Lecture held at the Austrian Central Bank ( …, 2006
Prof. Dr. Uwe Schmock PRisMa Lab Financial and Actuarial Mathematics (FAM) Institute for Mathemat... more Prof. Dr. Uwe Schmock PRisMa Lab Financial and Actuarial Mathematics (FAM) Institute for Mathematical Methods in Economics Vienna University of Technology Wiedner Hauptstraße 810/105-1 A-1040 Vienna, Austria ... 1Updates on www.fam.tuwien.ac.at/∼schmock/notes/ ...
International Congress on Insurance: Mathematics and Economics, Jun 17, 2010
We are interested in the performance of different estimators of dependence measures. Our work is ... more We are interested in the performance of different estimators of dependence measures. Our work is motivated by the fact, that for elliptical distributions there is a unique connection between the linear correlation coefficient and the rank-based measure Kendall's tau. This offers two ways for estimation: using the standard estimator of the linear correlation directly or estimating Kendall's tau and transforming it into linear correlation. The second way is promising as one has the advantages of a rank-based measure like eg robustness ...
In this article we derive and present in detail the Austrian annuity valuation table AVÖ 2005R, w... more In this article we derive and present in detail the Austrian annuity valuation table AVÖ 2005R, which is the successor to the AVÖ 1996R table to be used for the valuation of standard annuity contracts in Austria. Its form is similar to the AVÖ 1996R in many respects: The table is a two-dimensional dynamic life table comprised of a base table for the year 2001 and age-dependent yearly trends for extrapolation into the future. The table is derived from current statistical data of the Austrian population (census of 2001, earlier censuses and yearly data since 1972), as well as from comparisons with similar German and Swiss annuity valuation tables. In contrast to the AVÖ 1996R, the AVÖ 2005R includes security margins for model and parameter risk to account for a possible adverse development of future mortality. Thus, first- and second-order tables are available. No explicit term for the risk of random fluctuations is included in the table, as
In this paper, we discuss a generalization of the collective risk model and of Panjer’s recursion... more In this paper, we discuss a generalization of the collective risk model and of Panjer’s recursion. The model we consider consists of several business lines with dependent claim numbers. The distributions of the claim numbers are assumed to be Poisson mixture distributions. We let the claim causes have certain dependence structures and prove that Panjer’s recursion is also applicable by finding an appropriate equivalent representation of the claim numbers. These dependence structures are of a stochastic non-negative linear nature and may also produce negative correlations between the claim causes. The consideration of risk groups also includes dependence between claim sizes. Compounding the claim causes by common distributions also keeps Panjer’s recursion applicable.
We give a unified mathematical framework for reduced-form models for portfolio credit risk and id... more We give a unified mathematical framework for reduced-form models for portfolio credit risk and identify properties which lead to positive dependence of default times. Dependence in the default hazard rates is modeled by common macroeconomic factors as well as by inter-obligor links. It is shown that popular models produce positive dependence between defaults in terms of association. Implications of these results are discussed, in particular when we turn to pricing of credit derivatives. In mathematical terms our paper contains results about association of a class of non-Markovian processes.
Stochastics An International Journal of Probability and Stochastic Processes, 1990
For Te [0, oo) and a Brownian motion path j3eQ:={0eC ([O, oo), IR): 0 (O)= O} denote by Cj (P) th... more For Te [0, oo) and a Brownian motion path j3eQ:={0eC ([O, oo), IR): 0 (O)= O} denote by Cj (P) the length of the compact interval C7 {0):={/?(t): fe [O, r]}. Fix v 0 and define Pj (A)=(lexp (vT| Cr|))/E (exp (vT| Cr|)) for all Borel sets A of Q. These measures favour those Brownian motion paths which spread over a small interval up to time T without being too improbable. It is proved that (Pr) reto, oo) converges weakly to a measure P as T co. The limiting measure P is characterised as a mixture of Brownian motions with a drift, forcing them to remain in ...
The three annual 2¼% interest coupons of the Winterthur Insurance convertible bond (face value Ch... more The three annual 2¼% interest coupons of the Winterthur Insurance convertible bond (face value Chf 4 700) will only be paid out if during their corresponding observation periods no major storm or hail storm on one single day damages at least 6 000 motor vehicles insured with Winterthur Insurance. Data for events, where storm or hail damaged more than 1 000 insured vehicles, are available for the last ten years. Using a constant-parameter model, the estimated discounted value of the three Wincat coupons together is Chf 263.29. A conservative evaluation, which accounts for the standard deviation of the estimate, gives a coupon value of Chf 238.25. However, fitting models which admit a trend or a change-point, leads to substantially higher knock-out probabilities of the coupons. The estimated discounted values of the coupons can drop below the above conservative value; a conservative evaluation as above leads to substantially lower values. Hence, already the model uncertainty is higher...
In this contribution we analyze two models for the joint probability of defaults of dependent cre... more In this contribution we analyze two models for the joint probability of defaults of dependent credit risks that are based on a generalisation of Polya urn scheme. In particular we focus our attention on the problems related to the maximum likelihood estimation of the parameters involved, and to this purpose we introduce an approach based on the use of the Expectation-Maximization algorithm. We show how to implement it in this context, and then we analyze the results obtained, comparing them with results obtained by other ...
Probabilistic approaches have played a prominent role in the study of complex physical systems fo... more Probabilistic approaches have played a prominent role in the study of complex physical systems for more than thirty years. This volume collects twenty articles on various topics in this field, including self-interacting random walks and polymer models in random and non-random environments, branching processes, Parisi formulas and metastability in spin glasses, and hydrodynamic limits for gradient Gibbs models. The majority of these articles contain original results at the forefront of contemporary research; some of them include ...
Lecture held at the Austrian Central Bank ( …, 2006
Prof. Dr. Uwe Schmock PRisMa Lab Financial and Actuarial Mathematics (FAM) Institute for Mathemat... more Prof. Dr. Uwe Schmock PRisMa Lab Financial and Actuarial Mathematics (FAM) Institute for Mathematical Methods in Economics Vienna University of Technology Wiedner Hauptstraße 810/105-1 A-1040 Vienna, Austria ... 1Updates on www.fam.tuwien.ac.at/∼schmock/notes/ ...
International Congress on Insurance: Mathematics and Economics, Jun 17, 2010
We are interested in the performance of different estimators of dependence measures. Our work is ... more We are interested in the performance of different estimators of dependence measures. Our work is motivated by the fact, that for elliptical distributions there is a unique connection between the linear correlation coefficient and the rank-based measure Kendall's tau. This offers two ways for estimation: using the standard estimator of the linear correlation directly or estimating Kendall's tau and transforming it into linear correlation. The second way is promising as one has the advantages of a rank-based measure like eg robustness ...
In this article we derive and present in detail the Austrian annuity valuation table AVÖ 2005R, w... more In this article we derive and present in detail the Austrian annuity valuation table AVÖ 2005R, which is the successor to the AVÖ 1996R table to be used for the valuation of standard annuity contracts in Austria. Its form is similar to the AVÖ 1996R in many respects: The table is a two-dimensional dynamic life table comprised of a base table for the year 2001 and age-dependent yearly trends for extrapolation into the future. The table is derived from current statistical data of the Austrian population (census of 2001, earlier censuses and yearly data since 1972), as well as from comparisons with similar German and Swiss annuity valuation tables. In contrast to the AVÖ 1996R, the AVÖ 2005R includes security margins for model and parameter risk to account for a possible adverse development of future mortality. Thus, first- and second-order tables are available. No explicit term for the risk of random fluctuations is included in the table, as
In this paper, we discuss a generalization of the collective risk model and of Panjer’s recursion... more In this paper, we discuss a generalization of the collective risk model and of Panjer’s recursion. The model we consider consists of several business lines with dependent claim numbers. The distributions of the claim numbers are assumed to be Poisson mixture distributions. We let the claim causes have certain dependence structures and prove that Panjer’s recursion is also applicable by finding an appropriate equivalent representation of the claim numbers. These dependence structures are of a stochastic non-negative linear nature and may also produce negative correlations between the claim causes. The consideration of risk groups also includes dependence between claim sizes. Compounding the claim causes by common distributions also keeps Panjer’s recursion applicable.
We give a unified mathematical framework for reduced-form models for portfolio credit risk and id... more We give a unified mathematical framework for reduced-form models for portfolio credit risk and identify properties which lead to positive dependence of default times. Dependence in the default hazard rates is modeled by common macroeconomic factors as well as by inter-obligor links. It is shown that popular models produce positive dependence between defaults in terms of association. Implications of these results are discussed, in particular when we turn to pricing of credit derivatives. In mathematical terms our paper contains results about association of a class of non-Markovian processes.
Stochastics An International Journal of Probability and Stochastic Processes, 1990
For Te [0, oo) and a Brownian motion path j3eQ:={0eC ([O, oo), IR): 0 (O)= O} denote by Cj (P) th... more For Te [0, oo) and a Brownian motion path j3eQ:={0eC ([O, oo), IR): 0 (O)= O} denote by Cj (P) the length of the compact interval C7 {0):={/?(t): fe [O, r]}. Fix v 0 and define Pj (A)=(lexp (vT| Cr|))/E (exp (vT| Cr|)) for all Borel sets A of Q. These measures favour those Brownian motion paths which spread over a small interval up to time T without being too improbable. It is proved that (Pr) reto, oo) converges weakly to a measure P as T co. The limiting measure P is characterised as a mixture of Brownian motions with a drift, forcing them to remain in ...
The three annual 2¼% interest coupons of the Winterthur Insurance convertible bond (face value Ch... more The three annual 2¼% interest coupons of the Winterthur Insurance convertible bond (face value Chf 4 700) will only be paid out if during their corresponding observation periods no major storm or hail storm on one single day damages at least 6 000 motor vehicles insured with Winterthur Insurance. Data for events, where storm or hail damaged more than 1 000 insured vehicles, are available for the last ten years. Using a constant-parameter model, the estimated discounted value of the three Wincat coupons together is Chf 263.29. A conservative evaluation, which accounts for the standard deviation of the estimate, gives a coupon value of Chf 238.25. However, fitting models which admit a trend or a change-point, leads to substantially higher knock-out probabilities of the coupons. The estimated discounted values of the coupons can drop below the above conservative value; a conservative evaluation as above leads to substantially lower values. Hence, already the model uncertainty is higher...
In this contribution we analyze two models for the joint probability of defaults of dependent cre... more In this contribution we analyze two models for the joint probability of defaults of dependent credit risks that are based on a generalisation of Polya urn scheme. In particular we focus our attention on the problems related to the maximum likelihood estimation of the parameters involved, and to this purpose we introduce an approach based on the use of the Expectation-Maximization algorithm. We show how to implement it in this context, and then we analyze the results obtained, comparing them with results obtained by other ...
Probabilistic approaches have played a prominent role in the study of complex physical systems fo... more Probabilistic approaches have played a prominent role in the study of complex physical systems for more than thirty years. This volume collects twenty articles on various topics in this field, including self-interacting random walks and polymer models in random and non-random environments, branching processes, Parisi formulas and metastability in spin glasses, and hydrodynamic limits for gradient Gibbs models. The majority of these articles contain original results at the forefront of contemporary research; some of them include ...
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