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Insurance benefits which are dependent on the joint mortality of two lives, typically a married couple, form an important part of many insurance portfolios. In this chapter we develop the concepts and models from previous chapters to examine joint life insurance policies. There are also important applications in pension design and valuation, as spousal benefits are a common part of a pension benefit package.
We describe typical benefits offered and introduce standard notation for actuarial functions dependent on two lives. We develop an approach for pricing and valuing these policies, based on the future lifetime random variables, and making the strong assumption that the two lives are independent with respect to mortality.
Next, we show how joint life mortality can be analyzed using multiple state models. This creates a flexible framework to introduce dependence between lives, and we can apply the methods of Chapter 8 to calculate probabilities and value benefits.
In this chapter we represent the future lifetime of an individual as a random variable, and show how probabilities of death or survival can be calculated under this framework. We then define the force of mortality, which is a fundamental quantity in mortality modelling. We introduce some actuarial notation, and discuss properties of the distribution of future lifetime. We introduce the curtate future lifetime random variable, which represents the number of complete years of future life, and is a function of the future lifetime random variable. We explain why this function is useful and derive its probability distribution.
In this chapter we introduce equity-linked insurance contracts. We explore deterministic emerging costs techniques with examples, and demonstrate that deterministic profit testing cannot adequately model these contracts.
We introduce stochastic cash flow analysis, which gives a fuller picture of the characteristics of the equity-linked cash flows, particularly when guarantees are present, and we demonstrate how stochastic cash flow analysis can be used to determine better contract design.
Finally we discuss the use of quantile and conditional tail expectation reserves for equity-linked insurance.
In this chapter we introduce emerging costs, or cash flow analysis for traditional insurance contracts. This is often called profit testing when applied to life insurance.
We introduce profit testing in two stages. First we consider only those cash flows generated by the policy, then we introduce reserves to complete the cash flow analysis.
We define several measures of the profitability of a contract: internal rate of return, expected present value of future profit (net present value), profit margin and discounted payback period. We show how cash flow analysis can be used to set premiums to meet a given measure of profit.
We restrict our attention in this chapter to deterministic profit tests, ignoring uncertainty. We introduce stochastic profit tests in Chapter 15.
In this chapter we introduce some actuarial approaches to estimation and inference used to construct the life tables and survival models that we have been using in previous chapters. We start with a discussion of typical characteristics of lifetime data for actuarial applications. We then show how to use lifetime data to fit survival models, including parametric and non-parametric approaches.
We next move to the Markov models from Chapter 8. Starting with the alive--dead model, and assuming a piecewise constant force of mortality, we derive the maximum likelihood estimator for the force of mortality for each age year. We then extend the methodology to multiple state models with piecewise constant transition intensities.
In this chapter we develop formulae for the valuation of traditional insurance benefits. In particular, we consider whole life, term and endowment insurance. For each of these benefits we identify the random variables representing the present values of the benefits and we derive expressions for moments of these random variables. The functions we develop for traditional benefits will also be useful when we move to modern variable contracts.
We develop valuation functions for benefits based on the continuous future lifetime random variable and the curtate future lifetime random variable from Chapter 2. We introduce a new random variable, the 1/m-thly curtate future lifetime, which we use to value benefits which depend on the number of complete periods of length 1/m years lived by a life age x. We explore relationships between the expected present values of different insurance benefits.
We also introduce the actuarial notation for the expected values of the present value of insurance benefits.
In this chapter we lay out the context for the mathematics of later chapters, by describing some of the background to modern actuarial practice, as it pertains to long term, life contingent payments. We describe the major types of life insurance products that are sold in developed insurance markets, and discuss how these products have evolved over the recent past. We also consider long term insurance that is dependent on the health status of the insured life, rather than simply survival or death. Finally, we describe some common pension designs.
We give examples of the actuarial questions arising from the risk management of these contracts.
In this chapter we review the basic financial mathematics behind option pricing. First, we discuss the no arbitrage assumption, which is the foundation for all modern financial mathematics. We present the binomial model of option pricing, and illustrate the principles of the risk neutral and real world measures, and of pricing by replication.
We discuss the Black--Scholes--Merton option pricing formula, and, in particular, demonstrate how it may be used both for pricing and risk management.
Longevity models allow for stochastic variation in the underlying force of mortality, so that instead of assuming, for example, that mortality follows a Gompertz model, we now assume thait changes with time, and can be modelled as a stochastic process. In this chapter we introduce the Lee-Carter and Cairns-Blake-Dowd models for longevity. We illustrate some of the structural assumptions of the models, and demonstrate key features. We also discuss briefly how the models are applied in actuarial risk management.
In this chapter we define a life table. For a life table tabulated at integer ages only, we show, using fractional age assumptions, how to calculate survival probabilities for all ages and durations.
We discuss some features of national life tables from Australia, England & Wales, and the United States.
We then consider life tables appropriate to individuals who have purchased particular types of life insurance policy and discuss why the survival probabilities differ from those in the corresponding national life table. We consider the effect of selection of lives for insurance policies, for example through medical underwriting. We define a select survival model and we derive some formulae for such a model.
We consider heterogeneity in populations, exploring how combining lives with different underlying mortality impacts the mortality experience of the group as a whole.
Finally, we present some methods for constructing survival models which allow for trends in underlying population mortality rates.
In this chapter we introduce the concept of a policy value for a life insurance policy. Policy values are used to determine the economic or regulatory capital needed to remain solvent, and also to determine the profit or loss for the company over any time period.
We define the policy value as the expected value of future net cash flows for a policy in force, and distinguish gross premium policy values, which explicitly allow for expenses and for the full gross premium, from net premium policy values, where expenses are excluded from the outgoing cash flows, and only the net premium is counted as income.
We show how to calculate policy values recursively, and how to analyse profit by source. We derive Thiele's differential equation for policy values -- the continuous time equivalent of the recursions for policies with annual cash flows.
We consider how policy values can be used as the basis for policy alterations.
We show how a retrospective valuation has connections both with asset shares and with the policy values . Finally, we define modofied net premium policy values, and consider specifically the Full Preliminary term reserve.
Universal life insurance is a form of whole life (or endowment) insurance, with some profit sharing incorporated in the design, and which also has more flexible payment schedules than traditional insurance. We demonstrate how to use the profit testing techniques from Chapter13 to analyze a universal life insurance contract, and we consider the impact of different types of death and surrender benefits.
In this chapter we introduce some of the notation and concepts of pension plan valuation and funding. We discuss the difference between defined benefit (DB) and defined contribution (DC) pension plans. We introduce the salary scale function, and show how to calculate an appropriate contribution rate in a DC plan to meet a target level of pension income.
We define the service table, which is a summary of the multiple state model appropriate for a pension plan. Using the service table and the salary scale, we can value the benefits and contributions of a pension plan, using the same principles as we have used for valuing benefits under an insurance policy.
We introduce benefit design, valuation and accruals based funding for final salary and career average earnings DB plans. In the final section, we apply the principles of pension funding and valuation to retiree health benefits.