# School of Mathematical and Computer Sciences

## F70LA Life Insurance Mathematics A

Lecturer: Tom Fischer

##### Aims

This module aims:

• to consider some more general models for mortality,
• to introduce life insurance policies,
• to introduce and develop the calculation of premiums,
• to introduce and develop the calculation of policy values.
##### Summary
• Selection and select life tables,
• actuarial functions using ultimate and select life tables,
• equations of value,
• impaired lives,
• with-profits policies,
• expenses and bonuses,
• net and gross premium policy values,
• recursive relationship between policy values,
• Thiele's differential equation and its numerical solution.
##### Learning outcomes

By the end of the module students should be able to:

• demonstrate an understanding of select mortality rates;
• construct a select-life mortality table;
• derive financial functions for non-select and select lives;
• express the variance of the present value of a stream of payments in terms of compound interest and life table functions, and evaluate the expression;
• describe (for a single life) the cash flows implied by pure endowments, level annuities, level whole life, endowment, and term assurances;
• derive expressions for the present value and accumulation of the contracts described above;
• calculate financial functions for benefits payable more frequently than annually;
• list the types of expenses incurred in writing a life insurance contract;
• describe the different types of bonus on a with-profits contract;
• calculate net and gross premiums for different types of life insurance and annuity contracts;
• describe how reserves arise, under long-term insurance contracts covering mortality risk;
• define the policy value as the expected future loss, and calculate the net and gross policy values for non-profit and with-profits contracts;
• derive the recursive relationship between policy values at different durations, and use it to calculate policy values at non-integer durations;
• derive and explain Thiele's differential equation in the two-state continuous-time model;
• use an Euler scheme to solve Thiele's differential equation numerically;
• use the Central Limit Theorem to show why risk reserves are needed, and to calculate risk reserves for insurance portfolios of different sizes;
• state and prove Lidstone's theorem, and use it to describe the traditional with-profits model of implicit risk reserving.
##### Prerequisites

F72ZB2 and F72ZD3.