship between the economic environment and the sums of loans, which involves the differentiation of equity accounting assets and default correlations, thereby implying that there are several additional market risk parameters (e.g., the loss given default, likelihood of default and exposure to default).

In the next section, we explain these rules and their effects on interested parties, including the common weaknesses of the models used.

2.1. Regulations

According to Das (2007), the Basel II accord increased the focus on credit risk, where an approach based on ratings was replaced by a relatively adequate form with an increased consideration of the counterparties that use internal models. The main conceptual basis of Basel II is related to an explicit consideration of correlations in a portfolio's total risk, thereby providing a better demonstration of the influence of correlated risks on capital requirements.

The basic variables required to develop a credit risk model are (Altman, 2006): (i) the probability of default (known as PD); (ii) LGD, which is equal to one minus the recovery rate (RR) in case of default; and (iii) exposure at default (EAD). The production and handling of these variables is explained in greater detail in the following sections. Altman (2006) also stated that there have been many previous studies related to estimates of PD, but few have considered the estimation of RR and the relationship between these components. According to Altman (2006), the  PD is related to systemic risk, i.e., to market-related factors, which is argued by Battaglia et al. (2014) as well. Furthermore, the most common models have utilized the RR as a variable that depends on idiosyncratic factors, i.e., metrics obtained from within the company, and thus they do not depend on the PD.

In particular, Jorion and Zhang (2009) asserted that quantifying PD, LGD and EAD may be used to infer economic capital, where the sum of funds owned by an institution should increase to absorb a great loss over a certain time horizon with a high confidence level. In the traditional form of economic capital creation, it is assumed that the rate of LGD (equal to one minus the RR) is known and not stochastic.

When criticizing this conservative stance, Bade et al. (2010) noted that previous studies and current banking practice do not mention several credit risk properties. Indeed, PD, RR and their correlations are often modelled as constants, thereby eliminating their randomness. Moreover, conditional parameters such as recoveries, which are conditioned to the occurrence of default, are modelled using unconditional linear regression models with the ordinary least squares method, which does not consider conditionality, and this could lead to a bias in the estimated parameters.

The basic Basel II model for calculating the minimum required capital specified internally by banks is very simple, where it only measures the systematic losses in a credit portfolio, i.e., the portfolio losses due to external influences, and thus it cannot be diversified (Giese, 2005). Das (2007) also criticized this model because it does not consider the specific portfolio risk, which tends to be very different according to the nature of each institution, even when considering the class of assets, similar leverage and maturity.

Botha and Van Vuuren (2009) showed that the impacts of the Basel II credit risk methodologies when calculating bank capital are still unknown, and studies are required to confirm their accuracy, by subjecting them to empirical tests to demonstrate their efficiency when calculating risk.

The adequacy of the model used is the main issue because it determines the minimum capital required for risk protection by banks. Das (2007) proposed the traditional Value at Risk (VaR) model as an information system for the risks found in transactions, where the loss distribution is modelled explicitly and the result obtained is the best estimate of capital adequacy, provided that the calculations involved and the modelling assumptions are reasonably accurate. However, this is not so simple in practice.

Basel II agreed that institutions may employ three models to evaluate the minimum capital requirements: the standard model using the same methodology employed in Basel I; the internal ratings-basic approach (IRB), which allows banks to develop their own formulae, although under supervision; and the advanced IRB, which is a more sophisticated treatment of risk measurement, similar to the basic IRB. Das (2007) explained that the change to the IRB approach gave banks greater flexibility when employing various assumptions to estimate parameters and to meet their internal expectations connected with capital levels. To avoid obtaining estimates that are biased in their favour, financial institutions should consider the following.

1. Awareness of more intense inspections because the minimization of economic capital should comply with predefined rules;
2. Awareness that the IRB approach recognizes that banks have employed risk-based capital for almost two decades, and thus these rules do not appear to be obstacles to the creation and handling of more adequate models than those used in the market; and
3. Awareness that this new approach is much more consistent with internal risk management in terms of making the business more reliable and consistent because institutions will aim to comply with the Basel II accord.

