Modelling Credit Risk: Estimation of Asset and Default ...1220556/FULLTEXT01.pdf · internal models to estimate the asset correlation used in the RWA formula. Default correlation,

Modelling Credit Risk: Estimation of Assetand Default Correlation for an SME

PortfolioCredit Risk Department at Handelsbanken

Master Thesis 30 hp

Authors:YAXUM CEDENOREBECCA JANSSON

Supervisors:BUJAR HUSKAJ

OLOW SANDE

Master Thesis 30 hpMSc. Industrial Engineering and Management – Risk Management 300 hp

Spring 2018

AbstractWhen banks lend capital to counterparties they take on a risk, known as credit riskwhich traditionally has been the largest risk exposure for banks. To be protectedagainst potential default losses when lending capital, banks must hold a regulatorycapital that is based on a regulatory formula for calculating risk weighted assets(RWA). This formula is part of the Basel Accords and it is implemented in the le-gal system of all European Union member states. The key parameters of the RWAformula are probability of default, loss given default and asset correlation. Bankstoday have the option to estimate the probability of default and loss given defaultby internal models however the asset correlation must be determined by a formulaprovided by the legal framework.

This project is a first approach for Handelsbanken to study what would happenif banks were allowed to estimate asset correlation by internal models. We assesstwo models for estimating the asset correlation of a portfolio of Small and MediumEnterprices (SME). The estimates are compared with the asset correlation given bythe regulatory formula and with estimates for another parameter called default cor-relation. The models are validated using predicted historical data and Monte-CarloSimulations. For the studied SME portfolio, the models give similar estimates forthe asset correlations and the estimates are lower than those given by the regulatoryformula. This would imply a lower capital requirement if banks were allowed to useinternal models to estimate the asset correlation used in the RWA formula. Defaultcorrelation, if not used synonymously with asset correlation, is shown to be anothermeasure and should not be used in the RWA formula.

Keywords: Basel Capital Accord, Capital Requirements, SME, Portfolio CreditRisk, Monte-Carlo Simulations, Risk Weighted Assets (RWA).

1 31 may 2018

Sammanfattning

När banker lånar ut kapital till motparter tar de en risk, mer känt som kreditrisksom traditionellt har varit den största risken för banker. För att skydda sig motpotentiella förluster vid utlåning måste banker ha ett reglerat kapital som byggerpå en formel för beräkning av riskvägda tillgångar (RWA). Denna formel ingår iBasels regelverk och är implementerad i rättssystemet i alla EU-länder. De vikti-gaste parametrarna för RWA-formeln är sannolikheten att fallera, förlustgivet fal-lissemang och tillgångskorrelation. Bankerna har idag möjlighet att beräkna de tvåvariablerna sannolikheten att fallera och förlustgivet fallissemang med interna mod-eller men tillgångskorrelation måste bestämmas med hjälp av en standardformelgivet från regelverket.

Detta projekt är ett första tillvägagångssätt för Handelsbanken att studera vad somskulle hända om banker fick beräkna tillgångskorrelation med interna modeller. Vianalyserar två modeller för att skatta tillgångskorrelation i en portfölj av Små ochMedelstora Företag (SME). Uppskattningarna jämförs sedan med den tillgångskor-relation som ges av regelverket och jämförs även mot en parameter som kallas fallis-semangskorrelation. Modellerna som används för att beräkna korrelationerna valid-eras med hjälp av estimerat data och Monte-Carlo Simuleringar. För den studeradeSME portföljen ges liknande uppskattningar för de båda tillgångskorrelationsmod-ellerna, samt visar det sig att de är lägre än den korrelationen som ges av regelverket.Detta skulle innebära ett lägre kapitalkrav om bankerna fick använda sig av internamodeller för att estimera tillgångskorrelation som används i RWA-formeln. Om fal-lissemangskorrelation inte används synonymt till tillgångskorrelation, visar det sigatt fallisemangskorrelation är en annan mätning än tillgångskorrelation och bör inteanvändas i RWA-formeln.

Nyckelord: Basel Kapitalavtal, Kapitalkrav, SME, Portfölj Kreditrisk, Monte-CarloSimuleringar, Riskvägda Tillgångar (RWA).

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AcknowledgementsFirst, we would like to give great thanks to Bujar Huskaj and Olow Sande for beingour supervisors throughout this master thesis, you have contributed to invaluablesupport and engagement. Further, we would like to acknowledge Handelsbankenand especially the Modelling group of the Independent Credit Risk Department fortaking extraordinary care of us during this semester. We would also like to acknowl-edge everyone else that has been involved in our project in some way, it has been atruly inspiring period.

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CONTENTS

Contents

1 Introduction 21.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Asset Correlation vs Default Correlation . . . . . . . . . . . 31.2 Project Description . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5.1 SME Portfolio . . . . . . . . . . . . . . . . . . . . . . . . 51.5.2 Macro-Economics . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Credit Risk and Regulations 82.1 Basel Committee of Banking and Supervisory . . . . . . . . . . . . 82.2 Internal Rating-Based System . . . . . . . . . . . . . . . . . . . . 92.3 Probability of Default . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Risk Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Asset Correlation . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 Dependence of Asset Correlation, PD and RW . . . . . . . 11

3 Mathematical Models of Credit Default 143.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Merton’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Derivation of the Risk Weight Formula . . . . . . . . . . . . . . . . 153.4 Asset Correlation Models . . . . . . . . . . . . . . . . . . . . . . . 16

3.4.1 Binomial Likelihood . . . . . . . . . . . . . . . . . . . . . 173.4.2 Large Portfolio Approximation . . . . . . . . . . . . . . . 17

3.5 Default Correlation Model . . . . . . . . . . . . . . . . . . . . . . 193.5.1 Joint Default Probability . . . . . . . . . . . . . . . . . . . 19

4 Statistical Methods 214.1 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . 214.2 Multivariate Regression Analysis & Ordinary Least Squares . . . . 224.3 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . 234.4 Monte-Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . 234.5 Validation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.5.1 Augmented Dickey-Fuller Test . . . . . . . . . . . . . . . . 244.5.2 Mean Absolute Error . . . . . . . . . . . . . . . . . . . . . 254.5.3 Root Mean Square Error . . . . . . . . . . . . . . . . . . . 25

5 Method 265.1 Predicting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1.1 PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.1.2 Regression Analysis . . . . . . . . . . . . . . . . . . . . . 29

5.2 Estimating Asset Correlation . . . . . . . . . . . . . . . . . . . . . 30

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CONTENTS

5.2.1 Binomial Likelihood . . . . . . . . . . . . . . . . . . . . . 305.2.2 Large Portfolio Approximation . . . . . . . . . . . . . . . . 31

5.3 Estimating Default Correlation . . . . . . . . . . . . . . . . . . . . 315.4 Simulation of Default Correlation . . . . . . . . . . . . . . . . . . 325.5 Validation of Likelihood Models . . . . . . . . . . . . . . . . . . . 32

6 Results 336.1 Data Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 Estimated Asset and Default Correlation . . . . . . . . . . . . . . . 366.3 Correlation Coefficient in the RW-Formula . . . . . . . . . . . . . 376.4 Asset and Simulated Default Correlation . . . . . . . . . . . . . . . 386.5 Validation of Likelihood Functions . . . . . . . . . . . . . . . . . . 39

6.5.1 Monte-Carlo with Different Maturity . . . . . . . . . . . . 396.5.2 Monte-Carlo with Different Number of Simulations . . . . . 40

7 Discussion 427.1 Modeling Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.2 Analysis of Estimated Correlation . . . . . . . . . . . . . . . . . . 427.3 RW - formula as a benchmark . . . . . . . . . . . . . . . . . . . . 437.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.5 Further development . . . . . . . . . . . . . . . . . . . . . . . . . 45

8 References 46

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CONTENTS

Abbreviations

BCBS Basel Committee of Bank Supervision

BL Binomial Likelihood

IRB Internal Rating-Based Approach

CPI Consumer Price Index

IAPI Inflation-adjusted property index

JDP Joint Default Probability

LGD Loss Given Default

LPA Large Portfolio Approximation

MAE Mean Absolute Error

MCS Monte-Carlo Simulation

MLE Maximum Likelihood Estimation

OLS Ordinary Least Square Method

PBI Price Base Index

PC Principal Component

PCA Principal Component Analysis

PD Probability of Default

RMSE Root Mean Square Error

RW Risk Weight

RWA Risk Weighted Assets

SME Small Medium Enterprises

TCW TCW-index

UL Unexpected Loss

UR Unemployment

VaR Value at Risk

10Y 10Y Government Bond

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1 INTRODUCTION

1 Introduction

This Section aims to introduce the reader to the project and enlighten importantknowledge for an understanding throughout the article. The introduction coversbackground, project description and the purpose of the project. An explanation ofthe two key parameters default and asset correlation that are emphasized throughoutthe article is provided. Furthermore the underlying data of the project is presented.

1.1 Background

Deposits provide capital for the banks which they can in turn lend to others at inter-est. This way loans is a source of payoff for the bank but also a potential source oflosses. There is always a risk that a borrower may not be able to repay the principalof a loan or the interest associated with it. This is known as credit risk (Investope-dia, 2018a). Governments across the world want the financial sector to be stable andthus financial institutions are heavily regulated. The central bank regulators requirebanks to hold a certain amount of capital in order to protect themselves towardsrisk exposures by following the Basel Capital Accords Supervision. Credit risk hastraditionally been the greatest risk a bank faces and it is usually the risk where mostregulatory capital is required (Hull, 2015).

The Basel Accords are recommendations of laws and regulations for internationallyactive banks issued by the Basel Committee on Banking Supervision (BCBS). TheBasel Committee does not have the power to enforce these recommendations, how-ever the European Union has incorporated the Basel Accords into the legal systemto ensure that the European banking specificities are appropriately addressed (BIS,2018). The main objective of bank regulation is to prevent banks from defaulting byholding enough capital for its risk exposures. It is obviously not possible to elimi-nate all risks, but governments aim to reduce the probability of default for banks.

