ANNUAL REPORT 21 JANUARY 1, 2005 – DECEMBER 31, 2005

LABORATORY OF ACTUARIAL MATHEMATICS UNIVERSITY OF COPENHAGEN

LABORATORY OF ACTUARIAL MATHEMATICS ANNUAL REPORT 2005

1 Introduction

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2 Faculty and Staff at the Laboratory

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3 Teaching and Supervision

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3.1 The study program for the Master’s degree in Actuarial Sciences . . .

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3.2 Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3 Graduates and their Master’s theses . . . . . . . . . . . . . . . . . . .

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3.4 Number of students over the past five years

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. . . . . . . . . . . . . .

4 Research

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4.1 Research interests . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2 Conferences and other research visits . . . . . . . . . . . . . . . . . .

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4.3 Colloquia at the Laboratory . . . . . . . . . . . . . . . . . . . . . . .

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5 Publications

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5.1 Refereed articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.2 Other articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.3 Preprints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6 Professional and administrative activities

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7 Other

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7.1 Consulting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7.2 Honors and awards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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1

Introduction

The Laboratory of Actuarial Mathematics is a division of the Institute for Mathematical Sciences within the Faculty of Science at the University of Copenhagen. The objective of the Laboratory is to: (i) educate actuaries qualified to perform risk-theoretical and financial analysis in life and non-life insurance and in related areas; and (ii) to maintain research in insurance mathematics and other fields of applied probability theory.

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Faculty and Staff at the Laboratory

Faculty: Professor Thomas Mikosch Associate Professor Jeffrey F. Collamore Associate Professor Jesper Lund Pedersen Associate Professor Mogens Steffensen Ph.D. Students: Ali Mahmoud Eltaybe Abdelrahman Mikkel Hindkær Dahl Tine Buch-Larsen Peter Holm Nielsen Administrative staff: Overassistent Rikke Helge Overassistent Mette B. Jensen Overassistent Ulla Nielsen Part-time scientific staff: Chartered Accountant Birger Berg Nielsen Skattedirektør Cand.jur. Leif Normann Jeppesen Kontorchef Cand.jur. Johan Jessen Cand.jur. Henning J¨onsson Teaching Assistants: Torben Zabel Brandt Anders Hedegaard Jessen

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3 3.1

Teaching and Supervision The study program for the Master’s degree in Actuarial Sciences

The study program consists of a compulsory part and an optional part. The compulsory part consists of courses in mathematical analysis, linear algebra, computer science, mathematical statistics, probability theory, life- and non-life insurance mathematics, financial mathematics, law and economics. The optional part is selected from courses in mathematics, statistics, insurance mathematics, computer science, law and economics. After the third year, the Bachelor’s degree is awarded to those students who have, in addition, completed two projects in insurance mathematics. After five years, the study program ends with a Master’s thesis. The Ph.D. program typically lasts three additional years.

3.2

Courses

The following courses were given during the period of January–December 2005. Compulsory Courses, Spring 2005: Insurance and Law by H. J¨ onsson and M. Steffensen Topics in General Insurance: Insurance Law by J. Jessen and H. J¨ onsson Topics in General Insurance: Accounting by B. B. Nielsen Basic Non-Life Insurance Mathematics (FM0-S) by T. Mikosch Life Insurance Mathematics, Part A (FM1A) by J.L. Pedersen Non-life Insurance Mathematics, Part B (FM2B) by J.F. Collamore Compulsory Courses, Fall 2005: Basic Life Insurance Mathematics (Liv) by M. Steffensen Stochastic Processes in Life Insurance (LivStok) by J.L. Pedersen Stochastic Processes in Non-life Insurance (SkadeStok) by J.F. Collamore Ruin Theory (Ruinteori) by J.F. Collamore Optional Courses, Spring 2005: Introduction to Extreme Value Theory by T. Mikosch Topics in Life Insurance by M. Dahl and P.H. Nielsen

