F79AG Risk Theory

Lecturer: I. D. Currie

The course runs over two terms.

Aims

Given a portfolio of insurance policies the course aims

• to develop models for the amount of a single claim from one policy in the portfolio, models for the number of claims from one policy in the portfolio, models for the total claim amount arising from the whole portfolio,
• to study the probability that a portfolio will be insolvent,
• to estimate premiums using both claims experience and general information, and
• to develop models for no claims discount.

Summary

• Models for insurance losses (loss distributions): the gamma, the Pareto, the log normal distribution. Models for claim numbers: the Poisson distribution, the negative binomial distribution. Models for aggregate claims: the compound Poisson distribution, the individual risk model.
• Ruin theory, Lundberg's inequality, the adjustment coefficient, reinsurance and ruin.
• Models for future claim amounts for claims that have occurred but are not settled.
• Estimation of premiums using both claims experience and general information, credibility theory, empirical Bayes methods.
• Models for no claim discount (NCD). Assessing the effectiveness of NCD schemes. Stable distributions.
• The use of generalized linear models in life and non-life insurance.

Learning outcomes

At the end of studying this module, students should be able to:

• Calculate the mean, variance and skewness of certain loss distributions.
• Calculate the effect of reinsurance on the distribution of claims.
• Estimate the parameters of a loss distribution using maximum likelihood.
• Calculate the mean, variance and moment generating function for the collective risk model.
• Use the compound Poisson distribution to describe aggregate claims.
• Use Panjer's recursion to compute the exact distribution of aggregate claims.
• Calculate the mean and variance of aggregate claims in the individual risk model and use these values to set premiums.
• Calculate the probability that a portfolio becomes insolvent using the adjustment coefficient and Lundberg's inequality.
• Assess the effect of reinsurance on the probability of ruin.
• Assess future claims using chain ladder, average cost per claim and Bornhuetter-Ferguson methods.
• Use credibility theory to estimate premiums by (i) the normal/normal model (ii) empirical Bayes methods.
• Obtain the transition matrix for an NCD system and find the stable distribution.
• Use a generalized linear model in problems of actuarial interest.

Reading

A course booklet is provided. It contains all tutorial material and copies of past examination papers.

Assessment

The two modules making up F79AG3 are examined synoptically in a single three hour written examination paper.

Help

If you have any problems or questions regarding the module, you are encouraged to contact the lecturer by email at i.d.currie@ma.hw.ac.uk.

Detailed syllabus

• Loss distributions:
  • log normal, Pareto, gamma distribution
  • reinsurance
  • fitting loss distributions to data
• Models for aggregate claims:
  • Poisson distribution, negative binomial distribution
  • collective risk model: mean, variance, moment generating function
  • compound Poisson distribution
  • Panjer's recursion
  • Individual risk model: mean, variance, security loading
• Ruin theory
  • probability of ruin
  • Lundberg's inequality, adjustment coefficient
  • reinsurance
• Run-off triangles
  • estimating outstanding claims by chain ladder, average cost per claim, Bornhuetter-Ferguson methods
• Credibility theory
  • Bayesian methods, loss functions
  • mixture distributions
  • Bayesian credibility: normal/normal model, Poisson/Gamma model
  • empirical Bayes credibility: Model I (constant volume), Model II (varying volume)
• No claims discount:
  • NCD scales
  • transition matrix, stable distribution
  • bonus hunger
• Generalized linear models
  • Gompertz model as a GLM
  • definition of GLM, examples
  • interpreting fitted models