Loss given default (LGD) and probability of default (PD) are well known in deter-mination of credit risk while asset correlation has not received as much attention.The reason for this is mainly due to the lack of available historical data and becausethe regulatory has focused more on PD and LGD (Moody’s, 2008). Although, inrecent years the interest in studying asset correlation has increased significantly,which is a result of the enhanced acknowledgement of credit portfolio managementand capital framework together with the development of credit default swaps (CDS)and collateralized debt obligation (CDO). The three parameters PD, LGD and assetcorrelation are the key drivers in the risk weight formula, see Equation 3 in Section2.4, which determines the capital requirement.

It is of high importance for banks to possess the ability to correctly estimate thecredit risk exposure. If banks hold excessive capital they might lose investment op-portunities whereas if they have a deficit of capital they are not protected in case ofa large counter-party default. Therefore, it is necessary for banks to use appropriate

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1 INTRODUCTION

parameters in the calculation of capital requirements. Today, banks have the oppor-tunity to estimate PD and LGD with internal models however, the asset correlationmust be calculated by a formula given by the legal framework, see Equation 5.

1.1.1 Asset Correlation vs Default Correlation

In this article we will study the measurement correlation. Two types of correlationswill be examined, asset and default correlation which refer to the dependence be-tween two enterprises. Asset correlation is one of the key drivers in the calculationof capital requirement and it is defined by how one firm’s value depends on the valueof another firm. Asset correlations may be calculated directly if the asset values areknown, for example using stock prices as estimates for asset values. However, assetcorrelations are often estimated using default data, as we do in this paper. Becauseof this, asset correlation is sometimes called "default correlation" in the literature,for example in the paper of Demey et al (2004) titled "Maxium likelihood estimateof default correlations".

Default correlation is defined as the dependence between the default of one firmand the default of another (Moody’s, 2008). This default correlation is studied infor example "Default correlation and credit risk analysis" by Lucas (1995). To avoidconfusion we will always call the correlation of asset values "asset correlation" andthe correlation of default events "default correlation". In this article we mainly studyasset correlation since this is the parameter used in the credit risk formula and thusof interest. However, we will also consider default correlation, as it was proposed byHandelsbanken. For the interest in comparing default and asset correlations we willuse historical default data as well as Monte-Carlo Simulations (MCS). In section3.1, the mathematical definition of correlation is presented to fully understand theconcepts of asset and default correlation.

1.2 Project Description

This article will investigate asset and default correlation for a Small Medium En-terprise (SME) portfolio. The project is proposed by the Credit Risk Departmentat Handelsbanken and boils down to two main objectives. On the one hand, theaim is to estimate asset and default correlation by using different models to makea comparison between the parameters. The second objective is to evaluate how therisk weight formula behaves when implementing the estimated asset correlation, incomparison to the asset correlation in the framework.

We will estimate the asset correlation by implementing two different models and thedefault correlation will be estimated by another model. The models are describedin Section 3 as well as the mathematical model of defaulting loans underlying theBasel risk weight formula.

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1 INTRODUCTION

1.3 Purpose

It is of interest for banks to utilize internal models to estimate the parameters in thecapital accords to achieve an optimal reflection of a portfolio. Thus, this thesis is afirst approach if Handelsbanken in the future are allowed to estimate the asset cor-relation in the risk weight formula by internal models. The second aim of this thesisis to ascertain that asset and default correlation for a SME portfolio differentiate.

1.4 Delimitations

To ensure that the project is completed by deadline some delimitations are made.There are various approaches for estimating correlation and for this thesis threemodels are selected. This limitation is necessary due to the time frame. The SMEportfolio consists of different risk classes however, due to complexity and the timelimitation, the portfolio is assumed homogeneous throughout the article. All oblig-ors are aggregated into one risk class and the migration is from non-default to de-fault. Hence the estimated asset and default correlation is assumed constant betweenall obligors in the portfolio.

Another factor to consider is the lack of knowledge about the portfolio. The onlyknowledge is that the analyzed portfolio comprises of small and medium enter-prises, beyond that there is no revealed information about which country or industrythe enterprises operate in. To be able to reflect the portfolio by the general state ofeconomy the SME portfolio is assumed to comprise of Swedish companies. Thislimitation is considered throughout the project and the authors are aware of that thislimitation might affect the results.

1.5 Data

In this article we will utilize internal default data of the SME portfolio providedby Handelsbanken to estimate the asset and default correlation. The data is con-fidential information and is therefore anonymized in the article. In Table 1 theconfiguration of the data is presented and we can perceive that the portfolio onlycomprises eleven years of data, 2005-2015. This is a common problem for financialinstitutions and hence it is of interest to predict longer time series for an enhancedanalysis. One approach to describe the behavior of enterprises is macro-economicfactors (Investopedia 2018b) and thus it is taken into consideration in the predictionof new data. In Table 4 we present the selected macro-economic variables and inSection 5.1 the modeling of the data prediction is explained.

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1 INTRODUCTION

1.5.1 SME Portfolio

By definition a SME portfolio is a portfolio consisting of small and medium enter-prises with less than 50 Million Euros in annual sales. The SME portfolio providedby Handelsbanken comprises the amount of counterparties, see Table 1, and theamount of defaulted counterparties, see Table 2, on a yearly frequency over thetime period 2005-2015. We can see that the counterparties are divided into differentrisk classifications depending on their creditworthiness. The AAA classification isthe highest rated securities which implies highest credit worthiness, while D has thelowest creditworthiness. With a higher probability to default for the borrower, thelarger credit risk exposure for the bank.

Table 1: The SME portfolio, containing number of counterparties each year.

Risk class 2005 2006 · · · 2015AAA • • · · · •AA • • · · · •A • • · · · •BBB • • · · · •BB • • · · · •B • • · · · •CCC • • · · · •D • • · · · •Total companies • • · · · •

Table 2: The SME portfolio, containing number of counterparties that defaultedeach year.

Risk class 2005 2006 · · · 2015AAA • • · · · •AA • • · · · •A • • · · · •BBB • • · · · •BB • • · · · •B • • · · · •CCC • • · · · •D • • · · · •Total defaulted companies • • · · · •

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1 INTRODUCTION

Although the SME portfolio is divided into different risk classes, as presented in Ta-ble 1 and Table 2, the portfolio is assumed homogeneous. This means that all oblig-ors are aggregated into a single risk class and the migration is from non-default todefault, shown in Table 3, and this is the portfolio that will be analyzed throughoutthe article.

Table 3: The SME portfolio, containing number of total companies and defaultedcompanies each year.

2005 2006 · · · 2015Total companies • • · · · •Total defaulted companies • • · · · •

1.5.2 Macro-Economics

It is no revealed knowledge of the enterprises for the SME portfolio and hence itwould not be feasible to use company specific information to explain the data set.However, Handelsbanken proposes to apply macro-economic data as the underlyingexplanation for the SME portfolio and thus ten variables are selected. The macro-economic variables are presented in Table 4 and are selected by availability as wellas the variables are implemented in the European Banking Authorities stress test,which is an authority that test how well banks in Europe manage their economy.The ten variables covers yearly observations of the time series 1990-2015 and arefree to collect from NASDAQ, Ekonomifakta and Statistics Sweden.

Table 4: Macro-economic data from Sweden.

Variables Unit

Export SEKImport SEKGDP SEK10Y Government Bond, (10Y) (%)Consumer Price Index, (CPI) SEKPrice Base Index, (PBI) SEKOMX30 SEKUnemployment, (UP) (%)Inflation-adjusted property index, (IAPI) SEKTCW-index, (TCW) SEK

The development of export and import explains the country’s relationships withother countries. These factors are sensitive to changes in the currencies and theeconomy. Gross domestic product (GDP) is one of the most important econom-ical measurements. The measurement describes how the country’s economy haschanged over time. A 10-year government bond is a financial product the gov-ernment sells to individuals and institutions which usually yields a higher return

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1 INTRODUCTION

than placing the money at a savings account although yields low return compare tostocks. The benefit of the product is the low risk and this factor is an indicator ofhow the market is moving (SCB, 2018).

Consumer price index, CPI, is a measure to evaluate how the development of prod-uct prices have changed. It describes the annual average price of a product com-pared to a product with the price of 100 SEK in 1980. The price base index, PBI,represents the cost of a basket of necessary products for one person. OMX30 is anindex that represents the 30 most digested companies on the Stockholm stock ex-change (NASDAQ, 2018). Inflation-adjusted property price index is retrieved fromthe value defining the countries property price and compares one- or two familyhouse or terraced house by the development of the price from start value 100 at1986 (Ekonomifakta, 2018). The TCW-index is a clear way to see how the SEKhas developed compared to a basket of currencies. A high value of the TCW-indexindicates that the SEK is weak (Sveriges Riksbank, 2018).

1.6 Outline

This article emphasis the estimation and analysis of asset and default correlation ona SME portfolio provided by Handelsbanken. To achieve this we apply statisticalmodeling based on the same underlying theory as the Basel risk weight regulation.The project disposition is as follow, Section 2 covers how the regulations and mod-eling of credit risk under the Basel Accords have developed over time. In Section3 the mathematical theories behind the three models that are implemented to esti-mate asset and default correlation are described, Binomial Likelihood (BL), LargePortfolio Approximation (LPA) and Joint Default Probabilty (JDP). Section 4 ex-plains the statistical methods that are applied in this article to achieve the objectivesas well as the data prediction models. Section 5 covers the implementation of themodeling along with the assumptions for the approaches. In Section 6 the resultsare presented which are then analyzed and discussed in Section 7.

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2 CREDIT RISK AND REGULATIONS

2 Credit Risk and Regulations

In this Section we aim to introduce the reader to credit risk management as wellas the development and explanation of the Basel Accords. In the determination ofcapital requirements a risk weight (RW) formula is used where asset correlation isone of the key drivers together with probability of default and loss given default,see Section 2.4. The risk weight formula for a SME portfolio is presented, Equation3, and further the dependence between asset correlation, probability of default andrisk weight are described.

2.1 Basel Committee of Banking and Supervisory

The Basel Committee of Banking and Supervisory Practices (BCBS) was foundedat the end of 1974 because of the instability in the international banking markets.Its main purpose was to enhance financial stability worldwide and increase banksupervision of their own business. The Basel Committee has since then establishednew directives, called Basel Accords, which are banking regulations. The purposeof the Basel Accords is to make sure that banks have sufficient capital for managingdifferent types of negative economic impacts that could lead to unexpected losses.The Basel Committee always endeavor to improve regulations on the present eco-nomic environment. As of today three regulations have been established which areBasel I, Basel II and Basel III accords. Almost all countries with internationallyoperating banks follow these frameworks (BIS, 2018).

The Basel I accord was announced in 1988 and introduced the minimum capitalrequirement. The aim of the regulation was to prevent banks from defaulting dueto credit risk exposure and this by having a risk weighted asset (RWA) of 8% asa buffer. In 2004 The New Capital Framework was implemented as Basel II andconsisted of three pillars. The first pillar defined the minimum capital requirementsas in the previous Basel Accord with an expansion to cover operational risk andmarket risk as well. The second pillar consists of Supervisory Review and the thirdpillar of Market Discipline (BIS, 2018). Basel III enhanced the Basel II frameworkand additional regulations are progressively being implemented into the banks until2019 (BIS, 2017).

Since the Basel Committee does not have authority in any country, the frameworkfrom Basel has been applied in the different countries’ legal systems as well as theEuropean Union. Therefore the Swedish banks have the obligation to follow boththe Swedish and the European Union’s financial legal system. In order to controlthis, the Swedish banks frequently report to the Swedish financial controlling au-thority, Finansinspektionen (BIS, 2018).

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2.2 Internal Rating-Based System

When Basel II was implemented into the system banks got the possibility to cal-culate their own credit risk and use their own models. There are three levels forcalculating the credit risk components:

1. Standardized Approach (SA)2. Foundation Internal Rating-Based Approach (FIRB)3. Advanced Internal Rating-Based Approach (AIRB)

Under the standardized approach banks have to follow Basel’s suggested methodsand models to calculate the credit risk. However, Foundation IRB gives banks thepossibility to estimate the PD with internal models. With the advanced IRB thebanks are, in addition to this, allowed to estimate LGD and exposure at default(EAD) and other parameters that are linked to RWA. Before banks are permitted touse internal estimations they need to get an approval from the financial supervisors(BIS, 2017). The relation between the risk measurements PD, EAD and LGD isgiven by:

EL = PD ·EAD ·LGD (1)

The formula displays how banks calculate expected loss (EL) which is mostly cov-ered by risk premiums and interest rates. However, there is also unexpected loss(UL) which is described by being larger than the expected loss as seen in Figure 1.

Figure 1: The line represents a loss distribution.

Unexpected loss is an important measure for banks since it is always a risk of un-expected events in the market which banks must be aware of (Coen, 2000). This

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is why banks and financial institutions are requested to hold a buffer against unex-pected losses.

2.3 Probability of Default

Default is the failure to pay principal or interest on a loan or security. It occurswhen a debtor is unable to perform the legal obligation for repayment (Investopedia,2018c). When lenders decide whether to issue a loan they must consider the risk ofdefault for the counterpart. The PD is described as the likelihood that a repaymentwill not be paid on time (Financial Times, n.d).

2.4 Risk Weight

The risk weight (RW) is a key parameter in the calculation of RWA which deter-mines the minimum capital requirement a bank must hold when lending money.In the calculation of RWA each individual asset within the different risk typesis weighted and the formula to determine the Risk Weighted Assets is given by(BCBS, 2005):

RWA = 12.5 ·RW ·EAD (2)

Where RW is the risk weight of the security expressed as a percentage of the expo-sure value and EAD the exposure at default. Depending on what kind of securitiesthat are assessed the risk weight formula is modified. The formula to determine RWfor securities such as enterprises is derived in Section 3.2 and it is given by (TheEuropean Unions Official Paper, 2013):

RW =(

LGD ·Φ(Φ−1(PD)+

√ρΦ−1(0.999)√

1−ρ)−LGD ·PD

)· 1+(M−2.5) ·b

1−1.5 ·b·12.5 ·1.06

(3)

Where Φ(x) is the cumulative distribution function for a standardized normal dis-tributed random variable and Φ−1(Z) is the inverse cumulative distribution functionfor a standardized normal distributed random variable. M is the maturity time forthe instruments of the credit portfolio and b is the maturity factor, calculated by:

b = (0.11852−0.05478 · ln(PD))2 (4)

.

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Furthermore, ρ is the asset correlation which is a function of PD with a minimumvalue of 0.12 and a maximum value of 0.24. It is calculated by:

ρ = 0.12 · 1− e−50·PD

1− e−50 +0.24 ·(

1− 1−e−50·PD

1−e−50

)(5)

2.4.1 Asset Correlation

In 2001 the Basel committee proposed the second consultative document (CP2)for the calculation of RWA. In the proposal the asset correlation parameter wasset to 0.2 for all firms. Criticism and extensive discussions of the AIRB approacharose and the reason for this was that especially SME were afraid of higher capitalcosts for banks that would lead to higher costs of the credits for these companies.The AIRB approach was very questionable since it gave a much higher risk weightin many cases compared to the standard approach. Enterprises with a very goodcredit rating resulted in a very low risk weight in comparison to companies with arating worse than BB- resulted in a significant higher risk weight. Since SME rarelyobtain a rating higher than BB- banks would have to hold a larger capital for suchcompanies according to CP2.

As a consequence for the criticism of the second consultative document the BaselCommittee proposed a refined version of the RWA function for the AIRB approachin 2004. Some changes for the final version of the AIRB approach were made. Theparameter asset correlation was modified to ρ(PD) as presented in Equation 5.

2.4.2 Dependence of Asset Correlation, PD and RW

As shown in Figure 2 the asset correlation ρ(PD) declines with increased valuesof PD by applying Equation 5. It equals 0.12 for the highest value of PD whilefor the lowest value of PD the asset correlation equals 0.24. The Basel II approachproposes that small firms have low asset correlation. This can be supported bytwo arguments. First, larger companies can be thought of as a portfolio of smallcompanies and therefore have a better diversification and consequently, lower PD.This means that the idiosyncratic risk would be smaller compare to the systematicrisk, and the systematic risk is related to the asset correlation. Secondly, businesssectors that are known to be very cyclical and hence depend more on the systematicrisk shows a majority of large firms. This indicates that firm size, i.e. large firmshave high asset correlations (Henneke & Trück, 2006).

11 31 may 2018


Figure 2: Dependence between ρ and PD with the formula retrieved from the regu-lations.

In Figure 3 the dependence between PD and RW is presented, by applying the riskweight formula, Equation 3. The parameter LGD is predetermined to 30%, thematurity time M is 1 and the PD takes on values between 1%-100%. The maturityfactor b and the asset correlation varies depending on the value of PD which aredetermined by applying Equation 4 and Equation 5. Increased values of PD up to31% gives an increased value of RW which can be explained by the fact that if anobligor is likely to default the creditor must hold a larger buffer in order to protectitself. It is also shown that RW is trending down with PD values over 31%. This is adrawback of the risk weight formula and the article will only focus on the increasingcurve of the risk weight since a loan to enterprises with PD values larger than 31%would be most unlikely.

12 31 may 2018


Figure 3: Dependence between RW and PD calculated by the formula retrievedfrom the legal framework.

13 31 may 2018

3 MATHEMATICAL MODELS OF CREDIT DEFAULT

3 Mathematical Models of Credit Default

This chapter describes the mathematical theories of the core models applied in thisarticle which are the risk weight formula and the three models of asset and de-fault correlation. Merton’s model is a well known one factor model which is thefoundation of the risk weight-formula defined in Section 2.4 and in this chapter wepresent the derivation of Equation 3, see Section 3.3. Further, the two models forestimating asset correlation are derived from Merton’s model, described in Section3.4 and lastly, the joint probability model to estimate default correlation is derivedin Section 3.5.

3.1 Correlation

In order to fully understand the concepts of asset and default correlation a briefreview of the definition of correlation is needed. Let X and Y be random variableswith expected values µX and µY and standard deviation σX and σY . The correlationof X and Y is defined by:

ρXY =E[(X−µX)(Y −µY )]

σX σY(6)

Note that if X =Y then the correlation is ρXY = E[(X−µX)2]/σ2

X = 1 and similarlyif X = −Y then the correlation is ρXY = −E[(X − µX)

2/σ2X = −1. On the other

hand, if X and Y are independent then ρXY = E[(X−µX)]E[(Y −µY )] = 0. In otherwords, correlation is a measure of how much two variables depend on each other,with 1 being complete positive dependence, -1 being complete negative dependenceand 0 being independence (Lucas & Douglas J., 1995).

3.2 Merton’s Model

Merton’s model is a commonly used model to describe the value of a firm, includingthe parameter asset correlation. It is an appropriate model to apply for this thesissince the risk weight formula is based on the model, derived in the next Section3.3. Another reason for the appropriateness is that this article analyzes default ofloans and from the model we can derive default likelihood functions as described inSection 3.4 and thus we can estimate asset correlations.

Conditionally independent credit risk models established from the factor-based ap-proach are popular to use by risk managers since they are among the few modelsthat can reflect a realistic interrelationship on the default motion and still enoughtractable to follow in an analytical way (De Servigny & Renault, 2002). The factormodels are adjusted with different constraints and in this article two factor modelsare applied to describe the likelihood of defaults, derived in Section 3.4.1 and 3.4.2.

