Modelling Asset Correlations of Revolving Loan Defaults in South Africa

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How to cite this thesis

Surname, Initial(s). (2012). Title of the thesis or dissertation (Doctoral Thesis / Master’s Dissertation). Johannesburg: University of Johannesburg. Available from: http://hdl.handle.net/102000/0002 (Accessed: 22 August 2017). Modelling asset correlations of revolving loan defaults in South Africa

by BONGANI MHLOPHE

MINOR DISSERTATION

Submitted in partial fulﬁllment of the requirements for the degree

MAGISTER COMMERCII

FINANCIAL ECONOMICS

in the

FACULTY OF ECONOMICS AND ECONOMETRICS

at the

UNIVERSITY OF JOHANNESBURG

SUPERVISOR: Professor J.W. Muteba Mwamba

December 18, 2019 Dedication

I would like to dedicate this dissertation to our Lord and Saviour Jesus Christ for granting me the intellectual ability to be able to embark on this journey. I would also like to dedicate this work to my wife, Thandi Mataboge, and my son, Simnikiwe Mhlophe, for being patient with me and supporting me throughout the late nights spent away from them. In closing, I would also like to thank my grandmother, aunt and siblings for the words of encouragement and, lastly, my late parents.

1 Acknowledgments

I would like to give thanks to the following people/institutions:

• My employer, for providing the ﬁnance for my Master’s degree.

• Bank X and the SARB for providing me with the data for the research.

• Professor J.W. Muteba Mwamba, my supervisor, for guiding and providing me with support throughout this journey.

2 Contents

1 Introduction1 Background...... 3 The BCBS...... 6 The Basel Capital Accords...... 8 The introduction of Basel I as a ﬁrst attempt at capital regulation ...... 8 The introduction of Basel II as a replacement to Basel I . . . 9 The Basel III Accord: "A global regulatory framework for more resilient banks and banking systems" ...... 16 The South African Economic Landscape ...... 19 The South African Credit Card Landscape ...... 21 Problem statement and objectives ...... 22 Dissertation Outline ...... 24

2 Literature Review 25

3 Methodology 34 3.1 The Methodology and Parameter estimations ...... 34 3.1.1 The ASRF approach: The mathematics behind the model . . 34 3.1.2 Distributions ...... 35 3.1.2.1 The Beta Distribution ...... 35 3.1.2.1.1 Empirical extraction of the asset correlation based on the beta distribution ...... 37 3.1.2.2 Vasicek distribution ...... 39 3.1.2.2.1 Empirical extraction of the asset correlation based on the Mode approach of the Vasicek distribution ...... 40 3.1.2.2.2 Empirical extraction of the asset correlation based on the Percentile approach of the Va- sicek distribution ...... 42

4 Empirical analysis 44 4.1 Data used in the study ...... 44 4.2 Empirical results ...... 46

5 Conclusions, Research Limitations and Recommendations for Fu- ture Investigations 60 5.1 Conclusion ...... 60 5.2 Limitations ...... 62

3 5.3 Recommendations...... 62

References 64

4 List of Tables

1.1 Basel II Capital Calculation Approaches ...... 12 1.2 Asset correlations descriptions ...... 16 1.3 The extent of bank failures in South Africa post democracy ...... 20

4.1 Asset classes for our data and their Basel II asset classiﬁcation . . . . 45 4.2 LGD data for the banks that participated in the BCBS’ 5th Quanti- tative Impact study ...... 46 4.3 Credit Card Empirical correlations compared to Basel II correlations 50 4.4 Comparison of the South African commercial loans empirical asset correlations and the BCBS-speciﬁed asset correlations ...... 51 4.5 Comparison of Bank X’s Credit Card Capital Charge (Relative to Basel Capital Charges ...... 52 4.6 Comparison of South African Commercial Loans Ratio of the Capital charge (Relative to the ratio of the Basel Capital Charge) ...... 53

5.1 The average asset correlation determined using the default data . . . 60 5.2 The average asset correlation determined using the write-oﬀ data . . 61 5.3 A presentation of the correlation results based on publications . . . . 61

5 List of Figures

1.1 The loss distribution that contains a total loss at the 99.9th percentile. Source: Stoﬀberg & van Vuuren, 2015 ...... 13

4.1 (a) The cumulative, and (b) the density function for Credit Card Actual Default Rate losses from February 2006 to September 2015. . 47 4.2 (a) The cumulative, and (b) the density function for Credit Card Write-Off data from January 2007 to May 2017...... 48 4.3 (a) The cumulative, and (b) the density function for the South African Commercial Loans from June 2008 - December 2016...... 49 4.4 A comparison of the empirical asset correlations compared to Basel specified asset correlations using the actual default data ...... 50 4.5 A comparison of the empirical asset correlations compared to Basel specified asset correlations using the Write-Off data ...... 50 4.6 South African commercial loans empirical correlations compared to Basel II correlations ...... 52 4.7 Comparison of Bank X’s Credit Cards Ratio of the Capital charge (Relative to the ratio of the Basel Capital Charge) ...... 52 4.8 Comparison of South African Commercial Loans Ratio of the Capital charge (Relative to the ratio of the Basel Capital Charge) ...... 54 4.9 A comparison of the five-year rolling empirical asset correlations to the Basel specified rolling asset correlations for the South African commercial loans data ...... 56 4.10 A comparison of the five-year rolling empirical asset correlations to the Basel specified rolling asset correlations for Bank X’s credit card default data ...... 56 4.11 A comparison of the 5-year rolling empirical asset correlations to the Basel specified rolling asset correlations for Bank X’s Credit Card Write-off Data...... 58

6 Abbreviations

ASRF Asymptotic Single Risk Factor

BCBS Basel Committee on Banking Supervision

BIS Bank of International Settlements

CDF Cumulative Density Function

EAD Exposure at Default

IRB Internal ratings-based

LCR Liquidity Coverage Ratio

LGD Loss Given Default

NSFR Net Stable Funding Ratio

PD Probability of Default

PDF Probability Density Function

RWA Risk-weighted assets

SA Standardised Approach

SARB South African Reserve Bank

7 Abstract This study examines the extraction of the empirical asset correlation for three datasets using both the Beta and Vasicek distributions over a static period of time, as well as a rolling period of time. The computed empirical asset correlations are thereafter used to determine the economic capital. The first two datasets relate to a sample of credit card accounts from a South African bank1. The first dataset contains monthly defaulted data which spans nine years (i.e February 2006-September 2015) and was calculated by taking yearly cohorts of actual defaulted customers as a percentage of open, performing customers at the beginning of each yearly cohort. The second dataset spans ten years (i.e January 2007-May 2017) and was calculated by taking the actual monthly write-off amount as a percentage of the monthly total exposure on the balance sheet. The third dataset contains data for all loans issued in South Africa2 which spans some nine years of monthly data (i.e June 2008- January 2017). This data was collected from the SARB (Venter, 2017) by dividing the monthly impaired advances by the monthly total exposure on the balance sheet. Two distributions have been selected for this study, the Beta and Vasicek distributions, however two different calculation approaches (mode and percentile) are used for the Vasicek distribution assumption. We first use these three distinct calculation approaches to empirically estimate the asset correlation over a static period of time and compare them to the BCBS prescribed asset correlations. The computed empirical asset correlations are thereafter used to determine the economic capital and compare it to the economic capital determined using the BCBS prescribed asset correlations. Secondly, we use these three distinct calculation approaches to empirically estimate the asset correlation over a rolling five-year period and compare them to the BCBS’ prescribed asset correlations. For both the static and five-year rolling empirical asset correlations, we show that the BCBS’ prescribed asset correlations are much higher than the empirical asset correlations for the South African loans dataset. However, the opposite is found for both the credit card default and write- off datasets which had higher empirical asset correlations. The economic capital charge calculated using the computed empirical asset correlations is lower than the economic capital calculated using the BCBS’ prescribed asset correlations for the South African loans dataset, while the opposite result is found for both the credit card default and write-off datasets. This result implies that the BCBS’ prescribed asset correlation is not as conservative as intended for South African bank specific credit cards and that the required capital charge stipulated by the BCBS is not sufficient to cover unexpected losses. This may have dire consequences to the South African banking system through systemic risk. Therefore, we recommend that the capital levels be raised to match the capital levels determined in this study3.

Keywords: Asset correlation, Basel, economic capital, Vasicek & Beta distributions

1Due to the proprietary nature of bank-speciﬁc data within the South African banking sphere, it is not possible to reveal the name of the bank which gave us access to their data 2This is a consolidated view of all South African loans advanced i.e Residential Mortgage, Retail Revolving, SME Retail, Retail Other, Corporate, SME corporate, Banks and securities ﬁrms, Securitisation. This data was collected from the SARB (Venter, 2017) 3The SARB doesn’t have the authority to change the BCBS’ prescribed asset correlations. Chapter 1

Introduction

The implementation of Basel II regulations introduced two calculation approaches for the regulatory capital. They are the standardised approach and the advanced internal ratings based (IRB) approach. Banks using the standardised approach are only allowed to calculate capital based on risk weights speciﬁed by the Basel Com- mittee on Banking Supervision (BCBS), while banks using the IRB approach are allowed to use their own estimates of key risk parameters for each obligor (Resti, 2016). The advanced IRB approach utilises an asymptotic single risk factor (ASRF) calculation method that has a simple, analytical closed form solution, which is a function of multiple parameters (Vasicek, 1987). The risk parameters are among others the probability of default (PD), loss given default (LGD) and most impor- tantly, the asset correlation.

The asset correlation may be described as the specific contribution that is made by each and every asset to the systematic risk of the whole portfolio. Thus, it can be used as an indicator of how sensitive each exposure is to systematic risk and has a significant effect on the estimation the credit risk of a portfolio. Thus, the Basel committee naturally proposed the asset correlation as one of the parameter’s in the calculation of the minimum regulatory capital requirements (Duchemin et al., 2003).

To maintain a suﬃcient capital level, banks must comply with the regulatory rules that have been set by the BCBS and to achieve that they must use the asset correlation (a measure of the strength of the link between obligors in a portfolio and the economy) speciﬁed by the BCBS. The economic capital provides banks with a more accurate estimate of their capital adequacy as it uses the banks’ own crite- ria of estimating capital, so asset correlations that are embedded in their empirical loss data are of considerable interest (Smit & van Vuuren, 2009). Banks trust their own internal models more as they have complete control over some of the input parameters used in the economic capital calculation. Thus, empirically estimating the empirical asset correlation is important for banks that want to use it in their economic capital calculation and determine their own correct level of capital. The determination of economic capital is important for banks as it is a critical focus in the global banking sector, especially after the credit crisis.

If a bank determines from using its own estimates of asset correlations that the Basel prescribed capital requirements are too lenient, they may choose to increase

1 the economic capital reserves to match up to the determined level. If the opposite is found, the banks may assess the prevailing economic conditions and make a decision as to whether a decrease in their capital reserves is warranted. Either case is valuable as it provides a bank with information on their level of capital reserves.

The 2008 credit crisis has revealed to the global banking community the importance of both liquidity and capital. This means that the role of capital regulation becomes increasingly important with increased risks being experienced by banks, which is particularly relevant in a country like South Africa that has already experienced many failed banks, e.g VBS Mutual bank. This study seeks to introduce concepts related to the capital and the quality of liquid assets that should be kept aside in times of crisis. The ﬁndings reveal how critical it is for banks to understand the risks they are facing and how to mitigate such risks. This study demonstrates methods that can be used by any bank that follows the IRB approach to empirically estimate their own empirical asset correlation using their internal loss data. Doing so will assist any bank, regardless of size or sophistication, in the determination of the correct level of capital charge that should be applied to the credit risks experienced in their own internal loss data.

Research on the empirical asset correlation has in the past mostly been explored from a corporate perspective, with the main pioneers of such research being Lu- cas (1995) who explored the empirical asset correlation of company assets. Sironi and Zazzara (2001), conducted a study using corporate default rates released by the Bank of Italy, Henneke and Trück (2006) explored the relationship between the asset correlation, PD and the firm size, and Bystrom (2011) proposes a new way of modeling the dynamics of the asset value of firms based on financial securities prices. These are however studies that are all related to the empirical asset correlation of corporates.

Banks also need to empirically estimate asset correlations of retail loans. However, unlike corporates, banks cannot directly observe changes in asset value, stock prices or other indicators that are commonly used on corporate loans to determine financial condition. Therefore, researchers are forced to come up with new approaches to estimating the asset correlation. This is examined in Bellotti and Crook (2012) who conduct a study into the estimation of the empirical asset correlation of two UK datasets using a Merton one-factor model. They find that the BCBS’ prescribed asset correlations are much higher than the empirical asset correlations, which were below 1%. A study by Hansen et al (2008) on the empirical asset correlation of retail loans, as well as credit cards found an empirical correlation of 1.3%, while Rosch and Scheule (2004) use charge-off rates from all United States of America (USA) commercial banks to estimate empirical asset correlations for credit cards, residential mortgages and other retail loans. They find asset correlations of around 1% for credit cards.

2 The main problem is that the technique has never been applied to a South African bank specific loan loss dataset. This technique has also never been applied from a beta distribution of loan loss data from South African market and South African bank-specific data. Stoffberg and van Vuuren (2015) find an empirical asset correlation of 7% for credit cards, which is not consistent with previous research.

This study seeks to assist South African banks to minimise unexpected losses that may arise from credit card defaults. To the best of our knowledge, there is no published study that has reviewed bank specific asset correlations for South African credit cards, which is one of the aims of this study. Empirically estimating the asset correlations for a South African bank specific credit cards default and write- off data and comparing them to the Basel prescribed asset correlations is the first contribution made by this study. The computed empirical correlations are used in the calculation of economic capital. We observe higher empirical asset correlations, which suggest that the Basel required capital needs to be increased to cover potential unexpected losses. This is only consistent with the results obtained in Stoffberg & van Vuuren (2015). Other studies had found empirical asset correlations which were way lower than the BCBS’ prescribed asset correlations. The second contribution is applying the Beta distribution to both South African commercial loans and bank-specific credit cards in the estimation of the empirical asset correlations. This method reveals varying results of 2.99% and 8.47% for the default data and write-off data respectively. These results are on the opposite sides of the stipulated BCBS’ prescribed asset correlation of 4%. The final contribution is exploring South African bank-specific empirical asset correlations over a rolling period to assess it over different economic conditions. This reveals that the Basel’s intended conservatism is not always applicable.

Background

One of the most regulated sectors is the banking sector. Pivotal to such regulation are the rules that have been put in place for each bank’s risk capital. This importance emanates from the central role that banks play as financial intermediaries, the co-ordination between the global banks in adopting common capital standards, and the importance of banks’ risk capital in achieving a stable and sound global banking system (Santos, 2001). In virtually all the economies around the world, banks have always played a financial intermediary service provider role, which is mainly due to their position of producing information and providing monitoring services and liquidity insurance. The importance of bank capital regulation is derived from its role in the improvement in the soundness of banks through regulating risk-taking incentives and improving the banking corporate governance. The modern theory on financial intermediation states that the two central roles performed by banks in the economy are creating liquidity and transforming risk (Berger & Bowman, 2007). These roles may be referred to as the qualitative asset transformation functions of banks (Stoffberg & van Vuuren, 2015). This liquidity-creation role of banks is exactly what this study seeks to investigate. To investigate this role, the regulatory capital, economic capital, and the banks’ liquidity are concepts that need to be thoroughly understood, as they give information on the functioning of banks, their risk-taking activities, and how those risks are mitigated (Farag et al., 2013).

3 Firstly, the regulatory capital is known as the minimum capital amount needed to absorb sufficient unexpected losses, such that it protects a bank from bankruptcy using shareholders’ equity rather than using deposits from customers (Martin, 2013). The capital amount that should be set aside has been determined by the Basel Com- mittee on Bank Supervision (BCBS). Local regulators are responsible for ensuring that banks follow these regulations, and may impose stricter capital requirements to cater for risks that may be inherent in that particular economy. The BCBS aims to equate the regulatory capital to the bank’s own internal (or economic) capital requirements, or, in other words, the BCBS aims to ensure that the capital amount required by banks to stay afloat is aligned as close as possible to the capital amount that banks themselves believe they require (Smit & van Vuuren, 2009). There are two important concepts that are related to the regulatory capital, which are the Risk-Weighted Assets (RWA) and the tiers of capital (DeChesare, 2012). The capital requirements are set based on how risky each individual asset is, rather than assuming that all assets are equally risky (i.e. RWA). As these assets’ riskiness is not equal, they have been segmented into different tiers of capital, as not all capital is able to equally protect banks. Therefore, to accurately determine the correct capital levels, both of these concepts must be considered. A strong capital buffer helps contain leverage in the banking system that arises from banks’ risk-taking activities. It also limits risk concentrations at banks and reduces the risk contained in large banks with embedded systematic risk from becoming insolvent, and thus impacting other institutions adversely (Wellink, 2008). Therefore, as a requirement, banks must have a capital buffer that will sufficiently cover their risk in order to protect money that belongs to depositors and other creditors.

Capital adequacy and liquidity are distinct but related concepts in the banking sector, and both reveal a lot of information about a bank’s sustainability and solvency. Liquidity may be defined as a measure of the minimum amount of assets (bank’s or other financial institution’s) that can easily and quickly be converted into cash (Martin, 2013). Liquid assets in a bank include cash, central bank deposits, and government bonds. Liquidity issues arise due to interplay between the asset side and funding of the bank’s balance sheet. This usually occurs when a bank does not have enough cash reserves or assets that it can quickly and easily convert into cash to honour payments to their depositors and creditors (Farag et al., 2013). The misconception in relation to capital is to think of it as literally being "held" or "set-aside" by a bank as it is not an asset. However, the capital is one of the forms of funding required to absorb losses that have a potential to threaten the bank’s ability to be solvent. For financial institutions to remain afloat, they must have sufficient liquid assets to honour their near-term (usually thirty days) obligations without requiring refunding from central banks (Martin, 2013). Thus, banks need to ensure that they keep regulatory capital that is liquid enough to be able to absorb short-term shocks that may be caused by withdrawals of deposits that are the most prone to bank runs (Nyaundi, 2015). The higher the bank capital, the lower the chance of financial distress.

The main cause of bank liquidity issues is the fact that they receive funding from deposits, which are subject to change and are often unpredictable (Whittlesey, 1945).

4 Banks can also face liquidity problems due to borrowers failing to honour their loan re-payments or when banks themselves fail to secure reﬁnancing when their existing funds are withdrawn (Rossouw, 2014). For banks, capital adequacy or increased liquidity can be seen as instruments that limit them from excessive risk-taking with limited liability, thus promoting a culture of optimal risk sharing between banks and depositors (Nyaundi, 2015). In contrast, the regulations around the capital requirements are often viewed as a protection against arising insolvency issues, thus limiting the costs associated with ﬁnancial distress by reducing the bank’s probability of insolvency (Caggiano & Calice, 2011). Even though this is the case, there is an opportunity cost associated with holding capital reserves. Thus, there must be a delicate balance between the capital reserves set aside and lending activities to ensure that banks take risks within their risk appetites.

