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 Statistical Methods in Actuarial Science
Code MATH374
Coordinator Dr L Rojas Nandayapa
Mathematical Sciences
leorojas@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 6 FHEQ First Semester 15

Aims

  1. Provide a solid grounding in analysis of general insurance data, Bayesian credibility theory and the loss distribution concept.

  2. Provide an introduction to statistical methods for managing risk in non-life insurance and finance.

  3. Prepare the students adequately to sit for the exams of CT6 subject of the Institute of Actuaries.


Learning Outcomes

Be able to apply the estimation methods described in (b) of the Syllabus for thedistribution described in (a) of the Syllabus, be able to make hypothesis testingdescribed in (b) of the Syllabus for the distribution described in (a) of the Syllabus.

Be able to estimate the parameters of the loss distributions when data complete/incomplete
using the method of moments and the method of maximum likelihood, be able to calculate the loss elimination ratio.

Understand and use the Buhlmann model, the Buhlmann-Straub model, be able to state the assumptions of the GLM models – normal linear model, understand the properties of the exponential family.

Be able to express the values of the life assurances in (d) of the Syllabus and the life annuities in (f) of the Syllabus in terms of the life table functions. Be able to use approximations for the evaluation of the life assurances in (d) of the Syllabus and the life annuities in (f) of the Syllabus based on a life table.

Be able to describe the properties of a time series using basic analytical and graphical tools.

Understand the definitions, properties and applications of well know time series
models, fit time series models to practical data sets and select the suitable models, be able to perform simple statistical inference (forecasting) based on the fitted models, estimate and remove possible trend and seasonality in a time series, analyse the residuals of a time series using stationary models.


Syllabus

(a) Review of probability theory and Statistical inference
Estimation, method of moments, unbiasedness, the likelihood function and maximum
likelihood estimation, confidence interval for the population mean difference, hypothesistesting, the exponential, gamma, log-gamma, Pareto, generalized Pareto, normal, lognormal,Weibull, Burr, log-logistic, Benktander II distributions, comparison of the tails ofdistributions (asymptotic techniques), mixing distributions.

(b) Loss distribution
Moment generating functions (if exists), moments, the variance and alternative methods for the calculation of moments when the moment generating function does not converge for the distributions described in (a) of the Syllabus, the Kolmogorov – Smirnov test, the likelihood ratio test.

(c) Special types of insurance schemes & Reinsurance
Define simple insurance schemes due to deductible s and retention of limits, policy limits, proportional reinsurance, excess of loss reinsurance, statistical inference for the previous insurance/reinsurance schemes with the methods in the in (b) of the Syllabus.

(d) Bayesian statistics and credibility theory
The Bayes Theorem and conditional probabilities, the prior and the posterior distribution, conjugate prior distributions and the linear exponential family, the loss function, the credibility premium, the credibility factor, the Buhlmann model, the Buhlmann-Straub model, exact credibility.

(e) GLM
Linear regression, the multiple linear regression, the normal linear model, GLM: the exponential family of distributions for the binomial, Poisson, exponential, Gamma, normal distribution, the Pearson and deviance residuals, application of tests in model fitting.

(f) Simulation
The basic of simulation, the simulation approach, the truly random and the pseudo random numbers, derivation/generation of the pseudo random numbers, applications to sets of simulation, Monte Carlo simulation, the number of simulations .

(g) Time Series with applications
Serial dependence (autocorrelation coefficients, correlograms), stationary and non-stationary processes, filters and operators, Moving Average (MA) process (properties and applications), Autoregressive (AR) process (properties and applications), Autoregressive moving average (ARMA) process (properties and applications), extension to integrated (ARIMA) processes, non-linear processes, such as ARCH and GARCH, the multivariate autoregressive model,


Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
Explanation of Reading List:

Key Texts

H. S., Klugman, H., Panjer & G., Willmot, (1998), Loss Models: From Data to Decisions, Wiley Series in Probability and Statistics, Wiley .


Recommended Texts

[1] A., Agresti,. And B., Finlay (1997), Statistical Methods for the Social Sciences (3rd ed.)New Jersey: Prentice Hall.  

[2] A. J., Dobson (1983),   An introduction to statistical modelling, Chapman & Hall.

[3] Formulae and Tables, (2002), The Faculty of Actuaries & The Institute of Actuaries.


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

MATH162; MATH263; MATH264; MATH101; MATH102; MATH103  

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

Programme(s) (including Year of Study) to which this module is available on a required basis:

Programme:NG31 Year 3

Programme(s) (including Year of Study) to which this module is available on an optional basis:

 

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Unseen Written Exam  2.5 hours   1st semester  100  Yes  Standard UoL penalty applies  Assessment 1 Notes (applying to all assessments) All 5 questions carry equal weight.  
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes