Risk Theory

Junior-Prof. Dr. Zakhar Kabluchko

Teaching Assistant

Judith Schmidt

Time and place


Monday 16-18 in H14 (Attention: we start at 16:00 and make a very short break)

Wednesday 8-10 in H12 (we start at 8:30 and make no break)

Exercise session:

Friday 10-12 in H3

First lecture: Mo, 16th April

First exercise session: Fr., 27th April


4 hours lecture + 2 hours exercises

The lecture can be held in German or in English


Introduction to Probability Theory, Calculus. 

Intended audience

Master students in Mathematics, Business Mathematics and Finance


This course provides an introduction to the mathematical models of non-life insurance with emphasis on

  1. Models for the claim arrival process
  2. Distribution of claim sizes
  3. Distribution of the number of claims
  4. Distribution of the aggregate claim amount
  5. Compound Poisson processes
  6. Ruin probabilities
  7. Simulation
  8. Premium calculation
  9. Reinsurance
  10. Risk reserves
  11. Credibility theory


Requirements to obtain the certificate (Übungsschein and/or DAV-Schein)

In order to obtain the certificate of the lecture (Übungsschein), one has to earn 50% of all homework credits and pass the written final exam.

At the end of this term, a certificate of the German Actuarial Society (DAV-Schein Schadensversicherungsmathematik) can be earned by passing the written final exam. In order to obtain the DAV certificate it is not necessary to earn 50% of home credits.

Lecture notes

Lecture notes of Evgeny Spodarev can be downloaded here

Lecture notes of Klaus Schmidt can be downloaded here

Lecture notes of Hanspeter Schmidli can be downloaded here

Lecture notes of Anders Martin-Löf and Anders Sköllermo can be downloaded here

Final exam

First exam took place on Wednesday, 25th July 2012, 12:00-14:00.
The results of the first exam are now online in the SLC: Link
To determine your grade please use the following table (please don't round up your points):

Grade Points 

The inspection of the exam takes place on Friday, 27th July, 8:30-11:30, in room E60 (Helmholtzstr. 18).

The second exam on Monday, the 8th October, 2012, 12:00-14:00 will be an oral exam. The duration of the exam is 30 minutes per student. The participants will receive an e-mail with more information. 

However, in order to receive the DAV certificate you have to pass the first exam. In the second exam you can get the certificate of the university only. In order to participate in the second exam you don't need to have participated in the first exam (that is, the exam is open).

If you wish to participate in the final exam and you are not registered in the "Hochschulportal", please send a short email containing your name with subject "Risk theory" to Renate Jäger: link. We need this information to know the exact number of participants in the final exam. It is not necessary to send the email if you register this exam in the "Hochschulportal", and the email does not replace registering there.

Exercise Sheets

In order to receive credit points, a registration at SLC is required.

If you have no account at the SLC, please follow this link and choose "Weiter zur Online-Registrierung" there.

Exercise Sheet 1

Exercise Sheet 2

Exercise Sheet 3

Exercise Sheet 4

Exercise Sheet 5

Exercise Sheet 6 In Exercise 1 the collective model can be assumed. In Exercise 2 the ruin probability is close to 1. The values 0.1 and 0.9 can be interchanged to obtain a more realistic ruin probability. Solutions in which the values were interchanged will be graded according to the same criteria as the solutions of the original problem.

Exercise Sheet 7

Exercise Sheet 8

Exercise Sheet 9

 Exercise Sheet 10

Exercise Sheet 11


  • Asmussen, S.
    Ruin probabilities
    World Scientific, Singapore, 2000
  • Beard, R.E., Pentikäinen, T., Pesonen, E.
    Risk Theory
    Chapman and Hall, London - New York, 1984
  • Embrechts, P., Klüppelberg, C., Mikosch, T.
    Modelling extremal events
    Appl. Math., 33, Springer, Berlin, 1997
  • Heilmann, W.
    Grundbegriffe der Risikotheorie
    Verlag Versicherungswirtschaft, Karlsruhe, 1987
  • Hipp, C., Michel, R.
    Risikotheorie: Stochastische Modelle und Statistische Methoden
    Schriftenreihe Angewandte Versicherungsmathematik, Heft 24, Verlag Versicherungswirtschaft, Karlsruhe, 1990
  • Kaas, R., Goovaerts, M., Dhaene, J., Denuit, M.
    Modern actuarial risk theory
    Kluwer, Boston, 2001
  • Mack, T.
    Schriftenreihe Angewandte Versicherungsmathematik, Heft 28, 2. Auflage, Verlag Versicherungswirtschaft, Karlsruhe, 2002
  • Mikosch, T.
    Non-life insurance mathematics
    Springer, 2004
  • Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.
    Stochastic Processes for Insurance and Finance
    J. Wiley & Sons, Chichester, 1998
  • Schmidt, K.
    Lectures on risk theory
    Teubner, Stuttgart, 1996
  • Straub, E.
    Non-life insurance mathematics
    Springer, Zürich, 1988

Further Literature

  • Daykin, C.D., Pentikäinen, T., Pesonen, M.
    Practical Risk Theory for Actuaries
    Chapman & Hall, London, 1994
  • Farny, D., Helten, E., Koch, P., Schmidt, R.
    Handwörterbuch der Versicherung
    Verlag Versicherungswirtschaft, Karlsruhe, 1988
  • Gerber, H.U.
    An Introduction to Mathematical Risk Theory
    Richard D. Irwin, Homewood, 1979
  • Wolfsdorf, K.
    Versicherungsmathematik. Teil 2: Theoretische Grundlagen, Risikotheorie, Sachversicherung
    Teubner, Stuttgart, 1988
  • Schwepcke, A.
    Rückversicherung. Grundlagen und aktuelles Wissen
    Swiss Re, Verlag Versicherungswirtschaft, Karlsruhe, 2001
  • Goovaerts, M.J., de Vylder, F., Haezendonck, J.
    Insurance premiums
    Elsevier, Amsterdam, 1984
  • Müller, A., Stoyan, D.
    Comparison methods for stochastic models and risks
    Wiley, 200
  • Klugman, S. A., Panjer, H. H., Willmot, G. E.
    Loss models. From data to decisions
    Wiley, 1998