# School of Mathematical and Computer Sciences

## F78AB Actuarial and Financial Mathematics B

Lecturer: Dr C. Donnelly

#### Aims

To introduce the student to more advanced mathematical models of cashflows accumulated or discounted at interest, and to develop skill in applying these models to real financial contracts and transactions.

To introduce simple survival models and associated life tables and moments of cashflows.

#### Summary

This course builds on and extends the ideas contained in the related course F78AA Actuarial and Financial Mathematics A as well as introduces the core concepts of survival modelling essential for life insurance modelling and pricing.

In financial mathematics, the concepts of a continuously-payable cashflow and the force of interest will be considered, leading to a wider discussion of what is called the term-structure of interest rates and the yield curve. Rates of return can be random and we will see how to model and measure this risk. We will see how interest-rate risk and share-price risk can be managed through the use of Redington's immunisation theory, and through the no-arbitrage principle to price so-called forward contracts.

Life insurance is one of the core areas of actuarial science. We will see how to model the future lifetime of an individual can be modelled as a random variable. This gives an introduction to survival models and the life table. As an example, the payments under a life insurance contract are normally linked to the future date of death of an individual, so we need a survival model to help us to value these contracts.

We will study some of the functions belonging to (or describing) survival models. These functions are essential knowledge required for the third year courses in survival models and life insurance mathematics.

#### Learning outcomes

On completion of this course, the student should be able to:
• Value and accumulate continuously-payable cash flows and calculate internal rates of return for transactions with such cash flows.
• Define the duration and convexity of a cash flow sequence and illustrate how these may be used to estimate the sensitivity of the value of the cash flow sequence to changes in the rate of interest.
• Know how duration and convexity are used in the immunisation of a portfolio of liabilities.
• Show an understanding of the term structure of interest rates and of the main factors influencing this structure.
• Calculate the delivery price and the value of a forward contract, using arbitrage-free pricing methods and to explain what is meant by hedging in the case of a forward contract.
• Know how to calculate the value of various types of forward contracts at any time during their duration.
• Explain the concept of a stochastic interest rate model.
• Calculate the mean value and the variance of the accumulated amount of a single premium for a stochastic interest rate model in which the annual rates of return are independently and identically distributed (and also do this for other simple models).
• Calculate the mean value and the variance of the accumulated amount of a level annual premium for a stochastic interest rate model in which the annual rates of return are independently and identically distributed.
• Explain the concept of a survival model.
• Derive the survival function from the definition of a random variable measuring the time until exit from a population.
• Use the survival function to evaluate probabilities of events defined in terms of the time until exit.
• Show how a survival table for integer values of x can be constructed using discrete rates of decrement.
• Define the probability of survival, the radix of a single decrement table, and the survivorship group at duration t.
• Define and develop relationships between the life functions l, q, p, d, mu and the expectation of life.
• Describe the typical shapes of the curves for q, l and mu for human mortality.
• Define the concepts of uniform distribution of decrements and a constant hazard rate, and use them to derive certain approximate relationships.
• Write down expressions in terms of life table functions for the probability function of the curtate future lifetime and also the probability density function of the complete future lifetime of a life subject to a given life table.
• Write down expressions in terms of simple life table functions for the mean and the variance of both the curtate and the complete future lifetime of a life subject to a given life table and evaluate these expressions in simple cases.

• McCutcheon, J.J. & Scott, W.F. (1995): An Introduction to the Mathematics of Finance. Published for the Institute and the Faculty of Actuaries.
• Hull, J.C. (2000): Options, Futures and Other Derivatives. 4th edition, Prentice-Hall.
• Dickson, D.C.M, Hardy, M.R. & Waters, H.R. (2009): Actuarial Mathematics for Life Contingent Risks. Cambridge University Press.
• Formulae and Tables for Actuarial Examinations (``Yellow Tables''). Published for the Institute and the Faculty of Actuaries.

#### Assessment

There will be a two-hour end-of-course examination, contributing 90% of the total mark. During the semester, there will be continuous assessment counting for 10% of the total mark.

#### Help

If you have any problems or questions regarding the course, you are encouraged to contact the lecturer.

#### Course web page

Further information and course materials are available on VISION.

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