### 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 I Code MATH273 Coordinator Dr M Boado Penas Mathematical Sciences Carmen.Boado@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2021-22 Level 5 FHEQ First Semester 15

### Aims

Provide a solid grounding in the subject of life contingencies for single life, and in the subject of the analysis of life assurance and life annuities, including pension contracts.

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

Develop skills of calculating the premium for a certain life insurance contract, including allowance for expenses and profits.

Prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CM1 subject of the Institute and Faculty of Actuaries.

### Learning Outcomes

(LO1) Be able to explain and analyze the factors that affect mortality, simple life assurance and life annuity contracts.

(LO2) Understand the concept (and the mathematical assumptions) of the future life time random variables in continuous and discrete time

(LO3) Be able to derive the distributions and the moment/variance of the aforementioned future lifetimes, be able to make graphs of these future life times.

(LO4) Be able to define the survivals probabilities and the force of mortality of the (c) section of the Syllabus, explain these types of probabilities and the force of mortality intuitively, be able to calculate the different types of the survival probabilities in theoretical and numerical examples. Understand the concept of the De Moivre, Makeham, Gompertz, Weibull and the exponential law (constant force of mortality) for modelling fractional ages, explain the basic difference between the laws above, be able to use these laws to calculate the survival probabilities of (c) of the Syllabus in numerical examples. Understand, define/calculate and derive the expected present values of all types of the life assurances of (d) of the Syllabus.

(LO5) Derive relations between life assurances both in continuous and discrete time, be able to use recursive equations for the calculation of the expected present value of different types of life assurances, calculate the variance of the present values for basic forms of life assurances.

(LO6) Be able to derive the distributions and the moment/variance of the aforementioned future lifetimes, be able to make graphs of these future life times.

(LO7) Be able to calculate premium and policy values with and without expenses.

(LO8) Be confident in using R and excel in simple life insurance problems.

### Syllabus

(a) Review of Survival models:

The future lifetime random variable in continuous time, the future lifetime random variable in discrete time, the 1/m future lifetime random variable, moments and distributions of the future lifetimes, the survival function.

(b) Survival probabilities:

Survival probabilities, the force of mortality, versions of the aforementioned survival/death probabilities in terms of the force of mortality, Fractional age assumptions

(c) Life Assurances:

Introduction to contracts of life assurances, expected present values if life assurances payable at the moment of death (in continuous time) and at the end of the year of death (in discrete time) for the following cases: term, whole life, endowment, pure endowment, deferred, term and deferred and their combinations, relations between discrete and continuous time assurances, increasing and decreasing life assurances, life assurances for variable insurance benefits, basic monthly life ass urances, recursive equation for the expected present value of different types of life assurances.

(d) Life annuities:

Introduction to annuities, expected present values (in discrete and continuous time) of an whole life annuity due/immediate, term annuity due/ immediate, deferred term annuity due/ immediate, whole life annuity, term annuity deferred continuously payable, pure endowment, temporary annuity, relations between different types of annuities, relations between annuities and life assurances in discrete and continuous time, fractional annuities, guaranteed annuities, increasing (arithmetically / geometrically) annuities.

(e) Life tables:

Introduction to life tables, the life table functions (select and ultimate). Relations between the life functions and the variables defined in (a).

(f) Net premium calculation and policy values:

The present value of the future loss random variable, the equivalence principle (net premiums), premiums for differ ent types of annuities (payable monthly, semi-quarterly, annually and continuously), prospective and retroprospective reserves.

(g) Benefits, bonuses and expenses:

Mortality profit, profit contracts, surrender values, reserves for contracts with benefits/profit contracts, gross premiums using the equivalence principle for different types of benefits.

### Recommended Texts

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

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

MATH162 INTRODUCTION TO STATISTICS; MATH102 CALCULUS II; MATH101 Calculus I

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final Assessment on campus  60 minutes    50
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Class test 1  around 60-90 minutes    20
class test 2 to be completed in excel  around 60-90 minutes    30