The main aim of this study is to empirically estimate the asset correlation factor used in the Basel II capital calculation. These empirical asset correlations are thereafter used to determine the economic capital. This will allow banks of all sizes and varying complexities to not only estimate the empirical asset correlations using their own internal risk data, but to also be able to use those empirically estimated asset correlations to determine the economic capital. Knowing these values will reveal to banks whether the BCBS was too conservative or lenient in its setting of the asset correlation ﬁgures.

The BCBS aims to ensure that banks closely align the regulatory capital that is required to keep them afloat with the amount of capital that banks themselves believe they should be keeping (i.e. the economic capital) (Smit & van Vuuren, 2009). Currently, banks that are sophisticated enough are able to determine their own economic capital amounts independent of the BCBS’ guidance, while smaller banks that lack such sophistication still heavily rely on the BCBS for guidance on the determination of the economic capital, as they lack the quantitative resources and systems needed to determine their own economic capital. Economic capital is different from the regulatory as it is solely dependent on a bank’s internal determination (or estimation) of the capital they need and would be defined as the capital amount that a bank would keep if there were no capital regulations. Regulatory capital is the amount of capital prescribed by a regulator that an institution needs to hold to maintain an adequate level of liquidity and solvency (Lang, 2009).

Many debates that took place leading up to the publication of the second Accord (Basel II) have drawn attention to the need to bring the regulatory capital required closer to the economic capital. This was the main aim of introducing the Pillar 1 of Basel II by the Basel Committee (Gordy & Howells, 2006).

There is a lot more literature covering regulatory capital compared to that on economic capital, indicating to the relative attention the two tend to receive. As economic capital is, by deﬁnition, proprietary and bank-speciﬁc, it makes sense that banks would be unwilling to share their methodologies and procedures as that would reduce their competitive advantage. In BCBS (2005), it is clearly stated that economic capital determination practices still continue to evolve, and it recognises that there remain major methodological and implementation issues related to the appli-

5 cation of the economic capital in banks, especially if banks want to use economic capital measures as an internal measurement of their capital adequacy. The BCBS recognises the need for economic capital. However, it does not nullify the existence of Basel II regulations in relation to the regulatory capital, thus banks are still required to calculate both.

Capital regulations are needed to create sound ﬁnancial institutions, protecting consumers from being exploited by banks (Smit & van Vuuren, 2009). Such protection includes protecting customer deposits from being lost and minimising systemic risk. Systemic risk is the risk that an economic shock in an institution has of causing not only its own failure but also failure in other institutions, eventually threatening the economy and economic growth (Smaga, 2014).

Two key players that have played a massive role in capital regulations are the BCBS and the Basel Accords. Each is discussed in this chapter to provide better insight into the world of global bank capital regulations. An overview of these role players will provide background on how banks can use the methods that are introduced in this study to empirically estimate their own asset correlation and determine their credit risk economic capital (which are the main aims of the study).

The BCBS has produced three documents that have been the key to global bank capital regulations. These documents are referred to as the Capital Accords and have been named Basel I, II, and III. The ﬁrst Basel Accord (Basel I) was introduced in 1988. This document was eventually replaced in 2008 by the second Accord (Basel II). Based on lessons learnt from the global credit crisis (recession), the Basel II Accord was enhanced through the introduction of the Basel III Accord. These Accords have been widely used, debated, and amended over the years. They have been the source of major global banking regulations.

The BCBS The harmonisation of financial markets, central banks and regulatory authorities occurred due to globalisation. This is a process that necessitated the need for a central vehicle to co-ordinate participants’ efforts (Bieri, 2008). Therefore, a global framework for integrating regulatory and supervisory processes has developed into what is collectively known as the Basel regulations process (Bieri, 2008), which com- prises different committees based in the city of Basel. These different committees play a vital role in making sure that there is better global financial stability, which is used as a platform for discussion. These discussions are mainly based on current threats that impact global financial stability (Bieri, 2008).

Central to the Basel processes are the four main Basel committees. These Basel committees have been segmented according to global financial industries. The seg- mentations are the financial markets and foreign exchange issues, monitoring of financial cross border issues, the insurance industry and bank capital regulation.

The Basel process presents an opportunity to the global ﬁnancial community to explore good governance when dealing with various supervisory and regulatory is-

6 sues in forums that facilitate discussions supported by analysis which is often highly sophisticated (Yoshikuni, 2002).

The most prominent aspect of the Basel regulation process is the establishment and monitoring of bank capital regulations which is governed by the BCBS. This is done through the implementation and monitoring of Basel I and II, soon to be dis- played in the implementation of Basel III. These regulations, which have impacted almost all the banks in the world, makes the BCBS the most prominent and inﬂu- ential committee in the Basel process (Bieri, 2008).

The BCBS, which has headquarters at the Bank of International Settlements (BIS) in the city of Basel, was formed in 1974 by a group of ten (G-10) international central bank governors’ due to a series of banking markets and international currency disturbances (BIS, 2001). The most notable disturbance was the collapse of Bankhaus Herstatt in West Germany. This was a (smallish) bank that mainly dealt in foreign exchange markets. It was not a major player in the European markets, but had taken out large positions in the foreign exchange markets. It had a total of $620 million of foreign exchange trades unsettled. It collapsed when it experienced liquidity issues due to its counter-parties attempting to re-claim their outstanding currency without success, resulting in the default of several parties (Schenk, 2014). The bank was eventually closed on the 26th of June 1974 by the German Super- visory office. This closure took place at the close of business in Frankfurt, which happened to coincide with the beginning of the working day in the offices in New York, thereby resulting in a number of operations remaining uncleared. This gave birth to the popular term known as the Herstatt risk, which is the risk a business takes by conducting business in companies that operate in different time zones (Mourlon-Druol, 2015).

While Bankhaus Herstatt was a small bank (relatively), the systemic shocks that resulted from its collapse had quite a signiﬁcant impact due to the damage caused to the trust that existed in inter-bank relations. This debacle prompted the urgent formation of the BCBS in 1974. The main aim of the formation of this committee was to foster collaboration between banking regulators to prevent the repetition of the Bankhaus Herstatt debacle and other similar incidents.

Therefore, this debacle forcibly reminded the global community of the fragilities obviously inherent in the Euromarkets and international capital transfers structures upon which the world’s economy was largely reliant (Goodhart, 2011). Con- sequently, beyond the considerations of the Herstatt risk, the BCBS’ aim was to enhance ﬁnancial stability through the enhancement of the quality of global banking supervision (BIS, 2001).

Interestingly, however, the Herstatt risk (as it was known) that was related to issues inherent within the structure of the foreign exchange markets was not ﬁnally settled until almost thirty years later with the formation of the continuously linked settlement (CLS) system.

Since its ﬁrst meeting in February 1975, the Basel committee has held regular

7 meetings three or four times a year, and has increased the number of members it had from ten to forty-ﬁve institutions emanating from twenty-eight diﬀerent ju- risdictions. Over the past four decades of consultations, the BCBS’ most notable outcomes are the formulation of its prominent publications of the Basel accords on the minimum capital requirements, which are Basel I, II, and, most recently Basel III (BIS, 2018).

The Basel Capital Accords Established by the BCBS, the Basel Accords are very important in the global capital regulations process. Their importance in this paper is highlighted by exploring the history of the formulation of the Accords, the introduction of Basel I, which is the ﬁrst Basel Accord, the introduction of Basel II and ﬁnally the formation of Basel III as an enhancement to Basel II.

The rules pertaining to Bank capital are important aspects of banking regulations. There are two main objectives that spurred the Basel Committee’s work on the regulatory rules. The ﬁrst was that the new framework should improve the soundness and stability of the global banking system. The second was that the framework must be applied fairly and consistently in all banks across the globe so as to decrease the existence of competitive inequality amongst banks (Santos, 2001).

With the foundations of bank supervision laid, the committee soon moved their focus to capital adequacy, spurred on by the 80’s crisis that occurred in Latin America that resulted in escalated concerns that international banks had capital ratios that were deteriorating while increasing risks globally. This situation resulted in the committee working together towards creating harmonised standards for the measurement of the adequacy in capital, which is commonly known as the Capi- tal Accords (BIS, 2001). As of 2018, three Basel Accords have been released, with only two fully implemented (Basel I and Basel II). The Basel III Accord is in the implementation process and should be fully implemented at the end of 2019.

The introduction of Basel I as a ﬁrst attempt at capital regulation The Basel I Accord was published and oﬃcially made available to all banks in July 1988 following a consultative paper that was published for comments in December 1987. It was approved by the G-10 central bank governors and was formally implemented in all internationally active banks by end of December 1992 (Gehrig & Iannino, 2017). These set of rules were aimed at regulating credit activity of banks and set out a common standard of measuring the capital adequacy and the minimum standards to be applied by all global banks (BIS, 2001).

The Basel I Accord introduced a new concept for the capital requirement: the Cooke ratio. The Cooke ratio, named after the Chairman of the Basel Commit- tee, is a ratio that expresses the minimum sum of capital as a percentage of its RWA that a bank should allocate for its credit activity (BCBS, 1988). The ratio was arbitrarily set at a minimum of 8%, meaning that the minimum sum of capital to be allocated by Basel-compliant banks to secure their credit activity should not be lower than 8% of the total amount of credit issued to its customers (Martin, 2013).

8 As the first framework to introduce global credit risk regulation, the Basel I Ac- cord was far from perfect, and had its limitations as expected. As this was a living document that had to be applied to banks with an ever-changing level of risk and technology at their disposal, it makes sense that it also needed to evolve with time. The first amendment was made in November 1991, in an effort to clarify the definition of the general provisions that could be included in the capital calculations. Later amendments came in April 1995 to take into account the impact of bilateral netting of banks’ credit risk exposures when dealing with derivative products, and in April 1996, a detailed definition of the impact of multilateral netting was provided. The most important amendment was the addition of the treatment of risks other than credit risk. This amendment, called the Market Risk Amendment, was drafted in 1996 by the Basel Committee following two consultative processes. It was released at the end of 1997. The main contribution that came from this amendment was the guidelines on how banks should set aside capital to cover the risk that may arise from potential losses due to the movement of market prices (Gehrig & Ian- nino, 2017). The main innovation arising from this amendment was that it allowed banks to choose between using the standard approach and their own internal loss models to estimate the capital amount required for market risk (Santos, 2001). This amendment incentivised banks to use their own internal models as a way to reduce their regulatory burdens and capital charges (Gehrig & Iannino, 2017).

Proposals for the Basel Committee to release an updated capital accord were spurred on by the formation of a consultation process to formulate a revised capital framework in June 1999. This process resulted in the three-pillar framework of Basel II in June 2006. The details around the formation of Basel II will be provided in the next section.

The introduction of Basel II as a replacement to Basel I The Basel Committee released a proposal for new capital adequacy standards to replace the 1988 Basel I Accord. This led to the 2004 release of a revised capital framework, commonly known as Basel II (BCBS, 2006). This revised framework was introduced as an effort to strengthen the soundness and the stability of the global banking system. Even though the Basel I Accord had made important contributions to bank capital regulations, it had limitations that resulted from banks coming up with “financial innovations” that made it easier to find “cosmetic” adjustments to boost their reported capital ratios without enhancing their capital soundness. A further criticism was that the risk weights did not accurately reflect the riskiness of obligors, thus bucketing them on the same risk level even though they differed in terms of risk (BCBS, 1999). Therefore, Basel I needed to be revised to account for these shortcomings.

This revised framework was met with a series of challenging inputs for almost six years from supervisory authorities and banking and other industry experts following its publication in 1999. This prompted the committee to subsequently release additional proposals for broad consultation and debate in January 2001 and again in April 2013 to supplement the initial release of the Basel II Accord. The committee also developed three quantitative impact studies that were related to their published

9 proposals (BCBS, 2006). These resulted in several improvements in developing a more risk-sensitive capital requirement framework. This framework was eventually approved by the G-10 authorities in 2004.

The Basel II Accord is focused not only on the measurement but also the management of the main risks faced by the bank such as credit risk, operational risk, and market risk (Himino, 2004). The in-country banking regulators are in charge of ensuring that Basel II is implemented correctly and that the minimum capital requirements are maintained – in the case of South Africa, the South African Reserve Bank (SARB) is the regulator in charge.

The new Basel framework was formulated to not only enhance banks’ risk management capabilities but to also improve their risk reporting using the three-pillar approach.

The ﬁrst pillar, which stipulates the new capital requirements, is formulated to ensure that the capital held by banks is reﬂective of the underlying risks inherent in their business. The Basel committee also developed capital requirements calculations for new risk types such as operational risk, banking book risk, and other risk types not captured in the Basel I Accord (Santos, 2001). The committee enhanced the standardised approach and also introduced the IRB approach.

The second pillar, which addresses the supervisory review process of bank capital adequacy, is intended to ensure that banks measure their capital position correctly relative to the underlying risks inherent in their business. In doing so, this requirement seeks to ensure that the banks’ capital amount is above the regulatory minimum standards and regulators should intervene should that not be the case (Illing & Paulin, 2004).

The last pillar encourages banks to disclose market information to create sound information standards among all participants in the market. It stipulates that banks must disclose information such as the banks’ regulatory capital and its risk proﬁle (Stoﬀberg & van Vuuren, 2015).

As the main goal of this study is to estimate the empirical asset correlation, which will subsequently be used to determine the credit risk economic capital, only the literature covering credit risk will be explored in detail. As such, because the capital calculation requirements fall under Pillar I, Pillars II and III will not be discussed in detail.

The most crucial aspects that one needs to comprehend in relation to Pillar 1 are: the deﬁnition and calculation of the capital requirements, the deﬁnition of capital, the basics of the Basel capital adequacy, and the subsequent approaches used to calculate risk in Pillar I (Smit & van Vuuren, 2009). These are critical to note here because the calculations used in this study are mainly based on Pillar I requirements.

A bank is exposed to credit risk when it lends money to a customer. This is usually money received from customer deposits. The credit risk in that instance arises

10 from the possibility of the bank not being able to recover the money it is owed, while the depositor would still have the expectation of receiving back their deposit. Therefore, banks need to keep capital aside for such credit risk events to protect depositors against these losses (RBNZ, 2007).

The BCBS’ definition of capital has been segmented into three different tiers, commonly known as tier I, tier II, and tier III capital. These have been separated based on the quality of each instrument. Tier I may be defined as the capital that is permanent and free to absorb potential losses without impacting the banks’ daily trading activities. Tier I instruments have the highest quality, including disclosed reserves and ordinary share capital. Tier I is very important to the bank as it provides protection for both the survival and stability of the financial system (BCBS, 2006).

Tier II capital refers to capital which only absorbs losses in the event of a bank being dissolved and therefore provides a lower level of protection against losses. This type of capital only comes into play when the tier I capital has already been depleted by the bank (RBNZ, 2007). It includes instruments such as general provisions and hybrid capital instruments (Smit & van Vuuren, 2009).

The ﬁnal tier of capital is known as tier III. It consists of short-term subordinated debt and is used to only provide cover for market risk losses in cases where Tiers I and II capital have already been used up. It is not as commonly referred to as frequently as other tiers of Basel II capital (RBNZ, 2007).

The initial guidelines related to the capital adequacy ratio were presented in the 1988 Basel I Accord. The key concept related to capital adequacy as defined in Basel I is that banks have to keep a level of capital that is reflective of the underlying credit risks contained in their portfolios. This definition has been updated to include market and operational risk in Basel II. The capital adequacy ratio is a ratio of a banks’ core capital as a percentage of the risk faced by the bank, commonly referred to as the RWA (Białas & Solek, 2010). This core capital acts as a “cushion” that protects the bank against potential risks. The calculation has been outlined below:

Capital Base Capital Ratio = ≥ 8% (1.1) Risk W eighted Assets where:

• The capital ratio shows how the capital relates to the RWA. The total capital ratio must not be lower than 8% (Białas & Solek, 2010).

• The RWA have been determined by multiplying each asset category’s value by a conversion factor. These various conversion factors are a reﬂection of the underlying risk attached to each category of assets. The resulting values of in each category of assets are then added together (NCAF, 2015).

11 • The capital base is the same capital that includes diﬀerent tiers of capital as already deﬁned.

A major innovation that accompanied the introduction of Basel II is the calculation of the capital requirement for not only the credit risk but additional risk types such as the operational and market risks. Pillar I requirements provide banks with options to select different calculation approaches, with varying levels of complexity and flexibility in the capital calculation of each risk type. Banks can select these approaches based on their operational structure and complexity (NCAF, 2015). These approaches have been outlined for the different risk types in Table 1.1:

Table 1.1: Basel II Capital Calculation Approaches

Credit-Risk Approach Operational-Risk Approach Market-Risk Approach A Standardised Basic Indicator Standardised Approach B Foundation IRB Standardised Internal Models C Advanced IRB Advanced Measurement

Table 1.1 provides an outline of the diﬀerent Basel capital calculation approaches for credit, operational, and market risk. These are applied by diﬀerent banks, depending on how sophisticated and advanced their systems are. The simplest approach is the standardised approach, which was introduced for credit risk in the 1988 Basel I Accord. The other two risk types were only introduced with later amendments of the Basel I Accord (BIS, 2018).

The standardised approach uses risk weights that are dependent on the rating given by an external agency, and in cases where this rating is not available, the risk weights are then imposed by the regulator (Białas & Solek, 2010). The Foundation IRB approach gives permission to banks to utilise their internal rating systems to assess the borrowers’ risk and estimate their likelihood of defaulting on their loans with the exception of the parameters used to calculate the loss given default (LGD) which are provided by supervisory authorities (BCBS, 2006).

The advanced IRB approach is the only approach that allows a bank to set and calculate all of its own risk parameters (Białas & Solek, 2010). Banks are only allowed to use the IRB approach if they qualify through meeting certain conditions and explicit regulatory approval. These banks must also have systems in place that can adequately evaluate risk. The advanced IRB approach utilises an ASRF calculation method that has a simple, analytical closed form solution, which is a function of multiple parameters (Vasicek, 1987). These risk parameters are the determination of a customer’s PD (the probability that a customer will default over a yearly period), the estimation of the LGD (an estimate of the percentage of the exposure at default that a bank will lose when a customer defaults on his credit obligations), the EAD (exposure amount that a bank stands to lose upon a customer going into default), and the eﬀective Maturity (M) (BCBS, 2005). These risk parameters are used as part of the determination of risk weights and the capital requirements using risk-weighted formulas that have been provided in the Basel Accord. The eﬀective maturity is only used for portfolios that consist of corporate assets. Furthermore,

12 the IRB approach contains measures such as the unexpected loss (UL) and expected loss (EL).

As banks take risks when they lend money to customers, they are often presented with issues related to customers not honouring their repayment obligations. That presents an issue for the bank in determining exactly how much it may stand to lose from such activities, but it can easily estimate the loss amount it expects to have incurred from such losses at the end of the year. That measure is called the EL, which may be deﬁned as the amount a bank expects to lose over a certain time period as a result of its credit risk (Banco Central del Uruguay, 2007). It is easily calculated by multiplying the customers’ PD, EAD, and the estimation of the LGD. As such, these are losses that the bank can expect to incur as one of the costs of doing business, thus they should be covered by the provisions banks make on every exposure and pricing (Jain, 2018). The UL, however, represents large losses (that seldom occur), which are larger than the EL. These losses are covered by the capital as they are larger than the expected level and provisions provided for such (BCBS, 2005).

Figure 1.1: The loss distribution that contains a total loss at the 99.9th percentile. Source: Stoﬀberg & van Vuuren, 2015

Figure 1.1 provides an illustration of the portfolio loss distribution. The EL is represented by the mean of the distribution. UL however, may be defined as the difference between the EL line and the confidence interval. Basel II has selected a confidence interval of 99.9% for losses that occur due to credit risk, which means that there only exists a 0.1% chance that the bank will lose an amount larger than their capital. The BCBS has fixed the confidence level at 99.9%, i.e. a bank is only expected to experience losses exceeding its tier 1 & 2 capital levels on an average of once in a thousand years. This specified confidence level might seem quite high, but it was chosen taking into consideration that only tier 1 capital is loss absorbing, while tier 2 capital does not contain any loss-absorbing capacity at all. This high

13 conﬁdence interval was also set to ensure that banks are protected from estimation errors that might occur in their internal estimation of the PD, EAD and LGD, as well as other modeling uncertainties (BCBS, 2005). Losses that exceed this point are referred to as catastrophic losses (Smit & van Vuuren, 2009).

Banks that make use of the IRB approach are supposed to group exposures into different categories called asset classes, classified by exposure to different risk pro- files. Therefore, in the calculation of capital requirements, the classes have been divided into five asset classes, which are (a) banks, (b) corporates, (c) sovereigns, (d) equity, and (e) retail (BCBS, 2006). This classification of exposures is generally consistent with the general banking community. However, the Basel Committee has made an allowance for banks to utilise different definitions in line with their own risk measurement and management systems. Basel II’s introduction is not meant to force banks to deviate from the process they are currently using to manage risks. However, it requires banks to apply the appropriate treatment to each and every exposure to enable them to calculate the capital charge, coupled with the requirement that banks must be able to demonstrate to the relevant regulator that their methodology used for the classification of exposures is appropriate (BCBS, 2006). The rules pertaining to the asset classes and their applications have been documented in detail in the Basel II Accord. Asset classes mainly influence the structure of the portfolio and risk weight.

The Basel II framework stipulates that the portfolio invariance condition of the capital requirements is a very important property that has a massive influence on how the portfolio model is structured. Gordy and Howells (2006) provided proof of the portfolio invariance condition of ASRF models. They show that the portfolio invariance condition is based on the law of large numbers. When a credit portfolio contains a large number of obligors, it is assumed that the individual risks are di- versified away, thus leaving only the systematic risk to have an effect on the losses of the portfolio. These are system-wide risks such as industry-wide or regional risks. This is the reason why it is called a single factor model as it is modeled with only one single factor, which is the systematic risk factor (Martin, 2013).

The BCBS states that their usage of the ASRF in the capital calculations is not owing to any preference towards one model or the other but it was selected because it was suitable for this portfolio type. They however encourage banks to employ credit loss models that provides a good ﬁt to their data(BCBS, 2005).

The ASRF model utilises the average PD, which is reﬂection of default rates under normal (non-stressed) business and economic conditions. Banks use these PDs to determine the conditional expected loss through a supervisory mapping function commonly known as the Merton model (BCBS, 2005). The function used in the derivation of conditional PDs from average PDs is loosely based on an adaptation of Merton’s single-asset factor model to a banks’ credit portfolio (Paudel, 2007). In that model, Merton deﬁnes default as an obligor’s failure to meet their credit obligations over a yearly period due to the value of their assets depreciating below the amount due (BCBS, 2005).

14 Similarly, the LGD must reﬂect adverse economic conditions due to increased defaults during economic downturns (recession) when compared to non-downturn economic conditions (BCBS, 2005). This prompted the Basel Committee to prescribe that banks use a downturn LGD measure in the Basel capital calculation. This downturn LGD measure is used because it considers the fact that the Basel Committee ignored the correlation between the PD and LGD in Basel II (Miu & Ozdemir, 2006).

Two approaches were considered by the Basel Committee for deriving the downturn LGD. The ﬁrst approach entails extrapolating downturn LGD’s from average LGD’s using a mapping function. The second approach stipulates that banks can use their own internal data to determine the downturn LGD based on their own assessment of a downturn (recession) period (BCBS, 2005).

Vasicek (Vasicek, 2002) demonstrated that under certain conditions, we can naturally extend Merton’s model to be used in a speciﬁc ASRF credit model. With a view on both the work done by Vasicek and Merton, the Basel committee made a decision to adopt the assumptions that ASRF portfolios follow a normal distribution in their link to the idiosyncratic and systematic risk factors.

The ASRF model contains a single systematic factor that reﬂects the macroeconomic conditions of the economy (local or global). The level of dependency of an obligor to this systematic factor is what we call the asset correlation, which gives a measure of how strong of a link there is between the asset values of each obligor to another obligor in the same portfolio (Martin, 2013). Alternatively, the asset correlation may be described as a measure of the dependence between the borrowers’ asset value and the general state of the economy (local or global). All borrowers are interconnected to each other through this single systematic risk factor (BCBS, 2005). The asset correlation is impossible to observe directly, and is often very dif- ﬁcult to estimate due to a lack of historical data.

In the IRB approach, the shape of the risk-weighted formulas is determined by the asset correlations. As different obligors show different levels of dependency to the economy, the asset correlations are also different for each asset class. All banks use the different levels of asset correlations in the capital calculations despite the knowledge that these asset correlation values were arbitrarily chosen and may not necessarily match the correlation levels that are suitable for their own credit portfolios (Martin, 2013). These Asset correlations have been illustrated in Table 1.2.

People usually use the asset correlation and default correlations interchangeably without realising the differences between these two concepts. The default correlation provides a measure of the extent to which the default of an obligor is related to the default of another obligor. This correlation is quite difficult to observe directly. For example, if one car company defaults, and the other car company does not, it does not necessarily imply that their default correlation will be zero. However, we can attempt to infer the default correlation of two obligors by taking a measure of their individual default probabilities and their asset correlations (Zhang et al, 2008). The idea of a default correlation is very basic in nature, and quite intuitive such that the default of an obligor will likely result when his asset value is insufficient to pay

15 Table 1.2: Asset correlations descriptions

Asset class Correlation Mortgages Fixed (15 percent)

Qual. revolving Fixed (4 percent)

1−e(−35·PD) 1−e(−35·PD) Other retail Varies with PD 0.03 · 1−e(−35) − 0.16 · 1 − 1−e(−35)

1−e(−50·PD) 1−e(−50·PD) High-volatility Varies with PD 0.12 · 1−e(−50) − 0.30 · 1 − 1−e(−50) commercial real estate Source: BCBS, 2005 his obligations, therefore the joint probability of the two obligors can be determined by taking the likelihood of both obligors’ asset values falling below their respective obligations during a certain period. This probability can be determined from simply having knowledge of the correlation between the two obligors’ asset values and their respective probabilities (Antwi, Joseph & Gyekye, 2015).

Applying the default correlation for the calculation of the economic capital would mean each and every obligor would contain their own unique asset correlation that can be used to determine credit losses. This would be infeasible and practically impossible to model from a regulatory capital calculations perspective. Therefore, Basel II proposed equations that make use of standard asset correlations that are either ﬁxed or PD-dependent, based on the respective asset classes (Smit & van Vuuren, 2009).

This is the ﬁnal parameter used in the ASRF model. All the parameters already discussed in this section are used in the calculation of credit risk capital in the advanced IRB approach. The next section provide an outline of the enhancements made to the Basel II framework through the introduction of Basel III.

The Basel III Accord: "A global regulatory framework for more resilient banks and banking systems" This section will outline the enhancements made to the Basel II framework through the introduction of Basel III. The BIS (BCBS, 2017) deﬁnes the Basel III framework as a central element in the response by the Basel Committee to the shortcomings caused by the ﬁnancial crisis. It addresses many of the shortcomings that were in the Basel II framework and strengthens the foundation and resilience of the banking system, enabling it to avoid being susceptible to economic stresses through all economic cycles.

The improvements brought on by the introduction of Basel II (2006) were short- lived and could not be evaluated properly due to a collapse of the ﬁnancial system and economic slowdown that occurred in the period 2007-2009 (van Dyk, 2018). This

16 event brought about a realisation that ﬁnancial and banking regulations needed to be reviewed and improved. In this frenzy of demands for changes in the ﬁnan- cial and banking sector regulations, bank capital was the main topic of discussion (Greenspan, 2009).

The failures observed during this period was due to banks excessively growing their balance sheets by taking on a significant number of derivative products while they did not have sufficient liquidity buffers in place. These issues were accompanied by poor governance structures and a lack of proper risk management processes. These factors were demonstrated by the lack of proper pricing of the liquidity and credit risks accompanying the excessive rise in credit (BIS, 2018). In response to these shortcomings, the BCBS started a process to update the Basel II Accord in order to ensure protection of the financial and banking system so that it does not again experience a collapse similar to the 2008 financial crisis.

In September 2008, which is the same month as the Lehman Brothers’ collapse, the Basel Committee released a document named “Principles for sound liquidity risk management and supervision”. They subsequently issued follow-up documents meant to improve the shortcomings of the Basel II Accord, in particular regarding the treatment of complex securitisation positions, exposures in the trading book, and the oﬀ-balance sheet instruments (BIS, 2018).

On the 12th of September 2010, after a receiving large a number of review comments, the Group of Central Bank Governors and Supervision Heads (GHOS), which is an oversight body of the BCBS, announced their approval of the design of the capital and liquidity reform package, commonly known as Basel III. This was followed by an endorsement by the G-20 leaders’ summit in November 2010. They also agreed to the long periods of transition (the implementation still in progress) that it will take to fully implement Basel III (van Dyk, 2018). Basel III has been phased in since 2013 and will fully be implemented in 2019.

The updated Basel framework is an extension of Basel II and serves to strengthen and expand on the already established Basel II pillars. It is meant to build on Basel II and the lessons learned from the “sub-prime” crisis (BIS, 2018). The reforms will be achieved through the introduction of the following concepts: The Liquidity ratio, solvency ratio, leverage ratio, capital conservation buﬀer and the countercyclical buﬀer. These ratios have been described in detail below:

The liquidity ratio

The most crucial elements that need to be understood in relation to the liquidity ratio are the NSFR and the LCR. These are ratios that ensure that banks remain liquid and provides a guideline on what resources banks should use for funding (BCBS, 2010). The LCR took full eﬀect in 2015, while the NSFR was applied fully from January 2018 (van Dyk, 2018).

The LCR is a ratio that provides an indication of a bank’s ability to survive a liquidity crunch within a thirty-day period without requiring the assistance of the

17 central banks. This ratio stipulates the quality of assets that may be owned by a bank such that it has enough liquidity even under stressed conditions (Martin, 2013).

Liquid Assets LCR = (LCR should be greater than 1) (1.2) Net Cash Outflows where;

• liquid assets denote assets such as cash, bonds and central bank deposits; and

• net cash outﬂows denote cash outﬂows from loans granted subtracting losses incurred from a stress situation

The NSFR on the other hand is a ratio that is meant to encourage banks to avoid funding their businesses using short-term resources. This requirement stipulates that they own stable resources that have an initial maturity greater than a year. The implications of NSFR are that the liabilities that make up the stable resources must be weighted according to the type of customer and product with which they are dealing. Similarly, the type of funding used has to take into account the weighting of the assets relative to the bank’s liquidity levels (Martin, 2013).

The solvency ratio

The BCBS has changed the solvency ratio in Basel III to enhance the quality of assets that can be considered for the economic capital calculation. They increased the minimum ratio of the total capital to the RWA from 8% to 10.5% to ensure that assets of better quality are used to calculate the banks’ capital (Martin, 2013).

The leverage ratio

During the 2008 ﬁnancial crisis, banks had a tendency to keep a low amount of quality assets in terms of funding. In essence, it was observed that the prices of some assets depreciated at an alarming rate during the credit crisis. The leverage ratio has been established to ensure that banks upgrade the quality of assets (tier I capital) at their disposal and reduce the potential of systemic risk that could occur in the economy due to their failure. It is expressed as: T ier 1 Capital Assets LR = (1.3) T otalExposure

Capital conservation buﬀer

Basel III sets a new prudential ratio called the capital conservation buffer at 2.5% of the RWA. This buffer is applied to the tier I core capital in addition to the already stipulated 4.5%, thus increasing the minimum tier I core capital needed to cover credit risk to 7% of the RWA, in addition to a rate of seasonality, which is between 0% and 2.5% (Martin, 2013). This buffer has been set to ensure that banks maintain sufficient capital amounts to absorb losses during different economic periods

18 including downturns. The seasonality rate will allow banks to be able to draw on the buffer during stressful times and vice versa (van Dyk, 2018). It was phased in from January 2016 by increasing the buffer by 0.625% of RWA on the 1st of January 2016 and incrementing it by the same percentage each year until the 1st of January 2019. These yearly increments will have a final effect of getting the buffer to the stipulated 2.5% (van Dyk, 2018).

The countercyclical buﬀer

The countercyclical buffer is a tool used by the BCBS to protects banks from acting irrationally in different economic cycles. One can observe that banks have a tendency of issuing more credit in good economic times while the opposite occurs in difficult times. Excess lending may lead to a build-up of risk for the whole banking system. Therefore, this buffer has been put in place to reduce over-lending during good economic periods and encourages banks to rather increase lending during tough economic times. The buffer may be within the range 0-2.5% based on the regulators’ discretion. This results in an increase to the cost of credit and a decrease in the demand (Martin, 2013). It began to be phased in from 2016 and is projected to be completely implemented by 2019 (van Dyk, 2018).

The South African Economic Landscape

South Africa has quite a history with liquidity issues and failed banks, having until today (August 2018) had thirteen banks put under curatorship in the past thirty years (Tjiane & van Heerden, 2015). Bad management and liquidity have been the most prevalent reasons for failed banks in South Africa. The list of these banks and reasons for failure are presented in Table 1.3.

In a discussion of the latest bank failures (post the millennium), we will review the failures of four banks, namely: Saambou Bank, BOE Bank, African Bank, and VBS Mutual Bank.

Saambou Bank’s demise was caused by unfavourable investor sentiments with regards to the micro lending space. This was just after the failure of Unifer and 20Twenty businesses that were committing fraud by siphoning cash out of their respective businesses. This led to a run on Saambou Bank, with around R1 billion withdrawn over a two-day period. Saambou’s R1.4 billion housing ﬁnance bo ok was eventually sold to FirstRand’s First National Bank (FNB) for a discounted R984 million (Makhubela & Phiri, 2006).

A month after Saambou’s liquidation, BOE ran into ﬁnancial diﬃculty. Prior to its demise, BOE had approximately R16 billion in deposits from its retail clients, around R30 billion in wholesale deposits, and an exposure of R1.5 billion in advances. They had also acquired NBS’ home loan book a few years prior to that, which resulted in a yearly loss of R39 million in the year leading up to September 2001, owing to a deterioration in the secured property space. This loss resulted in market ru- mours that there was a possible sale of the home loan division to FNB and Standard

19 Table 1.3: The extent of bank failures in South Africa post democracy

BANK YEAR CAUSE OF FAILURE (RISK TYPES) PRIMA 1994 MARKET CREDIT LIQUIDITY SEC HOLD 1994 LIQUIDITY AFRICAN BANK 1995 OPERATIONAL CREDIT LIQUIDITY COMMUNITY BANK 1996 CREDIT LIQUIDITY ISLAMIC BANK 1997 CREDIT LIQUIDITY NEW REPUBLIC 1999 REPUTATIONAL CREDIT LIQUIDITY FBC FIDELITY 1999 REPUTATIONAL CREDIT LIQUIDITY REGAL TREASURY 2001 REPUTATIONAL CREDIT LIQUIDITY SAAMBOU 2002 LIQUIDITY REPUTATIONAL BOE 2002 LIQUIDITY AFRICAN BANK 2014 CREDIT LIQUIDITY VBS MUTUAL 2018 LIQUIDITY

Bank (Makhubela & Phiri, 2006). This pressure caused an increase in withdrawals from mainly corporate money market clients, amounting from about R200 million to around R500 - R600 million a day. The run on the deposits eventually led to a sale of the NBS home loan book to FirstRand’s FNB, and the unfortunate deregistration of BOE limited (Tjiane & van Heerden, 2015).

African Bank is the only South African bank to have failed twice, the first instance was in 1995 due to a classic case of growing “too big, too soon”. African Bank’s deposits grew from R273 million to R600 million between September 1991 and March 1994, with advances also growing from R196 million to R359 million in the same period. Over sixty percent of their loan book had been in the lower end of the housing market. They ran into provision and liquidity issues due to difficult economic conditions affecting the ability of their customers to make timeous payments (Makhubela & Phiri, 2006).

This Bank was subsequently bought and renamed to African Bank Investments Ltd. in 1999 (Sanchez, 2014). The revived African Bank diﬀered from its predecessor by the fact that it was not a deposit taking institution, but purely an unsecured lender

20 earning income from the interest and insurance products related to the unsecured loans. In 2008, It also acquired one of the country’s biggest furniture shops, Ellerines Holdings Ltd., which was not only loss making, but also offered unsecured loans on their furniture (Carrim, 2016). Much like its predecessor, African Bank experienced an aggressive growth of over 30% a year between 2010 and 2012, and offered loans to mainly low income earners. Their issues began when their core market experienced “severe deterioration” due to protracted strikes in the mining sector that began in the year 2012. This caused the bank to be put under curator-ship due to inadequate provisions for non-performing loans, which resulted in the bank needing R8.5 billion of additional capital to survive (Sanchez, 2014). African Bank eventually emerged out of curator-ship and is currently repositioning themselves as a full banking unit including offering deposit taking services (Ziady, 2017).

The latest bank to be put under curator-ship is VBS Mutual Bank which is the 13th bank to be put under curator-ship in South Africa and the ﬁrst since African Bank’s failure in 2014 (Tjiane & van Heerden, 2015). VBS is a mutual bank which had a stronghold in ﬁnancing rural land. It fell to its demise when the Bank’s management failure to align the business with its sudden rate of growth. The VBS balance sheet grew from about R200 million to R2 billion when they started accepting deposits from municipalities. They used short term municipal deposits to make long-term loans. The issue came when the Bank was made aware that the Municipal Finance Management Act (2003) (MFMA, 2004) does not allow municipalities to bank with a mutual Bank. This resulted in liquidity problems for the Bank when the municipalities came to withdraw their funds (National Treasury, 2018).

It seems like the case of “bad management and liquidity problems” is a recurring theme in the South African banking landscape. Therefore, this motivates the need for Banks to actively manage their liquidity and capital. In this study we update the previous South African correlation data used in Stoffberg & van Vuuren (2015). We use this data to empirically estimate the empirical asset correlation over a static period of time, as well as a rolling period of time. The computed empirical asset correlation is thereafter used to calculate the economic capital. Consistent with previous research, we find that the Basel asset correlations are significantly conservative.

The South African Credit Card Landscape According to the Experian Consumer Default Index (Experian, 2017), more than 14.7 South African consumers make use of credit cards and/or vehicle, personal, and/or home loans. The report states that South African consumers are in debt amounting to 1.53 trillion Rands, which is an unbelievably high figure. It also revealed that in the three months of June-August (2017), first time defaulters accounted for 13 billion Rands in debt with credit card facilities, accounting for about 2 Billion Rands of new outstanding debt. Another astounding figure revealed is that 66.48% of the accounts in debt are credit facilities, of which credit card debt is the highest contributor. This is mainly due to an unquestionable mismanagement of credit accounts, especially credit cards, which have landed a large number of South Africans in debt.

21 A study conducted by Moody’s Analytics (2013) on the comparison of card usage in 56 countries revealed that the total contribution of credit cards to the South African GDP is 7.8 billion USD. It also asserted that South Africa has an average payment per card per person of 1.25 times more than other African countries such as Kenya, Morocco, Nigeria, and Egypt. It noted that as the credit card market expands and reaches segments that are deemed to be risky, more issues associated with credit cards emerge. This study was conducted in the Pretoria North area using customers of the following banks: Nedbank, FNB, ABSA, Capitec, Standard Bank, and African Bank.

A large number of credit card consumers default on their debts, especially those in low-income segments. This could be due to easy access to credit and over-borrowing. One of the main reasons for making use of this credit is the increasing cost of living, and the usage of credit cards to pay for daily expenses (Kruger, 2014). Over- indebtedness has become a major concern for both the state and banks. Credit cards have not dealt with the issue as over-indebtedness has not diminished at all. The issue of misusing credit cards and subsequently falling into debt has been also attributed to a lack of ﬁnancial literacy. The terms and conditions that accompany credit agreements have not made it easy for consumers to fully comprehend how credit cards work (Thipe and Musvoto, 2016).

This study seeks to assist South African banks to minimise the impact they may experience due to unexpected losses that may arise from credit card defaults. There is no published study that has reviewed bank specific asset correlations for South African credit cards, which is the aim of this paper. We estimate the empirical asset correlation for both default and write-off data over a stable period of time, as well as a rolling period of time. Since South Africa is a country that has high debt in credit products especially in credit cards, it is wise for banks to consider a forward-looking approach and estimate their own empirical asset correlations using their unique internal loss data, which reflects the South African credit experience. We find that the empirical asset correlations are mostly higher than the Basel prescribed asset correlation. The implications of a higher empirical asset correlation are that the capital charge produced using the Basel asset correlation must be increased because it won’t be sufficient to cover all capital requirements. We also find that the Basel’s intended goal of conservatism is not always achieved over a rolling period.

Problem statement and objectives

Botha and van Vuuren (2010) proposed and tested a method of extracting asset correlations using a Vasicek and Beta distribution of gross loan loss data from international markets, while Stoffberg and van Vuuren (2015) applied a Vasicek distribution of gross loan loss data from both international and South African markets. The main problem is that the technique has never been applied to a South African bank specific loan loss dataset. This technique has also never been applied from a beta distribution of loan loss data from South African market and South African bank-specific data.

22 This study seeks to examine the extraction of the bank-specific credit card empirical asset correlations and compare it to the credit card (retail revolving) asset correlation specified by the BCBS. In addition to that, we aim to update the South African model used in Stoffberg and van Vuuren (2015). Doing so will equip us with a solution to the problem of not knowing the value of the South African credit card empirical asset correlation and how it compares to credit card empirical asset correlations that have been calculated in international markets and the BCBS’ prescribed asset correlation. These empirically determined asset correlations are then used to determine the economic capital and are compared to an economic capital calculated using the BCBS’ prescribed asset correlation. The second part of the paper explores rolling asset correlations for both bank-specific credit card and South African loan loss datasets and compares them to the rolling BCBS’ prescribed asset correlations. The main goals for the dissertation have been described below.

Part 1:

The main goals of Part 1 of this study are to:

1. provide an evaluation of the empirical asset correlation based on South African bank-speciﬁc loss data and the South African commercial loan loss data;

2. compare these empirical correlations with those speciﬁed by the BCBS; and

3. calculate the economic capital using these empirically determined asset correlations; and

4. compare this economic capital with the economic capital calculated using the BCBS’ prescribed asset correlations.

Part 2:

The main goals of Part 2 of this study are to:

1. estimate and compare the ﬁve-year rolling asset correlations of the South African bank-speciﬁc loan loss data and South African commercial loss data; and

2. assess the impact of changing these values for diﬀerent economic periods and comparing them to the rolling asset correlations used by the regulator;

23 Dissertation Outline

This research study contains the following chapters:

• Chapter 1: Introduction Chapter 1 provides the introduction, which includes the background of the study. An outline of the literature related to the development of Basel regulations, focusing two key role players – the BCBS and the Basel II framework is also explored. The concepts around the economic capital are discussed, as are the historical development and functioning of the Basel Accords (Basel I, II & III). The second Basel Accord’s (Basel II) three-pillar approach is explored in detail, including the deﬁnition of minimum capital requirements in Pillar I and its usage in credit risk. An outline of the South African economic landscape and Credit Cards Landscape is also provided.

• Chapter 2: Literature Review Chapter 2 provides an outline of the literature related to the calculation of asset correlation and economic capital.

• Chapter 3: Methodology Chapter 3 provides an outline of the methodologies used in the empirical estimation of asset correlations and its usage in the calculation of economic capital. This is explored by providing an overview of the Mathematics behind the ASRF model. A review of the Beta and Vasicek distributions will also be provided, including how the diﬀerent calculation (Beta, Mode and percentile) approaches can be used in the extraction of the empirical asset correlations.

• Chapter 4: Empirical analysis Chapter 4 contains presents the data and the empirical results.

• Chapter 5 – General conclusion Finally, Chapter 5 provides conclusions. It outlines the limitations of the study, makes some recommendations for future studies, provides a summary of the contribution, and oﬀers a ﬁnal statement.

24 Chapter 2

Literature Review

This chapter provides a review of previous studies that have been conducted on the empirical estimation of the asset correlation using both the ASRF approach and multifactor models. A review of the relationship between the PD and asset correlation, and the usage of the asset correlation in the calculation of the economic capital

The importance of the asset correlation emanates from the fact that if there is a change in the correlation between assets, statistics dictates that this will results in a transfer of the risk which centred around the mean towards the tail of the loss distribution (Smit & van Vuuren, 2015). This means that an increase in the correlation between assets results in a fattening of the tail of the loss distribution thus causing an increase in the capital amount that a bank needs to cover potential losses (BIS, 2018). The asset correlation also plays a crucial role in credit risk, as it is one of the inputs in the determination of the PD of a credit loss (Smit & van Vuuren, 2015).

Estimation of each unique obligor’s asset correlation becomes quite unfeasible to attempt in the case of large credit portfolios that contain a significantly large number of obligors. Thus, the Basel committee rather proposed that an average correlation be used as a representation of all obligors in the minimum regulatory capital requirements. This correlation was thereafter set at 20% for corporates by the Basel committee in 2001 (BIS, 2001). Asset correlations have been segmented by asset class due to different asset classes exhibiting different levels of dependence on the state of the economy (local or global). Therefore, logically a large corporate loans portfolio would exhibit a higher asset correlation than a retail loans portfolio.

The initially set average asset correlation of 20% was a subject of a lot of scrutiny from several researchers who challenged the assumptions made in the underlying model through empirical testing and eventually pointing out that the ratio of the risk weights was too high. One of the champions of this cause was, among other researchers, Sironi and Zazzara (2001), who conducted a study using corporate default rates released by the Bank of Italy. Their study employs a two-state Merton-type model to empirically estimate the asset correlation for loans granted to family businesses and non-ﬁnancial companies. These companies were segmented according to their geographical area and size of the facility. Sironi and Zazzara (2001), found that Italian banks were penalised due to their high corporate default rates, thus had to maintain a higher capital requirement in the IRB approach than when they

25 were using the standardised approach. Their empirical results revealed a much lower empirical asset correlation than the suggested 20%, which made them argue that they would only adopt the IRB approach if the Basel prescribed asset correlations are lowered in order to be closely aligned to their empirical results.

Based on these results, the Basel committee eventually proposed a replacement of the 20% average asset correlation with an alternative statistical formula which deﬁned the asset correlation as a decreasing function of the PD (BIS, 2001). A the- oretical contribution by Resti (2002) provides support to this proposal by showing that the usage of a lower asset correlation coeﬃcient for borrowers with a higher risk is a more reasonable way of decreasing the cost of credit. The author asserted that this was a step in the right direction, which would decrease the risk that banks would face if they had to adopt an overly conservative approach.

However, empirical research provides results which have been a lot less support- ive. On one hand, Lopez (2002) provides an investigation of empirical relationship between the firms’ PD, asset correlation and firm size. They analyse year end 2000 data for credit portfolios of US, European and Japanese firms using the book value of assets through ASRF approach obtained in the Moody’s KMV correlation model, which is used in the calculation of credit risk regulatory capital. The results of Lopez (2002) reveal that the asset correlation is indeed a decreasing function of the PD, and an increasing function of the firm’s size, which is a confirmation of the logical reasoning that large firms tend to be relatively more sensitive to systematic risk than SME’s. The PD result is also an intuitive result. For example, a higher PD indicates a higher borrower specific risk, based more on firm specific factors and less subject to the existing economic conditions, the common systematic factor, thus a lower asset correlation.

However, on the other hand Dietsch and Petey (2003), empirically estimate the asset correlation for a sample of 280 000 German and 440 000 French SMEs. They use default data and employ the ASRF model over the periods 1995 - 2001 and 1997 - 2001 in France and Germany respectively. Their study finds asset correlation values as low as 1% for both countries. Moreover, contrary to previous research, they find a positive relationship between the asset correlations and the PD. In fact, their results show a u-shaped relationship for both countries. The authors argued, that based on these findings, SMEs should be given a more favourable treatment when compared to larger firms due to their low sensitivity to the systematic risk factor. Secondly, they propose that the riskiness of SMEs should be ignored as they to have a low correlation to the systematic risk factor (global economy) when compared to large corporates, thus suggesting that smaller SMEs should be treated as retail loans in the consideration of credit risk and the regulatory capital calculation.

Henneke and Trück (2006) also explored the relationship between the asset correlation, PD and the firm size in the Basel regulatory capital approach and compares them to results obtained in previous research studies. Their results suggest that the Basel prescribed asset correlations to be a lot more conservative than the empirical asset correlations. They also find that there is a decreasing relationship between the firm size and asset correlation. However, in contrast to Lopez (2002), they find

26 that the asset correlation has a positive relationship with the PD, which is also a contradiction of the assumptions imposed by the IRB risk weight functions. This prompted more studies to be done, especially on the relationship between the asset correlation and the PD.

Düllmann et al. (2007) provides a direct extension of the study done by Lopez (2002). They do this by estimating a time-series of asset correlations, including an investigation into the sector-specific differences observed in asset correlations and analyse the impact of a change in the asset correlations on the value of the economic capital. In detail, they estimate asset correlations based on monthly time series data using Moody’s KMV model for about 2000 European firms for the period 1996 - 2004. They compare the impact of utilising individual firm asset correlations and sector- specific (multi-firm) asset correlations on the economic capital (Value-at-Risk) in a market (one-factor) model and a sector (multi-factor) model. In contrast to previous research, they find a complex relationship between the asset correlation and the PD, which has important implications for credit risk modelling processes in banks, especially when employing multi-factor models. Düllmann et al. (2007) also observed a considerable fluctuation of asset correlations over time, which suggested that further research should be undertaken to review their stability over time. This current study provides a review of the asset correlation over time for both a South African bank specific credit cards dataset and South African commercial loans. Their stability is reviewed and they are also compared to the Basel prescribed asset correlations over time.

In another research study from the software providers Moody’s KMV, Zhang et al. (2008) explores the relationship between the asset correlation, the default correlation and the portfolio credit risk. They make use of both correlation and default data. The employ asset correlation data from the Moody’s KMV global correlation factor model, while the default data is from the US public firm defaults for the period 1981-2006. They firstly examine the default-implied asset correlations and compare them to the asset correlations reported in previous studies. Secondly, they compare their forecasted default correlations with the realised default correlations. They firstly find that the default-implied asset correlation is significantly higher when compared to previous research studies. Secondly, they find that their results on the default-implied asset correlations are aligned to the Basel prescribed asset correlations for large corporates.

Chernih, Henrad & Vanduffel (2006) analysed corporate defaults and their impact on asset correlations. They used asset value data used from the Moody’s KMV Credit Monitor. They used a sample of companies’ asset returns for the period 1997 - 2006. They divided the data into those using default data and those using asset data. They firstly estimate default correlations directly and then use assumptions regarding the joint asset value movements to back out the asset correlations. Sec- ondly, they use the asset returns data to directly estimate asset correlations and convert them to asset correlations. Chernih, Henrad & Vanduffel (2006) results reveal that asset correlations are affected by the assumption that LGD’s are independent of the PD, and that the asset correlations estimated from default data must be increased in order to avoid underestimating the dependence between the PD and

27 LGD. They deduce that default data is the best source of default correlations as there are no intermediate processes that need to be assumed in such a case. They however highlighted the challenges of acquiring such data, which makes such estimations diﬃcult. The scarcity of such data is a result of the unwillingness of banks to avail their internal loss data. In this study, we employ retail default data from a South African bank’s credit card portfolio to determine empirical asset correlations. Our results are compared to the BCBS’ prescribed asset correlations and results in previous literature. The empirical asset correlations are subsequently used to calculate a fair level of the South African bank’s economic capital.

Over the years, banks have been developing complex IRB models that suit their internal loss data needs and have also collected a substantial amount of BCBS’ input IRB data, which includes data on the correlation parameters, (Gore, 2006). Thomas & Wang (2005) conduct research on the Basel IRB formula (see equation 3.28), which is based on the Vasicek formula. From this research, they highlighted the observation that the IRB capital formula is not fully aligned to what may be considered best practice in the banking industry, but however represents a hybrid between a negotiated compromise meant to bring about simplicity to the formula, portfolio invariance and easier acceptance of the Basel prescribed capital levels.

Borio (2011), highlights that one of the concerns regarding the Basel regulatory capital rules relates to the usage of the ASRF model that the Basel II framework is based on. They argue that it is unrealistic to assume that a single measure of systematic risk is applicable for all purposes. The credit crisis in 2008 alerted the banking community and other financial institutions on the deficiencies in the Basel II framework by revealing alternative sources of systemic risk. For example, a common shock that was due to the simultaneous defaults of financial institutions. Another example is the “information spill overs” where the spreading of news of the default of one bank results in an impact on other banks through an increase in the cost of refinancing (Co-Pierre, 2011). It however needs to be noted that the Basel II framework had not been fully implemented when the credit crisis hit the global economy.

Bystrom (2011) proposes a new way of modeling the dynamics of the asset value of firms based on financial securities prices. This new way is different to the commonly used ASRF model. The author uses it to compute the asset correlations in multi- variate credit risk models. They combine credit spreads taken from credit default swap markets with leverage ratios and stock prices to construct a proxy for the asset value. They use this proxy in the calculation of the asset correlations for major Eu- ropean banks to determine the impact caused by the credit crisis. Bystrom (2011) finds that the features of the banks in the study in such as location, size and default risk, had an influence on the impact of the crisis on the empirical asset correlations, and that these banks had higher correlations during the crisis. This current study performs an analysis of the effect of rolling the asset correlations over time, however, it will not be able to provide an assessment of the impact of the credit crisis on the empirical asset correlations due to a data limitation, as the data does not go back far enough, which is one of the limitations of this study.

Bams et al (2012) provide a generalisation of the ASRF model in an eﬀort to address

28 issues such as default clustering, heterogeneity of industries and the uncertainty in the capital requirement’s parameters for a US retail loan dataset. They conduct an investigation into the distribution of common risk factors across the economy and which firm characteristics would be relevant to enable diversification. They do this by inspecting US small businesses for the period 2005 - 2011, and compare their empirical estimates of the capital with the capital adequacy requirements specified by the Basel committee over the 2008 credit crisis. Their empirical results reveal that retail exposures have a better credit risk than the regulator suggests, thus they should have a lowered capital. Their conclusions state that this could be due to the overly simplistic way that is used to estimate asset correlations for retail loans. Bams et al (2012) also found that small businesses have a lower empirical asset correlation than Basel prescribed asset correlations regardless of the small business’ size, industry, or riskiness.

Lee et al. (2011) explores the behaviour of Basel asset correlations with the market returns based on the ASRF approach. They use a sample period of 1988 to 2007 to explore a similar relationship of the asset correlation, PD and the firm size. Similar to Lopez (2002), they find that find that the empirical asset correlations have a positive relationship with the firm size and a negative relationship with the firm’s PD. This result is consistent with the assumptions imposed by the IRB risk weight functions. They also found that industry-specific asset correlations such as companies in media, semiconductor, pharmaceutical, and transportation industries have higher empirical asset correlations than retail and consumer staples industries. They also identified that the asset correlations tend to be asymmetric in nature and have a procyclical impact on the economy. This means they tend to rise during periods of economic downturns (recessions), but tend to decline during non-downturns periods, thus implying that asset correlations might be underestimated during recessions. This is an important result, which can assist banks in their planning for future downturns. The Basel committee stipulates that banks have to increase their capital reserves in cases of enhanced risks. This however weakens the lending activities during recessionary periods, as the increased capital reserves will limit the amount that banks can lend out. Thus, a recession can lower the ability of banks to stimulate of the economy, thus deepening the impact of the recession and weakening the economy even more. Therefore, banks can lower the impact of a recession on their institutions by using their internal loss data to estimate the empirical asset correlation and using those results to determine their own estimates of the economic capital. Knowledge of this will assist them in determining their own measure of how much capital they should reserve for unexpected losses, which is the aim of this study.

There has been extensive research that has been ventured into for examining corporate loans and loans to banks, however research into retail loans such as consumer loans and mortgages is not as extensive, and thus necessitating further review. In the case of retail loans, banks cannot directly observe changes in asset value, stock prices or other indicators that are commonly used on corporate loans to determine ﬁ- nancial condition. Therefore, researchers are forced to come up with new approaches to estimating the asset correlation. In 2003, Duchemin et al. (2003), examined the asset correlation of 50 000 vehicle lease exposures between the periods 1990 – 2001 issued by two of the major European vehicle leasing ﬁnancial institutions operating

29 in very distinct markets. To measure the asset correlation, they employed a unit systematic factor, ordered probit model, where the credit performance of an obligor was limited to a default or non-default status. This particular model made use of a restricted form of the CreditMetrics model (Gordy, 2000), which is a model that is utilised in the Basel capital requirement calculation. They find empirical asset correlations which are significantly lower than Basel prescribed asset correlations. The authors of this paper also suggested that a more accurate estimation of the empirical asset correlation may be established if the volatility of the PD is taken in to account. Secondly, they observe a positive relationship between the PD and the asset correlation, which is a contradiction to the Basel’s proposals. This means that the Basel’s proposal is not fully reflective of the risk embedded in vehicle lease portfolios. The results of this study necessitated a need for more research into retail loans to ascertain whether the Basel committee’s assertions on the shape of IRB risk weights are reflective of both corporate and retail loans.