However, these proposals aim to reduce internal regulations and long-term risk management costs, although in the short term it is expensive to produce the reports required under both Basel I and Basel II. Credit risk, as defined previously by Neal (1996), specifies the likely losses or non-payment of a credit granted to the borrower.

Schuermann (2004) asserted that Basel II allows banking organizations with international activities to calculate their credit risk capital needs using an IRB, subject to supervisory review. Das (2007) explained that Basel II considers two types of loss for determining the credit risk: (a) expected loss (EL) and (b) unexpected loss (UL). Thus, any notions of regulatory capital or economic capital must be related to the concepts of EL and UL.

In order to determine the quality of companies when undertaking credit obligations, the ratings agencies (e.g., Fitch, Standard & Poor's and Moody's) employ their own methodologies to list companies in an orderly manner, thereby providing the market with an additional tool for studying good and poor performers.

Boot et al. (2005) stated that many market participants consider risk ratings to be an important factor in their analysis, and they also suggested that this has changed investor behaviour, i.e., companies must consider one more performance indicator related to their risks for credit transactions and the influence that this has on the external environment.

The agencies prepare seasonal reports about the behaviour of each rating level and their changes, as well as the PD per level due to default events among the companies included in each of these levels.

If we look at the ratings, they should, theoretically, show higher PD levels in the lower ranges, since higher ranking is expected when the classified company is more reliable. Regarding LGD, the classification is more complicated since the default event may have some correlation with the extent of the losses (which is suggested by many studies and has been increasingly discussed regarding how it is measured), but the size of losses is still questionable given the default and its connection with the rating levels. However, the rating agencies seek to generate reports about the frequency of defaults that occurred in their own rating disclosures of previous periods, and, therefore, the actual events do not always confirm the PD measures estimated theoretically, which can present abnormalities in this process.

3. Measurement model and simulation procedures

Accordingto Altman et al. (2005), credit risk models may be divided into three key classes: (1) the "first generation" of structural models; (2) the "second generation" of structural models; and (3) the reduced form models. The first class is based on the Merton (1974) model, where a company's assets play an important role as well as the risk of default. It was also stated that the intuition behind this model is simple, i.e., a default event occurs when a company's asset market value is below the value of its liabilities. Many new models were created based on this model by Merton(see Altman et al., 2005). The second class also covers models based on Merton (1974), but it reconsiders several assumptions: that default may occur before a liability expires, that the default event occurs when a borrower's indebtedness is greater than its assets, that the RR in case of default is exogenous and does not depend on a company's equity value, and that LGD does not depend on PD if the previous assumption is valid(Altman, 2006). Finally, according to Altman (2006), reduced models assume that the RR does not depend on the PD and that the model is able to price credit risk based on stochastic processes. The difference between the structural form and the reduced form is related to the default event's predictability, where the difference is greater in reduced form models.

Due to this division, numerous models are used by corporations and in academia. As discussed earlier, the models defined under Basel II are very simple and they may allow banks to allocate a greater amount of capital than required. To improve these models, ratings agencies and banks have developed their own models related to risk management. With respect to rating agencies, we may consider Standard & Poor's and Moody's as the key credit risk rating agencies. Among the banks, JP Morgan is the creator of CreditMetrics and Credit Suisse developed CreditRisk+. However, Sundmacher and Ellis (2011) stated that due to the lack of a unified theory, there has been an explosion in the number of empirical methods employed to foresee business failure in different markets.

Using the basic Basel II model, the minimum capital required by credit risk is supported by EL and the UL. According to Crouhy et al. (2000), the calculation of EL is based on three variables: EAD, PD and LGD. Under this definition, EL may be described according to Equation 1.

EL= PD × LGD × EAD                                                                                    (1)

where PD is the probability of default; LGD is the loss given default; and EAD is the exposure at default. The model employed in the present study assumes that PD and LGD are independent factors, which explicitly fix the LGD, and EAD is equal 1 without loss of generality. Thus, the value of EL is expressed in terms of PD and LGD exclusively. Given that the loss distribution is based only on the probability distribution of the PD variable, and that this variable's outcome is determined by a Bernoulli event (whether or not default occurs), EL is the mean of the distribution and (UL) represents the standard deviation

measure for EL (for further details on this outcome see Ong (1999)).

