 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).

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, th e 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 the
R(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 are 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 examples (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).

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 r isk 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 compound 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)

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