An important part when managing credit risk is to assess the probability of default of different counterparties. Increases and decreases in such probabil- ities are central components in the assessment, and this is where transition matrices become useful. These matrices are commonly used tools when as- sessing counterparty credit risk, and contain the probability of default, as well as the probability to migrate between different predefined rating classifica- tions. These rating classifications are used to reflect the risk taken towards different counterparties. Therefore, it is important for financial institutions to develop accurate transition matrix models to manage predicted changes in credit risk exposure. This is because counterparty creditworthiness and prob- ability of default indirectly affect expected loss and the capital requirement of held capital. This thesis will analyze how two specific models perform when used for generating transition matrices. These models will be tested to investigate their performance when predicting rating transitions, including probability of default. / En viktig del vid hanteringen av kreditrisk är att bedöma sannolikheten för fallissemang för olika motparter. Ökningar och minskningar i dessa sanno- likheter är centrala komponenter i bedömningen, och det är här migrations- matriser blir användbara. Dessa matriser är vanligt förekommande verktyg vid bedömning av kreditrisk mot olika motparter och innehåller sannolikheten för fallissemang samt sannolikheten att migrera mellan olika fördefinierade be- tygsklassificeringar. Dessa betygsklassificeringar används för att återspegla den risk som tas mot olika motparter. Det är därför viktigt för finansinstitut att utveckla träffsäkra migrationsmatris modeller för att hantera förväntade förändringar i kreditriskexponering. Detta beror på att kreditvärdigheten hos motparter samt sannolikheten för fallissemang indirekt påverkar expected loss och kapitalkrav. Detta examensarbete kommer att analysera hur två specifika modeller presterar när de används för att generera migrationsmatriser. Dessa mod- eller kommer att testas för att undersöka hur de presterar när de används för att förutsäga övergångar inom betygsklassificering, inklusive sannolikheten för fallissemang.

Having just survived what is arguably the worst financial crisis in time, it is expected that the focus on regulatory capital held by financial institutions such as banks will increase significantly over the next few years. The probability of default is an important determinant of the amount of regulatory capital to be held, and the accurate calibration of this measure is vital. The purpose of this study is to propose the use of the Shannon entropy when determining the parameters of the prior bivariate beta distribution as part of a Bayesian calibration methodology. Various bivariate beta distributions will be considered as priors to the multinomial distribution associated with rating categories, and the appropriateness of these bivariate beta distributions will be tested on default data. The formulae derived for the Bayesian estimation of Shannon entropy will be used to measure the certainty obtained when selecting the prior parameters. / Dissertation (MSc)--University of Pretoria, 2010. / Statistics / unrestricted

The main purpose of this paper is to provide evidence of financial leverage, liquidity, profitability, and firm size ability to predict bankruptcy of Swedish small and medium-sized enterprises (SMEs), and to create a bankruptcy prediction model for Swedish SMEs. The sample consists of 1086 Swedish SMEs, among which 543 did go bankrupt between 2015 and 2019. The paper employs logistic regression and Mann-Whitney U-test to test the hypotheses. The independent variables are derived from previous research and further filtered in a selection process, resulting in a final set of six variables. Financial leverage, liquidity, profitability, and firm size is found to have significantly predictive abilities to determine SME bankruptcy. The model has an overall classification accuracy of 77.6% out-of-sample and is able to classify 82.2% of the bankruptcies correctly out-of-sample.