Therefore, according to the simpler Basel II criteria, it can be concluded that UL is obtained as described by Equation 2.

UL2 = Var(EL) = LGD2 × PD (1 − PD) = PD × LGD × (LGD − PD × LGD)                                                (2)

By replacing 1 in 2, UL can be expressed based on EL, thereby yielding Equation 3.

UL2 = EL × (LGD − EL)                                                                                     (3)

Economic capital is determined based on a proportion of UL because the credit risk is predicted based on the precise amount of capital invested that the financial institution does not expect to lose. Figure 1 illustrates the loss distribution for an asset portfolio, where the measures that comprise part of the economic capital measure, i.e., EL and UL, are represented as a graphic visualization based on the frequency of the losses that occur. Given these considerations, the only way of minimizing the economic capital is to find the PD and LGD measures that will lead to this outcome because they are the basic criteria related to losses. Based on this specific difficulty, methods for estimating each have also been devised with the same purpose.Altman et al. (2005) explained that the key risk components may be adapted adequately for an approach based on IRB in order to include specific methods for estimating the PD structure, LGD distribution and EAD calculation.

3.1. Simulation procedures

The present study aims to analyse changes in the economic capital when an institution intends to enter into a credit transaction and to adopt a better risk rating position due to the counterparty's current position. This is possible when the economic capital is considered using a borrower's data. According to Sundmacher and Ellis (2011) and Ong (1999), economic capital is associated with the counterparty at a certain confidence level and time horizon. The confidence level is defined precisely by the quality of credit associated with the rating desired by the lender, which is related directly to its PD. Thus, Sundmacher and Ellis (2011) named this value the solvency rate because having a lower default rate means that the company enjoys greater market credibility, and this value becomes increasingly closer to 1 when the lender seeks a better rating.

Figure 1: Loss distribution shape in terms of the frequency. In financial studies, this curve is useful for
describing the behaviour of losses in default events. The higher loss expectations increased as the
frequency declined, but only for very high expected losses. Economic capital can be visualized as the
proportion of the UL beyond the level of the EL.

Given that the loss distribution fits better to a Beta distribution, EL measures acquire the distribution's mean (µ) property. In the case of the Beta distribution, the probability density function is based on a gamma function (Γ(x)) and its form is based on the value of two parameters called α and β, which determine the distribution's form. The Beta model's probability density function (PDF) is expressed by Equation 4.

where Γ (·) represents Gamma function; x, in practice, determines the size of loss; α and β are shape parameters (α > 0; β > 0).

By applying the average and variance concepts to an ongoing distribution, measures for EL and UL can be defined that are close to the Beta distribution, where the result is shown in Equation 5.

The variance (UL2) is determined based on the distribution's parameters, as shown in Equation 6. 

Using Equations 5 and 6, we can obtain the distinct values of α and β based on the values of UL and EL. Hence, the following are obtained,

and

The EL measures may be obtained by reviewing ratings agency reports, which contain PD values. The PD data employed in our simulations were based on Moody's annual reports (Moody's Investors Service, 1999, 2003), as shown in Table 1. The LGD values are more difficult to calculate Frye and Jacobs Jr. (2012), because these data are disclosed rarely, while their incidence is difficult to measure. However, the Society of Actuaries (2002, 2006) reports provided sufficient LGD measures to prepare our simulation in the same period of PD data. Table 1 shows the LGD data gathered from the aforementioned reports as well.

Table 1: Probability of default (PD) data collected from Moody's reports based on the accrued
averages for the last 20 years in terms of each period. Note that the AA rating had a higher PD than the
A rating in 1998. LGD data collected from the reports of Society of Actuaries for the years 1998 and
2002 for the same ratings in Moody's reports. The AA rating was not presented for the first year of our analysis.
Source: Moody's Investors Service (1999, 2003), and Society of Actuaries (2002, 2006).

Table 1 highlights the decline in the LGDs based on the AA rating from the first to the second period. This change caused important changes in the economic calculation values, which were explored based on the outcomes of the simulations. Carey and Gordy (2004) commented that LGD was very high in 1998 and this is not common. Regarding the period between 1999 and 2002, which is considered a period of financial crisis, Altman (2006) explains that, in times of economic distress, the LGD value may fluctuate due to the tendency of an increase in default events.

Objectively, the first procedure can be summarized as:

Step 1: We collected PD and LGD data observed in two different periods. The data were provided by Moody's Investors Service (1999, 2003) and theSociety of Actuaries (2002, 2006). EL and UL measures are calculated for each rating level. Taking into consideration a beta distribution, the parameters α and βare estimated for all ratings from the 1998 data. The same estimation is repeated for the 2002 data. The measurements of EL, UL, α, and β follow the Equations 1, 3, 7, and 8. Without loss of generality, EAD = 1. Moreover, Table1suggests the existence of anomalies, in which, for instance, PD and LGD levels of AA rating are higher than the ones for the A rating. There are a number of potential explanations for this phenomenon. First, Caouette et al. (2008) suggest that the hierarchy of default rates, especially in investment grade ratings, may not be observed due to specific events, citing Texaco bankruptcy filing in 1987 and WorldCom defaulted bonds in 2002. Supporting the empirical data used in our study, Hanson and Schuermann (2006) built confidence intervals for probability of default using Standard & Poor's data from S&P from 1981 to 2002, which covers our research time frame, and found that it is difficult to distinguish PD from AA– and A+ ratings.

Second, estimation of PD using frequency of actual defaults may be vulnerable to the very small number of firms/defaults in higher ratings. The occurrence of an unlikely default in higher ratings, for instance, AA, may considerably influence PD and LGD estimates.Carey (1998) analyzed data from 1988 to 1992 and identified that the average AA rating loss rate was higher than the one for the A rating. According toCarey (1998), severities for the AA grade are likely noisy due to the small number (3) of credit risk events in comparison to the A grade that presented a higher number (10) of credit events. It is also important to highlight that the data for LGD in a specific rating considers all losses related to actual default events and not to all a priori sample. Therefore, LGD data is conditional to the event of default and, especially for higher ratings with a small number of observations, empirical data may be subject to the very distinct characteristic of a specific default.

Third, the relationship between ratings and credit parameters such as PD and LGD may not be robust or even expected.Carey (1998, p. 1372) argues that "No clear relationship between loss severities and ratings is evident". FitchRatings (2014, p. 11) explicitly affirms that the limitations of the rating scale include the fact that "ratings do not predict a specific percentage of default likelihood or expected loss over any given time period". Therefore, empirical data and rating methodology do not direct related rating levels with PD and LGD. In fact, Altman and Rijken (2004) and Cantor (2004) suggest that rating agencies are slow to react to changes in credit quality of a borrower leading to failures in short-term measures of PD, which is reinforced by Athanasoglou et al. (2014).

After this analysis, a second step can be done:

Step 2: Lender's PD provides the confidence level of the distribution. When the lender's rating gives a PD = 0%, we arbitrary use a very low number (i.e., 0.0005%) in order to being able to compute the value of x in Equation 10. In particular, for AAA lenders, this measure is not necessary (see Step 4).

A lender will assess the circumstances under which it is possible to obtain a competitive advantage in order to determine the minimum required capital measure while complying with Basel II according to its basic model, thereby obtaining a better rating with the same volume of economic capital.

In the proposed method explained in this study and presented bySundmacher and Ellis (2011), we stress the change in the LGD for borrowers with rating levels between A and B, and we provide feasible solutions for lenders to obtain an increase in the credit quality. The economic capital measures comply with these changes, which are determined by the k proportion of UL, as illustrated in Figure1. According to the method devised by Ong (1999), we can see that:

where UL is the unexpected loss, expressed in terms of percentage. This measure is given by the average PD and LGD of the borrower, according to its rating; and, k measures the UL from the lender's perspective to determine the minimum required capital.

The latter is expected to improve the quality of the credit risk rating based on the borrower's position, so this proportion of UL (the value of k) was assessed under a rough distribution of the borrower's average PD and variance parameters by applying fixed data (shown in Table1), and inserting them into Equations 1 and 3:

where x is the exact value associated with the volume of capital where P (x ≤ EL + k × UL). Its value may be found using the inverse of the cumulative distribution function, i.e., x = f −1(P, α, β). According to Sundmacher and Ellis (2011), the average P is expressed by 1 − PD to determine the solvency required by the lender. It is important to remember that PD value is extracted from the desired lender rating available in Table 1, while EL and UL are given by the borrower's data in the Table 1. In this case, they were fixed to determine α and β. On the contrary, UL from Equation9is computed to each LGD value, resulting in different levels of economic capital for each borrower analyzed.

