Actuarial Mathematics
PROF. NINO SAVELLI; PROF. PAOLA BIFFI
[The 'Life Insurance' module is borrowed from the Financial and Insurance Consultant
profile entitled 'Actuarial Math' (Module I).]
PROPERTY-CASUALTY MODULE (PROPERTY-CASUALTY INSURANCE): Prof. Nino Savelli
COURSE AIMS
The aim of the course is to analyse main technical aspect regarding non-life
insurance focusing on the theory, methods and empirical implementation of
actuarial models. Starting from the main forms of insurance cover, the program
will focus on the valuation of premiums and technical reserves for short-term
Property and Casualty Insurance. In particular, actuarial models for claim losses
are discussed by modelling, through empirical and probabilistic approaches,
separately the two components, namely claim frequency and claim severity.
Moreover the course covers the basic methodology used in experience rating
system (both a priori and a posteriori premium customization) and the valuation of
safety and expenses loadings. Different reinsurance treaties will be described by
analysing the effect on the aggregate claim amount and pricing. Finally, a historical
review and a description of the actual solvency system will be accomplished.
Besides the theory lectures, the course includes a few practice sessions. At the end
of the course, students will be able to calculate rate premiums for non-life
insurance contracts and to assess the balance in order to ensure a technical and
capital equilibrium.
COURSE CONTENT
Prerequisites
Before entering the course, the student should:
– be able to use continuously discrete and continuous random variables;
– know the concepts of probability function, density function and distribution
functions;
– be able to obtain main moments and cumulants of a probability distribution;
– be familiar with the main discrete and continuous probability distributions.
Course outline and detailed learning goals
Non-Life insurance covers and main Italian market statistics
At the end of this section of the course the student will:
– be able to define the basic short-term contracts in non-life insurance;
– know the main non-life lines of business;
– be able to understand the main characteristics of the Italian insurance
market.
Calculation of risk premium:
At the end of this section of the course the student will be able to:
– describe the random variable aggregate claim amount;
– construct appropriate models for short term insurance contracts in terms of
numbers of claims and amount of individual claims;
– obtain a fair premium and a fair rate for non-life contracts with both
probabilistic and empirical approach;
– evaluate risk premium by considering policy limits and deductibles;
– adjust risk premium in order to take into account the effect of inflation,
changes in policy limits and in technical assumptions.
Calculation of gross premium
At the end of this section of the course the student will be able to:
– understand the need of safety loading;
– evaluate the safety loading by different criteria;
– understand the main source of costs in non-life insurance;
– evaluate expenses loading;
– obtain a gross premium;
Premium customization
At the end of this section of the course the student will be able to:
– describe the basic methodology used in rating general insurance business;
– implement a priori premium customization;
– understand a simple experience rating system,
– adjust rate premiums by using credibility theory
Technical reserves
At the end of this section of the course the student will:
– know non-life technical provisions;
– be able to evaluate the unearned and total premium reserve;
– be able to construct and understand a run-off triangle by the accounting
reports;
– be able to understand the claims reserve, the IBNR reserve and a valuation
case by case of outstanding claims reserve;
Deterministic methods for claim reserves
At the end of this section of the course the student will be able to:
– evaluate claims reserve using the paid chain-ladder method;
– evaluate claims reserve using the Fisher-Lange method.
Risk retention and reinsurance methods
At the end of this section of the course the student will:
– know different form of reinsurance agreements;
– be able to compare different treaties;
– be able to investigate the methods used to choose reinsurance cover and its
effects on the equilibrium of an insurer.
Solvency
At the end of this section of the course the student will:
– know a historical review of solvency system;
– be able to evaluate the minimum solvency margin and solvency ratio;
– be able to understand main limits of actual solvency system.
READING LIST
N. SAVELLI-G.P. CLEMENTE, Lezioni di Matematica Attuariale Danni (slides made available
online on Blackboard platform).
In-depth study Readings/Lecture notes will be indicated during the course in relation to
classroom interventions by experts in the field.
TEACHING METHOD
Lectures.
ASSESSMENT METHOD
Oral examination.
NOTES
Further information can be found on the lecturer's webpage at
http://docenti.unicatt.it/web/searchByName.do?language=ENG or on the Faculty notice
board.
LIFE MODULE (LIFE INSURANCE): Prof. Paola Biffi
COURSE AIMS
This course develops mathematical methods used in defining life insurance
contracts. The student can expect to learn about the tools used for the pricing of
insurance products and the valuing of the commitments of insurance companies to
policyholders via present value considered in a framework of market equilibrium.
COURSE CONTENT
1. The main biometric functions: biometric functions - survival functions mortality tables.
It is possible to assign probabilities regarding life and death starting from
statistical findings about the number of living in a population.
2. Mortality trends: projection methods.
This part of the course looks at the variability of lifespan over time, and several
methods for assigning the probability of mortality and survival in future years.
3. Traditional life insurance contracts: technical bases - insurance in the event of
survival - in the event of death and mixed investment/insurance policies for
single individuals or groups of individuals.
This part of the course looks at the quantification of the commitments of
insurance companies, taking into account the demographic and financial
framework.
4. Annuities. Present value of a life annuity for single individuals and groups of
individuals - annuities included in retirement plans.
While looking at the various types of contracts, a special emphasis will be
placed on the role of annuity products within retirement plans.
5. Life insurance premiums: means for computation of premiums, premium
loading and premiums - return of premiums.
This part of the course discusses the services provided by insurance companies,
as well as the services received by the contracting party.
6. Formation of actuarial reserves: prospective reserve, retrospective reserve and
recurring reserve - risk premium and savings premiums.
This part of the course highlights the difference in timing between the
insurance Firm's obligations and the payment of premiums. With the technical
bases already outlined, the student will learn how to value provisions to the
reserves.
READING LIST
Text adopted
P. MAZZOLENI, Matematica Attuariale-assicurazioni sulla vita, EDUCatt Università Cattolica, 2014.
Support material available online through the Blackboard platform.
Recommended reading
E. PITACCO, Elementi di matematica delle assicurazioni, Ed. LINT, Trieste, 2002.
Readings for more in-depth study of the material will be indicated during the course.
TEACHING METHOD
Lectures.
ASSESSMENT METHOD
Written examination (exercises and essay questions)
NOTES
Further information can be found on the lecturer's webpage at
http://docenti.unicatt.it/web/searchByName.do?language=ENG or on the Faculty notice
board.