Download Risk Theory I

Risk Theory I
PROF. NINO SAVELLI
COURSE AIMS
The course is designed to supply the methodological tools for analysing random
phenomena in the operation of an insurance company, particularly with reference
to a non-life insurance company.
The following topics will be covered: the definition of the basic structure of the
individual and collective approaches to the estimation of the cost of losses, the
valuation of the main aggregates that influence the aggregate cost of losses
(number and severity losses), the determination of the moments and the cumulants
in compound Poisson process (simple or mixed), the determination of the
distribution function of the aggregate cost through the use of different
methodologies, the definition and the valuation of the probability of collapse and
the capital at risk (CaR), the determination of CaR through estimation formulas,
and the long-term analysis of the risk borne by the insurance company.
In addition to the lectures, class time will be dedicated to exercises and to
multimedia presentations of several practical cases with the use of simulation
models.
By the end of course, the students should be able to analyse the problems linked to
the evaluation of an insurance company's solvency, and to determine the capital
requirement needed over pre-established time horizons and confidence intervals.
COURSE CONTENT
COURSE PROGRAMME WITH DETAILED INSTRUCTIONAL OBJECTIVES
Instructional objectives that students must meet before taking the course
Before taking the course, the student should be able to:
– understand the concepts of discrete and continuous random variables;
– understand the concepts of the probability function, density function and
distribution functions;
– understand the concepts of mean value, variance and asymmetry and methods
for the determination of moments;
– understand the principal distributions of discrete and continuous probabilities.
Instructional objectives of the course
Moments, generating functions and probability distributions
After completing the study of the topic, the student should be able to:
– exploit basic notions of probability;
– understand the main probability distributions that can be used within the
framework of risk theory;
– choose the proper probability distribution to describe a random variable;
– use the generating functions for the aggregation of random variables and for the
determination of moments and cumulants.
Risk theory: individual approach and collective approach
After completing the study of the topic, the student should be able to:
– understand the approaches that can be used for risk analysis in terms of life
insurance and non-life insurance;
– evaluate the principal random variables that make up the aggregate cost of nonlife losses both from an individual and collective approach;
– understand the concept of frequency-severity modelling.
The moments of the aggregate cost of losses
After completing the study of the topic, the student should be able to:
– model the behaviour of the random variable (number) of losses;
– understand the problems associated with the use of pure Poisson or mixed
Poisson processes and the introduction of disturbance factors;
– model the behaviour of the random variable (cost) of the individual loss;
– understand the exact moments (mean value, variance and asymmetry) of the
aggregate cost of the losses identified by compound Poisson processes (simple
or mixed);
– determine the distribution function for the aggregate cost of losses;
– use appropriate formulas for approximation and recursive methods for the
determination of the distribution function for the aggregate cost of losses.
The risk reserve and its relationships with the cost losses
After completing the study of the topic, the student should be able to:
– understand and describe the behaviour of the risk reserve over an annual
period;
– understand the concept of probability of collapse and capital-at-risk (CaR);
– determine the probability of collapse and CaR for an insurance company;
– determine the CaR through approximation formulas;
– understand the analogies between the concept of CaR and the minimum
solvency margin required of insurance companies under prevailing regulations
(Solvency I and Solvency II).
The risk reserve over a period of two or more years
After completing the study of the topic, the student should be able to:
– evaluate the relationship that describes the risk reserve with time horizons
extending over two or more years;
– evaluate the aggregates that can be adjusted by an insurance company in order
to limit risk and CaR;
– verify the effect of reinsurance policies on CaR;
– understand the asymptomatic levels and formulas for time horizons to infinity.
READING LIST
Required reading:
N. SAVELLI – CLEMENTE G.P., Fondamenti di Teoria del Rischio per le Assicurazioni, Draft of the
book for the students is available online on Blackboard platform, 2013.
Recommended reading for further study:
C.D. DAYKIN - T. PENTIKAINEN - M. PESONEN, Practical risk theory for actuaries, Chapman & Hall,
Londra, 1994.
N. SAVELLI, La legislazione italiana e comunitaria sui requisiti patrimoniali di una compagnia di
assicurazioni: problematiche e proposte per un loro superamento, Finanza, Imprese e
Mercati, Four-monthly journal of corporate finance, Il Mulino, anno VI, n. 2, Agosto, 1994.
T. PENTIKAINEN - H. BONSDORFF - M. PESONEN - J. RANTALA - M. RUOHONEN, Insurance solvency and
financial strength, Finnish Insurance Training and Publishing Company, Helsinky, 1989.
TEACHING METHOD
Lectures. Presentation of several practical cases with the use of simulation models.
ASSESSMENT METHOD
Oral exam.
NOTES
Further information can be found on the lecturer's webpage
http://www2.unicatt.it/unicattolica/docenti/index.html, or on the Faculty notice board.
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