So, the next tasks are:

Step 3: We set the desired rating by the lender and the rating of the borrower, we calculate the economic capital from the Equation9. The inputs used in the first factor, k (from Equation 10), are: (i) x, by using lender's data, and (ii) EL and UL given by the borrower's rating. The second factor (UL) inputs are: (i) LGD, given by the simulation, and (ii) borrower's PD (available in Table1). Note that the only part of economic capital associated to the lender is x; and

Step 4: We follow Sundmacher and Ellis (2011) in the case of (i) lenders in the AA rating and (ii) LGD value is lower than the average LGD borrower's rating, the beta distribution is given by a constant value of 100%. If the LGD used in the simulation is equal or higher than the borrower's rating average LGD, we apply the step 3 procedure normally. For AAA ranked lender, the beta distribution is always constant and therefore the economic capital is given by 1 − PDborrower × LGDsimulated for any level of simulated LGD. Based on this mathematical formalization, the economic capital was determined for lenders rated at AAA, AA, A, BBB and BB levels, where the borrowers were allocated to A, BBB, BB and B. In the case of borrowers, the LGD was varied between 10% and 50% in order to confirm the economic capital dimensions

for these cases. This variation was applied to the UL calculation employed explicitly in Equation 9. Please note that this variation does not apply to the value of k, where the LGD parameter was fixed according to Table1, considering only information about the borrower's rating. This part of the methodology can be resumed in the following step:

Step 5: We repeat the previous procedure changing only the rating of the borrower. The result is the amount of economic capital of the lender for suggested values of LGD (10% to 50%) to four rating levels of the borrower (A, BBB, BB and B).

Due to the level AAA solvency, the value of UL was questioned because the LGD events are practically unknown. Thus, this situation required that the economic capital did not change with UL, but instead it was based only on EL and the percentage of allocated capital (x) applied to a proportion of UL, (k), which depended on the borrower's features.

As we are focused on credit risk measures and their influence on the rating goals of financial institution, the endogeneity issue is not discussed in depth. First, this research does not test regression, and it is also susceptible to endogeneity. It is important to mention that the average PD and LGD were collected from two different sources, adopting distinct methodologies. Second, if it is a concern that PD and/or LGD are influenced by any bank's characteristics, many authors suggest using one-year lagged variables to estimate them. In addition, Chalermchatvichien et al. (2014) examined the relationship between bank risk-taking and ownership concentration, which are similar variables to those presented in this work, and found that their results were not influenced by unobservable bank characteristics. While we recognize that endogeneity can be present, we believe this problem is alleviated in light of Chalermchatvichien et al. (2014), and Sundmacher and Ellis (2011).

Intending to make the analysis more evident and clear, we organize all results as follows:

Step 6: We sort the results, aiming to compare the desired rating levels by the lender considering each borrower credit rating. This step allows us to identify strategies to achieve better ratings with lower economic capital levels for different borrowers that are in the same rating. This step can also help identify ways to minimize the economic capital and, at the same time, improve the lender's rating itself.

4. Analysis and results

The results shown in Table 2 and Figure 2 reflect the economic capital values for borrowers rated at A and lenders that wished to achieve AAA, AA, A and BBB levels based on assessments using data gathered during two different periods: 1998 and 2002 (data from Society of Actuaries (2002, 2006)) reports and from Moody's reports (Moody's Investors Service, 1999, 2003).

Table 2: Economic capital calculated in percentage using data from years 1998 and 2002 for A, BBB, BB and B-rated borrowers.

(a) 1998

(b) 2002
Figure 2: Economic capital for a desired lender's rating, based on the variation
in A-rated borrower LGD and for different counterparty ratings.

