In their article the authors derive some new asymptotic relations between the multivariate tail c... more In their article the authors derive some new asymptotic relations between the multivariate tail conditional expectation and Value-at-Risk for heavy-tailed scale mixtures of multivariate distributions. This complements earlier work of Li and Sun (2009), whose results require the evaluation of univariate integrals of tail dependence functions, which are not always known. See also Asimit et al. (2011) for related results. The authors also show the appropriateness of their asymptotic results by comparing these with known explicit results for the multivariate elliptical case. In this discussion we provide comment on Sections 2 and 3. In particular, we show how a proof for a key result can be simplified by conditioning first. Unless otherwise mentioned, we assume the notations and conventions from the article. All expectations and statistical quantities mentioned are tacitly assumed to exist.
We investigate the influence of the dependence between random losses on the shortfall and on the ... more We investigate the influence of the dependence between random losses on the shortfall and on the diversification benefit that arises from merging these losses. We prove that increasing the dependence between losses, expressed in terms of correlation order, has an increasing effect on the shortfall, expressed in terms of an appropriate integral stochastic order. Furthermore, increasing the dependence between losses decreases the diversification benefit. We also consider merging comonotonic losses and show that even in this extreme case a strictly positive diversification benefit will often arise.
Test-Aankoop haalde midden maart haar slag thuis in een rechtszaak tegen ziekteverzekeraar DKV. D... more Test-Aankoop haalde midden maart haar slag thuis in een rechtszaak tegen ziekteverzekeraar DKV. Deze laatste had in september 2003 aangekondigd zijn tarieven te verhogen. Hierbij moesten de statistisch betere risico’s minder (of niets) bijbetalen, de slechtere risico’s daarentegen wel. Dit betekent dat de verzekeraar de verhoging differentieerde in functie van de leeftijd van de verzekerde. De verbruikersorganisatie zag hierin een vorm van discriminatie met “de consument” als slachtoffer, spande een rechtzaak aan en kreeg gelijk van de rechter: DKV moet de leeftijdsafhankelijke premieverhogingen stopzetten. 12
ABSTRACT The Basel II Accord outlines a general framework for determining regula-tory capital req... more ABSTRACT The Basel II Accord outlines a general framework for determining regula-tory capital requirements for credit risk portfolios. Different obligors usually operate in dependent socio-economic environments and these structural cor-relations are the main reason why regulatory capital is needed. Therefore, it is not surprising that an important component of the regulatory regime for capital is the asset correlation between obligors. Basel II has set a range for corporate asset correlations from 8 to 24%, the exact value depending on several individual firm characteristics. We use monthly asset value data to calculate asset correlations and compare these with Basel II as well as results from other papers. Our results are in line with literature but a clear difference is found between the majority of these results and the results from Basel II and some major software providers. We discuss these differences and offer some explanations as an attempt to recon-cile the differences: we argue that assuming independent loss given defaults 1 can and should affect the asset correlations used. The impact of horizon is considered as well.
We consider a continuous-time Markowitz type portfolio problem that consists of minimizing the di... more We consider a continuous-time Markowitz type portfolio problem that consists of minimizing the discounted cost of a given cash-fl ow under the constraint of a restricted Capital at Risk. In a Black-Scholes setting, upper and lower bounds are obtained by means of simple analytical expressions that avoid the classical simulation approach for this type of problems. The problem is easily extended to cope with more general discount processes.
Research reports do not necessarily represent the views of the Canadian Institute of Actuaries. M... more Research reports do not necessarily represent the views of the Canadian Institute of Actuaries. Members should be familiar with research reports. Research reports do not constitute standards of practice and therefore are not binding. Research reports may or may not be in compliance with standards of practice. Responsibility for the manner of application of standards of practice in specific circumstances remains that of the members.
