## Quantitative models for prudential credit risk management

Doctoral Thesis

2021

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The thesis investigates the exogenous maturity vintage model (EMV) as a framework for achieving unification in consumer credit risk analysis. We explore how the EMV model can be used in origination modelling, impairment analysis, capital analysis, stress-testing and in the assessment of economic value. The thesis is segmented into five themes. The first theme addresses some of the theoretical challenges of the standard EMV model – namely, the identifiability problem and the forecasting of the components of the model in predictive applications. We extend the model beyond the three time dimensions by introducing a behavioural dimension. This allows the model to produce loan-specific estimates of default risk. By replacing the vintage component with either an application risk or a behavioural risk dimension, the model resolves the identifiability problem inherent in the standard model. We show that the same model can be used interchangeably to produce a point-in-time probability forecast, by fitting a time series regression for the exogenous component, and a through-the-cycle probability forecast, by omitting the exogenous component. We investigate the use of the model for regulatory capital and stress-testing under Basel III, as well as impairment provisioning under IFRS 9. We show that when a Gaussian link function is used the portfolio loss follows a Vašíček distribution. Furthermore, the asset correlation coefficient (as defined under Basel III) is shown to be a function of the level of systemic risk (which is measured by the variance of the exogenous component) and the extent to which the systemic risk can be modelled (which is measured by the coefficient of determination of the regression model for the exogenous component). The second theme addresses the problem of deriving a portfolio loss distribution from a loan-level model for loss. In most models (including the Basel-Vašíček regimes), this is done by assuming that the portfolio is infinitely large – resulting in a loss distribution that ignores diversifiable risk. We thus show that, holding all risk parameters constant, this assumption leads to an understatement of the level of risk within a portfolio – particularly for small portfolios. To overcome this weakness, we derive formulae that can be used to partition the portfolio risk into risk that is diversifiable and risk that is systemic. Using these formulae, we derive a loss distribution that better-represents losses under portfolios of all sizes. The third theme is concerned with two separate issues: (a) the problem of model selection in credit risk and (b) the problem of how to accurately measure probability of insolvency in a credit portfolio. To address the first problem, we use the EMV model to study the theoretical properties of the Gini statistic for default risk in a portfolio of loans and derive a formula that estimates the Gini statistic directly from the model parameters. We then show that the formulae derived to estimate the Gini statistic can be used to study the probability of insolvency. To do this, we first show that when capital requirements are determined to target a specific probability of solvency on a through-the-cycle basis, the point-in-time probability of insolvency can be considerably different from the through-the-cycle probability of insolvency – thus posing a challenge from a risk management perspective. We show that the extent of this challenge will be greater for more cyclical loan portfolios. We then show that the formula derived for the Gini statistic can be used to measure the extent of the point-in-time insolvency risk posed by using a through-the-cycle capital regime. The fourth theme considers the problem of survival modelling with time varying covariates. We propose an extension to the Cox regression model, allowing the inclusion of time-varying macroeconomic variables as covariates. The model is specifically applied to estimate the probability of default in a loan portfolio, where the experience is decomposed the experience into three dimensions: (a) a survival time dimension; (b) a behavioural risk dimension; and (c) calendar time dimension. In this regard, the model can also be viewed as an extension of the EMV model – adding a survival time dimension. A model is built for each dimension: (a) the survival time dimension is modelled by a baseline hazard curve; (b) the behavioural risk dimension is modelled by a behavioural risk index; and (c) the calendar time dimension is modelled by a macroeconomic risk index. The model lends itself to application in modelling probability of default under the IFRS 9 regime, where it can produce estimates of probability of default over variable time horizons, while accounting for time-varying macroeconomic variables. However, the model also has a broader scope of application beyond the domains of credit risk and banking. In the fifth and final theme, we introduce the concept of embedded value to a banking context. In longterm insurance, embedded value relates to the expected economic value (to shareholders) of a book of insurance contracts and is used for appraising insurance companies and measuring management's performance. We derive formulae for estimating the embedded value of a portfolio of loans, which we show to be a function of: (a) the spread between the rate charged to the borrower and the cost of funding; (b) the tenure of the loan; and (c) the level of credit risk inherent in the loan. We also show how economic value can be attributed between profits from maturity transformation and profits from credit and liquidity margin. We derive formulae that can be used to analyse the change in embedded value throughout the life of a loan. By modelling the credit loss component of embedded value, we derive a distribution for the economic value of a book of business. The literary contributions made by the thesis are of practical significance. The thesis offers a way for banks and regulators to accurately estimate the value of the asset correlation coefficient in a manner that controls for portfolio size and intertemporal heterogeneity. This will lead to improved precision in determining capital adequacy – particularly for institutions operating in uncertain environments and those operating small credit portfolios – ultimately enhancing the integrity of the financial system. The thesis also offers tools to help bank management appraise the financial performance of their businesses and measure the value created for shareholders.

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Malwandla, M. 2021. Quantitative models for prudential credit risk management. . ,Faculty of Commerce ,Department of Finance and Tax. http://hdl.handle.net/11427/33779