On some aspects of coherent risk measures and their applications

Le sujet principal de cette thèse porte sur les mesures de risque. L'objectif général est d'investiguer certains aspects des mesures de risque dans les applications financières. Le cadre théorique de ce travail est celui des mesures cohérentes de risque telle que définie dans Artzner et al (1999). Mais ce n'est pas la seule classe de mesure du risque que nous étudions. Par exemple, nous étudions aussi quelques aspects des "statistiques naturelles de risque" (en anglais natural risk statistics) Kou et al (2006) et des mesures convexes du risque Follmer and Schied(2002). Les contributions principales de cette thèse peuvent être regroupées selon trois axes: allocation de capital, évaluation des risques et capital requis et solvabilité. Dans le chapitre 2 nous caractérisons les mesures de risque avec la propriété de Lebesgue sur l'ensemble des processus bornés càdlàg (continu à droite, limité à gauche). Cette caractérisation nous permet de présenter deux applications dans l'évaluation des risques et l'allocation de capital. Dans le chapitre 3, nous étendons la notion de statistiques naturelles de risque à l'espace des suites infinies. Cette généralisation nous permet de construire de façon cohérente des mesures de risque pour des bases de données de n'importe quelle taille. Dans le chapitre 4, nous discutons le concept de "bonnes affaires" (en anglais Good Deals), pour notamment caractériser les situations du marché où ces positions pathologiques sont présentes. Finalement, dans le chapitre 5, nous essayons de relier les trois chapitres en étendant la définition de "bonnes affaires" dans un cadre plus large qui comprendrait les mesures de risque analysées dans les chapitres 2 et 3.

A Representation of Risk Measures

We provide a representation theorem for risk measures satisfying (i) monotonicity; (ii) positive homogeneity; and (iii) translation invariance. As a simple corollary to our theorem, we obtain the usual representation of coherent risk measures (i.e., risk measures that are, in addition, sub-additive; see Artzner et al. [2]).

On the design of customized risk measures in insurance, the problem of capital allocation and the theory of fluctuations for Lévy processes

Dans cette thèse, nous étudions quelques problèmes fondamentaux en mathématiques financières et actuarielles, ainsi que leurs applications. Cette thèse est constituée de trois contributions portant principalement sur la théorie de la mesure de risques, le problème de l’allocation du capital et la théorie des fluctuations. Dans le chapitre 2, nous construisons de nouvelles mesures de risque cohérentes et étudions l’allocation de capital dans le cadre de la théorie des risques collectifs. Pour ce faire, nous introduisons la famille des "mesures de risque entropique cumulatifs" (Cumulative Entropic Risk Measures). Le chapitre 3 étudie le problème du portefeuille optimal pour le Entropic Value at Risk dans le cas où les rendements sont modélisés par un processus de diffusion à sauts (Jump-Diffusion). Dans le chapitre 4, nous généralisons la notion de "statistiques naturelles de risque" (natural risk statistics) au cadre multivarié. Cette extension non-triviale produit des mesures de risque multivariées construites à partir des données financiéres et de données d’assurance. Le chapitre 5 introduit les concepts de "drawdown" et de la "vitesse d’épuisement" (speed of depletion) dans la théorie de la ruine. Nous étudions ces concepts pour des modeles de risque décrits par une famille de processus de Lévy spectrallement négatifs.

Different risk measures: different portfolio compositions?

Traditionally, the measure of risk used in portfolio optimisation models is the variance. However, alternative measures of risk have many theoretical and practical advantages and it is peculiar therefore that they are not used more frequently. This may be because of the difficulty in deciding which measure of risk is best and any attempt to compare different risk measures may be a futile exercise until a common risk measure can be identified. To overcome this, another approach is considered, comparing the portfolio holdings produced by different risk measures, rather than the risk return trade-off. In this way we can see whether the risk measures used produce asset allocations that are essentially the same or very different. The results indicate that the portfolio compositions produced by different risk measures vary quite markedly from measure to measure. These findings have a practical consequence for the investor or fund manager because they suggest that the choice of model depends very much on the individual’s attitude to risk rather than any theoretical and/or practical advantages of one model over another.