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Optional Courses, Fall 2005: Convergence of Stochastic Processes by T. Mikosch Modelling Dependence in Discrete Time by T. Mikosch Bachelor Projects: Bachelor Project in Non-Life Insurance Mathematics by Mette Havning (SB Aktuarr˚ adgivning)

3.3

Graduates and their Master’s theses

Stine Breiner Andersen: “Optimale dækningsstrategier i livsforsikring” (Peter Holm Nielsen and Mogens Steffensen). Jacob Bille: “Værdifastsættelse i livsforsikring baseret p˚ a højere ordens momenter” (Mogens Steffensen). Christina H. Schipper Bjaaland and Merete Nortoft Jensen: “Dynamical and Stochastic Longevity” (Thomas Mikosch). Asger Reinholdt Christensen: “Modeling Dependence with Copulas” (Thomas Mikosch). Peter Franck: “Bootstrap for Shot Noise” (Thomas Mikosch). Anina Grell: “Stopping Time Problems and Stopping Games in Life Insurance” (Mogens Steffensen). Julie Have: “Corporate Bond Prices in a Market with Correlated Defaults” (Mogens Steffensen). Lasse Jensen: “Aktiv Passiv modellering i forsikring” (Mogens Steffensen). Jens Wissing Jensen: “Unit-link produkter baseret p˚ a ekstrema og fraktiler” (Jesper Lund Pedersen and Mogens Steffensen). Rikke Schl¨ uter Justesen: “Optimale pensionsopsparingsstrategier” (Peter Holm Nielsen and Mogens Steffensen). Jeanette Halle Larsen: “Fair fordeling af overskud i livsforsikring” (Mogens Steffensen). Kirsten Lendrick Sasady: “Hedging the Unexpected” (Thomas Mikosch). Helle Simonsen: “Price Indexed Pension Contracts” (Mogens Steffensen). Lars Keyper Winkel: “Prissætning af Unit-link forsikringer med særlige stiafhængeige garantier” (Jesper Lund Pedersen and Mogens Steffensen).

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3.4

Number of students over the past five years

The following table lists the number of students who finished the program with a master’s degree (“Graduates”), followed by the number of students admitted to the program (“Admissions”), and finally the total number of students in the program (“Total”). Year

Graduates

Admissions

Total

2001 2002 2003 2004 2005

15 11 9 11 15

15 17 22 54 39

132 121 110 148 152

4 4.1

Research Research interests

Below is a brief summary of the research topics studied by the scientific staff during the past year. Thomas Mikosch (Professor): Continued studying models in insurance, finance and telecommunications which allow unusually large values. The main focus has been on more realistic applied probabilistic models for telecommunications and insurance. Studied the probabilites of rare events under heavy-tailed input distributions. Those include probabilities of large deviations for sums of multivariate stochastic processes and statistical work for financial time series models with heavytailed innovations. Collaborated with Richard A. Davis (Colorado State Univ.), Barbara Gonzalez (Univ. Louisiana–Lafayette), Henrik Hult (Cornell Univ.), Dimitrios Konstantinidis (Samos), Filip Lindskog (KTH, Stockholm), Sidney Resnick (Cornell Univ.), Gennady Samorodnitsky (Cornell Univ.), Catalin St˘aric˘a (Chalmers Univ. and Univ. Gothenburg), and Gilles Fa¨y (Lille). Jeffrey F. Collamore (Associate Professor): General research interests in the areas of large deviations and the theory of general state-space Markov chains; applications to risk theory and risk management. Some specific themes during the past year have been: “small-time” ruin estimates for the sums of heavy-tailed, Markov-dependent financial processess (joint with A. H¨oing, ETH Z¨ urich); ruin in the presence of Markov-dependent investments (joint with H. Nyrhinen, Univ. Helsinki); and general characterizations for the behavior within a regenerative cycle of a Harris recurrent Markov chain. 6

Mogens Steffensen (Associate Professor): Portfolio optimization in corporate bond markets: jump risk, recovery risk, credit risk, contagion risk, rating models. Optimal insurance decisions: Microeconomic insurance decisions, optimal consumption and life insurance. Jesper Lund Pedersen (Associate Professor): In a joint project with Bjarke Jensen (Nordea), a new type of employee option plan has been introduced for lowand mid-level employees. The option plan is of American type and is a combination of an accrued account and a digital option with a strike price. Two cases of the strike have been considered: a fixed strike price and an indexed strike price. In both cases the optimal exercise strategies are obtained and valuation formulae derived. Mikkel Dahl (Ph.D. student): Finished Ph.D. dissertation on the following topics: Finance in insurance: A fair distribution of the assets between the insured and the owners of an insurance company has been determined using the no-arbitrage criterion from finance. Finance: Models for the reinvestment risk in bond markets have been studied in both continuous and discrete time. Stochastic mortality in life insurance: Market reserves and hedging strategies have been determined in the presence of stochastic mortality, and “mortality-linked” insurance contracts have been introduced. Peter Holm Nielsen (Ph.D. student): Finished Ph.D. dissertation. Specifically, a survey of the literature on optimal investment theory has been included in the dissertation. Moreover, a study of optimal long-term investment strategies in markets without long-term bonds has been initiated, and a chapter with the current results has been included.

4.2

Conferences and other research visits

Thomas Mikosch: Invited presentations at conferences and workshops: 4th Conference on L´evy Processes, Manchester, January 2005. Conference on Statistics and Dependence, Paris, January 2005. ASTIN Day, Copenhagen, April 2005. Workshop on Heavy Tails and Long Range Dependence, Cornell, April 2005. Workshop on Risk Measures and Risk Management, Eindhoven (EURANDOM), May 2005. Conference on Functional Analysis, Dubrovnik, June 2005. Conference on Mathematical Finance, Kruger National Park, South Africa, August 2005. 4th International Conference on Extreme Value Analysis, Gothenburg, August 2005. Workshop on Finance and Extremes, Eindhoven (EURANDOM), December 2005.

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Seminar talks: University P. Sabatier Toulouse; Joint Statistics Seminar of the Universities of Paris; Ecole Normale Sup´erieure de Cachan; Paris X; University of Bochum; SAS Institute Johannesburg; Chalmers University Gothenburg; University of Hamburg. Invited research visits: Cornell University (April 2005); Paris X (May 2005); Toulouse (October 2005). Courses: Modelling Financial Time Series and their Extremes, University of Helsinki, March 2005. Heavy-Tailed Modelling for Finance, Insurance and Telecommunications, University of Paris X, May 2005. Heavy-Tailed Modelling for Finance, Insurance and Telecommunications, Charles University of Prague, November 2005. Organization of workshops and conferences: Chair of the Scientific Programme Committee (jointly with H. Rootz´en, Chalmers Univ.), 4th International Conf. on Extremes and Risk, Gothenburg, August 2005. Member of the Scientific Programme Committee, Meeting on Stochastic Processes and their Applications, Santa Barbara, June 2005. Member of the Scientific Programme Committee (jointly with M. Sørensen, R. Poulsen, A. Rahbek, M. Steffensen), 3rd Thiele Symposium on Stochastic Volatility, Copenhagen, December 2005. Jeffrey F. Collamore: Presentations at conferences and workshops: Joint Meeting of the Finnish Actuarial Society and the Univ. Helsinki, April 2005. 4th International Conference on Extreme Value Analysis, Gothenburg, August 2005. Participant, International conference on Finance, Copenhagen, September 2005. Seminar talks: Univ. Helsinki, April 2005. Invited research visits: Univ. Helsinki, March-April 2005. Mogens Steffensen: Presentations at conferences and workshops: International Conference on Finance, Copenhagen, September 2005. Seminar talks: Tanaka Business School, Imperial College, London, November 2005. London School of Economics and Political Science, November 2005. Invited research visits: London School of Economics and Political Science, November-December 2005. Organization of workshops and conferences: Member of the Scientific Programme Committee (jointly with T. Mikosch, M. Sørensen, R. Poulsen, A. Rahbek), 3rd Thiele Symposium on Stochastic Volatility, Copenhagen, December 2005. 8

Jesper Lund Pedersen: Presentations at conferences and workshops: Conference on Mathematical Finance, Kruger National Park, South Africa, August 2005. Participant, 3rd T.N. Thiele Symposium on Stochastic Volatility, Copenhagen, December 2005. Mikkel Dahl: Presentations at conferences and workshops: AFIR Meeting, Z¨ urich, September 2005. Participant, 3rd T.N. Thiele Symposium on Stochastic Volatility, Copenhagen, December 2005. Peter Holm Nielsen Seminar talks: London School of Economics. Participant, 3rd T.N. Thiele Symposium on Stochastic Volatility, Copenhagen, December 2005.

4.3

Colloquia at the Laboratory

March 1, Chitro Majumdar, ETH Z¨ urich. “Stochastic volatility model consequences with multi-fractal statistics: FIGARCH(1,d,0) nomenclature”. March 4, Allan Gut, Uppsala University. “The moment problem”. (Joint seminar with Dept. Statistics.) September 6, Alexander Melnikov, University of Alberta. “Quantile pricing methodology applied to equity-linked insurance”. October 11, Daryl Daley, The Australian National University. “Some problems in applied probability”. (Joint with Dept. Statistics.) December 7, Tertius de Wet, University of Stellenbosch. “Optimality of weighted L2-Wasserstein goodness-of-fit statistics.” Other guests at the laboratory during this period included Prof. Laurens de Haan, Erasmus Univ., Rotterdam and Univ. Lisbon (in August 2005), and Prof. Richard A. Davis, Colorado State Univ. (in October 2005).

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5 5.1

Publications Refereed articles

Alili, L., Patie, P. and Pedersen, J.L. (2005). Representations of first hitting time density of an Ornstein-Uhlenbeck process. Stoch. Models 21, 967–980. Hult, H., Lindskog, F., Mikosch, T. and Samorodnitsky, G. (2005). Functional large deviations for multivariate regularly varying random walks. Ann. Appl. Probab. 15, 2651–2680. Konstantinides, D. and Mikosch, T. (2005). Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Probab. 33, 1992–2035. Mikosch, T. (2005). How to model multivariate extremes if one must? Stat. Neerlandica 59, 324–338. Nielsen, P.H. (2005). On Optimal Bonus Strategies in Life Insurance: The Markov Chain Interest Rate Case. Scandinavian Actuarial Journal, 81–102. Norberg, R. and Steffensen, M. (2005). What is the Time Value of a Stream of Investments. J. Appl. Prob. 42, 861-866.

5.2

Other articles

Pedersen, J.L. (2005). Optimal stopping problems for time-homogeneous diffusions: a review. Recent advances in applied probability, Springer, 427–454. Steffensen, M. (2005). A Note on the Free Policy Reserve. Bl¨atter der DGVFM, XXVII,2.

5.3

Preprints

Collamore, J.F. and H¨ oing, A. Small-time ruin for a financial process modulated by a Harris recurrent Markov chain. Working paper No. 205, May 2005. Dahl, M. A continuous-time model for reinvestment risk in bond markets. Working paper No. 206, May 2005. Dahl, M. A discrete-time model for reinvestment risk in bond markets. Working paper No. 207, November 2005. Dahl, M. and Møller, T. Valuation and hedging of life insurance liabilities with

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systematic mortality risk. Working paper No. 203, March 2005. Fa¨ y, G., Gonz´ alez-Ar´ evalo, B., Mikosch, T. and Samorodnitsky, G. Modeling teletraffic arrivals by a Poisson cluster process. Working paper No. 208, November 2005. Kraft, H. and Steffensen, M. How to invest optimally in corporate bonds: a reduced-form approach. Working paper No. 204, May 2005. Mikosch, T. Copulas: tales and facts. Working paper No. 209, November 2005. Mikosch, T. and St˘ aric˘ a, C. Stock market risk-return inference. An unconditional non-parametric approach. Working paper No. 202, January 2005.