14 31 may 2018


In credit risk models it is common to model default events as a discrete randomvariable Z that follows a Bernoulli law. Thereby Z can take on the value 0 or 1 andhere, Z = 1 indicates default of a firm.

Merton’s formula is a well known one-factor model to describe the value of a firm,proposed by Robert C. Merton in 1974. The value of the assets of an obligor isdriven by a common systematic risk factor Y and an obligor-specific risk factor εithat is independent across the obligors and independent of Y . Both Y and εi arestandard normally distributed and the asset value of firm i is given by:

Vi(T ) =√

ρY +√

1−ρεi (7)

Where Vi(T ) is the value of the assets at a given maturity time T . Note that bydefinition Vi(T ) is a standard normally distributed random variable, since the sum oftwo independent normally distributed random variables also is normally distributedand the coefficients are chosen such that E[Vi(T )] = 0 and E[Vi(T )2] = 1.

Furthermore, given the definition of correlation in Section 3.1, it is easy to see thatthe correlation of two asset values Vi and Vk is ρViVk = ρ while the correlation of oneasset value Vi and the systematic risk factor Y is ρViY =

√ρ . Note that in this model,

the correlation of two asset values is the same for every pair of assets. Because ofthis ρ is simply called the asset correlation. Following Gordy (2010) we will oftencall√

ρ a "factor loading".

In the setting of Merton’s model a firm will default if Vi(T ) falls below a givendefault threshold γ . If Zi is the random variable that models the default of the firmi then P(Zi = 1) = P(Vi(T ) < γ). In other words, since Vi(T ) is standard normallydistributed the probability of default is given by PD = P(Vi(T )< γ) = Φ(γ), whereΦ is the cumulative distribution function of the standard normal distribution. Ifthe systematic risk factor is known the individual conditional default probability isgiven by:

p(y) = P[Vi(T )< γ|Y = y] = Φ(γ−√ρy√

1−ρ) (8)

3.3 Derivation of the Risk Weight Formula

Let Li denote the loss of the bank if firm i defaults on its loan, then Li = LGD ·Zi,where LGD is loss given default as usual. The expected loss for the bank is thengiven by E[Li] = LGD ·PD. However, the aim of the risk weight formulas in theBasel framework is that a bank should have enough capital to cover it’s losses in99.9% of states of the economy. Let α denote the contrary quantile for Y , indicatingthat the chance for an inferior outcome of the systematic risk factor Y is 0.1%.In other word P(Y < α) = 0.001. Compare this to Figure 1, depicting expected

15 31 may 2018


loss, unexpected loss and stress loss. A bank should hold enough capital to coverexpected losses and normal unexpected losses. Conditional on this worse state ofeconomy for a single loan the probability of default is:

P[Vi(T )< γ|Y = α] = Φ(γ−√ρα√

1−ρ) (9)

Since Vi(T ) is standard normally distributed from Equation 7, we get:

PD = P(Vi(T )< γ) = Φ(γ) (10)

Hence the default threshold is given by γ = Φ−1(PD). By the same reasoning weget α = Φ−1(0.001) but using that the standard normal distribution is symmetricwe get that α = −Φ−1(0.999). Then the expected loss of one loan, given this badstate of the economy, is given by:

E[Li|Y = α] = LGD ·Φ(Φ−1(PD)+

√ρα√

1−ρ) (11)

Prior to June 2004 the risk weight formula was based on Equation 11. However, theformula was modified to only use unexpected loss, given by:

E[Li|Y = α]−LGD ·PD (12)

Furthermore the risk weight was calibrated by multiplying the unexpected loss witha maturity factor, a function depending on the maturity time M and the maturityfactor b, which resulted in the final Basel AIRB function to determine the regulatorycapital (Henneke & Trück, 2006), as seen in Section 2.4:

RW =(

LGD ·Φ(Φ−1(PD)+

√ρΦ−1(0.999)√

1−ρ)−LGD ·PD

)· 1+(M−2.5) ·b

1−1.5 ·b·12.5 ·1.06

(13)

3.4 Asset Correlation Models

In this Section we describe the two default likelihood models Binomial Likelihoodand Large Portfolio Approximation that are derived from Merton’s model. Themodels are based on different constrains and they are applied in this thesis to esti-mate the asset correlation.

16 31 may 2018


3.4.1 Binomial Likelihood

In Equation 7 we studied the asset value of a single obligor. Recall that the valueVi(T ) of firm i depends on the systematic risk factor Y and a firm specific randomvariable εi. Let us denote the number of obligors as n and defaulted obligors as d foreach year t. Each obligor’s probability of default is modeled as in Equation 8 and itis assumed that the default threshold γ and asset correlation ρ is constant across allobligors in the portfolio and over time. Where Z follows a Bernoulli law for eachcompany i. Then conditional on the systematic risk factor Y = y the probability ofd number of defaults for the portfolio is binomially distributed:

P[n

∑i=1

Zi = d|Y = y] =(

nd

)p(y)d(1− p(y))n−d (14)

Where p(y) is given by Equation 8. The probability density function of the numberof defaults is thus the expected value of the probability of d defaults conditional onthe systematic risk, by the law of iterated expectations. This implies the followinglikelihood function for the parameter ρ:

L(ρ|d,γ,n) =∫

R

(nd

)Φ(

γ−√ρy√1−ρ

)d(1−Φ(γ−√ρy√

1−ρ))n−ddΦ(y) (15)

The parameters in the Binomial Likelihood (BL) function can be estimated by max-imizing the likelihood (Gordy and Heitfield, 2010). In this article all parametersexcept the asset correlation is known and hence Equation 15 will be applied as oneof the two models to estimate the asset correlation.

3.4.2 Large Portfolio Approximation

This model is based on the theory of the law of large numbers, which implies that theactual default rate of an entire portfolio equals the individual probability of defaultfor the obligors in the portfolio. From Merton’s model the likelihood function ofdefaults can be derived, as seen in Equation 15. Conditional on the systematicrisk factor y the defaults happen independently from each other. Therefore, in avery large portfolio the fraction of the defaulted obligors X equals the individualprobability of defaults, ensured by the law of large numbers:

P[X = p(y)|Y = y] = 1 (16)

Where p(y) is given by Equation 8. If Y is known, the fraction of obligors that willdefault can be predicted by certainty. Although Y is yet unknown the probabilitydistribution function of X can be reached by the law of iterated expectations. Theprobability of the loss fraction X to result in the value x is defined as:

17 31 may 2018


P(X ≤ x) =∫

RP[X = p(y)≤ x|Y = y]φ(y)dy (17)

Where φ is the probability density function of the standard normal distribution. Thelower bound of the integral is substituted with a parameter denoted as −y∗ which isan expression corresponding p(−y∗) = x, hence:

P(X ≤ x) =∫

∞

−y∗φ(y)dy = Φ(y∗) (18)

Recalling the definition of p(y) in Equation 8 we get:

y∗ =1√

ρ(√

1−ρΦ−1(x)− γ) (19)

By combining Equation 18 and Equation 19 we can derive the probability distribu-tion function of the loss fraction X :

F(x) := P(X ≤ x) = Φ(y∗) = Φ(1√

ρ(√

1−ρΦ−1(x)− γ)) (20)

And hence, taking the derivative with respect to x, the default probability densityfunction is derived. The function depends on three parameters x, ρ and γ . Thedefault probability density function for the yearly fraction x is obtained by:

f (x) =

√1−ρ

ρ· exp{1

2(Φ−1(x))− 1

2ρ(γ−

√1−ρΦ

−1(x))2} (21)

Which then can be written as the likelihood function:

L(ρ|x,γ) =

√1−ρ

ρ· exp{1

2(Φ−1(x))− 1

2ρ(γ−

√1−ρΦ

−1(x))2} (22)

The parameters in the Large Portfolio Approximation (LPA) can be estimated bymaximizing the likelihood (Schönbucher, 2000). Here, all parameters are known aswell except the asset correlation and hence Equation 22 will be applied as a model toestimate the asset correlation. The two models BL and LPA, presented in Equation15 and Equation 22, will be the models for estimating asset correlation on the SMEportfolio, described in Chapter 5.

18 31 may 2018


3.5 Default Correlation Model

This Section describes the model for estimating default correlation which is basedon joint default probabilities. In the article we compare asset correlation with de-fault correlation and since the model is well known in modeling portfolio credit riskit is selected for the estimation of default correlation.

3.5.1 Joint Default Probability

Default correlation is built on the approach to estimate bivariate transition proba-bilities from given default rates. It is defined by discrete events of survival or non-survival for two obligors over one year. The default correlation ρd is then calculatedbased on the standard definition of correlation between two random variables as de-scribed in Equation 6 (Lucas & Douglas J., 1995):

ρd =pd− pd pd√

pd(1− pd)pd(1− pd)(23)

Where pd is the average joint probability for the number of creditors migrating todefault for the entire time period and pd equals the average default rate over theentire time period. D is the number of obligors migrating to default and N is thetotal number of obligors each year, then the joint probability for a given year iscalculated by:

D2/N2 (24)

In practice, default data is given on a yearly basis and joint probabilities are cal-culated with the same frequency. When the joint probabilities are determined theyare aggregated into an average joint probability for the observation period, with theassumption of each year to be an independent data set (Servigny & Renault, 2002).The formula to aggregate the probabilities to calculate the average joint probabilityis given by:

pd =n

∑t=1

Nt

∑ns=1 Ns

(Dt)2

(Nt)2 (25)

Where each year is weighted by its relative size. The formula weights the num-ber of counterparties each year to the total number of counterparties of the entiretime period. Thereafter the weight is multiplied with the joint default probabilityand finally the terms are summed. When the average joint probability has beendetermined the default correlation is calculated by applying Equation 23. In thisderivation of default correlation only one risk class is considered since the analyzedSME portfolio is assumed homogeneous. But one should notice that the model isapplicable for portfolios with different risk classes as well (Servigny & Renault,

19 31 may 2018


2002). We will apply Equation 23 as the model for estimating default correlation inthis article which is described in Chapter 5.

The three models described in Section 3.4 and 3.5 will be applied as the modelsfor estimating asset and default correlation in this article. To achieve an optimalestimation of asset and default correlation the models are combined with statisticalmethods which are described in the next chapter.

20 31 may 2018

4 STATISTICAL METHODS

4 Statistical Methods

In this chapter we emphasize the statistical models applied in this article which canbe seen as a complement to the models described in the previous chapter in orderto achieve optimal results. As depicted in Section 1.5 default data is scarce andthus we execute Principal Component Analysis and Regression Analysis to predictlonger time series, presented in Section 4.1 and 4.2.

Maximum Likelihood Estimation is described in Section 4.3 which is applied toapproximate asset correlation in Equation 15 and 22. Further, we execute Monte-Carlo Simulations for two purposes. First we apply it for validation of the twomodels and secondly to evaluate the difference of asset and default correlation andthe method is explained in Section 4.4. The chapter also includes three validationtests which are presented in Section 4.5.

4.1 Principal Component Analysis

Principal Component Analysis (PCA) is a dimension reducing method to obtain alower dimensional representation of a data set. It finds a simplified representation ofthe data by tracing where it contains as much variation as possible. In this article weapply PCA to simplify the modeling of the ten macro-economic variables describedin Section 1.5. By conducting the method it is easier to estimate new data sets withthe multivariate regression analysis which is described in Section 4.2.

To explain the principal components (PC), let a data set contain n observations ina set of p features, X1,X2, · · · ,Xp. Not all p dimensions are of equal interest, thatis why the first PC, Z1, has a normalized linear combination of X1,X2, · · · ,Xp, withthe largest variance. The first PC, Z1, is presented below:

Z1 = φ11 ·X1 +φ21 ·X2 + · · ·+φp1 ·Xp (26)

The components φ11,φ21 · · ·φp1 are referred as loadings for the first principal com-ponent, i.e eigenvectors of ordered sequence of the matrix XT X and the eigenvaluesare the variance of the components. The loadings are a loading vector for the firstprincipal, φ1 = (φ11,φ21 · · ·φp1)

T . By normalizing the loadings we avoid promptlylarge loadings, which in turn could lead to promptly sizable variance, hence:

p

∑j=1

φ2j1 = 1 (27)

21 31 may 2018


The components z11 · · ·zn1 are referred as scores of the first principal component:

Z1 =

zi1 = φ11 · x11 +φ21 · x12 + · · ·+φp1 · x1p...

zn1 = φ11 · xn1 +φ21 · xn2 + · · ·+φp1 · xnp

, (28)

After the first Z1 is estimated the second Z2 can be estimated. The second principalcomponent is uncorrelated with the first principal component and it is also a linearcombination of X1,X2, · · · ,Xp that has the most variance compared to all other linearcombinations. This process is repeated until all principal components for the dataset are estimated, at most min(n− 1, p) for Zi. However the first few principalcomponents are the most representative to the data (James et al., 2013).

4.2 Multivariate Regression Analysis & Ordinary Least Squares

To predict longer time series of default data we apply Multivariate Regression Anal-ysis. It is a model for describing the dependence between two or more variableswith a principal to construct a model that best suits the observed data. The methodis commonly used for making forecast and the linear regression model is given by:

Yt = α +Xt ·β + εt , t = 1, · · · ,T. (29)

Where Yt is the response on the forecaster variable Xt at time t. In this article we willassign Xt to represent macro-economic variables as described in Section 1.5, both byutilize principal components by applying Equation 26 and by implement individualmacro-economic variables which is described in Section 5.1.2. The parameter β isthe slope term coefficients, α is the intercept constant coefficient and εt is the errorterm at time t. The form of the variables are:

Y =

y1...

yn

,X =

x1,1 · · · x1,k... . . . ...

xn,1 · · · xn,k

,β2 =

α

β1...

βk−1

,ε =

ε1...

εn

(30)

Where k is the number of macroeconomic factors and n is the number of observa-tions. The coefficients α and β are estimated by the ordinary least squares (OLS)method. A new matrix X∗ is first considered with an extra column of an one-vectorfor the OLS calculation:

X∗ =

1 x1,1 · · · x1,k...

... . . . ...1 xn,1 · · · xn,k

, (31)

22 31 may 2018


With the new matrix X∗ the OLS calculation can be estimated by the formula (Mc-Neil et al., 2005):

(α

β

)= (XT

∗ ·X∗)−1 ·XT∗ ·Y. (32)

4.3 Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a general method for parametric esti-mation. The main idea behind the method is to find the parameters of a statisticalmodel that maximizes a likelihood function given a set of observations. In this arti-cle we will maximize the two likelihood models in Equation 15 and 22 to estimatethe asset correlation assigned as the parameter ρ .

Consider a random vector X = (X1,X2, · · · ,Xn) of iid components, describing a setof data where n is the number of observations, with joint probability density func-tion fX(x;θ) indicated by a parametric vector θ = (θ1,θ2, · · · ,θp), p number ofparameters, for some unknown value of θ .

Given the data, the likelihood function for the unknown parameters θ is L(θ ;X) =fX(x;θ) and the maximum likelihood estimator θ is the value that maximizes thelikelihood function. Equivalently by maximizing the log-likelihood function l(θ ;X)=ln L(θ ;X). For large n the estimated parameters are expected to be close to the realvalue of θ . For the vector X with univariate density f the log-likelihood function isgiven by (McNeil et al., 2005):

ln L(θ ;X) = lnn

∏i=1

f(Xi;θ) =n

∑i=1

ln L(θ ;Xi) (33)

The implementation of Equation 33 is described in Section 5.2 where we applyMaximum Likelihood Estimation to approximate the asset correlation on the SMEportfolio.

4.4 Monte-Carlo Simulation

Monte-Carlo Simulation (MSC) is the general name for mathematical algorithmsthat are based on random numbers. In this paper we will execute MSC to performa validation of the two likelihood models in Equation 15 and 22. It will also beutilized for evaluation of the difference between the parameters default and assetcorrelation which is described in Section 5.4.

Monte-Carlo Simulation is the general name for mathematical algorithms that arebased on random numbers. Let X be a random vector, X = (X1, · · · ,Xn) with den-

23 31 may 2018


sity function f (x1, · · · ,xn) and a given function g(x). Suppose the objective is toestimate the expected value of the function, calculated by:

E[g(X)] =∫ ∫

· · ·∫

g(x1, · · · ,xn) f (x1, · · · ,xn)dx1 · · ·dxn (34)

To estimate E[g(X)], a random vector X(1) is generated with joint density function f ,and Y (1) = g(X(1)) is calculated. The procedure is repeated i times, until r numberof iid random variables Y (i) = g(X(i)) where i = 1, ..,r, have been generated. Thestrong law of large numbers (SLLN) is given by:

limr→∞

Y (1)+ · · ·+Y (r)

r= E[Y (i)] = E[g(X)] (35)

Where the expected value of g(X) thus can be estimated by the mean value of thegenerated Y (1) · · ·Y (r) (Ross, 2010).

4.5 Validation Tests

To enhance the credibility of the predicted default data, described in Section 5.1,we apply three different validation measurements. To test stationarity of the macro-economic variables the Augumented Dickey-Fuller test is applied and to verify thepredicted default data we use Mean Absolute Error and Root Mean Square Error.

4.5.1 Augmented Dickey-Fuller Test

The Augmented Dickey-Fuller test examines if a data set has a trend or not andin this article it is applied to evaluate if the macro-economic variables in Table4 are stationary. The purpose is that in the execution of the regression analysis,as described in Section 4.2, it is important to implement stationary variables, i.edata with no trend to achieve the most accurate results. The test assumes the nullhypothesis for each variable to have a unit root and the equation for testing this isgiven by:

∆yt = α +δ t +βyt−1 + c1∆yt−1 + · · ·+ cp∆yt−p + εt (36)

Where α is a constant, the β is the coefficient on a time movement and p the lagorder of the autoregressive process. When implementing the regression to the dataand finding β=0, it implies that the data have a random walk with a trend, hence thedata have a unit root (Dickey and Fuller, 1979). Contrary to this β=1 implies thatthe data is stationary and this is what we aim to achieve for the macro-economicvariables.

24 31 may 2018


4.5.2 Mean Absolute Error

The Mean Absolute Error (MAE) is a method that measures the spread betweenpredicted data y and the actual data y by taking the average absolute value of then observations. When estimating longer time series of default data, as we do inthis paper by conducting regression analysis described in Section 4.2, it is of highimportance to verify if the data is realistic. Hence, MAE is applied and the formulais given by (Chai & Draxler, 2014):

MAE =1n

n

∑j=1| y j− y j | (37)

4.5.3 Root Mean Square Error

As described in the previous Section 4.5.2, we have a corresponding method tomeasure the spread between predicted data y and the actual data y which is the RootMean Square Error (RMSE). This test is also applied for the predicted time series ofdefault data in this article. The formula measures the spread by taking the squaredroot of the mean of the squared discrepancy, for all data points n. The formula isgiven by (Chai & Draxler, 2014):

RMSE =

√1n

n

∑j=1

(y j− y j)2 (38)

25 31 may 2018

5 METHOD

5 Method

The aim of this chapter is to explain the model implementation in this thesis. Be-fore the asset and default models are implemented we predict new time series ofdata and the approach for this is explained. When the data is validated asset anddefault correlations are estimated and this chapter describes the assumptions of theimplementation for the three approaches. As a supported argument of the differ-ence between asset and default correlation we execute a Monte-Carlo Simulation toestimate default correlation. Lastly, in order to evaluate if the selected models arecredible, a validation is performed for the two likelihood models.

