Risk Theory

Prof. Dr. Evgeny Spodarev

Teaching Assistant
Wolfgang Karcher

Time and place

Tuesday 10-12 in H3
Wednesday 8-10 in H7

Exercise session
Thursday 10-12 in H9


4 hours lecture + 2 hours exercises


Probability Calculus

Intended audience

Graduate students in Mathematics and Business Mathematics


This course provides an introduction to the mathematical models of non-life insurance with emphasis on
  1. Distribution of claim sizes
  2. Distribution of the number of claims
  3. Distribution of the aggregate claim amount
  4. Simulation
  5. Premium calculation
  6. Reinsurance
  7. Risk reserves
  8. Ruin probabilities

Requirements to obtain the certificate (Übungsschein)

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

Lecture notes

The lecture notes can be downloaded here.

Final exam

The final exam is scheduled for July 17, 2008, 8:00 - 10:00 a.m.

Auxiliary material for the final exam: 1 DIN A4 sheet with own notes, non-programmable pocket calculator, pencils

Registration procedure for Master students:

  • Check if you have enough exercise points (170)
  • If yes, you may fill out the registration form by hand (NOT online) and give it to Mr. Imbacher, Studiensekretariat, M24, 224, until Monday

Please bring your student ID card to the final exam!

Last name starting with

  • A - N: H3
  • O - Z: H15

Exercise sheets

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

  • Sheet 1
  • Sheet 2
  • Sheet 3 (Useful functions for ex. #5: qqplot, mrlplot (package evd))
    Hint for exercise #1: The expression for the cdf is not correct. An updated version of the sheet is online.
    Hint for exercise #5: The parameters of the distributions to be used for the Q-Q plots are included in the online version of the exercise sheet.
  • Sheet 4
    Hint for exercise #2: X(t+h)-X(t) should be replaced by X(t+h)-X(t). An updated version of the sheet is online.
  • Sheet 5 (Useful functions for exercise 1: sum, ppois, pdf_N)
    Hint for exercise #2: In part (a), the transition matrices for both risks need to be computed. An updated version of the sheet is online.
  • Sheet 6
  • R solutions: exercise 1, exercise 3
  • Sheet 7
    Hint for exercise #2 (a): In order to estimate the expected value and the variance of the total claim amount, you can also use Wald's formula.
  • Sheet 8
    Hint for exercise #1: P(X >= 0) should be replaced by P(X > 0)
  • Sheet 9
  • Sheet 10
  • Sheet 11

An R tutorial may be found here: http://www.math.ilstu.edu/dhkim/Rstuff/Rtutor.html.
You may install additional R packages locally on the computers in the KIZ Linux pools.
Documentation for the R packages: http://cran.rakanu.com/web/packages/index.html.

Further information

At the end of this term, a certificate of the German Actuarial Society (DAV-Schein Schadensversicherungsmathematik) can be earned by passing the written DAV-exam.


  • Asmussen, S.
    Ruin probabilities
    World Scientific, Singapore, 2000
  • Beard, R.E., Pentikäinen, T., Pesonen, E.
    Risk Theory
    Chapman and Hall, London - New York, 1984
  • Bowers, N.L., Gerber, H.U., Hickman, J., Jones, D., Nesbitt, C.
    Actuarial Mathematics
    Society of Actuaries, Itasca, 1997
  • Daykin, C.D., Pentikäinen, T., Pesonen, M.
    Practical Risk Theory for Actuaries
    Chapman & Hall, London, 1994
  • Embrechts, P., Klüppelberg, C., Mikosch, T.
    Modelling extremal events
    Appl. Math., 33, Springer, Berlin, 1997
  • 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
  • Goovaerts, M.J., de Vylder, F., Haezendonck, J.
    Insurance premiums
    Elsevier, Amsterdam, 1984
  • 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
  • Klugman, S. A., Panjer, H. H., Willmot, G. E.
    Loss models. From data to decisions
    Wiley, 1998
  • Mack, T.
    Schriftenreihe Angewandte Versicherungsmathematik, Heft 28, 2. Auflage, Verlag Versicherungswirtschaft, Karlsruhe, 2002
  • Mikosch, T.
    Non-life insurance mathematics
    Springer, 2004
  • Müller, A., Stoyan, D.
    Comparison methods for stochastic models and risks
    Wiley, 2002
  • 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
  • Schwepcke, A.
    Rückversicherung. Grundlagen und aktuelles Wissen
    Swiss Re, Verlag Versicherungswirtschaft, Karlsruhe, 2001
  • Straub, E.
    Non-life insurance mathematics
    Springer, Zürich, 1988
  • Wolfsdorf, K.
    Versicherungsmathematik. Teil 2: Theoretische Grundlagen, Risikotheorie, Sachversicherung
    Teubner, Stuttgart, 1988



 Teaching Assistant


  • The DAV-Schein can be picked up at He 22, room 003 (Mrs. Moritz).
  • Registration for the retake / colloquium of the final exam: email to Wolfgang Karcher until 20th August.
  • Auxiliary material for the final exam: 1 DIN A4 sheet with own notes, non-programmable pocket calculator, pencils
  • Please bring your student ID card to the final exam!
  • Rooms for the final exam:
    Last names starting with
    • A - N: H3
    • O - Z: H15
  • Post-exam review: Thursday, July 24, 2:00 - 4:00 p.m. , room 145, He 18

Anonymous Feedback

Here you can send anonymous comments about the lecture and exercise classes.