Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title MATHEMATICAL RISK THEORY Code MATH366 Coordinator Dr A Papaioannou Mathematical Sciences A.Papaioannou@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2021-22 Level 6 FHEQ Second Semester 15

Aims

•To provide an understanding of the mathematical risk theory used in the study process of actuarial interest

• To provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities)

• To develop skills of calculating the ruin probability and the total claim amount distribution in some non‐life actuarial risk models with applications to insurance industry

• To prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth).

Learning Outcomes

(LO1) After completing the module students should be able to:
(a) Define the loss/risk function and explain intuitively the meaning of it, describe and determine optimal strategies of game theory, apply the decision criteria's, be able to decide a model due to certain model selection criterion, describe and perform calculations with Minimax and Bayes rules.
(b) Understand the concept (and the mathematical assumptions) of the sums of independent random variables, derive the distribution function and the moment generating function of finite sums of independent random variables.
(c) Define and explain the compound Poisson risk model, the compound binomial risk model, the compound geometric risk model and be able to derive the distribution function, the probability function, the mean, the variance, the moment generating function and the probability generating function for exponential/mixture of exponential severities and gamma (Erlang) severities, be able to calculate the distribution of sums of independent compound Poisson random variables.
(d) Understand the use of convolutions and compute the distribution function and the probability function of the compound risk model for aggregate claims using convolutions and recursion relationships.
(e) Define the stop‐loss reinsurance and calculate the (mean) stop‐loss premium for exponential and mixtures of exponential severities, be able to compare the original premium and the stoploss premium in numerical examples.
(f) Understand and be able to use Panjer's equation when the number of claims belongs to theR(a, b, 0) class of distributions, use the Panjer's recursion in order to derive/evaluate the probability function for the total aggregate claims.
(g) Explain intuitively the individual risk model, be able to calculate the expected losses (as well as the variance) of group life/non‐life insurance policies when the benefits of the each person of the group a re assumed to have deterministic variables.
(h) Derive a compound Poisson approximations for a group of insurance policies (individual risk model as approximation),
(i) Understand/describe the classical surplus process ruin model and calculate probabilities of the number of the risks appearing in a specific time period, under the assumption of the Poisson process.
(j) Derive the moment generating function of the classical compound Poisson surplus process, calculate and explain the importance of the adjustment coefficient, also be able to make use of Lundberg's inequality for exponential and mixtures of exponential claim severities.
(k) Derive the analytic solutions for the probability of ruin, psi(u), by solving the corresponding integro‐differential equation for exponential and mixtures of exponential claim amount severities,
(l) Define the discrete time surplus process and be able to calculate the infinite ruin probability, psi(u,t) in numerical exam ples (using convolutions).
(m) Derive Lundberg's equation and explain the importance of the adjustment coefficient under the consideration of reinsurance schemes.
(n) Understand the concept of delayed claims and the need for reserving, present claim data as a triangle (most commonly used method), be able to fill in the lower triangle by comparing present data with past (experience) data.
(o) Explain the difference and adjust the chain ladder method, when inflation is considered.
(p) Describe the average cost per claim method and project ultimate claims, calculate the required reserve (by using the claims of the data table).
(q) Use loss ratios to estimate the eventual loss and hence outstanding claims.
(r) Describe the Bornjuetter‐Ferguson method (be able to understand the combination of the estimated loss ratios with a projection method). Use the aforementioned method to calculate the revised ultimate losses (by making use of the credibility factor ).

Syllabus

1(a) Decision Theory Optimum strategies, loss/risk functions, expected utility principle, rationality principles and the likelihood principle of optimal strategies, Minimax criterion, proper Bayes rules, model selection, the travel insurance example.
(b) Applications of Probability Theory to actuarial risk models Brief review of probability theory: moment generating functions and distribution functions for finite sums of independent random variables (obeying the Bernoulli, geometric, negative binomial, binomial, Poisson, Normal, and exponential distribution).
(c) The collective risk model (aggregate loss models) The compound risk model for aggregate claims, convolutions for the calculation of the distribution function and the probability function of the compound risk model, the moment generating function and the probability generating function of the aforementioned model, mean and variance calculation of the compound risk model, the compound Poisson risk model, the compoun d binomial risk model, the compound geometric risk model, sums of independent compound Poisson random variables, the compound risk model from the insurer/reinsurer point of view for simple forms of proportional and stop‐loss reinsurance (subject to a deductible), net stop‐loss premiums, the R (a, b, 0) family of distributions (satisfy Panjer's equation) for the random variable corresponding to the number of claims (frequency distribution), the probability function recursion (Panjer's recursion) of the total aggregate losses [for the class R (a, b, 0).
(d) The individual risk model (group insurance models) The aggregate loss models (individual risk model) for insurance contracts (finite sums of independent but not necessarily identically distributed random variables), the mean and the variance under the specific risk model, applications of the aforementioned model to groups of life insurance contracts (with certain probability of death within a year) and to groups of non‐life insurance contracts, the compound Poisson approximation for the individual risk model.
(e) Ruin Theory The general surplus process of an insurance portfolio, the Poisson process and waiting times for the number of the events in a given time interval, the classical compound Poisson surplus process and its moment generating function, the adjustment coefficient and Lundberg's inequality, the integro‐differential equation for the ruin probability psi ( u) , closed form expressions of the ruin probability, psi ( u) , as solution of an ordinary differential equation (which is a consequence of the aforementioned integro‐differential equation for specific claim amount distributions, eg. exponential and mixture of exponentials), definition of the discrete time surplus process and recursive evaluation of the finite ruin probability psi(u,t), the probability of ruin and the adjustment coefficient under a simple reinsurance schemes.
(f) Claim reserving methods Introduction to concept of IBNR claims and outstanding report claims reserve, delay or Run‐off triangles: the basic chain ladder method, inflation‐adjusted chain ladder method, the average cost per claim method, Loss ratios, the Bornjuetter‐Ferguson method for projecting run‐off triangles.

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH264 STATISTICAL THEORY AND METHODS II; MATH101 Calculus I; MATH103 Introduction to Linear Algebra; MATH162 INTRODUCTION TO STATISTICS

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
final assessment  60 minutes    50
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
class test open book and remote  around 60-90 minutes    50