5.1 Predicting Data

We know from Section 1.5 that the historical default data provided by Handels-banken is scarce and thus it is of interest to predict longer time series. Increasedtime series imply more accurate results in the estimation of correlation (Gordy &Heitfield, 2010). For the data prediction we conduct a Multidimensional Regres-sion Analysis as explained in Section 4.2 by using historical default rates of theSME portfolio. Before the regression model is implemented it is of high importanceto select relevant underlying variables and thus the ten macro-economic variablespresented in Table 4 in Section 1.5.2 are chosen for this purpose.

Before the new data sets are predicted we must assure that the macro-economicvariables are stationary. Data that is non-stationary are unpredictable and cannot beforecasted since the results obtained by such data may indicate time series with arelationship between two variables where one does not actually exist (Investopedia,2018d). In Table 4 the macro-economic variables are presented in different unitshowever, an adequate approach to retrieve stationary data is to transform it to per-centage differences. That is calculated by Pt−Pt−1

Pt−1, where Pt represents the value of

the variable at time t and Pt−1 represents the value with one lag.

We transform the ten macro-economic variables to percentage differences and applythe Augmented Dickey–Fuller test, described in Section 4.5.1, to test stationarity.As expected the variables reflects stationarity which is confirmed by the test, seeTable 5, where β=1 indicates stationary variables.

26 31 may 2018

5 METHOD

Table 5: Macro-economic variables tested with the Augmented Dickey-Fuller test.

Variables β StationaryExport 1 YesImport 1 YesGDP 1 Yes10Y Government Bond, (10Y) 1 YesConsumer Price Index, (CPI) 1 YesPrice Base Index, (PBI) 1 YesOMX30 1 YesUnemployment, (UP) 1 YesInflation-adjusted property index, (IAPI) 1 YesTCW-index, (TCW) 1 Yes

5.1.1 PCA

To simplify the management of the ten macro variables, a Principal ComponentAnalysis is conducted as described in Section 4.1. Before the implementation ofPCA the macro-economic data is normalized due to the fact that data is not oftenconsistent and thus, by using the normalization formula:

Z =(Y −U)

σ(39)

Where Y is the macro-economic data, U the mean value of for each macro-economicvariable and σ is the standard deviation of each variable. In Table 6 the normalizeddata of the ten macro-economic variables are presented.

27 31 may 2018

5 METHOD

Table 6: Normalized macro-economic data by applying Equation 39.

Year Export Import GDP 10Y CPI PBI OMX30 UP IAPI TCW1990 -0.37 -0.14 2.07 3.20 2.18 0.88 1.00 0.09 0.34 0.181991 -0.77 -1.42 0.99 2.77 3.11 -1.61 -0.09 2.39 -0.64 -0.151992 -0.97 -0.98 -1.50 0.10 1.33 0.08 -0.52 2.68 -1.66 -0.391993 1.32 0.39 -1.41 0.99 0.11 -0.08 0.42 2.05 -3.19 3.421994 1.53 1.46 0.82 0.06 0.23 1.55 -0.91 -0.12 -0.33 -0.001995 1.87 1.22 1.25 0.19 -0.19 -0.36 2.21 -0.53 -0.58 -0.051996 -0.92 -0.96 -0.65 -0.58 -0.20 0.27 -0.71 0.00 -0.70 -1.601997 0.92 1.03 0.03 -0.57 -0.73 0.86 -0.45 -0.17 0.30 0.421998 0.10 0.59 0.21 -0.80 -0.73 0.70 -0.38 -0.96 0.63 0.171999 -0.04 0.02 0.38 -0.60 -0.86 0.30 -0.86 -0.94 0.89 0.012000 1.16 1.54 0.68 -0.38 -0.60 2.08 1.91 -0.94 0.87 -0.222001 -0.27 -0.56 -0.11 0.16 -0.47 -1.00 -0.45 -0.80 1.04 1.282002 -1.03 -1.01 -0.25 0.04 0.41 -1.24 0.46 -0.22 -0.37 -0.472003 -0.72 -0.54 -0.08 -0.02 0.00 -1.93 -0.20 0.13 0.21 -0.912004 0.67 0.28 0.11 -0.61 -0.00 0.49 0.14 0.22 0.67 -0.402005 0.45 0.97 -0.27 -0.60 -0.74 0.14 -0.43 -0.17 0.66 0.102006 0.82 0.76 0.76 -0.24 -0.50 0.63 -0.29 -0.57 1.35 -0.312007 0.20 0.54 0.69 0.07 -0.15 0.32 0.80 -0.84 0.40 -0.462008 -0.09 0.17 -0.59 0.55 -0.04 -0.81 0.35 -0.36 0.35 0.042009 -2.66 -2.64 -2.60 -0.88 1.20 -1.79 -0.94 1.32 -0.90 1.462010 0.51 0.73 0.92 -0.28 -1.30 1.02 0.91 -0.14 0.90 -1.442011 0.12 0.14 -0.19 0.23 -0.42 0.37 0.09 -0.69 -0.25 -1.032012 -0.91 -0.94 -1.29 -0.42 0.45 -1.04 -1.54 -0.24 -1.22 -0.382013 -1.13 -1.11 -0.74 -0.78 -0.32 0.02 0.39 -0.24 -0.07 -0.682014 0.04 0.34 -0.00 -0.83 -0.97 0.18 1.32 -0.34 0.38 0.592015 0.18 0.06 0.79 -0.78 -0.75 -0.06 -2.22 -0.56 0.88 0.85

After the data is standardized as presented in Table 6, a principal component analy-sis is performed. Since we select ten macro-economic variables the PCA returns tenprincipal components (PC) where each component describes how much it representsthe macro-economic data. In Table 7 this is presented and we can see that the firstprincipal component represents the most and the tenth represents the least. Handels-banken suggest the use of two PC’s as underlying variables for the data predictionand thus PC1 and PC2 are selected since those represents the macro-economic datathe most with 41.64% respectively 21.20%.

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Table 7: The PC’s representation of the ten macro-economic variables.

PC RepresentationPC1 41.64%PC2 21.20%PC3 14.90%PC4 7.82%PC5 6.72%PC6 3.45%PC7 2.21%PC8 0.91%PC9 0.87%PC10 0.23%

5.1.2 Regression Analysis

The principal components together with the historical default rates are used for theimplementation of the Multidimensional Regression Analysis. The coefficients α ,β1 and β2 are estimated for the Multidimensional Regression Analysis based on thehistorical data and the error term εt is assumed zero. To achieve coefficients withpreferably small error marginal the OLS method is applied, see Section 4.2. The es-timated coefficients together with the principal components are used to predict newtime series of default data for 2005-2015, 1990-2004 and 1991-1994. The Multidi-mensional Regression Analysis is repeated two more times however, now by usingindividual macro-economic variables instead of principal components to predict thedata sets. One approach is to select GDP and unemployment as underlying variableswhile another approach is to select GDP and OMX30, proposed by Handelsbanken.The data is predicted for the same time series.

Although default data for the time period 2005-2015 is estimated, we only utilizeit as a validation to the historical data. The time period 1990-2004 is combinedwith the historical data in order to get the data set 1990-2015. The default dataof 1991-1994 is estimated to evaluate how the correlations are affected during afinancial crisis and when only using a few observations. Thus, in this article wewill analyze seven different data sets, described in Table 8, to estimate asset anddefault correlation. We refer the predicted data sets as "Data 1", "Data 2" and "Data3" depending on the two underlying variables utilized in the regression analysis andeach data set is assumed homogeneous.

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Table 8: The different data sets analyzed in this project sets.

Data sets Macro-variables Time periodHistorical Data - 2005-2015Data 1 -Full set PC1 & PC2 1990-2015Data 1 -Extreme PC1 & PC2 1991-1994Data 2 -Full set GDP & Unemployment 1990-2015Data 2 -Extreme GDP & Unemployment 1991-1994Data 3 -Full set GDP & OMX30 1990-2015Data 3 -Extreme GDP & OMX30 1991-1994

To validate the predicted data sets of 2005-2015 it is compared to the historicaldefault data. The spread between the different trajectories are measured by theMean Average Error (MEA) and Root Mean Square Error (RMSE), as describedin Section 4.5. The trajectories are also evaluated graphically and for a furthervalidation it is of interest to interpret how the trajectories move over a longer timeseries. Hence the "Full" data sets are graphically illustrated over the time period1990-2015 and compared to the number of bankruptcies for Swedish enterprisesover the same time period, see Figure 6 in Section 6.1.

5.2 Estimating Asset Correlation

In this article we estimate asset correlation on the SME portfolio by fitting the de-fault data to two likelihood functions with different constraints. The models arederived from Merton’s formula and presented in Equation 15 and 22 and by apply-ing Maximum Likelihood Estimation we can approximate the asset correlation.

5.2.1 Binomial Likelihood

In this approach the Binomial Likelihood (BL) function is applied. For this modelthe number of defaults and the number of obligors each year must be known, inorder to fit the likelihood function, as explained in Section 3.4.1. In the SME port-folio provided by Handelsbanken this is presented whereas for the predicted datasets from the regression analysis we only retrieve default rates. Therefore, the pre-dicted data sets are transformed into yearly number of obligors and defaults.

The likelihood function is an integral function including the parameters γ , assetcorrelation ρ and the realized number X in R. The default threshold γ is determinedas the average default rate and the start value of the factor loading

√ρ = 0.1. The

BL model is fitted into a maximum likelihood model to estimate the asset correlationρ which is approximated for all seven data sets.