Siarka (2014) conducted research on the asset correlation of ﬁnancial institutions operating in Poland. The aim of their study was to empirically estimate the asset correlation, and assess their implications on the economic capital. They estimated the empirical asset correlations for 108 000 retail vehicle loans granted in 1997-2002. They mainly use default data, and make the assumption that the PD does not change over time. Their study uses both the Vasicek and Beta distributions and ﬁnds an average asset correlation of 1.8% which is consistent with previous research. Thus, they conclude that the empirical asset correlation between vehicle loan assets will always be lower than the Basel correlations, irrespective of the country. The implication of their results is a lowered economic capital.

Rosch and Scheule (2004) use charge-oﬀ rates from all United States of America (USA) commercial banks to estimate empirical asset correlations for credit cards, residential mortgages and other retail loans. They employ a variant of the two-state one-factor CreditMetrics model, which is similar to the model used in the Basel II framework to calculate risk weights. They estimate asset correlations of 1% and 0.98% for for credit cards and residential mortgages respectively. The asset correlations became as low as 0.7% and 0.3% respectively when macroeconomic risk variables were included in the model. They also provide details as to how the in- clusion of lagged macroeconomic data into the credit risk model may improve loss forecasts and reduce the economic capital by a considerable amount. In this study, we extract empirical asset correlations and use them to calculate the economic capital, however, we will not include macroeconomic data in our model.

Smit & van Vuuren (2009) establish methodologies that can be used by banks to estimate empirical asset correlations for both market and credit risk from their own internal loss data. With the knowledge of the empirical asset correlation, the authors determine whether the BCBS was too lenient or too conservative in their setting of asset correlations by calculating the economic capital using the empirical asset correlations. They employ a Beta distribution and use loan loss data from 100 of the largest US banks. The data spans around 24 years for the period 1985-2009 and has been extracted from the US Federal Reserve Bank. They ﬁnd that the empirical asset correlations are much lower than the Basel prescribed asset correlations. Smit

30 & van Vuuren (2009) results show that the Basel prescribed asset correlations were about 3.2 times higher than the empirical asset correlations. The economic capital results also reveal that the Basel committee was too conservative in their initial setting of the economic capital.

In 2010, Botha and van Vuuren (2010) also extracted empirical asset correlations for 100 of the largest United States of America (USA) banks and compared them to the BCBS-prescribed asset correlations. They however employed both the Vasicek and beta distribution and introduced techniques to reverse-engineer asset correlations of gross loan losses from US based quarterly reports for the periods 1985 - 2009. Their findings show that the empirical asset correlations calculated using gross loan loss data are much lower than the Basel prescribed asset correlations. This implies that the asset correlations provided by the BCBS contain a certain level of conservatism. They also explored ways in which empirical asset correlations change over time. This study is beneficial to banks who are interested in calculating empirical asset correlations using their own internal data and subsequently using them to determine their own fair level of economic capital. Their research essentially introduced simple techniques that can be used to empirically estimate the asset correlations using only gross loan loss data. Studies such as this one are of great benefit to banks that use the advanced IRB approach and want to determine their own internal measure of the asset correlation for the economic capital calculation.

Further investigation into the fairness of empirical asset correlations was done by Stoffberg and van Vuuren (2015), whose study updated the USA correlation data used in Botha and van Vuuren (2010). These authors compared these empirical asset correlations to those experienced using South African (SA) loan loss dataset for the period 1985 - 2014. However, they only employed the Vasicek distribution and found that both a percentile approach and mode approach are needed to fully evaluate the empirical data and BCBS’ intended conservatism, as both those approaches closely fit the data when looking at their density and cumulative functions. They found similar results to Botha and van Vuuren (2010), with the main exception being the empirical asset correlation of the financing of agricultural production, which was higher than the Basel specified asset correlation for the 99.9th percentile approach, which would require a higher capital charge when compared to the capital charge produced when using the Basel specified asset correlation, which is not sufficient enough to cover all capital requirements. The other exception, was the "Qualifying revolving" asset class, mainly credit cards, which also had a higher empirical asset correlation, thus requiring a higher capital charge than the charge provided using the Basel specified asset correlations. The result on the Qualifying revolving asset class, credit cards in particular differs from previous researchers that found empirical asset correlations which were lower than 1%. Stoffberg and van Vuuren (2015) also found that the BCBS had introduced a higher level of conservatism for South Africa than for the US. Their study also explored in detail the effect of macroeconomic events on the empirical asset correlations over a rolling period. This current study seeks to update the previous South African correlation data used in Stoffberg and van Vuuren (2015) as well as calculate empirical asset correlations embedded in a bank-specific credit card dataset. It however employs both the Beta and Vasicek distribution including the usage of the percentile and mode approach. These empir-

31 ical asset correlations are subsequently used to calculate the economic capital. The macroeconomic eﬀects on the asset correlation will also be explored by rolling the empirical asset correlations over time.

The Basel prescribed asset correlations are applied in every bank that follows the IRB approach regardless of that country’s development level, which implies a burden on developing countries like South Africa in that they are measured against developed countries in their eﬀort to ensure that they have suﬃcient capital to carry them through various economic conditions. This provides a research opportunity to investigate whether applying these asset correlations rules universally achieves the necessary conservatism that the Basel committee intended.

Hansen et al (2008) also conducted research on the asset correlation of retail loans, as well as credit cards. They use a time-series of charge-rates which are published by the US Federal bank for all banks’ exposures, aggregated loss rates from collater- alised assets transactions contained within the Fitch-rated USA structured ﬁnancial transactions, and loss rates for all the UK banks as published by the Bank of Eng- land (BoE) for the period 1985 - 2007. They employ the Beta distribution and ﬁnd that the empirical asset correlations of retail loans, as well as those of credit cards come to around 1.3%, which is consistent with previous studies.

Bellotti and Crook (2012) conduct a study into the estimation of the empirical asset correlation of two UK datasets using a Merton one-factor model. The first dataset was a sample of credit card accounts from a UK financial institution. They used a combination of default and write-off data spanning from the 1990’s to the mid-2000’s. The second dataset was quarterly time-series data for all UK credit card issuers from 1990 to 2007. This dataset contains data that shows the number of accounts that were overdue on their accounts over various periods, and accounts that were written off per quarter. They find that the BCBS’ prescribed asset correlations are much higher than the empirical asset correlations, which were below 1%.

Since the advent of the ﬁrst Basel accord in 1988, and other amendments that were introduced in 1992 and 2008, banks have been developing complex IRB models which are meant to suit their own internal loss and performance data. A few research papers that explore the modeling of credit risk have been published. In the case of corporate loans, Fatemi and Fooladi (2006) found that one of the most important purposes that is served by the credit risk models they used was the identiﬁcation of counterparty default risk. In the case of retail loan portfolio risks models, there has only been a few research papers published. This is because account level data of EAD, LGD and PD’s collected by banks is often unavailable, especially for research purposes.

This study explores the extraction of empirical asset correlations for a South African bank’s credit card portfolio using both default and write-oﬀ data, which has not been explored in any study. It also seeks to update the previous South African correlation data used in Stoﬀberg & van Vuuren (2015) and estimate the empirical asset correlations embedded in this data. It uses the Vasicek & beta distributions and empirical asset correlation extraction methodologies introduced in Botha & van Vu-

32 uren (2010). These empirical asset correlations will be compared with the Basel prescribed asset correlations and previous studies. They will thereafter be used to calculate the economic capital.

33 Chapter 3

Methodology

This chapter provides an overview of the Mathematics behind the ASRF model. A review of the Beta and Vasicek distributions is also be provided, including how the diﬀerent calculation approaches can be used in the extraction of the empirical asset correlations. This data is also used in the Basel capital calculation methods to calculate the economic capital.

3.1 The Methodology and Parameter estimations

Banks that employ the standardised approach to calculate the capital required use a simple linear equation. The equation is expressed as:

Capital Requirement(K) = EAD × Risk W eight × 8% (3.1)

Where EAD represents the Exposure at Default. This is in contrast to the more complicated advanced IRB capital calculation, which expresses the risk weight as a factor of the loss given default (LGD), probability of default (PD), maturity (M), and asset correlation (ρ). To determine the required capital, the risk weight is multiplied by the EAD. This can be seen in equation 3.2 below:

Capital Requirement(K) = EAD × (f(LGD, P D, ρ, M) ×8% (3.2) | {z } Risk Weight

3.1.1 The ASRF approach: The mathematics behind the model The equation used for the Basel capital calculation for retail portfolios is given by: √ N −1(PD) + ρ · N −1(PD) Capital Requirement(K) = EAD · LGD · N √ − PD 1 − ρ (3.3)

Where N is the standard normal distribution that is applied to a conservative and

34 threshold value of a systematic factor, N −1 is the inverse of the standard normal distribution. No explicit maturity adjustment is required for a retail portfolio, thus it was not taken into account in this equation.

The prescribed Basel II asset correlations shown in Table 1.2 are compared with the empirical asset correlations derived from both a bank-speciﬁc credit card portfolio and a South African commercial loans dataset over a static period of time, as well as a rolling ﬁve-year period. The empirical asset correlations are compared to the Basel prescribed asset correlations to ascertain whether the Basel committee was too lenient, or too conservative in their initial assumptions. These results are thereafter used to calculate the economic capital, which will be compared to the economic capital calculated using the Basel prescribed asset correlations.

The mathematics behind the ASRF approach has already been introduced, thus the next step is outlining the diﬀerent distributions and approaches used to determine the empirical asset correlations. These are explored in the next section.

3.1.2 Distributions This section outlines suitable distributions that can be used to determine the empirical asset correlations from loan loss data. From the distribution, the average loss and the 99.9th percentile loss can be used (and hence we can use the UL) to determine the empirical correlations.

Tasche (2008) states that the Vasicek, Kumaramswamy and beta distributions do not have considerable diﬀerences when ﬁtting data that contains the same mean and standard deviation. He alluded to the fact that the Kumaramswamy and beta methods are much simpler than the Vasicek method. The only issue is that the Ku- maramswamy method has some implementation challenges as a result of moment matching for this distribution requiring complex solving of a two dimensional op- timisation problem. However, moment matching is straight- forward in the case of the beta distribution. The Basel formulation also employs the beta distribution in the calculation of IRB securitisation capital (Duponcheele, Perraudin & Totouom- Tangho, 2013).

Botha and Van Vuuren (2010) employed both the Beta and Vasicek distributions in their estimation of the empirical asset correlation and found that the Vasicek provided a better ﬁt to their data. Based on these two contrasting views on the distributions, this study will explore both the beta and Vasicek distributions for the calculation of the empirical asset correlations. Both these distributions have never been tested on a South African bank-speciﬁc credit portfolio. We will however not explore the Kumaramswamy distribution due to its implementation challenges.

3.1.2.1 The Beta Distribution The beta distribution has two main parameters, which are α and β. These parameters can easily be measured using the mean (µ), and standard deviation (σ) of the

35 empirical loan losses.

These two quantities, α and β are described using the following equations (Wol- fram Research, 2009):

µ · (1 − µ) α = µ · − 1 (3.4) σ2

µ · (1 − µ) β = (1 − µ) · − 1 (3.5) σ2 where µ is the mean of the gross loan losses and σ is the standard deviation of gross loan losses.

After calculating α and β (in equations 3.4 and 3.5 ) for the period under scrutiny. The beta distribution may be deﬁned by the the following probability density function: 1 f(x : α, β) = · xα−1 · (1 − x)β−1, (3.6) B(α, β)

Γ(α)·Γ(β) where B(α, β) is the beta function given by Γ(α+β) and where Γ(...) is the gamma function.

The cumulative beta distribution is given by:

R x(1 − t)β−1 · tα−1dt P (x) = 0 (3.7) B(α, β)

Γ(α + β) Z x P (x) = · (1 − t)β−1tα−1dt 1 ≥ x ≥ 0, α, β > 0. (3.8) Γ(α) · Γ(β) 0 where x is the distribution variable, Γ is a standard Gamma function and B(α, β) is the incomplete beta function which is a generalisation of the beta function. it is given by: Z x B(x, α, β) = (1 − t)β−1tα−1dt α, β > 0. (3.9) 0

The total loss (EL + UL) is defined as the value of the distribution variable x when P(x) = 99.9%, which was chosen to abide by the Basel’s definition of the value of the systemic risk factors. The UL is therefore defined in the following way:

Unexpected Loss (UL) = T otal Loss − Expected Loss (EL) (3.10)

The concept of the UL has already been explained in chapter 1 and illustrated in Figure 1.1.

36 3.1.2.1.1 Empirical extraction of the asset correlation based on the beta distribution

The key step introduced in this section is how to use the Beta Distribution of loan loss data to extract the empirical asset correlations.

We derive the empirical asset correlation by determining the correlation value that will be equal to the Basel Total Loss (UL + EL) at a conﬁdence level of 99.9%. The procedure of extracting empirical asset correlations using the Beta approach has originally been outlined in Botha and Van Vuuren (2010) and proceeds as follows:

1. Source gross loan loss data, which can be arranged as a time series.

2. Calculate the mean(µ), and standard deviation(σ), of this loan loss data. These values can easily be calculated by taking the simple average(µ) of the gross loss, and the standard deviation(σ) of the gross loss, over the period under observation.

3. Calculate α and β using equations 3.4 and 3.5.

4. Determine the value of the distribution variable x when P(x) = 99.9% using equation 3.8. This represents the total gross loss at the 99.9th percentile 99.9% (Ltotal ) of the ﬁtted beta distribution.

99.9% 5. Substitute the Ltotal value obtained in step 4 into the BCBS equation for the total gross loss value (which is measured at a conﬁdence interval of 99.9%), i.e

√ N −1(PD) + ρ · N −1(pd) UL99.9% = N · √ − EL (3.11) total 1 − ρ

√ N −1(PD) + ρ · N −1(PD) EL + UL99.9% = N · √ (3.12) total 1 − ρ

99.9% 99.9% EL + ULtotal = T otal gross loss = Ltotal (3.13)

√ N −1(PD) + ρ · N −1(PD) N −1(L99.9%) = √ (3.14) total 1 − ρ

−1 99.9% p −1 √ −1 N (Ltotal ) · 1 − ρ = N (PD) + ρ · N (PD) (3.15)

Setting: −1 99.9% ω = N (Ltotal ) (3.16) π = N −1(PD) (3.17)

37 ψ = N −1(0.999) (3.18)

Substitute into equation 3.15: √ ω ·p1 − ρ = π + ρ · ψ (3.19)

Squaring both sides of equation gives √ ω2 · (1 − ρ) = π2 + 2πψ · ρ + ψ2 · ρ (3.20)

√ 0 = (ω2 + ψ2) · ρ + 2πψ · ρ + (ω2 + π2) (3.21)

√ This is a quadratic formula in the asset correlation ρ with solutions: √ √ −b ± b2 − 4ac ρ = (3.22) 2a where

a = (ω2 + ψ2) (3.23) b = 2πψ (3.24) c = ω2 + π2 (3.25) so

√ −(2πψ) ±p(2πψ)2 − 4(ω2 + ψ2)(ω2 + π2) ρ = (3.26) 2(ω2 + ψ2) or

−(2πψ) ±p(2πψ)2 − 4(ω2 + ψ2)(ω2 + π2)2 ρ = (3.27) 2(ω2 + ψ2)

The asset correlation(ρ) is the only unknown in this equation, since the compo- nents π, ψ and ω are known.

The above calculation (equation 3.27) results in two values for the asset correlation (ρ). However, only one of the empirical asset correlations is correct. Botha and Van Vuuren (2010) calculated both ρ’s to establish which one provided the most economically logical UL. They showed that the smaller of the two ρ’s should be used, as the other asset correlation value (ρ) resulted in an unrealistically high capital charges, which does not make any sense economically (Guttler and Liedtke, 2007). Stoﬀberg and Van Vuuren (2015) also found similar results, thus only one ρ will be presented.

38 3.1.2.2 Vasicek distribution Botha and van Vuuren (2010) reverse-engineered the Vasicek distribution to calculate the retail asset correlation using the South African and USA empirical loss data. A Merton-type model was derived by Vasicek (1987, 1991, 2002) to extract an expression that can be used for the distribution of credit losses.

Vasicek (1987) makes the assertion that the cumulative probability of the portfolio loss L, given the state of the economy will be less than some arbitrary variable, x, given by:

√ 1 − ρ.N −1(x) − N −1(PD) P [L ≤ x] = N · √ (3.28) ρ where ρ represents the asset correlation, N[...] represents the standard normal distribution function; N −1 represents the standard inverse normal distribution; while PD is the average of the portfolios probability of default. This cumulative distribution provides a description of credit portfolio losses and is mainly dependent on (ρ and PD). It is valid over the interval 0 ≤ x ≤ 1 and is given by: √ 1 − ρ · N −1(x) − N −1(PD) F (x; PD; ρ) = N √ (3.29) ρ with ρ > 0 and 0 < ρ < 1. As the asset correlation ρ → 0, the distribution converges to a normal distribution, N(0, 1), with probability functions PD and 1 − PD. From the above function (equation 3.29), we can infer that F (x; PD; ρ) = 1 − F (1 − x; 1 − PD; ρ) and that as PD → 1 or PD → 0, then the loss distribution converges to L = 1 or L = 0 respectively.

The loss distribution presented in equation 3.28 has been illustrated in Figure 1.1. This is a highly skewed and leptokurtic loan loss distribution, which is described by the following density:

√ r1 − ρ 1 1 1 − ρ · N −1(x) − N −1(PD) f(x; PD; ρ) = · exp (N −1(x))2 − √ ρ 2 2 ρ (3.30) and has a single mode located at:

√ 1 − ρ L = N · · N −1(PD) (3.31) mode 1 − 2ρ

The α-percentile value of L, which is the inverse of the loss distribution is given by:

Lα = (x; PD; ρ) = 1 − F (1 − x; 1 − PD; ρ) (3.32)

39 This is a typical, skewed loan loss distribution with all the relevant features provided in Figure 1.1.

Botha and van Vuuren (2010) employed two distinct calculation approaches to the Vasicek distribution assumption, namely the mode and percentile approach. The mode approach is mainly calculated using the portfolio PD and mode of the portfolio losses to determine the empirical asset correlation, while the percentile approach uses the UL (UL99.9%) at the 99th percentile and portfolio PD to determine the empirical asset correlation. This study follows a similar path and employs both the the mode and percentile approach to the Vasicek distribution assumption. We expand on both approaches in the sections that follow.

3.1.2.2.1 Empirical extraction of the asset correlation based on the Mode approach of the Vasicek distribution

The key step introduced in this section is how to use the mode approach to extract the empirical asset correlations from loan loss data.

