### 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 Life Insurance Mathematics II Code MATH373 Coordinator Dr DK Falden Mathematical Sciences Debbie.Falden@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level Three First Semester 15

### Aims

1. Provide a solid grounding in the subject of life contingencies for multiple-life, and in the subject of the analysis of life assurance, life annuities, pension contracts, multi-state models and profit testing.

2. Provide an introduction to mathematical methods for managing the risk in life insurance.

3. Analyze problems of pricing and reserving in relation to contracts involving several lives.

4. Prepare the students to sit for the exams of CM1 subjects of the Institute and Faculty of Actuaries.

5. Be confident in solving life insurance problems using R programming language and excel.

### Learning Outcomes

(LO1) Be able to explain, define and analyse the joint survival functions.

(LO2) Understand the concept (and the mathematical assumptions) of the joint future life time random variables in continuous and discrete time and monthly. Be able to derive the distributions and the moment/variance of the joint future lifetimes.

(LO3) Be able to define the survivals probabilities/death probabilities of either or both two lives, explain these types of probabilities and the force of interest intuitively, be able to calculate the different types of the survival/death probabilities in theoretical and numerical examples.

(LO4) Understand, define and derive the expected present values of different types of the life assurances and life annuities for joint lives, be able to calculate the expected present values of the joint life assurances and life annuities in theoretical and numerical examples.

(LO5) Be familiar with R software and uses in actuarial mathematics

(S1) Problem solving skills

(S2) Numeracy

### Syllabus

( a) Revision of Life insurance I. Net premiums, gross premiums, benefits and expenses.

(b) Multiple-life actuarial functions. Joint life survival functions for the joint future life times , probabilities of death and survival of either or both two lives, relations between the joint survival probabilities and the corresponding probabilities of the one life, joint life and last survivor assurance and annuity functions and the corresponding 4/5/2019 programmeplan.liv.ac.uk/_layouts/PrintableInfoPath/html.aspx?sourceFile=http://programmeplan.liv.ac.uk/Modules2/MATH373.xml&O… programmeplan.liv.ac.uk/_layouts/PrintableInfoPath/html.aspx?sourceFile=http://programmeplan.liv.ac.uk/Modules2/MATH373.xml&OpenIn=Bro… 4/10 expected present value, extension to consideration of continuous and monthly frequencies, and to functions that depend on term as well as age, joint force of mortality, joint life table functions, numerical applications.

(c) Multiple life m odels (Markov models) The death-alive model, term insurance with increasing benefit on accident death, the permanent disability model, the disability income insurance model, the joint life and last survivor model, assumptions of the models, probabilities based on intensities using Kolmogorov’s equations, transition intensities based on probabilities, numerical evaluation of probabilities.

(d) Emergency costs for all contract types and pensions. Unit-linked contracts and disability long-term contracts, profit test annual premium contracts, the profit vector, the net present value, the profit signature, the profit margin, the profit test to price a product/determine reserves, construction and use of a multiple decrement service table for pension calculation, extension of the above techniques to calculate expected cash flows contingent risk other than human lives.

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

MATH102 CALCULUS II; MATH101 Calculus I; MATH162 INTRODUCTION TO STATISTICS; MATH273 Life Insurance Mathematics I

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final assessment on campus There is a resit opportunity.  120    70
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Class Test  60    30