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Professional and administrative activities

Thomas Mikosch: Defenses: Member and referee for Habilitation of Olivier Perrin, University P. Sabatier, Toulouse. Opponent for Licenciate Thesis of Erik Brodin, Department of Statistics, Chalmers Univ., Gothenburg. Referee for the Ph.D. thesis of Gilles Teyssiere, Dept. Mathematics, Univ. Paris VII. Committees: Member of Selection Committee for Appointments at the Dept. Mathematics, Univ. Cyprus. Chair of the Selection Committee for position of Associate Professor at the Laboratory of Actuarial Math., Univ. Copenhagen. Member of the Selection Committee in Mathematics for the Swedish Research Council (VR). Member of the Selection Committee for the SCOR Prize in Actuarial Science for Graduate Students in Germany. Member of the DFG Evaluation Panel for the Graduate School in Applied Mathematics, Univ. Kaiserlautern. Member of the Election Committee for President and Council for the Institute of Mathematical Statistics. Member of the following professional organizations: Bernoulli Society; Danish Actuarial Association; Association of Danish Theoretical Statistics; Royal Danish Academy of Sciences and Letters (Foreign Member); Institute of Mathematical Statistics (Fellow). Associate Editor for: Annals of Applied Probability; Bernoulli; Probability and Mathematical Statistics; Stochastic Processes and their Applications; Journal of Applied Probability/Advances in Applied Probability; Applied Stochastic Models in Business and Industry; Mitteilungen der Deutschen Gesellschaft f¨ ur Versicherungsmathematik; Lithuanian Mathematical Journal. Editor for the Birkhauser Book Series “Operational Research and Financial Engineering”. 11

Jeffrey F. Collamore: Member of the Institute of Mathematical Statistics; Member of the Danish Actuarial Association. Referee for various international journals. Mogens Steffensen: Member of the Danish Actuarial Association; Member of the Study Board. Jesper Lund Pedersen: Member of the Danish Society for Theoretical Statistics. Referee for various international journals. Mikkel Dahl: Member of the Danish Actuarial Association. Peter Holm Nielsen: Member of the Danish Actuarial Association.

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Other Consulting

During this period, Mogens Steffensen was a consultant for Nordea Investment Management, and Peter Holm Nielsen was a consultant for Edlund A/S.

7.2

Honors and awards

Thomas Mikosch was elected Fellow of the Institute of Mathematical Statistics in August 2005.

7.3

Miscellaneous

On December 21, Mikkel Dahl defended his Ph.D. dissertation, “On Mortality and Investment Risk in Life Insurance,” supervised by Thomas Møller, Mogens Steffensen, and Thomas Mikosch. The assessment committee consisted of: Associate Prof. Rolf Poulsen (Univ. Copenhagen), Prof. Christian Hipp (Karlsr¨ uhe), and Prof. Ragnar Norberg (LSE). On December 19-20, the department hosted the 3rd T.N. Thiele Symposium on Stochastic Volatility. Organisers included Thomas Mikosch and Mogens Steffensen (FML), and Rolf Poulsen, Anders Rahbek, and Michael Sorensen (Dept. Statistics). Speakers: T.G. Andersen (Northwestern Univ.), O.E. Barndorff-Nielsen (˚ Arhus Univ.),

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T. Bollerslev (Duke Univ.), M. Hoffmann (Marne la Vall´ee), J. Jacod (Univ. Paris VI), R. Lee (Univ. Chicago), A. Lunde (˚ Arhus Univ.), R. Norberg (LSE), A.J. Patton (LSE), R. Poulsen (Univ. Copenhagen), M. Richter (Danske Bank), and A.N. Shiryaev (Steklov Institute, Moscow). ——— The department received a grant for 2005-2007 from SNF, jointly with colleagues at ˚ Arhus University (as listed below). The details of the grant are as follows. SNF Grant: Multivariate Risk Models for Finance and Insurance. Participants: S. Asmussen (˚ Arhus), M. Steffensen (FML), J.L. Pedersen (FML), J.F. Collamore (FML), O.E. Barndorff-Nielsen (˚ Arhus), B.J. Christensen (˚ Arhus), ˚ F. Hubalek (Arhus).

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