The Binomial Likelihood function is a sensitive model and therefore some assump-tions are made. When data fitted into the distribution contains large numbers of

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either obligors or defaults the model is infeasible. This is because the binomial of nand d becomes very large, see Equation 14, which is a problem in numerical calcu-lations. Since the SME portfolio is a large portfolio we assume that the binomial ofn and d equals one. Because the integration is a linear operation and the binomial isconstant with respect to x and w, this assumption is appropreiate and makes it easierto handle large numbers. Another assumption for the model to work appropriatelyis to scale down the yearly number of obligors and defaults since too large numbersin the likelihood function imply values too close to zero to handle numerically in theintegral. MLE is implemented on the likelihood function to estimate the unknownparameter asset correlation ρ which is approximated for the seven data sets.

5.2.2 Large Portfolio Approximation

In this approach the Large Portfolio Approximation (LPA) is applied and in thismodel yearly default rates are used hence, the data is not transformed. Since theSME portfolio is a relatively large portfolio the assumption of a large uniform port-folio is appropriate to reflect the SME portfolio, see Section 3.4.2. The distributionfunction of LPA includes the parameters asset correlation ρ , yearly default rate xand the default threshold γ which is determined as the average default. The twolatter parameters are known.

For the predicted data sets the yearly default rates x are retrieved from the regressionanalysis and for the historical data provided by Handelsbanken x is calculated bythe number of defaults divided by number of obligors on a yearly frequency. Theaverage default rate p for each data set is calculated by the mean value of the defaultrates. MLE is implemented on the distribution function to estimate the unknownparameter asset correlation ρ which is approximated for the seven data sets.

5.3 Estimating Default Correlation

To estimate default correlation the joint default probability (JDP) model is appliedon the SME portfolio. In Section 3.5.1 it is explained that in order to calculate theaverage joint default probabilities the number of obligors and defaults each yearmust be known. Hence, the predicted data sets are transformed into yearly numberof obligors and defaults. The average default rate is calculated by the total numberof defaults over the entire time period divided by the total number of obligors. Thiscalculation is done for all seven data sets and accordingly the average joint probabil-ity is calculated by Equation 25. Thereafter we can estimate the default correlationfor all data sets by applying Equation 23.

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5.4 Simulation of Default Correlation

As one objective of this article is to verify the difference of asset and default cor-relation we emphasize this with a Monte-Carlo study as a further argument. Tocompare asset correlation with default correlation MSC is implemented where thebasic idea is to set the asset correlation to a predetermined value to simulate assetreturns with a random multivariate normal distribution.

The asset values are compared to the default threshold γ = 0 which implies defaultfor negative asset returns. A binomial matrix is created with values of one if theasset value is considered default and values of zero if the asset value is considerednon-default, hence default probabilities can be derived. The JDP model then cal-culates default correlation. The number of simulations are 10,000 and the test isexecuted five times with predetermined asset correlations of 0.1, 0.2,..., 0.5. Thestudy implies that asset correlation is higher than default correlation and the resultsare presented in Table 11 in the Section 6.4.

5.5 Validation of Likelihood Models

To validate the Binomial Likelihood function and Large Portfolio Approximationmodel, which are derived in Section 3.4, we implement MSC to create syntheticdata sets. Asset values are simulated by Merton’s model, see Section 3.2, where thefactor loading

√ρ is set to 0.45 and ε is assumed to be random normal distribution

variable. The Y is assumed to be a constant risk factor for all obligors which is anormally distributed random variable.

The Monte-Carlo Simulation is first implemented by changing values of the param-eter time to maturity, T = 20,40,60,80. The data sets are assumed to have 500companies each year and are simulated 500 times. To evaluate how many of thesimulated companies that defaults each year the asset values are compared to a de-fault threshold γ = Φ−1(0.05). The asset correlation is then estimated with the twomodels by MLE and in order to evaluate how the models react to different timeseries (Gordy & Heitfield, 2010).

Secondly, a Monte-Carlo Simulation is implemented by changing the number ofsimulations, S = 50,250,400,700 and set a fixed time to maturity, T = 20 with500 companies each year and the same default threshold as earlier to calculate thedefault rates. The asset correlation is estimated with the two likelihood functionsby MLE to evaluate how the models react to increased number of simulations. TheLPA model is sensitive to very small or zero number of defaults and thus, for theMonte-Carlo Simulation only years with number of defaults equal to or greaterthan one are assumed feasible for the estimation. This assumption is necessary forthe model to be able to execute the validation. The two MCS aim to achieve anestimated factor loading of

√ρ= 0.45.

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6 Results

This chapter presents the results of this project and first, it covers a validation of thedata prediction. In Section 6.3 the results of the estimated asset and default correla-tions are presented and thereafter the parameters are compared to the limitations ofthe asset correlation in the risk weight formula by applying Equation 3. The chapteralso includes the differences between asset and simulated default correlation as wellthe validation of the binomial likelihood function and large portfolio approximation.Since the SME portfolio contains confidential information some graphs and tablesare empty in respect to this, however the reader can still follow the chapter sincetrends and observations of graphs and tables are presented in the text.

6.1 Data Validation

In Figure 4, a comparison between the historical default data provided by Han-delsbanken and the predicted data is presented for the time series 2005-2015. Thedata is predicted by regression analysis and the three different data sets presentedin Section 5.1.2 are analyzed. "Data 1" is based on the underlying variables PC1 &PC2, "Data 2" is based on GDP & Unemployment and "Data 3" is based on GDP &OMX30. In Figure 4 we can see that the trajectories of the three predicted data setsare a clear representation of the historical trajectory.

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Figure 4: Comparing the data given by the bank against estimated data.

After a graphically inspection in Figure 4 we evaluate the comparison of the histor-ical default data and the three simulated data sets of the SME portfolio with RootMean Square Error (RMSE) and Mean Absolute Error (MAE), described in Section4.5. The models aims to estimate the error measures and in Table 9 the performanceof the three data sets are presented. The models are ranked by the order one to threewhere one represents the least error estimation.

Table 9: Error estimation of the predicted data sets.

Error Methods Data 1 Data 2 Data 3RMSE 3 1 2MAE 3 1 2

In Table 9 we can see that "Data 2" is the best performing data set since it has theleast error for both RMSE and MAE, although the difference between the three datasets is not significant. In Figure 5 the three trajectories of the "Full set" is presented.The "Full set" includes estimated default data from 1990-2004 combined with thehistorical data 2005-2015 to get the time series 1990-2015, see Section 5.1.2 for afurther description.

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Figure 5: The three trajectories of the "Full set" data, 1990-2015.

When assessing Figure 5, we can see that the three estimated data sets follow a sim-ilar pattern for the entire time series and that the difference is marginal. For a furthervalidation we compare Figure 5 to the statistics of the number of bankruptcies forthe same time series, as seen in Figure 6. This emphasizes that the trajectories of theestimated data sets in Figure 5 follow the trajectory of the number of bankruptcies inFigure 6. The two figures have similar movements and particularly in the beginningof the 1990s where default rates peak as well as the number of bankruptcies. "Data2" follows the pattern of the bankruptcies slightly closer however, the performanceof the three different data sets are significantly small.

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Figure 6: Number of enterprise bankruptcies in Sweden (Ekonomifakta, 2018).

6.2 Estimated Asset and Default Correlation

In this Section the estimated asset and default correlations for the SME portfolioare presented, see Table 10, which are represented by the different data sets. Thehistorical data covers 2005-2015, provided by Handelsbanken while the "Full set"covers 1990-2015 and the "Extreme" represents 1991-1994, see Section 5.1.2 fora further description. The models applied for the asset correlation estimations areBinomial Likelihood (BL) and Large Portfolio Approximation (LPA) and for thedefault correlation estimation the Joint Default Probability (JDP) model is applied.The implementation of the models can be find in Section 5.2 and 5.3.

Table 10: Estimated asset and default correlation based on different data sets.

Data sets Default correlation(JDP)

Asset correlation(BL)

Asset correlation(LPA)

Historical DataData 1 -Full setData 2 -Full setData 3 -Full setData 1 -ExtremeData 2 -ExtremeData 3 -Extreme

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In Table 10 it is shown that the JDP model generates similar default correlationsfor the historical, "Full" and "Extreme" data sets. The result from the LPA modelgenerates a wide range of estimated asset correlations although, for the "Full" setsthe values are similar. The same situation follows for the BL model however, the"Data 2-Full set" generates a significant higher asset correlation. It is seen that theLPA and BL model follows a similar trend depending on the data sets. When wecompare the three models in Table 10, it is emphasized that both LPA and BL resultin higher asset correlations compare to the default correlation of the JDP model,which is the case for all data sets.

6.3 Correlation Coefficient in the RW-Formula

In Figure 7 it is shown how the risk weight calculated by Equation 3 behaves whenthe correlation coefficient takes on different values, calculated with a constant LGDof 30% and historical default rates provided by Handelsbanken. We know fromEquation 4 that the asset correlation has a minimum value of 0.12 and a maximumvalue of 0.24 in the legal framework. This can be seen as a benchmark for a com-parison to the risk weight calculated with the estimated correlations in Table 10.

Even though it has been proven in this article that asset and default correlationare two different measurements, it is of interest from Handelsbanken to test bothcorrelations in the RW formula and compare it to the risk weight based on theasset correlation of 0.12 and 0.24. As seen in Table 10 the estimated asset anddefault correlation from the three models JDP, BL and LPA are significantly lowercompared to the benchmark of 0.12 and thus, it implies significantly lower riskweights which is shown in Figure 7.

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Figure 7: The risk weight of the SME portfolio calculated with historical defaultrates and different correlations.

6.4 Asset and Simulated Default Correlation

Asset correlation is predetermined to 0.1, 0.2, 0.3, 0.4 and 0.5 and from this assetvalues and default rates are generated using 10,000 simulations, see Section 5.4 fora further explanation. Default correlation is then calculated, see Table 11, and wecan see that default correlation is lower than asset correlation.