Master of Agribusiness / Department of Agricultural Economics / Allen M. Featherstone / Institutions of the Farm Credit System (FCS) focus on risk-based lending in accordance with regulatory direction. The rating of risk also assists retail staff in loan approval, risk-based pricing, and allowance decisions. FCS institutions have developed models to analyze financial and related customer information in determining qualitative and quantitative risk measures. The objective of this thesis is to examine empirical account data from 2006-2012 to review the probability of default (PD) rating methodology within the overall risk rating system implemented by a Farm Credit System association. This analysis provides insight into the effectiveness of this methodology in predicting the migration of accounts across the association’s currently-established PD ratings where negative migration may be an apparent precursor to actual loan default. The analysis indicates that average PD ratings hold relatively consistent over the years, though the distribution of the majority of PD ratings shifted to higher quality by two rating categories over the time period. Various regressions run in the analysis indicate that the debt to asset ratio is most consistently statistically significant in estimating future PD ratings. The current ratio appears to be superior to working capital to gross profit as a liquidity measure in predicting PD rating migration. Funded debt to EBITDA is more effective in predicting PD rating movement as a measure of earnings to debt than gross profit to total liabilities, although the change of these ratios over time appear to be weaker indicators of the change in PD rating potentially due to the variable nature of annual earnings of production agriculture operations due to commodity price volatility. The debt coverage ratio is important as it relates to future PD migration, though the same variability in commodity price volatility suggests the need implement multi-year averaging for calculation of earnings-based ratios. These ratios were important in predicting the PD rating of observations one year into the future for production agriculture operations. To further test the predictive ability of the PD ratings, similar regression analyses were completed comparing current year rating and ratios to future PD ratings beyond one year, specifically for three and five years. Results from these regression models indicate that current year PD rating and ratios are less effective in predicting future PD ratings beyond one year. Furthermore, because of the variation in regression results between the analyses completed for one, three and five years into the future, it is important to regularly capture ratio and rating information, at least annually.

Markov chains have been widely used to the credit risk measurement in the last years. Using these chains we can model movements and distribution of clients within rating grades. However, various types of markov chains could be used. The goal of the theses is to present these types together with their advan- tages and disadvantages. We focus our attention primarily on various parameter estimation methods and hypotheses testing about the parameters. The theses should help the reader with a decision, which model of a markov chain and which method of estimation should be used for him observed data. We focus our attention primarily on the following models: a discrete-time markov chain, a continuous-time markov chain (we estimate based on continuous- time observations even discrete-time observations), moreover we present an even- tuality of using semi-markov chains and semiparametric multiplicative hazard model applied on transition intensities. We illustrate the presented methods on simulation experiments and simu- lation studies in the concluding part. Keywords: credit risk, markov chain, estimates in markov chains, probability of default 1

In many applications, the success probability 𝜋 of a Bernoulli distributed variable 𝑌 is influenced by another variable 𝑋. For example for loans granted, it is necessary to rate debtors in different rating classes, where the probability of default (PD) 𝜋 of 𝑌 is assumed to be homogeneous within and heterogeneous between the rating classes. The PD of a debtor is largely influenced by macroeconomic and individual variables (𝑋). In this work, we study a Bernoulli mixture model for 𝑌, where the success probability of 𝑌 changes systematically at the breakpoints. We focus on cross-sectional data and our main objective is to estimate all 𝑘 breakpoints with 𝑘 either known or unknown and their corresponding success probabilities between each pair of neighbouring breakpoints. To the best of our knowledge, an estimator for estimating multiple breakpoints has not yet been developed in this context. Thus, we develop an approach with a view to closing this research gap. We show that our estimator works for independent and identically distributed (i.i.d.) 𝑋 as well as for a linear one-factor model for 𝑋. A theoretical foundation for this estimator is also presented. In practice, the number of breakpoints 𝑘 is often unknown a priori. As the multiple estimator is based on an iterative procedure, we propose stopping criteria for estimating 𝑘 correctly. We conduct a simulation study in the context of credit rating to demonstrate the performance of the developed estimator. Furthermore, we apply the new estimator on credit risk data from the Sächsische Aufbaubank, the Development Bank of Saxony. To simplify the use of the new estimator, we also develop an R package called MultipleBreakpoints.

Este trabalho tem por objetivo discutir as metodologias de estimação da PD utilizadas na indústria financeira. Além disso, contextualizar a aplicação do trabalho ao IFRS9 e seu direcionamento para o tema de Risco de Crédito. Historicamente os grandes bancos múltiplos utilizam variadas metodologias econométricas para modelar a Probabilidade de Descumprimento (PD),um dos métodos mais tradicionais é a regressão logística, entretanto com a necessidade do cálculo da Perda Esperada de Crédito através do IFRS9, se torna necessário mudar o paradigma de estimação para uma abordagem forward-looking, isto está sendo interpretado por muitas instituições e consultorias como a inclusão de fatores e variáveis projetadas dentro do processo de estimação, ou seja, não serão utilizados apenas os dados históricos para prever o descumprimento ou inadimplência. Dentro deste contexto será proposto uma abordagem que une a estimação da Probabilidade de Descumprimento com a inclusão de um fator foward-looking. / This paper aims to discuss the methodologies used to estimate the Probability Of Default used in the financial industry. In addition, contextualize the application of the work to IFRS9 requirements and its targeting to the Credit Risk theme. Historically large multi-banks use a variety of econometric methodologies to model the Probability of Default, one of the more traditional methods is logistic regression. However, with the need to calculate the expected credit loss through IFRS9, it becomes necessary to change the estimation paradigm to a forwardlooking approach, this is being interpreted by many institutions and consultancies companies as the inclusion of factors and variables projected within the estimation process, that is, not only historical data are used to predict the default. Within this context will be proposed an approach that joins the estimation of Probability of Default with the inclusion of a forward-looking factor.

När propositionen för ett reformerat studiestödssystem lades 1999 poängterades det att studiestödssystemet skulle bära sina egna kostnader. Trots det skrivs stora belopp av. Både Riksrevisionen och Riksgälden har visat att CSN inte använder vedertagna metoder vid beräkningen av det som förväntas gå förlorat på grund av misskötta betalningar. Uppsatsens syfte har varit att skatta vad misskötta betalningar väntas kosta staten i form av framtida avskrivningar samt beräkna vad det skulle innebära för individen att istället bära kostnaden. Som en del i det arbetet har även faktorer som påverkar sannolikheten för misskötta betalningar av studielån identifierats. Resultaten av denna uppsats har bland annat visat att sannolikheten för misskötta betalningar är lägre för individer med eftergymnasial utbildning, hög skuld och låg ålder. Statens kreditförluster på studielån för till exempel individer som blev återbetalningsskyldiga under 2012 förväntas bli mellan 100 och 338 miljoner kronor. Om denna kostnad istället skulle bäras av årskullen innebär det en kostnadsökning på 2,2-7,8 procent för en individ med genomsnittlig skuld.