The procedure of extracting empirical asset correlations using the mode approach has originally been outlined in Botha and Van Vuuren (2010) and proceeds as follows:

1. Source gross loan loss data, which can be arranged as a time series.

2. Calculate the mean loss (PD in equations 3.28 and 3.29) and the mode (Lmode in equation 3.31). These values can be calculated by taking the simple average of the gross losses (PD), and by determining the most frequent loss experienced (Lmode), over the speciﬁed time period. 3. Using equation 3.31, the asset correlation may now be determined as follows:

√ 1 − ρ L = N · N −1(PD) (3.33) mode 1 − 2ρ

√ 1 − ρ N −1(L ) = · N −1(PD) (3.34) mode 1 − 2ρ

√ N −1(L ) 1 − ρ mode = (3.35) N −1(PD) 1 − 2ρ

√ N −1(L )2 1 − ρ2 mode = (3.36) N −1(PD) 1 − 2ρ

N −1(L )2 1 − ρ mode = (3.37) N −1(PD) (1 − 2ρ)2

40 let

N −1(L )2 ξ = mode (3.38) N −1(PD) substituting into equation 3.37 gives: 1 − ρ ξ = , (3.39) (1 − 2ρ)2

ξ · (1 − 2ρ)2 = 1 − ρ (3.40)

ξ · (1 − 4ρ + 4ρ2) = 1 − ρ ξ − 4ξρ + 4ξ (3.41)

4ξρ2 + (1 − 4ξ)ρ + (ξ − 1) = 0 (3.42)

This is a quadratic formula in ρ (the asset correlation) which has solutions:

√ −b ± b2 − 4ac ρ = (3.43) 2a where a = 4ξ (3.44) b = (1 − 4ξ) (3.45) c = (ξ − 1) (3.46) so

(4ξ − 1) ±p(1 − 4ξ)2 − 4(4ξ)(ξ − 1) ρ = (3.47) 2(4ξ)

(4ξ − 1) ±p(1 − 8ξ + 16ξ2) − 16ξ2 + 16ξ ρ = (3.48) (8ξ)

(4ξ − 1) ±p(8ξ + 1) ρ = (3.49) (8ξ)

41 3.1.2.2.2 Empirical extraction of the asset correlation based on the Per- centile approach of the Vasicek distribution

The previous section introduced how to use the mode approach to extract the empirical asset correlations. This section shows how the percentile approach may used in the empirical extraction of the asset correlation from loan loss data.

To empirically measure the total loss at a conﬁdence level of 99.9%, we may combine equations 3.29 and 3.32, where a conﬁdence interval of 99.9% implies that α = 0.1%):

p1 − (1 − ρ) · N −1(α) − N −1(1 − PD) F (α; 1 − PD; 1 − ρ) = N · √ , (3.50) 1 − ρ and

√ N −1(PD) + ρ · N −1(α) Gross total loss = N √ . (3.51) 1 − ρ

As shown in Figure 1.1 and equation 3.10, the gross total loss L at a speciﬁed con- ﬁdence level is simply the sum of the expected and unexpected gross loss (EL + UL99.9%).

Thus, the UL at a 99.9th percentile is: √ N −1(PD) + ρ · N −1(99.9%) UL99.9% = N √ − EL. (3.52) 1 − ρ

But, the expected portfolio loss EL = PD, the portfolio’s probability of default, as gross loss data is used (and the LGD =1 since we assume no recoveries). Thus we can express the UL as:

√ N −1(PD) + ρ · N −1(99.9%) UL99.9% = N √ − PD (3.53) 1 − ρ

Equation 3.53 is dependent on three variables, which are the unexpected gross loss (UL99.9%), the average portfolio loss (PD) and the asset correlation (ρ). If we can empirically determine the UL99.9% and the PD from loan loss data, then we can manipulate equation 3.53 to get: √ N −1(PD) + ρ · N −1(99.9%) PD + UL99.9% = N √ (3.54) 1 − ρ

√ N −1(PD) + ρ · N −1(99.9%) N −1(PD + UL99.9%) = √ (3.55) 1 − ρ

√ N −1(PD + UL99.9%) ·p1 − ρ = N −1(PD) + ρ · N −1(99.9) (3.56)

42 with ρ the only unknown.

Letting ω = N −1(PD + UL99.9%) (3.57) π = N −1(PD) (3.58) ψ = N −1(0.999) (3.59) we get √ ω ·p1 − ρ = π + ρ · ψ (3.60) squaring both sides gives √ ω2 · (1 − ρ) = π2 + 2π · ψ · ρ + ψ2 · ρ (3.61)

√ 0 = (ω2 + ψ2) · ρ + 2π · ψ · ρ + (ω2 + π2) (3.62)

√ which is a quadratic formula in ρ which has solutions: √ √ −b ± b2 − 4ac ρ = (3.63) 2a where

a = (ω2 + ψ2) (3.64) b = 2πψ (3.65) c = ω2 + π2 (3.66) so

√ −(2πψ) ±p(2πψ)2 − 4(ω2 + ψ2)(ω2 + π2) ρ = (3.67) 2(ω2 + ψ2)

Which is an equation that can easily be solved since we know the solutions to the variables as π, ψ and ω. N −1(PD + UL99.9%) is simply an inverse of the normal distribution at the 99.9th percentile of total losses.

The amount of economic capital that must be reserved by banks is calculated using the empirical asset correlation extracted from loan loss data. To demonstrate this calculation methodology, this study shows a comparison between the economic capital calculated using the empirically extracted asset correlation and the asset correlation provided by the BCBS.

43 Chapter 4

Empirical analysis

This chapter presents an overview the estimation results and empirical analysis. The first section provides a discussion of the data used in the study. The second section provides the empirical results of the asset correlations and their corresponding economic capital charge using the different calculation approaches. The final section provides empirical asset correlation results over a five-year rolling period.

4.1 Data used in the study

Three datasets are used in the empirical estimation of the asset correlation. The ﬁrst two datasets relate to a sample of credit card accounts from a South African bank. The third dataset contains data for all loans issued in South Africa1. Due to the proprietary nature of bank-speciﬁc data within the South African banking sphere, it is not possible to reveal the name of the bank which gave us access to their data. For the purposes of this study, we will call it "Bank X".

The South African commercial loans loss data spans some nine years (i.e. June 2008 to January 2017), which covers 103 consecutive months. This data was collected from the SARB (Venter, 2017) by dividing the monthly impaired advances by the monthly total exposure on the balance sheet.

Two datasets were used for the Bank X’s credit card loan losses. The ﬁrst dataset spans some nine years of monthly data (i.e. February 2006 to September 2015), which covers 117 consecutive months. This data was calculated by taking yearly cohorts of actual defaulted customers as a percentage of open, performing customers at the beginning of each yearly cohort.

The second dataset spans ten years (i.e. January 2007 to May 2017), which covers 125 consecutive months. This data was calculated by taking the actual monthly write-oﬀ amount as a percentage of the monthly total exposure on the balance sheet.

The usage of both the write-oﬀ and default data as loan loss estimates has never been explored in previous studies, but previous authors have chosen to use one or

1This is a consolidated view of all South African loans advanced i.e Residential Mortgage, Retail Revolving, SME Retail, Retail Other, Corporate, SME corporate, Banks and securities ﬁrms, Securitisation.

44 the either as loss estimates. This study employs both these datasets to explore the impact of using an accounting loss estimate (write-off data) and regulatory loss estimate (default data) on the empirical asset correlation. The default data has however also been chosen as a loss estimate to match banking regulatory standards, as the long run actual default rates are used in the construction of through-the-cycle PD models. Default is defined as a borrower missing three consecutive monthly payments, or when a borrower’s account is in forbearance, which relates to a customer making payment arrangements to restructure their payments, consolidation of debts and other similar payment relief arrangements. The first month of default is taken as the month at which either event occurs. Write-offs in our case represent total loss as the bank only writes off an account when they have exhausted all prospects of collecting on the outstanding debt.

The South African commercial loans and Bank X’s loss data each represent a different Basel II asset class category for retail exposures, namely, qualifying revolving and the corporate, bank and sovereign. These asset types have been summarised in Table 4.1.

Table 4.1: Asset classes for our data and their Basel II asset classiﬁcation

Asset Class Basel II Classiﬁcation Bank X’s Credit card loans Qualifying Revolving exposures South African commercial loans The Corporate, Bank and Sovereign

The losses used in this study are at a gross loss level, thus need to be converted to net losses for the economic capital calculation. These gross losses were converted to net losses using:

Net Losses = Gross Losses · LGD (4.1)

The LGDs used are obtained from the results of the BCBS’ 5th impact study (BCBS, 2006), from the "G10 group 1: Including US" group, for Bank X’s credit cards loss data, while the "G10 group 1: Excluding US" was used for the South African commercial loans loss data. These LGD averages have been presented in Table 4.2. The LGD used for the Bank X’s credit cards loss data is 71.6% (QRE), while the LGD used for the South African commercial loans loss data is 39.8% (Corp.).

This study employs both the Vasicek and beta distributions for the extraction of the empirical asset correlations. The sequence of the investigation proceeds as follows.

First, the eﬀect of diﬀerent approaches (beta, mode and percentile approaches) of calculating the empirical asset correlations is explored for the South African commercial loans and Bank X’s credit card loan loss datasets.

Secondly, the empirical asset correlations from both Bank X’s credit cards and the South African commercial loans data – deducted using the Vasicek and beta distributions – are compared with the BCBS-speciﬁed asset correlations for the observation period.

45 Table 4.2: LGD data for the banks that participated in the BCBS’ 5th Quantitative Impact study

Retail IRB AIRB Whole-sale RM Q-R-E Others SME. Corp. Bank Sov SME Corp G-10 Group-1 (excl US) 39 40 33 35.0 G-10 Group-1 (incl US) 20 71.6 48.0 46.2 G-10 Group-2 26 57 43 31.1 CEBS Group-1 16 55 47.9 38.8 38.1 37.7 27.7 35.1 CEBS Group-2 21 51 42 31 35 39 38 26 Other non-G-10 Group-1 11.0 67.2 48.3 28.4 Other non-G-10 Group-2 40.4 55.7 45.1 49.6

Thirdly, the economic capital calculated using the empirical asset correlations from the diﬀerent calculation approaches is compared to the economic capital calculated using the Basel prescribed asset correlation for both the South African commercial loans loss data and Bank X’s credit card loan loss datasets.

Finally, the South African and Bank X’s credit card empirical asset correlations are explored over a rolling ﬁve-year period. These rolling empirical asset correlations are also compared to the BCBS-speciﬁed asset correlations over the period.

4.2 Empirical results

The BCBS-speciﬁed asset correlations have been presented in Table 1.2. The "Cor- porate, bank and sovereign" calculation is used for the South African commercial loans loss data, as conﬁrmed in Hill (2012). The "Retail revolving" asset correlation value is used as the Bank X’s credit card asset correlation.

Three datasets have been used in this study with one representing the South African commercial loans and the other two representing Bank X’s credit card loan loss data. The credit card data has been separated by the way it was calculated. The ﬁrst dataset was calculated using the actual default data as the loan loss data estimate and is denoted with a PD subscript. The second dataset was determined using the write-oﬀ percentage as a loan loss estimate and is denoted by an EL subscript.

Two distributions have been selected for this study, the Beta and Vasicek distributions, however two diﬀerent calculation approaches (mode and percentile) are used for the Vasicek distribution assumption. For Bank X’s data, V_mode_PD, V_percentile_PD and Beta_PD represent results using the Actual Default data as the loss data estimate, while the V_mode_EL, V_percentile_EL and Beta_EL represent results using the Write-Oﬀ data as the loss data estimate.

The distributions were statistically tested (see appendix I) to determine their goodness of ﬁt. The beta distribution was found to be ranked as one of the best in

46 all three datasets using the Kolmogorov-Smirnov and Anderson-Darling tests. A summary of the results is provided in appendix 1. Based on these results, the beta distribution (Beta_PD, Beta_EL, Beta) was on average ranked between 1-3 using the Kolmogorov-Smirnov test. Even though that was the case, all the other distributions had P-values below the critical value (see Appendix I) for all distributions at signiﬁcant levels 0.01 - 0.2 except for the V_Percentile_EL.

The CDF’s were also plotted for the diﬀerent datasets. For a discrete random variable, the CDF at a certain value x gives us the probability that the random variable will have at that value x. For a continuous random variable, the PDF describes the probability of ﬁnding a random variable in the area under the curve (Smit & van Vuuren, 2009). The CDF evaluated at a value x, describes the probability that a variable takes a value within a range which is less than or equal to x (Brown, 2005). These are shown in Figures 4.1 to 4.3.

Figures 4.1(a), 4.2(a) and 4.3(a) illustrate the cumulative density function for the diﬀerent approaches, plotted against their respective empirical loss data. Figures 4.1(b), 4.2(b) and 4.3(b) shows the density function for the diﬀerent approaches plotted against their respective empirical loss data.

Figure 4.1: (a) The cumulative, and (b) the density function for Credit Card Actual Default Rate losses from February 2006 to September 2015.

Visually seen in Figure 4.1(b), both the Vasicek (V_Mode_PD and V_Percentile_PD)

47 and beta (Beta_PD) distributions closely fits Bank X’s credit card default empirical loss data, except at the point of highest frequency of losses in the case of the Beta_PD. The cumulative densities in Figure 4.1(a) are also closely aligned for all the distributions. This was confirmed using the Kolmogorov-Smirnov (K-S) test for statistical goodness of fit (see appendix 1).

Figure 4.2: (a) The cumulative, and (b) the density function for Credit Card Write- Oﬀ data from January 2007 to May 2017.

Visually seen in Figure 4.2(b), both the V_Mode_EL and Beta_EL distributions closely fits Bank X’s Credit Card Write-Off empirical data, while the Beta_EL shifts slightly to the left at the point of the highest frequency of losses. The V_Percentile_EL provides a sub-optimal fit to the empirical loss data. The cumulative densities in Figure 4.2(a) are also closely aligned for all the distributions except for the V_Percentile_EL which seems to be an outlier. This is also supported by "V_Percentile_EL - Goodness of Fit - Summary" table in the appendix I, which validates statistically using the Kolmogorov-Smirnov Test that V_Percentile_EL is not a good this fit to the empirical loss data.

Visually seen in Figure 4.3(b), all three of these distributions provides a sub-optimal ﬁt to the South African commercial loans empirical loss data. The Beta and Vasicek distributions are both unimodal, thus it makes sense why they would provide a sub- optimal ﬁt to an (apparent) bimodal distribution of data. The V_Mode and beta

48 Figure 4.3: (a) The cumulative, and (b) the density function for the South African Commercial Loans from June 2008 - December 2016. distributions provide a good fit to the main body of the loss distribution but a poor fit to the tail region, while the V_Percentile provides a good fit to most of the loss data at low frequencies, while it aligns more to the left at the point of the highest frequency of losses. The cumulative densities illustrated in Figure 4.3(a) are also not as closely aligned, with the V_Percentile more closely aligned to the empirical loss data than the other cumulative densities. Even though these distributions do not visually portray a perfectly good fit to the data, the Kolmogorov-Smirnov Test results reveal that they are a good fit to the data (see appendix 1).

The BCBS’ specified asset correlations were compared with the empirical asset correlations calculated using all the gross loan loss data (from equations 3.27, 3.49 and 3.67). Table 4.3 and Figures 4.4 and 4.5 present a comparison of the empirical asset correlations against the Basel II-specified asset correlations for Bank X’s credit card portfolio using all three approaches, the percentile approach (V_percentile in equation 3.67), and the Mode approach (V_mode in equation 3.49) and the beta (Beta in equation 3.27) calculation. These calculation approaches were applied to different datasets, the Actual Default data (V_percentile_PD & V_mode_PD & Beta_PD) and the Write-Off (V_percentile_EL & V_mode_EL & Beta_EL) data.

These empirical results indicate that the BCBS-speciﬁed asset correlation for credit card data is on average less conservative than the empirically calculated asset cor-

49 Table 4.3: Credit Card Empirical correlations compared to Basel II correlations

Credit Card Loans Basel II Correlation Empirical Correlation Ratio a:b V_Mode_PD 4% 8.47% 0.47 V_Perc_PD 4% 5.83% 0.69 Beta_PD 4% 2.99% 1.34 V_Mode_EL 4% 5.53% 0.72 V_Perc_EL 4% 17.44% 0.23 Beta_EL 4% 8.47% 0.47

Figure 4.4: A comparison of the empirical asset correlations compared to Basel speciﬁed asset correlations using the actual default data

Figure 4.5: A comparison of the empirical asset correlations compared to Basel speciﬁed asset correlations using the Write-Oﬀ data

50 relation as shown in Table 4.3. The only exception is the Beta_PD which results in an empirical asset correlation of 2.99%. This outlier result by the Beta_PD is the expected result that demonstrates the BCBS’ intended conservatism and corre- sponds with previous studies of the Credit Card empirical asset correlation by Rosch & Scheule (2004) and Bellotti & Crook (2012). The diﬀerence though is that they found empirical asset correlation values below 1% for Credit Cards in the USA and the UK respectively.

In the case of the empirical asset correlations that are higher than the BCBS- specified asset correlations, we may take comfort in the fact that the data used by these researchers does not fully take the 2008 global credit crunch into account, as Bellotti and Crook (2012) used the time period 1990 Q2 - 2007 Q4, while Rosch and Scheule (2004) used the time period 1985 to 2001 for their research. The credit crunch dramatically increased credit losses (Newscientist, 2009), therefore it is assumed that their results would’ve differed had their research included the impact of the global recession on the loss rate. Our results do however correspond to the results obtained in Stoffberg & van Vuuren (2015), which had a credit cards empirical asset correlation as high as 7%. Further research should be established to determine whether credit card portfolios from other South African banks behave in a similar manner. One of the limitations of this study was accessing credit card and retail revolving data for the other South African banks to compare with the results acquired from Bank X’s loan loss data. Table 4.4: Comparison of the South African commercial loans empirical asset correlations and the BCBS-specified asset correlations

Commercial Basel II Correlation Empirical Correlation Ratio a:b South African Loans V_Mode 15% 0.81% 18.77 V_Percentile 15% 6.98% 2.18 Beta 15% 1.01% 15.11

Table 4.4 and Figure 4.6 compares the BCBS-specified correlations with the empirical correlations for the South African commercial loans using all three approaches, the percentile approach (V_percentile in equation 3.67), the Mode approach (V_mode in equation 3.49) and the beta (Beta in equation 3.27) calculation. For all three approaches, it was found that the Basel specified asset correlation is more conservative than the empirically estimated asset correlations by a factor of 2, 15 and 19 for the percentile, beta and mode approaches respectively. This is an expected result, which also agrees with the results found by Stoffberg and Van Vuuren (2015).

This study has thus far shown how to calculate the empirical asset correlation for Bank X and the South African Commercial loans loss data. However, knowledge of only the empirical asset correlation does not give us an indication of how much capital is needed to protect a bank against potential risks. This sparks a need for the calculation of the economic capital using the empirical asset correlations.

