Actuarial Mathematics I

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• analyse straightforward problems involving an equation of value, using a single decrement model to take account of the probability of the payments being made
• conduct a cash-flow analysis and profit test for a unit-linked contract or complex conventional contract.
• Calculate present values and accumulated values of cash flows at a specified rate of interest, using a single decrement model to take account of the probability of the payments being made;
• Describe the various factors that affect mortality, and how risk classification may be used to address the heterogeneity within a given population
• Define terms associated with single decrement models, such as radix, decrement rate (failure rate, mortality rate), probability of survival, expectation of life, hazard rate (force of mortality
• Analyse the liabilities of simple assurance and annuity contracts, both in terms of emerging costs and present values, in relation to product pricing, reserving, and their applications
• Analyse simple problems of emerging costs using simple single decrement models
• show how the relationship between the pricing basis and reserving basis affects the emergence of profit for simple assurance and annuity contracts
• Show how a single decrement model may be used to describe the evolution of a population subject to a single source of decrement
• define and use standard annuity and assurance functions for a single decrement model, including both select and ultimate functions, and demonstrate the relationships between corresponding annuity and assurance functions

• Factors affecting mortality. Selection. Risk classification. Mortality indices.
• Survival models, survival functions, survival tables. Temporary initial selection in mortality tables.
• Select, ultimate, and aggregate mortality rates.
• Single decrement functions and inter-relationships. Radix, decrement rate (failure rate, mortality rate), probability of survival, expectation of life, hazard rate (force of mortality). Human mortality functions.
• Probability (density) functions, mean, and variance for complete future lifetime and curtate future lifetime.
• Hazard rate functions due to Gompertz and Makeham. Laws of mortality.
• Cash flows under financial contracts dependent on death or survival. Concept and calculation of present value and accumulated value of a contingent stream of payments. Mean and variance of such a present value.
• Equations of value for given contingent payment streams and methods of solution for unknown payment amounts, times of payment, and yields.
• Cash flows for pure endowments, level and varying immediate annuities, level and varying whole life, endowment, and term assurances, including critical illness insurance, for a single life, and corresponding standard annuity and assurance functions. Present values and accumulated values for such contracts. Mean and variance of such present values. Extension to annuities payable more frequently than once per year and to assurances payable immediately on death. Definition and use of corresponding commutation functions, and their relationships to annuity and assurance functions.
• Equations of value for net and office premiums for assurance and annuity contracts, and their solution.
• Reserves and calculation of policy values by both retrospective and prospective methods.
• Expected and actual death strain, mortality profit.
• Paid-up policy values, surrender values, alterations.
• Emergence of profit from a contract, for a given reserving basis and discount rate. Profit testing, profit vector, profit signature, and measures of profitability.
• Choice of pricing and reserving bases, and why they may differ. Effect of choice of basis on emergence of profit.
• Cash-flow modelling and profit testing for unit-linked contracts and for complex conventional contracts.

Lectures, tutorials, assigned problems, private study

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.