When banks lend money to another party they face a risk that the borrower will not fulfill its obligation towards the bank. This risk is called credit risk and it’s the largest risk banks faces. According to the Basel accord banks need to have a certain amount of capital requirements to protect themselves towards future financial crisis. This amount is calculated for each loan with an attached risk-weighted asset, RWA. The main parameters in RWA is probability of default and loss given default. Banks are today allowed to use their own internal models to calculate these parameters. Thus hold capital with no gained interest is a great cost, banks seek to find tools to better predict probability of default to lower the capital requirement. Machine learning and supervised algorithms such as Logistic regression, Neural network, Decision tree and Random Forest can be used to decide credit risk. By training algorithms on historical data with known results the parameter probability of default (PD) can be determined with a higher certainty degree compared to traditional models, leading to a lower capital requirement. On the given data set in this article Logistic regression seems to be the algorithm with highest accuracy of classifying customer into right category. However, it classifies a lot of people as false positive meaning the model thinks a customer will honour its obligation but in fact the customer defaults. Doing this comes with a great cost for the banks. Through implementing a cost function to minimize this error, we found that the Neural network has the lowest false positive rate and will therefore be the model that is best suited for this specific classification task. / När banker lånar ut pengar till en annan part uppstår en risk i att låntagaren inte uppfyller sitt antagande mot banken. Denna risk kallas för kredit risk och är den största risken en bank står inför. Enligt Basel föreskrifterna måste en bank avsätta en viss summa kapital för varje lån de ger ut för att på så sätt skydda sig emot framtida finansiella kriser. Denna summa beräknas fram utifrån varje enskilt lån med tillhörande risk-vikt, RWA. De huvudsakliga parametrarna i RWA är sannolikheten att en kund ej kan betala tillbaka lånet samt summan som banken då förlorar. Idag kan banker använda sig av interna modeller för att estimera dessa parametrar. Då bundet kapital medför stora kostnader för banker, försöker de sträva efter att hitta bättre verktyg för att uppskatta sannolikheten att en kund fallerar för att på så sätt minska deras kapitalkrav. Därför har nu banker börjat titta på möjligheten att använda sig av maskininlärningsalgoritmer för att estimera dessa parametrar. Maskininlärningsalgoritmer såsom Logistisk regression, Neurala nätverk, Beslutsträd och Random forest, kan användas för att bestämma kreditrisk. Genom att träna algoritmer på historisk data med kända resultat kan parametern, chansen att en kund ej betalar tillbaka lånet (PD), bestämmas med en högre säkerhet än traditionella metoder. På den givna datan som denna uppsats bygger på visar det sig att Logistisk regression är den algoritm med högst träffsäkerhet att klassificera en kund till rätt kategori. Däremot klassifiserar denna algoritm många kunder som falsk positiv vilket betyder att den predikterar att många kunder kommer betala tillbaka sina lån men i själva verket inte betalar tillbaka lånet. Att göra detta medför en stor kostnad för bankerna. Genom att istället utvärdera modellerna med hjälp av att införa en kostnadsfunktion för att minska detta fel finner vi att Neurala nätverk har den lägsta falsk positiv ration och kommer därmed vara den model som är bäst lämpad att utföra just denna specifika klassifierings uppgift.

Submitted by Diego Peterlevitz Frota (theairguitar@yahoo.com.br) on 2015-09-18T04:44:47Z No. of bitstreams: 1 Dissertação de Mestrado v2.2.1 FINAL.pdf: 702750 bytes, checksum: c584d69fa5b1e9246d10622f4fad5e64 (MD5) / Approved for entry into archive by Renata de Souza Nascimento (renata.souza@fgv.br) on 2015-09-22T13:51:12Z (GMT) No. of bitstreams: 1 Dissertação de Mestrado v2.2.1 FINAL.pdf: 702750 bytes, checksum: c584d69fa5b1e9246d10622f4fad5e64 (MD5) / Made available in DSpace on 2015-09-22T14:21:01Z (GMT). No. of bitstreams: 1 Dissertação de Mestrado v2.2.1 FINAL.pdf: 702750 bytes, checksum: c584d69fa5b1e9246d10622f4fad5e64 (MD5) Previous issue date: 2015-08-24 / This paper covers the fundamentals of the relation between the risk-neutral measure and the real-world, exhibiting some known methods of transforming probability measure associated with each of these two contexts. We show how bonds can be used to estimate the probability of default by their issuers, explaining the reasons that cause it does not reflect, at first, the data observed historically. Using data from Brazilian companies, we estimate the ratio between the risk-neutral and actual probability of default. These results, when compared with other similar studies suggest that the risk premium of Brazilian companies is higher than that of American companies. / Este trabalho aborda os fundamentos da relação entre a medida neutra a risco e o mundo físico, apresentando algumas metodologias conhecidas de transformação da medida de probabilidade associada a cada um destes dois contextos. Mostramos como titulos de crédito podem ser utilizados para a estimação da probabilidade de inadimplência de seus emissores, explicitando os motivos que fazem com que ela não reflita, em um primeiro momento, os dados observados historicamente. Utilizando dados de empresas brasileiras, estimamos a razão entre a probabilidade de default neutra a risco e a probabilidade de default real. Tais resultados, quando comparados com outros trabalhos similares, sugerem que a razão do prêmio de risco de empresas brasileiras possui valor maior do que a de empresas americanas.