Several major issues were highlighted in these simulations: a comparison of both periods detected a subtle increase in EC, with a greater emphasis on lenders rated at AA, whereas the remainder was not perceived. This was supported by the LGD value for a lender with an AA rating in 1998, which declined abruptly compared with 2002, whereas PD increased. The likelihood of a lower capital requirement may still be found for lenders that desired an AA rating in the case of borrowers with an LGD above 17% (in 1998), where this required close to 35% of the regulatory capital. In the following period, i.e., 2002, the results were much improved because the A rating's LGD increased from one period to the next, thereby reflecting a capital requirement of roughly 61% for an LGD of 26%. By contrast, for lower rating targets such as BBB, the results reflected values below the 8% minimum defined by BCBS in both periods (Altman and Saunders, 2001), which was also the case for institutions that desired to be rated as A with borrowers with an LGD below 30%. At this time, an opportunity arose for those interested in an increased rating, i.e., it was possible for a creditor to increase its credit risk rating under this methodology simply by obtaining level A borrowers with an LGD below 30%. Yet, it is not indicated to lend to an A borrower if the lender seeks a higher rating, such as AA. The minimum economic capital is too high (near 34%) compared to other prior desired ratings (A and BBB). Furthermore, this was the only case where the economic capital was greater in 2002 compared with the preceding period.

Please note that 8% was required for BBB as well as for A, which means the economic capital was the same when a borrower's LGD was less than 30%. Logically, opting for the best rating obtained improved results for a lender.

Table 2 shows similar analysis for BBB target rating, which are also depicted in Figure 3. However, their quality was higher in terms of the economic capital. This was because the borrower's rating was below that assumed in the preceding part of the table.

(a) 1998

(b) 2002
Figure 3: Economic capital for a desired lender's rating, based on the variation in
BBB-rated borrower LGD and for different counterparty ratings.

In a simple inspection of Table 2 when the LGD values for a BBB level borrower were 25% and 35% in 1998 and 2002, respectively, the window of opportunity for achieving an AA rating by a lender was a more complex task, i.e., the economic capital increased to roughly 75% in 1998 and to 93% in 2002. Despite this, the minimum economic capital criteria followed the same trend as the previous case, where it dropped one rating level, i.e., for borrowers with an LGD below 30%, A rating could be obtained with an 8% economic capital requirement. Here, the rating improvement demands a substantial difference in LGD measure. More specifically, in 1998, the lender would change a borrower with LGD higher than 45% to another with LGD close by 20% lower without requiring an increment in the economic capital. The same artifice was less feasible in 2002, because 93% of economic capital was required for a borrower with LGD between 35% and 37% to achieve an AA classification. If the lender was looking for a ratings shift from BBB to A, it would have to find a borrower with a much smaller LGD (around 40% lower).

The most interesting outcomes of the simulation are confirmed in Table 2 and Figure 4. A lender that desired a BBB rating via a transaction with a BB borrower with an LGD greater than 16% required the same amount of economic capital (almost 27%) when they desired an A rating (relative to the 1998 period, the concept for 2002 was analogous to an LGD greater than 19%, resulting in a capital required close by 25%). Thus, because the economic capital was the same under both scenarios, it would be beneficial to find borrowers with an LGD, at least, 6% lower, because they could allow a higher rating to be achieved. The same could apply to an increase from A to AA, although with a smaller change in LGD, i.e., between 42% and 46% in 1998, and between 39% and 40% in 2002. If we consider values out of these intervals, the lender should search a AAA rating. The required capital vacancy to achieve AA was still possible when considering a borrower with an LGD bounded by 29%–31% for both periods.

(a) 1998

(b) 2002
Figure 4: Economic capital for a desired lender's rating, based on the variation
in BB-rated borrower LGD and for different counterparty ratings.

As shown in Table 2, the results agreed with those in the preceding part of the table, i.e., when observing an A or BBB desired rating. However, the distance between them is only 2% in the beginning of the simulations. For example, if the lender could choose between two borrowers, X or Y, where both are rated as BB and X's LGD is 10% whereas Y's LGD is 12%, the lender can pursue an A rating itself. This is because the economic capital required for Y to achieve its BBB desired rating (that is 23.86%) is a little bigger than the EC needed by X (23.74%), when seeking an A rating. The changes related to LGD had greater amplitude, and more capital was required, which is evidently related to a borrower's declining credit quality. Nevertheless, Figure 5 shows more clearly the results displayed in the end of Table 2.