Operativa. He maintained these responsibilities for 15 years. Later, he also served as dean of th... more Operativa. He maintained these responsibilities for 15 years. Later, he also served as dean of the Department of Statistics. In the fourth interview of the series, Dependence Modeling presents a conversation with Giorgio Dall'Aglio, an Italian mathematician and probabilist who is internationally acknowledged as one of the main contributors to the theory of Distributions with Given Marginals. In addition to describing his career path and his achievements in mathematics and probability, Giorgio Dall'Aglio portraits the several milestone mathematicians he met during his long career. In the following, our questions to Giorgio Dall'Aglio are typeset in boldface. Early career How did you start your academic career? Which topic did you enjoy most during your university studies? I completed my studies at the University of Pisa, starting in 1950, and at the same time I was a student of the Scuola Normale Superiore. It was a very good arrangement: we had room and board, which was very helpful since my parents were not rich (even more valuable considering the di cult times soon after the war), and could follow extra courses. It was an excellent education, aimed at an academic career and the fact that it included classes in humanities helped to open our minds and to give us a classical, as well as a scienti c preparation. For this, I am indebted to an excellent teacher, Elodia Santollino, whom I had in the last year of high school. She noticed my potential in mathematics and prompted me to apply for the competition at the Scuola Normale. In fact, she acted as a mentor for me and provided extra classes, which turned out to be very valuable to enter the school. At the university, I was especially interested in analysis
In this paper, we assess the magnitude of model uncertainty of credit risk portfolio models, i.e.... more In this paper, we assess the magnitude of model uncertainty of credit risk portfolio models, i.e., what is the maximum and minimum Value-at-Risk (VaR) of a portfolio of risky loans that can be justified given a certain amount of available information. Puccetti and Rüschendorf (2012b) and Embrechts et al. (2013) propose the rearrangement algorithm (RA) as a general method to approximate VaR bounds when the loss distributions of the different loans are known but not their interdependence. Their numerical results show that the gap between worst-case and best-case VaR is typically very high, a feature that can only be explained by lack of using dependence information. In this paper, we propose a modification of the RA that makes it possible to approximate sharp VaR bounds when besides the marginal distributions also higher order moments of the aggregate portfolio such as variance and skewness are available as sources of dependence information. A numerical study shows that the use of moment information makes it possible to significantly improve the (unconstrained) VaR bounds. However, VaR assessments of credit portfolios that are performed at high confidence levels (as it is the case in Solvency II and Basel III) remain subject to significant model uncertainty and are not robust. We provide some suggestions for strengthening future (capital) regulation.
Most decision theories, including expected utility theory, rank dependent utility theory and cumu... more Most decision theories, including expected utility theory, rank dependent utility theory and cumulative prospect theory, assume that investors are only interested in the distribution of returns and not in the states of the economy in which income is received. Optimal payoffs have their lowest outcomes when the economy is in a downturn, and this feature is often at odds with the needs of many investors. We introduce a framework for portfolio selection within which state-dependent preferences can be accommodated. Specifically, we assume that investors care about the distribution of final wealth and its interaction with some benchmark. In this context, we are able to characterize optimal payoffs in explicit form. Furthermore, we extend the classical expected utility optimization problem of Merton to the state-dependent situation. Some applications in security design are discussed in detail and we also solve some stochastic extensions of the target probability optimization problem.
Assuming that agents' preferences satisfy first-order stochastic dominance, we show how the Expec... more Assuming that agents' preferences satisfy first-order stochastic dominance, we show how the Expected Utility paradigm can rationalize all optimal investment choices: the optimal investment strategy in any behavioral law-invariant (state-independent) setting corresponds to the optimum for an expected utility maximizer with an explicitly derived concave non-decreasing utility function. This result enables us to infer the utility and risk aversion of agents from their investment choice in a non-parametric way. We relate the property of decreasing absolute risk aversion (DARA) to distributional properties of the terminal wealth and of the financial market. Specifically, we show that DARA is equivalent to a demand for a terminal wealth that has more spread than the opposite of the log pricing kernel at the investment horizon.
In this paper we examine and summarize properties of several well-known risk measures that can be... more In this paper we examine and summarize properties of several well-known risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of non-independent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. (2003).