Selection of the right risk measures for portfolio allocation

 An optimisation framework is proposed to enable investors to select the right risk measures in portfolio selection. Verification is deployed by performing experiments in developed markets (e.g., the US stock market), emerging markets (e.g., the South Korean stock market) and global investments. A preselection procedure dealing with large datasets is also introduced to eliminate stocks that have low diversification potential before running the portfolio optimisation model. Portfolios are evaluated by four performance indices, i.e., the Sortino ratio, the Sharpe ratio, the Stutzer performance index, and the Omega measure. Experimental results demonstrate that high performance and also well-diversified portfolios are obtained if modified value-at-risk, variance, or semi-variance is concerned whereas emphasising only skewness, kurtosis or higher moments in general produces low performance and poorly diversified portfolios. In addition, the preselection applied to large datasets results in portfolios that have not only high performance but also high diversification degree.

Multistep stochastic mirror descent for risk-averse convex stochastic programs based on extended polyhedral risk measures

We consider risk-averse convex stochastic programs expressed in terms of extended polyhedral risk measures. We derive computable con dence intervals on the optimal value of such stochastic programs using the Robust Stochastic Approximation and the Stochastic Mirror Descent (SMD) algorithms. When the objective functions are uniformly convex, we also propose a multistep extension of the Stochastic Mirror Descent algorithm and obtain con dence intervals on both the optimal values and optimal solutions. Numerical simulations show that our con dence intervals are much less conservative and are quicker to compute than previously obtained con dence intervals for SMD and that the multistep Stochastic Mirror Descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart. Our con dence intervals are also more reliable than asymptotic con dence intervals when the sample size is not much larger than the problem size.

Elicitable distortion risk measures: A concise proof

Elicitability has recently been discussed as a desirable property for risk measures. Kou and Peng (2014) showed that an elicitable distortion risk measure is either a Value-at-Risk or the mean. We give a concise alternative proof of this result, and discuss the conflict between comonotonic additivity and elicitability.

Risk measures with the CxLS property

In the present contribution, we characterise law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (Math. Finance 16:419–441, 2006), we show that these risk measures can be identified with a class of generalised shortfall risk measures. As a direct consequence, we are able to extend the results in Ziegel (Math. Finance, 2014, http://onlinelibrary.wiley.com/doi/10.1111/mafi.12080/abstract) and Bellini and Bignozzi (Quant. Finance 15:725–733, 2014) on convex elicitable risk measures and confirm that expectiles are the only elicitable coherent risk measures. Further, we provide a simple characterisation of robustness for convex risk measures in terms of a weak notion of mixture continuity.

Risk Measures for Classical and Perturbed Risk Processes - a Survey

2000 Mathematics Subject Classification: 60B10, 60G17, 60G51, 62P05.

Aligning off-balance sheet risk, on-balance sheet risk and audit fees: a PLS path modelling analysis

This study focuses on using the partial least squares (PLS) path modelling technique in archival auditing research by replicating the data and research questions from prior bank audit fee studies. PLS path modelling allows for inter-correlations among audit fee determinants by establishing latent constructs and multiple relationship paths in one simultaneous PLS path model. Endogeneity concerns about auditor choice can also be addressed with PLS path modelling. With a sample of US bank holding companies for the period 2003-2009, we examine the associations among on-balance sheet financial risks, off-balance sheet risks and audit fees, and also address the pervasive client size effect, and the effect of the self-selection of auditors. The results endorse the dominating effect of size on audit fees, both directly and indirectly via its impacts on other audit fee determinants. By simultaneously considering the self-selection of auditors, we still find audit fee premiums on Big N auditors, which is the second important factor on audit fee determination. On-balance-sheet financial risk measures in terms of capital adequacy, loan composition, earnings and asset quality performance have positive impacts on audit fees. After allowing for the positive influence of on-balance sheet financial risks and entity size on off-balance sheet risk, the off-balance sheet risk measure, SECRISK, is still positively associated with bank audit fees, both before and after the onset of the financial crisis. The consistent results from this study compared with prior literature provide supporting evidence and enhance confidence on the application of this new research technique in archival accounting studies.

