· to provide an understanding of the mathematical risk theory used in thestudy process of actuarial interest,

· to provide an introduction to mathematical methods for managing therisk in insurance and finance (calculation of risk measures/quantities),

· to develop skills of calculating the ruin probability and the totalclaim 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 orderto be ready to sit for the exams of CT6 subject of the Institute of Actuaries(MATH366 covers 50% of CT6 in much more depth).

(LO1) (a) Define the loss/risk function and explain intuitively the meaning ofit, describe and determine optimal strategies of game theory, apply thedecision criteria's, be able to decide a model due to certain model selectioncriterion, describe and perform calculations with Minimax and Bayes rules.

(LO2) (b) Understand the concept (and themathematical assumptions) of the sums of independent random variables, derivethe distribution function and the moment generating function of finite sums ofindependent random variables,

(LO3) (c) Define and explain the compoundPoisson risk model, the compound binomial risk model, the compound geometricrisk model and be able to derive the distribution function, the probabilityfunction, the mean, the variance, the moment generating function and theprobability generating function for exponential/mixture of exponentialseverities and gamma (Erlang) severities, be able to calculate the distributionof sums of independent compound Poisson random variables.

(LO4) (d) Understand the use of convolutions and compute the distributionfunction and the probability function of the compound risk model for aggregateclaims using convolutions and recursion relationships.

(LO5) (e) Define the stop‐loss reinsuranceand calculate the (mean) stop‐loss premium for exponential and mixtures of exponential severities, beable to compare the original premium and the stoploss premium in numericalexamples,

(LO6) (f) Understand and be able to usePanjer's equation when the number of claims belongs to theR(a, b, 0) class of distributions, use the Panjer's recursion in orderto derive/evaluate the probability function for the total aggregate claims,

(LO7) (g) Explain intuitively theindividual risk model, be able to calculate the expected losses (as well as thevariance) of group life/non‐life insurance policies when the benefits of the each person of thegroup are assumed to have deterministic variables,

(LO8) (h) Derive a compound Poissonapproximations for a group of insurance policies (individual risk model asapproximation),

(LO9) (i) Understand/describe the classicalsurplus process ruin model and calculate probabilities of the number of therisks appearing in a specific time period, under the assumption of the Poissonprocess,

(LO10) (j) Derive the moment generatingfunction of the classical compound Poisson surplus process, calculate andexplain the importance of the adjustment coefficient, also be able to make useof Lundberg's inequality for exponential and mixtures of exponential claimseverities,

(LO11) (k) Derive the analytic solutions for the probability of ruin, psi(u),by solving the corresponding integro‐differential equation forexponential and mixtures of exponential claim amount severities,

(LO12) (l) Define the discrete time surplusprocess and be able to calculate the infinite ruin probability, psi(u,t)in numerical examples (using convolutions),

(LO13) (m) Derive Lundberg's equation andexplain the importance of the adjustment coefficient under the consideration ofreinsurance schemes,

(LO14) (n) Understand the concept of delayedclaims and the need for reserving, present claim data as a triangle (mostcommonly used method), be able to fill in the lower triangle by comparingpresent data with past (experience) data,

(LO15) (o) Explain the difference and adjustthe chain ladder method, when inflation is considered,

(LO16) (p) Describe the average cost per claim method and project ultimateclaims, calculate the required reserve (by using the claims of the data table),

(LO17) (q) Use loss ratios to estimate theeventual loss and hence outstanding claims,

(LO18) (r) Describe the Bornjuetter‐Ferguson method (beable to understand the combination of the estimated loss ratios with aprojection method), use the aforementioned method to calculate the revisedultimate losses (by making use of the credibility factor).

(S1) Numeracy

(S2) Problem solving skills

(a) Decision Theory

Optimum strategies, loss/risk functions, expectedutility principle, rationality principles and the likelihood principle ofoptimal strategies, Minimax criterion, proper Bayes rules, model selection, thetravel insurance example.

(b) Applications of Probability Theory to actuarialrisk models

Brief review of probability theory: momentgenerating functions and distribution functions for finite sums of independentrandom variables (obeying the Bernoulli, geometric, negative binomial,binomial, Poisson, Normal, and exponential distribution).

(c) The collective risk model (aggregate lossmodels)

The compound risk model for aggregate claims,convolutions for the calculation of the distribution function and theprobability function of the compound risk model, the moment generating functionand the probability generating function of the aforementioned model, mean andvariance calculation of the compound risk model, the compound Poisson riskmodel, the c ompound binomial risk model, the compound geometric risk model,sums of independent compound Poisson random variables, the compound risk modelfrom the insurer/reinsurer point of view for simple forms of proportional andstop‐loss reinsurance(subject to a deductible), net stop‐loss premiums, the R (a, b, 0) family of distributions (satisfyPanjer''s equation) for the random variable corresponding to the number ofclaims (frequency distribution), the probability function recursion (Panjer''srecursion) 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 notnecessarily identically distributed random variables), the mean and thevariance under the specific risk model, applications of the aforementionedmodel to groups of life insurance contracts (with certain probability of dea thwithin a year) and to groups of non‐life insurance contracts, the compound Poisson approximation for theindividual risk model.

(e) Ruin Theory

The general surplus process of an insuranceportfolio, the Poisson process and waiting times for the number of the eventsin a given time interval, the classical compound Poisson surplus process andits 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), assolution of an ordinary differential equation (which is a consequence of theaforementioned integro‐differential equation for specific claim amount distributions, eg.exponential and mixture of exponentials), definition of the discrete timesurplus process and recursive evaluation of the finite ruin probability psi(u,t),the probability of ruin and the adjustment coefficient under a simplereinsurance schem es.

(f) Claim reserving methods

Introduction to concept ofIBNR claims and outstanding report claims reserve, delay or Run‐off triangles: the basicchain ladder method, inflation‐adjusted chain laddermethod, the average cost per claim method, Loss ratios, the Bornjuetter‐Ferguson method forprojecting run‐off triangles.