In their article the authors derive some new asymptotic relations between the multivariate tail c... more In their article the authors derive some new asymptotic relations between the multivariate tail conditional expectation and Value-at-Risk for heavy-tailed scale mixtures of multivariate distributions. This complements earlier work of Li and Sun (2009), whose results require the evaluation of univariate integrals of tail dependence functions, which are not always known. See also Asimit et al. (2011) for related results. The authors also show the appropriateness of their asymptotic results by comparing these with known explicit results for the multivariate elliptical case. In this discussion we provide comment on Sections 2 and 3. In particular, we show how a proof for a key result can be simplified by conditioning first. Unless otherwise mentioned, we assume the notations and conventions from the article. All expectations and statistical quantities mentioned are tacitly assumed to exist.
We investigate the influence of the dependence between random losses on the shortfall and on the ... more We investigate the influence of the dependence between random losses on the shortfall and on the diversification benefit that arises from merging these losses. We prove that increasing the dependence between losses, expressed in terms of correlation order, has an increasing effect on the shortfall, expressed in terms of an appropriate integral stochastic order. Furthermore, increasing the dependence between losses decreases the diversification benefit. We also consider merging comonotonic losses and show that even in this extreme case a strictly positive diversification benefit will often arise.
Test-Aankoop haalde midden maart haar slag thuis in een rechtszaak tegen ziekteverzekeraar DKV. D... more Test-Aankoop haalde midden maart haar slag thuis in een rechtszaak tegen ziekteverzekeraar DKV. Deze laatste had in september 2003 aangekondigd zijn tarieven te verhogen. Hierbij moesten de statistisch betere risico’s minder (of niets) bijbetalen, de slechtere risico’s daarentegen wel. Dit betekent dat de verzekeraar de verhoging differentieerde in functie van de leeftijd van de verzekerde. De verbruikersorganisatie zag hierin een vorm van discriminatie met “de consument” als slachtoffer, spande een rechtzaak aan en kreeg gelijk van de rechter: DKV moet de leeftijdsafhankelijke premieverhogingen stopzetten. 12
ABSTRACT The Basel II Accord outlines a general framework for determining regula-tory capital req... more ABSTRACT The Basel II Accord outlines a general framework for determining regula-tory capital requirements for credit risk portfolios. Different obligors usually operate in dependent socio-economic environments and these structural cor-relations are the main reason why regulatory capital is needed. Therefore, it is not surprising that an important component of the regulatory regime for capital is the asset correlation between obligors. Basel II has set a range for corporate asset correlations from 8 to 24%, the exact value depending on several individual firm characteristics. We use monthly asset value data to calculate asset correlations and compare these with Basel II as well as results from other papers. Our results are in line with literature but a clear difference is found between the majority of these results and the results from Basel II and some major software providers. We discuss these differences and offer some explanations as an attempt to recon-cile the differences: we argue that assuming independent loss given defaults 1 can and should affect the asset correlations used. The impact of horizon is considered as well.
We consider a continuous-time Markowitz type portfolio problem that consists of minimizing the di... more We consider a continuous-time Markowitz type portfolio problem that consists of minimizing the discounted cost of a given cash-fl ow under the constraint of a restricted Capital at Risk. In a Black-Scholes setting, upper and lower bounds are obtained by means of simple analytical expressions that avoid the classical simulation approach for this type of problems. The problem is easily extended to cope with more general discount processes.
Research reports do not necessarily represent the views of the Canadian Institute of Actuaries. M... more Research reports do not necessarily represent the views of the Canadian Institute of Actuaries. Members should be familiar with research reports. Research reports do not constitute standards of practice and therefore are not binding. Research reports may or may not be in compliance with standards of practice. Responsibility for the manner of application of standards of practice in specific circumstances remains that of the members.
