Management

ACTUARIAL MATHEMATICS FOR LIFE CONTINGENT RISKS

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In Chapter 8 we introduce multiple state models, which generalize the life–
death contingency structure of previous chapters. Using multiple state models
allows a single framework for a wide range of insurance, including benefits
which depend on health status, on cause of death benefits, or on two or more
lives.
In Chapter 9 we apply the theory developed in the earlier chapters to problems
involving pension benefits. Pension mathematics has some specialized
concepts, particularly in funding principles, but in general this chapter is an
application of the theory in the preceding chapters.
In Chapter 10 we move to a more sophisticated view of interest rate models
and interest rate risk. In this chapter we explore the crucially important
difference between diversifiable and non-diversifiable risk. Investment risk represents
a source of non-diversifiable risk, and in this chapter we show how we
can reduce the risk by matching cash flows from assets and liabilities.
In Chapter 11 we continue the cash flow approach, developing the emerging
cash flows for traditional insurance products. One of the liberating aspects
of the computer revolution for actuaries is that we are no longer required to
summarize complex benefits in a single actuarial value; we can go much further
in projecting the cash flows to see how and when surplus will emerge. This is
much richer information that the actuary can use to assess profitability and to
better manage portfolio assets and liabilities.
In Chapter 12 we repeat the emerging cash flow approach, but here we look
at equity-linked contracts, where a financial guarantee is commonly part of
the contingent benefit. The real risks for such products can only be assessed
taking the random variation in potential outcomes into consideration, and we
demonstrate this with Monte Carlo simulation of the emerging cash flows.
The products that are explored in Chapter 12 contain financial guarantees
embedded in the life contingent benefits. Option theory is the mathematics
of valuation and risk management of financial guarantees. In Chapter 13 we
introduce the fundamental assumptions and results of option theory.
In Chapter 14 we apply option theory to the embedded options of financial
guarantees in insurance products. The theory can be used for pricing and for
determining appropriate reserves, as well as for assessing profitability.
The material in this book is designed for undergraduate and graduate programmes
in actuarial science, and for those self-studying for professional
actuarial exams. Students should have sufficient background in probability to
be able to calculate moments of functions of one or two random variables, and
to handle conditional expectations and variances. We also assume familiarity
with the binomial, uniform, exponential, normal and log normal distributions.
Some of the more important results are reviewed in Appendix A. We also assume that readers have completed an introductory level course in the mathematics of
finance, and are aware of the actuarial notation for annuities-certain.
Throughout, we have opted to use examples that liberally call on spreadsheetstyle
software. Spreadsheets are ubiquitous tools in actuarial practice, and it is
natural to use them throughout, allowing us to use more realistic examples,
rather than having to simplify for the sake of mathematical tractability. Other
software could be used equally effectively, but spreadsheets represent a fairly
universal language that is easily accessible. To keep the computation requirements
reasonable, we have ensured that every example and exercise can be
completed in Microsoft Excel, without needing any VBA code or macros.
Readers who have sufficient familiarity to write their own code may find
more efficient solutions than those that we have presented, but our principle
was that no reader should need to know more than the basic Excel functions
and applications. It will be very useful for anyone working through
the material of this book to construct their own spreadsheet tables as they
work through the first seven chapters, to generate mortality and actuarial
functions for a range of mortality models and interest rates. In the worked
examples in the text, we have worked with greater accuracy than we record, so
there will be some differences from rounding when working with intermediate
figures.
One of the advantages of spreadsheets is the ease of implementation of numerical
integration algorithms.We assume that students are aware of the principles
of numerical integration, and we give some of the most useful algorithms in
Appendix B.
The material in this book is appropriate for two one-semester courses. The
first seven chapters form a fairly traditional basis, and would reasonably constitute
a first course. Chapters 8–14 introduce more contemporary material.
Chapter 13 may be omitted by readers who have studied an introductory course
covering pricing and delta hedging in a Black–Scholes–Merton model. Chapter
9, on pension mathematics, is not required for subsequent chapters, and could
be omitted if a single focus on life insurance is preferred.

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