Risk Theory

Lecturer
Prof. Dr. Evgeny Spodarev

Teaching Assistant
Wolfgang Karcher

Time and place

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

Exercise session
Thursday 10-12 in H9

Type

4 hours lecture + 2 hours exercises

Prerequisites

Probability Calculus

Content

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
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.

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

Place:
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))
claims.dat
solution
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.
counts.dat
solution
• Sheet 5 (Useful functions for exercise 1: sum, ppois, pdf_N)
pdf_N.R
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.
solution
• 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.

Literature

• 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.
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

News

• 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