51 Figure 4.6: South African commercial loans empirical correlations compared to Basel II correlations

Table 4.5: Comparison of Bank X’s Credit Card Capital Charge (Relative to Basel Capital Charges

Credit Card Loans Capital Charge us- Capital Charge us- Ratio a:b ing Basel II Correla- ing Empirical Corre- tions lations V_Mode_PD 9% 15.84% 0.58 V_Perc_PD 9% 11.96% 0.76 Beta_PD 9% 7.47% 1.22 V_Mode_EL 2% 2.67% 0.76 V_Perc_EL 2% 8.21% 0.25 Beta_EL 2% 3.93% 0.52

Figure 4.7: Comparison of Bank X’s Credit Cards Ratio of the Capital charge (Relative to the ratio of the Basel Capital Charge)

52 Table 4.5 presents a comparison between Bank X’s credit card capital charge relative to the Basel capital charge. Figure 4.7 presents a comparison of Bank X’s credit card capital charge ratio relative to the Basel capital charge.The BCBS speciﬁes that banks that employ the IRB approach should use a downturn LGD in the capital calculation, due to the correlation between the LGD and PD not being accounted for in the capital formula (Miu & Ozdemir, 2006). Therefore, they proposed that a principles-based approach be used. This approach requires banks to identify an appropriate downturn period and consider the adverse dependencies between default rates and recoveries.

The BCBS implicitly states that using a model that possesses a systematic correlation between the LGD and PD using LGD inputs over a long term gives comparative capital when compared to a similar credit risk model that does not possess correlated LGD and PD using downturn LGD inputs. In an eﬀort to compensate for the lack of correlation, mean LGDs need to be raised by about 35% to 41%. Therefore, downturn LGDs were obtained by increasing the LGDs in Table 4.2 by 38%, i.e. the LGDs were multiplied by 1.38. The calculated downturn LGDs were then used as inputs in the capital calculations presented in this section using the BCBS’ IRB approach. For the empirical asset correlation calculations, the unaltered LGDs as presented in table 4.2 were used.

Bank X’s capital charges were calculated using the empirical asset correlations, while the Basel capital charge was calculated using the BCBS-speciﬁed asset correlations (Table 1.2). Bank X’s capital charge is larger than the Basel-speciﬁed capital charge for the most part (see Table 4.5). The only exception is the Bank X’s Beta_PD capital charge, which is an advantageous result for Bank X, and provides the necessary conservatism that was intended by the BCBS.

The case where Bank X’s capital charges are larger than the Basel capital charges indicate that the bank’s liquidity may not be enough to carry them through all their losses. The maximum capital cover by the Basel capital charge compared to Bank X’s capital charge is 0.76 (see Figure 4.7), again excluding the Beta_PD capital charge. This implies that the Basel capital charge is not suﬃcient to ensure enough capital for Bank X’s credit card portfolio, and that the capital levels need to be reviewed and possibly increased.

Table 4.6: Comparison of South African Commercial Loans Ratio of the Capital charge (Relative to the ratio of the Basel Capital Charge)

Commercial Capital Charge us- Capital Charge using Em- Ratio a:b South African ing Basel II Corre- pirical Correlations Loans lations V_Mode 10% 1.18% 8.68 V_Percentile 10% 5.29% 1.94 Beta 10% 1.35% 7.59

53 Figure 4.8: Comparison of South African Commercial Loans Ratio of the Capital charge (Relative to the ratio of the Basel Capital Charge)

A comparison of the South African commercial loans capital charge relative to the Basel Capital charge has been presented in Table 4.6 and Figure 4.8. Unlike Bank X’s case, the Basel asset correlations were found to be very conservative for all three approaches (See Table 4.6). Figure 4.8 shows that the BCBS capital charge was 2 times more conservative using the V_Percentile approach, 7.6 times using the Beta approach and 8.7 times using the V_Mode approach.

The BCBS has consistently stressed that banks should always strive to have suf- ﬁcient capital cover to shield them against insolvency, placing emphasis on the need to always remain conservative (Carver, 2014). However, it is clear from the estimated empirical results that this is not always the case. As shown in Figure 4.7, the ratio of the BCBS-speciﬁed capital charge to the Bank X’s credit card empirical capital charge is lower than 1 for the most part. Using the BCBS’ current formulation, Basel II-compliant banks that analyse their credit risk using the IRB approach may alter a few parameters to increase the capital cushion. The LGD is one of the adjustable parameters, while the asset correlation is the other. It is clear that the manner in which the asset correlation is calculated has an impact on the regulatory capital used to protect banks against insolvency.

Banks with a large capital cushion have high costs, which are attributable to the cost of maintaining capital. Therefore, for a bank to have a competitive advantage there needs to be a balancing act between maintaining an adequate amount of capital reserves and limiting the opportunity costs associated with maintaining these capital reserves. Large Basel II retail banks gain a cost and ultimately a credit pricing advantage when the required capital reserves are lower. This would result in higher ROE. Such an advantage forces smaller community banks to reduce their pricing and also adjust their underwriting guidelines to enable them to compete with such banks. This could have an adverse impact on their proﬁtability and potentially

54 their viability. It is a possibility that the BCBS wished to avoid such possibilities by adjusting the only variable they had at their disposal, which is the asset correlation. This adjustment ultimately resulted in increased capital charges for most Basel- compliant banks, which is perhaps why the BCBS imposed the asset correlation for all participating banks, i.e. as a way to promote fairness and eliminate competitive advantages larger banks had compared to smaller banks. Any bank may use the methodology presented in this study to extract the empirical asset correlations from their internal empirical loss data. These empirical correlations may in-turn be used to calculate the economic capital to be held by the bank.

Exploring Empirical Asset Correlations over A Rolling Period This study has thus far empirically estimated the asset correlation for both the South African commercial loans and Bank X’s credit cards dataset over a constant time period. In this section we empirically extract the asset correlation over a rolling five-year period and compare them with the BCBS specified asset correlation. The same datasets are used for both the South African commercial loans and Bank X’s credit cards, over the same period, using the same extraction methodologies shown above. The difference in this section is that these empirical asset correlations have been empirically estimated over a rolling five year period instead of the total constant time period. These empirical asset correlations have been estimated over a rolling period to illustrate the impact of macroeconomic changes on the empirical asset correlations and to determine whether the BCBS’ specified asset correlations maintain their intended conservatism over different economic conditions.

The empirical asset correlation values are highly sensitive to the parameters estimated from portfolio loss data. Figure 4.9, provides an illustration of how loan losses and their associated implied asset correlations vary over a ﬁve-year rolling period for South African commercial loans. Figures 4.10 and 4.11 illustrate the same information for Bank X’s credit card default and credit card write-oﬀ data respectively.

The South African economy experienced the end of a benign period characterised by inflation rates within the targeted 3-6 percent, economic growth, and a man- ageable budget and current account deficit in mid-2007 (Padayachee, 2012). After this period, much like the rest of the world, South Africa experienced the effects of the global recession which resulted in the country experiencing a breach in the inflation target upper limit, increased interest rates, a drop in the country’s GDP growth rate, and a fall in the mining and manufacturing output. These macroeconomic changes resulted in increase in the default rates and number of delinquencies.

In Figure 4.9, it is clear that the worst losses were experienced around Septem- ber 2009 due to the effects of the global recession experienced in 2008. After this point, the effects of the global recession started to slowly wear-off thus resulting in a gradual decrease in portfolio losses which eventually stabilised around 2014. Due to a lack of data going back five years before the global recession, it was not possible to demonstrate rolling empirical asset correlations before the global recession. The first point represents the empirical asset correlation calculated using five years of data between June 2008 and June 2013. The empirical asset correlations values are

55 Figure 4.9: A comparison of the five-year rolling empirical asset correlations to the Basel specified rolling asset correlations for the South African commercial loans data calculated over a rolling five year period from that point up until January 2017. The results show that the empirical asset correlations were lower than the BCBS’ prescribed asset correlations for the whole period under scrutiny. This means the BCBS’ asset correlation has been extremely conservative throughout this period.

Figure 4.10: A comparison of the ﬁve-year rolling empirical asset correlations to the Basel speciﬁed rolling asset correlations for Bank X’s credit card default data

The same economic events that had an impact on the losses experienced by the South African commercial loan data also had an impact on Bank X’s loss data. Figure 4.10, Bank X’s credit card default data, shows a spike in losses much earlier than the spike seen in the South African commercial loss data (see ﬁgure 4.9). As a credit card is an unsecured product, it makes sense for it to experience an immediate

56 impact of the global recession, as opposed to a more secured product. Conventional wisdom has always been that consumers will likely pay their secured obligations ﬁrst, mortgages in particular, when faced with a ﬁnancial crisis (Bell, 2007).

The BCBS’ prescribed asset correlation is less conservative than the V_Percentile_PD empirical asset correlation from 2011 until they become equal around 2013, and con- tinued to be quite close to each other from there on. It is assumed that the empirical asset correlation is higher due to the effects of the global recession. These changes are not evident in the BCBS’ prescribed asset correlation as they prescribed a constant figure of 4%, which does not adapt to changes in the economic environment. This means the BCBS’ asset correlations do not always achieve their intended conservatism.The BCBS’ prescribed asset correlation is higher than the V_Mode_PD empirical asset correlation for the most part, except for a short period in 2014. The Mode approach of the Vasicek distribution produces very volatile asset correlations, as it is very sensitive to changes in the loss data. A change in the underlying data yields an exaggerated gross loss value than the previous mode. Data analysed over periods of great loss, and then periods of low losses, or vice versa, will result in the mode undergoing a “jump” to a high value and then a low value, (or other- wise) without jumping to intermediate values in-between. This will have the effect of extremely exaggerating changes in the empirical asset correlations estimated using this approach. The mode approach is an approach that takes on the most prevalent loss experienced in the data (Sharma, 2010). In contrast, the percentile approach aims to find the 99.9th percentile of the loan loss data, which produces a smoother result over time. The BCBS’ prescribed asset correlation is more conservative than the Beta_PD for the whole period under scrutiny (see Figure 4.10). The Beta_PD seems to coincidentally take on an average value of the V_Mode_PD and V_Percentile_PD’s empirical asset correlations. This is an advantageous result for banks that would like to adjust their economic capital depending on the prevailing economic conditions. It also achieves the BCBS’ intended level of conservatism over a long period.

Figure 4.11, which presents Bank X’s Credit Card Write-off loss data, shows much lower loss rates compared to both the South African commercial loan and Bank X’s credit card default losses. Bank X’s credit card write-off losses increased gradually during the crisis, with the bank experiencing their largest loss around October 2009, which is similar to the period that the South African commercial loans experienced their largest losses. Even though that is the case, the losses shown in Figure 13 are not as clear, nor as large, as the both the South African commercial loans and Bank X’s credit card default losses. This is due to the way this loss dataset was determined as it takes the monthly write-off data as a percentage of the total loans on the balance sheet.

A write-off usually happens when a customer has not made any payments for at-least six months or when the bank considers a customer’s debt as noncollectable. San- tucci (2016) tracks the disposition of revolving card balances that existed in March 2009, and found that only 27.8 percent of those balances had been written-off, while 72.2 percent were paid down. Thus, we can infer that the small percentage of losses resulting from the graph in figure 13 exhibits similar behaviour.

57 Figure 4.11: A comparison of the 5-year rolling empirical asset correlations to the Basel speciﬁed rolling asset correlations for Bank X’s Credit Card Write-oﬀ Data.

Figure 4.11 illustrates that the BCBS’ prescribed asset correlation is more conservative than the V_Mode_EL empirical asset correlation for the most part, except for a short period in 2014. The mode approach gave very volatile results, with the empirical asset correlations being high as the effects of the recession wore off. It is therefore evident that the credit crisis resulted in an increase in the number of highly correlated assets with regards to the mode approach, and thus this should be taken into consideration in the calculation of the BCBS’ capital calculation. The V_Percentile_EL and Beta_EL had much higher empirical asset correlations than the BCBS’ prescribed asset correlation up until the year 2015, where they then con- tinued on a downwards trend. These high correlations may be due to the effects of the recession wearing off. The V_Percentile_EL has a high correlation amount as expected because it produced a sub-optimal fit to the loss data as seen in Figure 4.2(b).

This chapter showed a trend of losses and their associated asset correlations over time. It demonstrated that the BCBS’ empirical asset correlations do not always achieve their intended level of conservatism, especially in relation to a Bank X’s credit cards population. This result indicates that Bank X must review the level of capital they have reserved for unexpected losses from credit cards and consider if it needs to be adjusted to the resulting economic capital levels. This result reveals that they would not have suﬃcient capital to cover all their unexpected losses in a stressed period.

The methods (Beta, Vasicek (Mode and percentile approach)) that have been used in this study can be used by banks to determine when to raise or lower levels of economic capital appropriately depending on the prevailing economic conditions. This means banks using these methods have the power to determine their own unique em-

58 pirical capital requirements without blindly accepting all of the obscured parameters given in the Basel capital calculations.

59 Chapter 5

Conclusions, Research Limitations and Recommendations for Future Investigations

5.1 Conclusion

This study had two main objectives which will be addressed below. The first objective was to determine the empirical asset correlation values for a South African bank-specific credit cards default and write-off data, as well as for a South African commercial loan loss data and comparing these to the BCBS’ prescribed asset correlations. These computed values are used to as direct inputs in the economic capital calculation and compared to the economic capital calculated using the BCBS’ stated asset correlations. We used both the Beta and Vasicek distribution. To fully evaluate the empirical asset correlation using the Vasicek distribution we employed a mode approach and percentile approach, as both of them provided a good fit when looking at their cumulative and density functions. Our results reveal that the BCBS’ stated asset correlations are much higher than the empirical asset correlations for the South African commercial loans dataset. However, our results on credit cards empirical asset correlations paint a different picture. It was found that the average empirical asset correlation estimated using default data was 5.76% (see table 5.1), while the average empirical asset correlation estimated using write-off data was 7% (see table 5.2).

Table 5.1: The average asset correlation determined using the default data

Approach used Correlation

V_Mode_PD 8.47% V_Perc_PD 6% Beta_PD 3% Default data average 5.76%

60 Table 5.2: The average asset correlation determined using the write-oﬀ data

Approach used Correlation

V_Mode_EL 5.53% Beta_EL 8% Write-oﬀ data average 7.00%

Table 5.1 presents Bank X’s credit cards default data’s empirical asset correlation results obtained using the three estimation approaches Beta, mode and percentile approach of the Vasicek distribution. Thus the average empirical asset correlation determined using the default data is 5.76%. Table 5.2 presents Bank X’s credit cards write-off data’s empirical asset correlation results obtained using only two estimation approaches Beta, mode approach of the Vasicek distribution. The write-off data’s empirical results estimated using the percentile approach were ignored from the final average as they provided a sub-optimal fit to the data, which resulted in an extremely high asset correlations. Thus the average empirical asset correlation determined using the write-off data is 7.00%.

Table 5.3: A presentation of the correlation results based on publications

Researcher Credit Cards

Mhlophe (Present paper): Default data 5.76% Mhlophe (Present paper): Write-off data 7.00% Stoffberg and van Vuuren (2015): Write-off data 7.00% Bellotti and Crook (2012): Default data 0.39% Hansen et al (2008): Write-off data 1.32% Rosch and Scheule (2004): Write-off data 0.66%

Table 5.3 presents correlation results based on different publications. In this table, we compare the average empirical asset correlations calculated in Table 5.1 and 5.2 for Bank X’s credit cards default and write-off data respectively with the empirical asset correlations obtained from previous research studies. We observe that most of the previous studies had obtained empirical asset correlations which were consid- erably lower than the Basel stipulated asset correlation of 4%, with most of them obtaining empirical asset correlations less than 2%. All three of these researchers do not take the global recession into account as they used loss data for periods before 2008. The credit crunch dramatically increased credit losses (Newscientist, 2009), therefore it is assumed that their results would’ve differed had their research included the impact of the global recession on the loss rate. This may be evident in the results obtained in Stoffberg & van Vuuren (2015), which had a credit cards empirical asset correlation as high as 7% for a US credit cards portfolio. Their result is consistent with the results obtained in our study. What is quite interesting is that they also use write-off data which gives exactly the same average empirical asset correlation as our credit cards empirical asset correlation, especially given that these calculations

61 have been conducted in both a developing economy (South Africa) and a developed economy (US). This reveals that the losses experienced during the 2008 credit crisis have dramatically increased the real asset correlations, which is one of considerations that separates these two studies from the previous research that obtained lower empirical asset correlations. This result however implies that the required credit cards capital charge stipulated by the BCBS is not suﬃcient to cover potential unexpected losses. This implication is consistent with the economic capital calculated using the empirical asset correlations in this study, which were higher than the economic capital calculated using the BCBS’ stated asset correlations. Thus, Bank X needs to increase their capital reserves to match those that have been estimated in this study.

The second objective of the study was to determine the asset correlations for a bank-specific credit cards default and write-off portfolio, as well as South African commercial loans dataset over a rolling five-year period, in order to the observe the sensitivity of the asset correlations over different economic conditions. We find that the South African commercial loans portfolio demonstrates the BCBS’ intended conservatism, as it has a lower empirical asset correlation over the whole period. However, the default data had correlation which were lower until they surpassed the Basel asset correlation in 2012, while the opposite was seen in the write-off data which had a higher empirical correlation for the whole period until 2015. This demonstrates that the BCBS’ intended conservatism is not always applicable especially in periods of enhanced risk.

5.2 Limitations

The limitations of this study include the difficulty experienced in an effort to gather credit card data for other South African banks. Even collecting summarised retail revolving (asset class representing credit cards and overdrafts) loss data proved to be a mammoth task. Having retail revolving data would have provided us with the ability to compare the Bank X’s credit card empirical asset correlation with the South African retail revolving asset class’ empirical correlation. The difficulty in accessing data is due to the fact that banks do not want to share their loan loss data.

Another limitation is that only one bank-speciﬁc product (credit cards) was explored, which provides a limited view of asset correlation levels of Bank X. Provid- ing a view of other products such as mortgages (which is a secured product) would provide a lot more insight into South African asset correlation levels, especially at a bank-speciﬁc level

5.3 Recommendations

This study has shown how to easily estimate empirical asset correlations from retail loan loss data. Using the beta, mode and percentile approaches of the vasicek distribution, this study has shown how to easily extract empirical asset correlations with only gross loss data and how these diﬀer from the BCBS’ prescribed asset correlations, and how they react to rolling them in diﬀerent economic conditions. The

62 computed empirical asset correlations are then used to calculate the economic capital. This study is quite beneficial and the methods used are easy to apply for banks who have their own internal loss data and are interested in determining their own estimate of the correlation for both economic and regulatory capital purposes. The credit cards empirical asset correlation results imply that the BCBS’ prescribed asset correlation is not as conservative as intended for South African bank specific credit cards and that the required capital charge stipulated by the BCBS (imposed by the SARB) is not sufficient to cover unexpected losses. A higher asset correlation results in higher unexpected losses thus requiring a higher capital charge. Unexpected losses larger than Bank X’s current capital buffer may have dire consequences for not only Bank X, but for the whole South African banking system through systemic risk. Therefore, we recommend that the Bank X raise their SARB imposed capital levels to match those determined in this study. Secondly, we recommend that the SARB should propose to other South African banks to use estimation approaches shown in this study to empirically estimate the asset correlations using their own internal loss data banks. These computed empirical asset correlations can thereafter be used to inform them of their real level of economic capital.

Finally, a study may consider research on empirical asset correlations and establish their impact on the Basel III capital regulations.

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70 Appendix I The best fit to the distribution of these loss data was the Beta distribution. In this appendix, the top 10 (10 best fits) fitting results are provided. Beta was – on average – ranked the best overall fit using the Kolmogorov-Smirnov, and Anderson-Darling.

1. Credit Cards Actual Default (V_Mode_PD)

V_Mode_PD - Descriptive Statistics Statistic Value Mean 0.04622069 Standard Error 0.001698612 Median 0.0409 Mode 0.0263 Standard Deviation 0.018294611 Sample Variance 0.000334693 Kurtosis -0.978579505 Skewness 0.506716536 Range 0.0611 Minimum 0.0257 Maximum 0.0868 Sum 5.3616 Count 116 Largest(1) 0.0868 Smallest(1) 0.0257 Conﬁdence Level(99,9%) 0.005736458

V_Mode - Goodness of Fit - Summary Kolmogorov Anderson # Distribution Statistic Rank Statistic Rank 1 Beta 0.02982 6 161.41 27 2 Burr 0.03784 17 108.09 18 3 Burr (4P) 0.03784 18 108.09 19 4 Cauchy 0.09421 35 29.819 12 5 Dagum 0.03675 16 100.37 15 6 Dagum (4P) 0.06804 29 166.22 36 7 Erlang 0.07442 31 164.6 34 8 Erlang (3P) 0.03111 14 163.89 33 9 Error 0.06504 28 138.64 22 10 Error Function 0.99243 53 1119.3 53

71 V_Mode - Fitting Results # Distribution Parameters a1=22,899 a2=3248,3 1 Beta a=-3,2283E-5 b=3,7779 2 Burr k=1,3158 a=7,6439 b=0,0273 k=1,3158 a=7,6438 3 Burr (4P) b=0,0273 g=3,2130E-7 4 Cauchy s=0,00331 m=0,02577 5 Dagum k=0,96765 a=8,427 b=0,02606 k=6,9033 a=5,3152 6 Dagum (4P) b=0,01662 g=1,0000E-6 7 Erlang m=22 b=0,00117 8 Erlang (3P) m=23 b=0,00115 g=9,9982E-7 9 Error k=1,6512 s=0,00555 m=0,02643 10 Error Function h=127,37

V_Mode - Goodness of Fit - Summary Kolmogorov-Smirnov Test Sample Size 102 Statistic 0.02982 P-Value 0.99997 Rank 6 a 0.2 0.1 0.05 0.02 0.01 Critical 0.10624 0.1211 0.134 0.1503 0.161 Value Reject? No No No No No

2. Credit Cards Actual Default (V_Percentile_PD)

V_Percentile_PD - Descriptive Statistics Statistic Value Mean 0.04622069 Standard Error 0.001698612 Median 0.0409 Mode 0.0263 Standard Deviation 0.018294611 Sample Variance 0.000334693 Kurtosis -0.978579505 Skewness 0.506716536 Range 0.0611 Minimum 0.0257 Maximum 0.0868 Sum 5.3616 Count 116 Largest(1) 0.0868 Smallest(1) 0.0257 Conﬁdence Level(99,9%) 0.005736458

72 V_Percentile_PD - Goodness of Fit - Summary Kolmogorov Anderson # Distribution Statistic Rank Statistic Rank 1 Beta 0.03363 11 223.64 36 2 Burr 0.03464 16 170.32 19 3 Burr (4P) 0.03464 15 170.37 20 4 Cauchy 0.12696 34 103.76 9 5 Dagum 0.03806 18 147.36 13 6 Dagum (4P) 0.03806 19 147.49 14 7 Erlang 0.15847 38 268.7 44 8 Erlang (3P) 0.06546 28 212.43 32 9 Error 0.13711 35 237.37 40 10 Error Function 0.71071 50 1065.3 51

V_Percentile_PD - Fitting Results # Distribution Parameters a1=3,8389 a2=42919,0 1 Beta a=5,6622E-7 b=516,91 2 Burr k=1,8839 a=2,7678 b=0,05611 k=1,884 a=2,7677 3 Burr (4P) b=0,05611 g=8,6007E-7 4 Cauchy s=0,01301 m=0,03975 5 Dagum k=0,73855 a=3,7014 b=0,04685 k=0,73851 a=3,7014 6 Dagum (4P) b=0,04685 g=9,9674E-7 7 Erlang m=3 b=0,01281 8 Erlang (3P) m=4 b=0,01201 g=9,4147E-7 9 Error k=1,0757 s=0,02433 m=0,04622 10 Error Function h=29,058

V_Percentile_PD - Goodness of Fit - Summary Kolmogorov-Smirnov Test Sample Size 116 Statistic 0.03363 P-Value 0.99891 Rank 11 a 0.2 0.1 0.05 0.02 0.01 Critical Value 0.09963 0.11355 0.126 0.14094 0.151 Reject? No No No No No

73 3. Credit Cards Actual Default (Beta_PD)

Beta_PD - Descriptive Statistics Statistic Value Mean 0.04622069 Standard Error 0.001698612 Median 0.0409 Mode 0.0263 Standard Deviation 0.018294611 Sample Variance 0.000334693 Kurtosis -0.978579505 Skewness 0.506716536 Range 0.0611 Minimum 0.0257 Maximum 0.0868 Sum 5.3616 Count 116 Largest(1) 0.0868 Smallest(1) 0.0257 Conﬁdence Level(99,9%) 0.005736458

Beta_PD - Goodness of Fit - Summary Kolmogorov Anderson # Distribution Statistic Rank Statistic Rank 1 Beta 0.03185 1 681.89 40 2 Burr 0.0437 13 432.43 20 3 Burr (4P) 0.0437 12 432.43 21 4 Cauchy 0.10892 34 153.37 9 5 Dagum 0.04967 15 327.49 17 6 Dagum (4P) 0.2178 39 72.031 6 7 Erlang 0.0967 32 669.24 39 8 Erlang (3P) 0.08708 28 657.27 37 9 Error 0.09512 31 967.96 44 10 Error Func- 0.83509 51 1625.6 53 tion

74 Beta_PD - Fitting Results # Distribution Parameters a1=6,0418 a2=124,68 1 Beta a=2,7878E-8 b=1,0 2 Burr k=2,4376 a=3,4294 b=0,06119 k=2,4377 a=3,4294 3 Burr (4P) b=0,06119 g=7,6516E-7 4 Cauchy s=0,01074 m=0,04305 5 Dagum k=0,61909 a=5,214 b=0,05073 k=0,88383 a=1,6927 6 Dagum (4P) b=0,04712 g=1,0000E-6 7 Erlang m=6 b=0,00724 8 Erlang (3P) m=6 b=0,00731 g=9,2336E-7 9 Error k=1,4892 s=0,01829 m=0,04622 10 Error Func- h=38,651 tion

Beta_PD - Goodness of Fit - Summary Kolmogorov-Smirnov Test Sample Size 116 Statistic 0.03185 P-Value 0.99956 Rank 1 a 0.2 0.1 0.05 0.02 0.01 Critical Value 0.09963 0.11355 0.126 0.14094 0.151 Reject? No No No No No

75 .

4. Credit Cards Write-Oﬀ Data (V_Mode_EL)

V_Mode_EL - Descriptive Statistics Statistic Value Mean 0.005964 Standard Error 0.000579125 Median 0.0045 Mode 0.003 Standard Deviation 0.00647482 Sample Variance 4.19233E-05 Kurtosis 55.63024684 Skewness 6.535892455 Range 0.0648 Minimum 0 Maximum 0.0648 Sum 0.7455 Count 125 Largest(1) 0.0648 Smallest(1) 0 Conﬁdence Level(99,9%) 0.00195207

V_Mode_EL - Goodness of Fit - Summary Kolmogorov Anderson # Distribution Statistic Rank Statistic Rank 1 Beta 0.05445 18 403.48 37 2 Burr 0.04299 12 260.57 20 3 Burr (4P) 0.04298 11 260.61 21 4 Cauchy 0.15251 32 184.77 9 5 Dagum 0.04555 15 229.6 15 6 Dagum (4P) 0.04555 14 229.64 16 7 Erlang 0.39488 48 536.97 47 8 Erlang (3P) 0.09502 27 451.65 43 9 Error 0.17652 37 416.79 40 10 Error Function 0.59648 52 1375.1 52

76 V_Model_EL - Fitting Results # Distribution Parameters a1=2,2658 a2=4,6086E+6 1 Beta a=9,9906E-7 b=12143,0 2 Burr k=1,6412 a=2,1611 b=0,0066 k=1,6422 a=2,1604 3 Burr (4P) b=0,0066 g=7,5536E-7 4 Cauchy s=0,00195 m=0,00448 5 Dagum k=0,81202 a=2,6951 b=0,00541 k=0,81153 a=2,6952 6 Dagum (4P) b=0,00542 g=7,7290E-7 7 Erlang m=1 b=0,00314 8 Erlang (3P) m=2 b=0,00261 g=1,0000E-6 9 Error k=1,0 s=0,00433 m=0,00596 10 Error Function h=163,49

V_Model_EL - Goodness of Fit - Summary Kolmogorov-Smirnov Test Sample Size 125 Statistic 0.05445 P-Value 0.83268 Rank 18 a 0.2 0.1 0.05 0.02 0.01 Critical Value 0.09597 0.10939 0.121 0.13577 0.146 Reject? No No No No No

77 5. Credit Cards Write-Oﬀ Data (V_Percentile_EL)

V_Percentile_EL - Descriptive Statistics Statistic Value Mean 0.005964 Standard Error 0.000579125 Median 0.0045 Mode 0.003 Standard Deviation 0.00647482 Sample Variance 4.19233E-05 Kurtosis 55.63024684 Skewness 6.535892455 Range 0.0648 Minimum 0 Maximum 0.0648 Sum 0.7455 Count 125 Largest(1) 0.0648 Smallest(1) 0 Conﬁdence Level(99,9%) 0.00195207

V_Percentile_EL - Goodness of Fit - Summary Kolmogorov Anderson # Distribution Statistic Rank Statistic Rank 1 Beta 0.24921 28 468.23 28 2 Burr 0.15044 10 361.94 19 3 Burr (4P) 0.1462 8 344.09 17 4 Cauchy 0.25992 32 335.48 16 5 Dagum 0.14898 9 325.48 15 6 Dagum (4P) 0.42074 41 37.791 2 7 Error 0.25908 30 681.36 40 8 Error Function 0.52689 45 1599.5 47 9 Exponential 0.143 5 701.98 41 10 Exponential (2P) 0.14305 6 702.23 42

78 V_Percentile_EL - Fitting Results # Distribution Parameters a1=0,50716 a2=8,3744 1 Beta a=1,0000E-6 b=0,1445 2 Burr k=2,2878 a=1,143 b=0,00944 k=2,0982 a=1,1571 3 Burr (4P) b=0,00871 g=2,6972E-7 4 Cauchy s=0,00207 m=0,00281 5 Dagum k=0,57853 a=1,7724 b=0,00603 k=0,35535 a=0,53216 6 Dagum (4P) b=0,47311 g=1,0033E-6 7 Error k=1,0 s=0,00939 m=0,00678 8 Error Function h=75,305 9 Exponential l=147,56 10 Exponential (2P) l=147,58 g=1,0000E-6

V_Percentile_EL - Goodness of Fit - Summary Kolmogorov-Smirnov Test Sample Size 125 Statistic 0.24921 P-Value 0.000000263 Rank 28 a 0.2 0.1 0.05 0.02 0.01 Critical Value 0.09597 0.10939 0.121 0.13577 0.146 Reject? Yes Yes Yes Yes Yes

79 6. Credit Cards Write-Oﬀ Data (Beta_EL)

Beta_EL - Descriptive Statistics Statistic Value Mean 0.005964 Standard Error 0.000579125 Median 0.0045 Mode 0.003 Standard Deviation 0.00647482 Sample Variance 4.19233E-05 Kurtosis 55.63024684 Skewness 6.535892455 Range 0.0648 Minimum 0 Maximum 0.0648 Sum 0.7455 Count 125 Largest(1) 0.0648 Smallest(1) 0 Conﬁdence Level(99,9%) 0.00195207

Beta_EL - Goodness of Fit - Summary Kolmogorov Anderson # Distribution Statistic Rank Statistic Rank 1 Beta 0.05401 3 97.681 31 2 Burr 0.05131 2 93.304 19 3 Burr (4P) 0.051 1 100.04 34 4 Cauchy 0.22106 40 97.557 30 5 Dagum 0.06096 15 82.091 10 6 Dagum (4P) 0.06093 14 82.103 11 7 Erlang 0.06483 22 96.709 26 8 Erlang (3P) 0.06954 28 110.19 38 9 Error 0.20271 38 114.12 40 10 Error Function 0.52648 50 331.42 50

80 Beta_EL - Fitting Results # Distribution Parameters a1=1,0922 a2=3,6060E+6 1 Beta a=1,0000E-6 b=20299,0 2 Burr k=8,475 a=1,1081 b=0,03946 k=8,1999 a=1,1122 3 Burr (4P) b=0,03778 g=1,0000E-6 4 Cauchy s=0,00253 m=0,00344 5 Dagum k=0,4752 a=2,1765 b=0,00738 k=0,47482 a=2,1768 6 Dagum (4P) b=0,00738 g=6,7834E-7 7 Erlang m=1 b=0,00602 8 Erlang (3P) m=1 b=0,00559 g=1,0000E-6 9 Error k=1,0 s=0,00608 m=0,00613 10 Error Function h=116,33

Beta_EL - Goodness of Fit - Summary Kolmogorov-Smirnov Test Sample Size 91 Statistic 0.05401 P-Value 0.94026 Rank 3 a 0.2 0.1 0.05 0.02 0.01 Critical 0.11064 0.1264 0.14 0.157 0.168 Value Reject? No No No No No

81 .

7. Commercial South African Loans (V_Mode)

V_Mode - Descriptive Statistics Statistic Value Mean 0.026429126 Standard Error 0.000643849 Median 0.0247 Mode 0.025 Standard Deviation 0.006534353 Sample Variance 4.26978E-05 Kurtosis -1.197755775 Skewness 0.448187695 Range 0.0216 Minimum 0.0168 Maximum 0.0384 Sum 2.7222 Count 103 Conﬁdence Level(99,9%) 0.002181669

82 V_Mode - Fitting Results # Distribution Parameters a1=22,899 a2=3248,3 1 Beta a=-3,2283E-5 b=3,7779 2 Burr k=1,3158 a=7,6439 b=0,0273 k=1,3158 a=7,6438 3 Burr (4P) b=0,0273 g=3,2130E-7 4 Cauchy s=0,00331 m=0,02577 5 Dagum k=0,96765 a=8,427 b=0,02606 k=6,9033 a=5,3152 6 Dagum (4P) b=0,01662 g=1,0000E-6 7 Erlang m=22 b=0,00117 8 Erlang (3P) m=23 b=0,00115 g=9,9982E-7 9 Error k=1,6512 s=0,00555 m=0,02643 10 Error Function h=127,37

8. Commercial South African Loans(V_Percentile)

V_Percentile - Descriptive Statistics Statistic Value Mean 0.026429126 Standard Error 0.000643849 Median 0.0247 Mode 0.025 Standard Deviation 0.006534353 Sample Variance 4.26978E-05 Kurtosis -1.197755775 Skewness 0.448187695 Range 0.0216 Minimum 0.0168 Maximum 0.0384 Sum 2.7222 Count 103 Conﬁdence Level(99,9%) 0.002181669

83 V_Percentile - Goodness of Fit - Summary Kolmogorov Anderson # Distribution Statistic Rank Statistic Rank 1 Beta 0.04323 17 187.99 36 2 Burr 0.03829 11 144.33 20 3 Burr (4P) 0.03829 10 144.36 21 4 Cauchy 0.14336 36 103.74 10 5 Dagum 0.04223 15 126.9 12 6 Dagum (4P) 0.04223 14 126.96 13 7 Erlang 0.13788 34 210.22 41 8 Erlang (3P) 0.13068 32 159.68 30 9 Error 0.16331 37 199.37 39 10 Error Function 0.62398 49 824.56 50

V_Percentile - Fitting Results # Distribution Parameters a1=2,649 a2=2756,3 1 Beta a=9,4780E-7 b=27,515 2 Burr k=1,9162 a=2,2545 b=0,03271 k=1,9164 a=2,2544 3 Burr (4P) b=0,03271 g=6,7741E-7 4 Cauchy s=0,00841 m=0,02095 5 Dagum k=0,73294 a=3,0356 b=0,02605 k=0,73284 a=3,0356 6 Dagum (4P) b=0,02605 g=9,3027E-7 7 Erlang m=2 b=0,01113 8 Erlang (3P) m=3 b=0,00995 g=9,9917E-7 9 Error k=1,0 s=0,01714 m=0,0264 10 Error Function h=41,246

V_Percentile - Goodness of Fit - Summary Kolmogorov-Smirnov Test Sample Size 102 Statistic 0.04323 P-Value 0.98688 Rank 17 a 0.2 0.1 0.05 0.02 0.01 Critical 0.10624 0.1211 0.134 0.1503 0.161 Value Reject? No No No No No

84 9. Commercial South African Loans (Beta)

Beta - Descriptive Statistics Statistic Value Mean 0.026429126 Standard Error 0.000643849 Median 0.0247 Mode 0.025 Standard Deviation 0.006534353 Sample Variance 4.26978E-05 Kurtosis -1.197755775 Skewness 0.448187695 Range 0.0216 Minimum 0.0168 Maximum 0.0384 Sum 2.7222 Count 103 Conﬁdence Level(99,9%) 0.002181669

Beta - Goodness of Fit - Summary Kolmogorov Anderson # Distribution Statistic Rank Statistic Rank 1 Beta 0.08607 2 2009.1 33 2 Burr 0.09723 17 1171 16 3 Burr (4P) 0.09723 16 1171 17 4 Cauchy 0.12234 30 276.75 8 5 Dagum 0.09688 14 913.17 15 6 Dagum (4P) 0.09688 13 913.12 14 7 Erlang 0.12259 31 2079.7 39 8 Erlang (3P) 0.11903 25 2071.4 37 9 Error 0.12289 32 1809.1 29 10 Error Func- 0.97865 51 2030.5 34 tion

85 Beta - Fitting Results # Distribution Parameters a1=16,278 a2=848,23 1 Beta a=-2,0265E-4 b=1,4141 2 Burr k=1,7156 a=5,9999 b=0,02922 k=1,7156 a=5,9997 3 Burr (4P) b=0,02922 g=5,1735E-7 4 Cauchy s=0,00394 m=0,02571 5 Dagum k=0,77828 a=7,6913 b=0,02717 k=0,77834 a=7,6916 6 Dagum (4P) b=0,02718 g=-1,9000E-6 7 Erlang m=16 b=0,00162 8 Erlang (3P) m=16 b=0,00162 g=9,1238E-7 9 Error k=1,7279 s=0,00653 m=0,02643 10 Error Func- h=108,21 tion

Beta - Goodness of Fit - Summary Kolmogorov-Smirnov Test Sample Size 116 Statistic 0.08607 P-Value 0.33714 Rank 2 a 0.2 0.1 0.05 0.02 0.01 Critical 0.09963 0.11355 0.126 0.14094 0.151 Value Reject? No No No No No