Aligning off-balance sheet risk, on-balance sheet risk and audit fees: a PLS path modelling analysis

This study focuses on using the partial least squares (PLS) path modelling methodology in archival auditing research by replicating the data and research questions from prior bank audit fee studies. PLS path modelling allows for inter-correlations among audit fee determinants by establishing latent constructs and multiple relationship paths in one simultaneous PLS path model. Endogeneity concerns about auditor choice can also be addressed with PLS path modelling. With a sample of US bank holding companies for the period 2003-2009, we examine the associations among on-balance sheet financial risks, off-balance sheet risks and audit fees, and also address the pervasive client size effect, and the effect of the self-selection of auditors. The results endorse the dominating effect of size on audit fees, both directly and indirectly via its impacts on other audit fee determinants. By simultaneously considering the self-selection of auditors, we still find audit fee premiums on Big N auditors, which is the second important factor on audit fee determination. On-balance-sheet financial risk measures in terms of capital adequacy, loan composition, earnings and asset quality performance have positive impacts on audit fees. After allowing for the positive influence of on-balance sheet financial risks and entity size on off-balance sheet risk, the off-balance sheet risk measure, SECRISK, is still positively associated with bank audit fees, both before and after the onset of the financial crisis. The consistent results from this study compared with prior literature provide supporting evidence and enhance confidence on the application of this new research technique in archival accounting studies.

An analysis of risk management disclosures: Australian evidence

- Purpose Communication of risk management practices are a critical component of good corporate governance. Research to date has been of little benefit in informing regulators internationally. This paper seeks to contribute to the literature by investigating how listed Australian companies in a setting where disclosures are explicitly required by the ASX corporate governance framework, disclose risk management (RM) information in the corporate governance statements within annual reports. - Design/methodology/approach To address our study’s research questions and related hypotheses, we examine the top 300 ASX-listed companies by market capitalisation at 30 June 2010. For these firms, we identify, code and categorise RM disclosures made in the annual reports according to the disclosure categories specified in Australian Stock Exchange Corporate Governance Principles and Recommendations (ASX CGPR). The derived data is then examined using a comprehensive approach comprising thematic content analysis and regression analysis. - Findings The results indicate widespread divergence in disclosure practices and low conformance with the Principle 7 of the ASX CGPR. This result suggests that companies are not disclosing all ‘material business risks’ possibly due to ignorance at the board level, or due to the intentional withholding of sensitive information from financial statement users. The findings also show mixed results across the factors expected to influence disclosure behaviour. Notably, the presence of a risk committee (RC) (in particular, a standalone RC) and technology committee (TC) are found to be associated with improved levels of disclosure. we do not find evidence that company risk measures (as proxied by equity beta and the market-to-book ratio) are significantly associated with greater levels of RM disclosure. Also, contrary to common findings in the disclosure literature, factors such as board independence and expertise, audit committee independence, and the usage of a Big-4 auditor do not seem to impact the level of RM disclosure in the Australian context. - Research limitation/implications The study is limited by the sample and study period selection as the RM disclosures of only the largest (top 300) ASX firms are examined for the fiscal year 2010. Thus, the finding may not be generalisable to smaller firms, or earlier/later years. Also, the findings may have limited applicability in other jurisdictions with different regulatory environments. - Practical implications The study’s findings suggest that insufficient attention has been applied to RM disclosures by listed companies in Australia. These results suggest that the RM disclosures practices observed in the Australian setting may not be meeting the objectives of regulators and the needs of stakeholders. - Originality/value Despite the importance of risk management communication, it is unclear whether disclosures in annual financial reports achieve this communication. The Australian setting provides an ideal environment to examine the nature and extent of risk management communication as the Australian Securities Exchange (ASX) has recommended risk management disclosures follow Principle 7 of its principle-based governance rules since 2007.

Generalized Systematic Risk

We generalize the concept of .systematic risk to a broad class of risk measures potentially accounting for high distribution moments, downside risk, rare disasters, as well as other risk attributes. We offer two different approaches. First is an equilibrium framework generalizing the Capital Asset Pricing Model, two-fund separation, and the security market line. Second is an axiomatic approach resulting in a systematic risk measure as the unique solution to a risk allocation problem. Both approaches lead to similar results extending the traditional beta to capture multiple dimensions of risk. The results lend themselves naturally to empirical investigation.