Table 11: Simulated default correlation derived from different asset correlations.

Asset correlation Default correlation10 % 3.39 %20 % 8.98 %30 % 14.78 %40 % 20.49 %50 % 27.31 %

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6.5 Validation of Likelihood Functions

In this Section the validation of the Binomial Likelihood and Large Portfolio Ap-proximation models is presented. It is performed by Monte-Carlo Simulations usingdifferent assumptions and in Section 5.5 the implementation of the validation is de-scribed in more detail.

6.5.1 Monte-Carlo with Different Maturity

Recall from Section 3.2 and Equation 7 that the parameter ρ is the asset correlationbetween two obligors. The sensitivity to the systematic risk factor

√ρ is often called

"factor loading". Figure 8 shows the distribution of factor loadings estimated by thelarge portfolio approximation and the binomial likelihood function for different timeseries. The distribution is a skewed normal distribution with true value 0.45 of thefactor loading. Figure 8 shows that data sets with higher time series approaches thereal value 0.45 for both the LPA and BL models.

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Figure 8: Distribution of estimated factor loadings for the LPA model (left figure)and BL model (right figure) with different time series and 500 simulations.

The average value of the factor loadings of the two models is presented in Table 12.This leads to a further indication that increased amount of yearly data converges tothe true value with a slightly underestimation. However, the Binomial Likelihoodmodel results in a slightly closer value then the Large Portfolio Approximation.

Table 12: Average asset factor loadings of different time series for LPA & BL.

T=20 T=40 T=60 T=80Factor loading (LPA) 0.4346 0.4398 0.4428 0.4457Factor loading (BL) 0.4438 0.4469 0.4482 0.4511

6.5.2 Monte-Carlo with Different Number of Simulations

Figure 9 shows the distribution of the factor loadings estimated by the Large Portfo-lio Approximation and the Binomial Likelihood function with different number ofsimulations. For this type of simulations the distribution is a skewed normal distri-bution with the true value of 0.45 and it implies that a larger number of simulationsapproaches the real value of the factor loadings for both models.

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Figure 9: Distribution of estimated factor loadings for the LPA model (left figure)and BL model (right figure) with different number of simulations and time seriesT=20.

In Table 13 the average value of the factor loadings for the different number ofsimulations are presented which is a further indication that the values converge to0.45 with an increased number of simulations. For this approach, the BL model ismarginally closer to the real value in comparison to the LPA model.

Table 13: Average asset factor loadings with different number of simulations forLPA & BL.

S=50 S=250 S=400 S=700Factor loading (LPA) 0.4182 0.4331 0.4345 0.4362Factor loading (BL) 0.4685 0.4406 0.4440 0.4501

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7 DISCUSSION

7 Discussion

This chapter emphasizes the advantages and disadvantages of the models applied inthis article. The results of asset and default correlation are discussed and explana-tions of trends are presented. Further, we assess how internal models for estimatingasset correlation affect the RW-formula. The reader is provided by a conclusion ofthe project as well as further developments.

7.1 Modeling Data

Multivariate linear regression analysis as described in Section 4.2 is a commonmethod for predicting data and it was selected for this thesis since the historicaldata is assumed to have a linear relationship. If too many coefficients are used inthe regression analysis it is a risk to over fit the model and hence two coefficientsreflecting macro-economic variables the most were chosen for a realistic estimation.Although the model is suitable for the SME portfolio the default data is scarce and itis therefore important to remember that it can imply biased results of the predicteddata.

From the validation of the predicted data in Section 6.1 it is shown in Figure 5and 6 that all data sets follow the pattern of the historical data as well as the num-ber of bankruptcies. This in combination with the well performance of the errormeasurements in Table 9 a conclusion can be drawn that all data sets reflects theSME portfolio. When analyzing the individual data sets we can see that "Data 2"performs slightly better compared to "Data 1" and "Data 3".

The explanation of the difference between "Data 2" and "Data 3" which are basedon individual macro-economic variables can be described by the fact that one of theunderlying macro-economic variables for "Data 3" is the index OMX30. The indexreflects the 30 largest companies in Sweden and thus, might not be the most suitablerepresentation of the SME portfolio since it contains small and medium enterprises.As mentioned, all three models follow the trajectory well however, "Data 2" and"Data 3" are only built on two macro-economic variables while "Data 1" representsten variables and thus, is a more realistic estimation since it reflects the general stateof economy.

7.2 Analysis of Estimated Correlation

The results of the estimated default and asset correlation for the SME portfolioconfirms that default correlation is lower than asset correlation, seen in Table 10.This is the situation for all data sets and is further confirmed by the Monte-Carlostudy in Section 5.4 together with Moody’s Modeling Methodology (2008). Theestimated asset correlation for the SME portfolio indicates generally low values.

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This can be explained by the fact that the portfolio contains small and mediumenterprises and as described in Section 2.4.2 the Basel II approach proposes thatsmall firms have low asset correlation. However, the highest asset correlation areestimated for the historical data and this can be explained by the subprime crisis of2007-2008. In a crisis, the asset correlation is known to increase (Koesterich, 2015).For the "Extreme" data sets the correlation is volatile and this can be explained bythe small data set, with only four years of data. This is an indication of that smalldata sets does not generate as reliable results.

In the validation of the Binomial Likelihood and Large Portfolio Approximationmodel in section 6.5 it is shown that both models somewhat underestimate the assetcorrelation although the BL model approaches the real value slightly faster withincreased time series and simulations compared to the LPA model. For the SMEportfolio the LPA model results in marginally higher estimated asset correlationscompared to the BL model, except for the "Data 2-Full set" where the BL modelgenerates a significantly higher value, seen in Table 10. The BL model is sensitiveto large numbers of obligors and defaults since it implies that the log-likelihoodfunction tends to infinity. Hence, the BL model is infeasible for large portfolios.The SME portfolio is a large portfolio and thus, in order to overcome this flawthe data must be scaled to fit the model. This has an impact of the results andaccordingly, an uncertainty in the estimated asset correlations. The high value ofthe asset correlation for the "Data 2-Full set" can be seen as an illustration of thisflaw.

The LPA model doesn’t have this restriction and the model assumes an infinitivelylarge portfolio. Although the SME portfolio doesn’t tend to infinity, it contains alarge number of obligors and hence, the model is suitable. This is also emphasizedin the result of the estimated asset correlation since the values behave in a consistentpattern shown in Table 10. It is important to remember that the two models under-estimates asset correlation which can have an impact of the results. Even thoughBL gives a better estimation in the validation with Monte-Carlo Simulations, theLPA model is the most suitable model for estimating asset correlation on the SMEportfolio provided by Handelsbanken due to the fact that it doesn’t have flaws inhandling large numbers, which is a central property of the analyzed portfolio. An-other factor to take into consideration is that both the BL and LPA model are derivedfrom Merton’s model which is only described by one systematic factor whereas thedata of this project is described by several systematic factors. Merton’s model isvery useful however one can contemplate the credibility of the model since assetvalues most often are affected by more than one systematic factor.

7.3 RW - formula as a benchmark

The estimated asset correlations are compared to the asset correlation limits fromthe legal framework to evaluate how the risk weight formula is affected by usingdifferent correlation models. As proposed from Handelsbanken, the estimated de-fault correlation is compared as well. From the results in Section 6.3 it is shown

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that the risk weight calculated by the estimated correlation using the models JDP,LPA BL are significantly lower compared to the lower bound of 0.12 in the legalframework. Likewise, this is recognized by several empirical studies suggestingthat asset correlation in the market tends to be lower than asset correlation calcu-lated by the legal framework (Henneke & Trück, 2006). If banks would be allowedto use internal models for estimating the correlation coefficient one can concludethat the risk weight could be adjusted downwards. This is of course in interest forbanks since with less capital requirements larger capital can be released for invest-ment strategies. On the other hand, it is important to not hold too little capital incase of large default events. Significantly low correlation coefficients imply verylow capital requirements which would not protect the bank against credit losses forthe SME portfolio.

7.4 Conclusion

The predicted data for this thesis using linear multivariate regression analysis wasof great importance to gain quality and credibility for analyzing the performance ofthe asset and default correlation models. In the validation of the three data predict-ing models of "Data 1", "Data 2" and "Data 3" we can conclude that all three modelsperform well and reflects the SME portfolio. This indicates the results of the esti-mated correlation as well since the three data sets generates similar results as wellas similar results of the estimated correlation compared to the historical data. Thus,the linear multivariate regression analysis is considered a suitable data predictionmodel for the SME portfolio.

The purpose of the project was to accomplish a comparison of default and assetcorrelation of the SME portfolio by using different models as well as performing acomparison of the asset correlation in the risk weight formula. This have been eval-uated throughout the project and it is verified several times that default correlationis lower than asset correlation. From the two models of estimating asset correlationit can be concluded that the SME portfolio has a low asset correlation and that itdoes not match the value of the parameter in the risk weight formula.

As of today, banks are not allowed to use internal models for the asset correlationin the calculation of capital requirements. However for the future, if they would beallowed to estimate the asset correlation, this thesis indicates that the risk weightcould be determined much lower by using internal asset correlation models. Allthree models applied are considered representative for the SME portfolio, howeverfor the estimated asset correlation the LPA model fits the data very well and achievesmost accuracy. It can be explained by the fact the LPA model manages larger port-folios very well which reflects the SME portfolio, whereas the BL model is not assuitable. However, the BL model is better on managing smaller portfolios.

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7.5 Further development

During this thesis some development opportunities have been observed for futuredevelopment. An approach is to not assume that the portfolio is homogeneous.By taking all risk classes into account the the estimated correlation would give amore optimal reflection of the portfolio. It is also interesting to test other modelsfor the estimation of default and asset correlation as well as other macro-economicvariables for the data prediction. Lastly it would be interesting to analyze differentportfolios to assess a comparison.

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8 REFERENCES

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