(a) 1998

(b) 2002
Figure 5: Economic capital for a desired lender's rating, based on the variation in
B-rated borrower LGD and for different counterparty ratings.

5. Final considerations

Our results demonstrate that there is an interesting opportunity for financial institutions when considering the credit risk rating attributed to their customers to improve their own positioning with rating agencies, while there is also no need for a larger capital requirement to achieve this goal. This was confirmed by our analysis, where we showed that a financial institution that lent to an A-rated company had the opportunity to improve its rating position and it required only a low LGD. In the cases where BBB and A ratings were desired by a lender, it should be noted that the capital required was the same over a suitable range of LGD values. These comparisons were visualized based on the results obtained with a BBB borrower, although the windows of opportunity for LGD were not very frequent. This demonstrates the decline in the economic capital required by the LGD's threshold when a lender desired an AA rating because the borrowers with an LGD below this limit had unexpected properties in this group.

For a BB borrower, there were two possible strategic outcomes: a leap in the LGD's economic capital with the respective rating when searching for an AA rating or using the same capital reserve for companies with a higher ranking, but with a substandard LGD. According to the same logic, for a B borrower compared with a BB borrower, the reasoning is similar to the BBB borrower compared with A and changing the players, i.e., the windows of opportunity are analogous but less frequent. However, the economic capital measure of a borrower's rating demonstrates that there is a feasible alternative for financial institutions that require an improved rating for themselves because this would obtain improved results. Thus, risk managers might assess their decisions more accurately with the aim of detecting such situations among their investment/credit granting alternatives.

The risk models are still considered to be an interesting topic and a huge field of study (Stoian and Balan,2012). Our contribution is based on verifying arbitrages on target ratings where often no increment is required in the economic capital using a study made bySundmacher and Ellis (2011), applying data of two different periods. So, we tested a practical method for assessing a loan option that, in addition to the expected return and the risk involved, may involve the leveraging of the lending institution's rating, thereby providing an improved market reputation and possibly attracting better investors and/or lenders.

An important issue considered in this study is the cooperation among the parameters in determining the outcomes of the simulations because the LGD and PD tended to move in the same direction; thus, when specifying one of these variables and balancing the other (which is likely to occur in practice), the economic capital value may lead to specific conclusions in practical applications because companies may change their own ratings based on their lender's position, which is extremely interesting. However, it is assumed that an improved credit quality is an indication that reduced capital requirements will appear in the future because improved positions will obtain this benefit.

Some limitations of this study may be addressed in future studies. In particular, the volume of LGD data is still insufficient to obtain results with high impact in the academic community. However, our study indicates that solutions to this problem may provide useful insights in this area. The availability of PD data has improved, where they are reported at sufficient intervals to discuss the matter, i.e., annually and at times monthly, but the latter is still considered to be a very small interval, which may cause excessive fluctuations or even erroneous assertions, thereby resulting in various biases. Another limitation is created by the endogeneity problem. To enter it in our model would change the nature of the discussion, making it about several borrowers, and would invite study of the credit portfolio, which would substantially change the focus of this research. We suggest that this could be explored in a future study on the level of credit portfolio, where it is possible to consider a bank's features. Further research may be possible based on this study, such as assessing the basic model by considering a proportion of PD and LGD dependency, thereby yielding fresh insights.

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This document was a collaborative effort. The authors want to thank Santander Bank, Coordination for Higher Education Staff Development (CAPES), Centro Paula Souza and Mackenzie Presbyterian University (all from Brazil) for financial supports.

1. Professor of Finance, Federal University of Uberlândia, School of Management and Business, Campus Santa Mônica. Corresponding author. Email: flmbarboza@ufu.br
2. Professor of Finance, University of Brasília, Department of Management, Campus Darcy Ribeiro, Brasília, Federal. Email: herbert.kimura@gmail.com

3. Professor of Finance, Mackenzie Presbyterian University, São Paulo, São Paulo 01302–907, Brazil. EMail: leonardobasso@mackenzie.br

4. Professor of Finance, University of Brasília, Department of Management, Campus Darcy Ribeiro, Brasília, Federal. Email: sobreiro@unb.br