Operativa. He maintained these responsibilities for 15 years. Later, he also served as dean of th... more Operativa. He maintained these responsibilities for 15 years. Later, he also served as dean of the Department of Statistics. In the fourth interview of the series, Dependence Modeling presents a conversation with Giorgio Dall'Aglio, an Italian mathematician and probabilist who is internationally acknowledged as one of the main contributors to the theory of Distributions with Given Marginals. In addition to describing his career path and his achievements in mathematics and probability, Giorgio Dall'Aglio portraits the several milestone mathematicians he met during his long career. In the following, our questions to Giorgio Dall'Aglio are typeset in boldface. Early career How did you start your academic career? Which topic did you enjoy most during your university studies? I completed my studies at the University of Pisa, starting in 1950, and at the same time I was a student of the Scuola Normale Superiore. It was a very good arrangement: we had room and board, which was very helpful since my parents were not rich (even more valuable considering the di cult times soon after the war), and could follow extra courses. It was an excellent education, aimed at an academic career and the fact that it included classes in humanities helped to open our minds and to give us a classical, as well as a scienti c preparation. For this, I am indebted to an excellent teacher, Elodia Santollino, whom I had in the last year of high school. She noticed my potential in mathematics and prompted me to apply for the competition at the Scuola Normale. In fact, she acted as a mentor for me and provided extra classes, which turned out to be very valuable to enter the school. At the university, I was especially interested in analysis
In this paper, we assess the magnitude of model uncertainty of credit risk portfolio models, i.e.... more In this paper, we assess the magnitude of model uncertainty of credit risk portfolio models, i.e., what is the maximum and minimum Value-at-Risk (VaR) of a portfolio of risky loans that can be justified given a certain amount of available information. Puccetti and Rüschendorf (2012b) and Embrechts et al. (2013) propose the rearrangement algorithm (RA) as a general method to approximate VaR bounds when the loss distributions of the different loans are known but not their interdependence. Their numerical results show that the gap between worst-case and best-case VaR is typically very high, a feature that can only be explained by lack of using dependence information. In this paper, we propose a modification of the RA that makes it possible to approximate sharp VaR bounds when besides the marginal distributions also higher order moments of the aggregate portfolio such as variance and skewness are available as sources of dependence information. A numerical study shows that the use of moment information makes it possible to significantly improve the (unconstrained) VaR bounds. However, VaR assessments of credit portfolios that are performed at high confidence levels (as it is the case in Solvency II and Basel III) remain subject to significant model uncertainty and are not robust. We provide some suggestions for strengthening future (capital) regulation.
Most decision theories, including expected utility theory, rank dependent utility theory and cumu... more Most decision theories, including expected utility theory, rank dependent utility theory and cumulative prospect theory, assume that investors are only interested in the distribution of returns and not in the states of the economy in which income is received. Optimal payoffs have their lowest outcomes when the economy is in a downturn, and this feature is often at odds with the needs of many investors. We introduce a framework for portfolio selection within which state-dependent preferences can be accommodated. Specifically, we assume that investors care about the distribution of final wealth and its interaction with some benchmark. In this context, we are able to characterize optimal payoffs in explicit form. Furthermore, we extend the classical expected utility optimization problem of Merton to the state-dependent situation. Some applications in security design are discussed in detail and we also solve some stochastic extensions of the target probability optimization problem.
Assuming that agents' preferences satisfy first-order stochastic dominance, we show how the Expec... more Assuming that agents' preferences satisfy first-order stochastic dominance, we show how the Expected Utility paradigm can rationalize all optimal investment choices: the optimal investment strategy in any behavioral law-invariant (state-independent) setting corresponds to the optimum for an expected utility maximizer with an explicitly derived concave non-decreasing utility function. This result enables us to infer the utility and risk aversion of agents from their investment choice in a non-parametric way. We relate the property of decreasing absolute risk aversion (DARA) to distributional properties of the terminal wealth and of the financial market. Specifically, we show that DARA is equivalent to a demand for a terminal wealth that has more spread than the opposite of the log pricing kernel at the investment horizon.
In this paper we examine and summarize properties of several well-known risk measures that can be... more In this paper we examine and summarize properties of several well-known risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of non-independent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. (2003).
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Papers by S. Vanduffel