Statistical Independence and Fractional Age Assumptions

Abstract

This paper considers in some detail the issue of statistical independence of the curtate future lifetime and the fractional part of the future lifetime of a general status.

Statistical independence is often employed in actuarial contexts, primarily because it leads to simple relationships between quantities of interest and statistical information that is of a discrete nature, such as a life table. The uniform distribution of deaths (UDD) assumption is the most commonly used because of its simplicity and intuitive appeal, but it can be somewhat restrictive. For example, all deaths or withdrawals may be assumed to be at a particular point in the year such as the middle; assumptions of this type are often made in a multiple decrement context. This paper attempts to unify these assumptions and extend their applicability in an actuarial context.

The conditions for independence need to be stated carefully, and the last-survivor status is cited as an example in which failure to do so can lead to erroneous conclusions.

The fractional independence (Fl) assumption is defined, and it is demonstrated that many of the formulas for life table functions that hold under the more restrictive UDD assumption are extended easily to the general Fl case. The simple relationship under UDD between insurances payable on other than an annual mode and those payable at the end of the year of death is extended to the Fl case as well. These results are then used to obtain results for annuities and reserves, again generalizing UDD relationships. It is then demonstrated that many contingent probabilities in the multiple life context are exactly the same under the Fl assumption as under the more restrictive UDD assumption. Finally, a very general result that holds in the multiple decrement context is shown to hold under the Fl assumption.     

Two Paradigms for The Market Value of Liabilities

Abstract

Asset/liability management (ALM) theory and practices of insurers have matured and developed from early applications to guaranteed investment contracts (GICs) to all annuity and insurance products today. An important and logical next step of inquiry is the definition of, and calculation procedures for, the market value of an insurance liability. Because all ALM strategies have as their goal the management of some value of assets in relation to some value of liabilities, this inquiry will provide at last a canonical basis for ALM: the management of relative market values.

To set the stage for this exploration, the theory and application of pricing in a complete market are reviewed, as are the practical limitations of this theory in the real, and far from complete, financial markets. The notion of an ad hoc pricing model is developed, and examples are reviewed and critiqued. These models, though imperfect compared with pricing in a complete market, bridge the gap between pricing theory and practice.

The current state of the liabilities market is also discussed, and this market is seen to naturally split into a “long” and a “short” submarket. Of particular interest is the theoretical possibility of these markets becoming broad-based, deep and active, and the conclusions are relevant to the issue of long/short price equalization.

Two paradigms are then explored for defining and subsequently calculating an insurance liability market value. A “paradigm” is a generalized model or framework for accomplishing the task at hand. Each paradigm reflects observable market trading activity, however infrequent, and each is based on methods of valuation consistent with finance-theoretic approaches that are routinely used for the market valuation of assets.

In addition, each paradigm allows for a sequence of ad hoc valuation methodologies, which differ in the extent to which various risks are explicitly modeled versus judgmentally reflected in a risk spread. These paradigms are discussed and contrasted, and arguments made for the potential evolution of the respective values if a “liability” market began trading actively. Practical constraints on the realization of this evolution are also noted.

The last section of this paper discusses a host of considerations related to the application of option-pricing theory to insurance company liabilities.     

A Model for Analyzing the Impact of Selective Lapsation on Mortality

Abstract

This paper presents a model for examining the effect of various relationships between mortality rates and lapse rates on the mortality experience of a cohort of insured lives. The approach is individual rather than the aggregate traditionally used in analyzing selective lapsation. The model assumes that insured lives are healthy at policy issue, but later may move to an impaired state from which the lapse rate is zero. Associated with each insured is an unobservable “risk level” random variable, which reflects the heterogeneity of the insured group. Individual mortality and lapse rates are functions of the risk level. A numerical illustration provides some interesting results obtained by using this model.     

Credibility Using Copulas

Abstract

Credibility is a form of insurance pricing that is widely used, particularly in North America. The theory of credibility has been called a “cornerstone” in the field of actuarial science. Students of the North American actuarial bodies also study loss distributions, the process of statistical inference of relating a set of data to a theoretical (loss) distribution. In this work, we develop a direct link between credibility and loss distributions through the notion of a copula, a tool for understanding relationships among multivariate outcomes.

This paper develops credibility using a longitudinal data framework. In a longitudinal data framework, one might encounter data from a cross section of risk classes (towns) with a history of insurance claims available for each risk class. For the marginal claims distributions, we use generalized linear models, an extension of linear regression that also encompasses Weibull and Gamma regressions. Copulas are used to model the dependencies over time; specifically, this paper is the first to propose using a t-copula in the context of generalized linear models. The t-copula is the copula associated with the multivariate t-distribution; like the univariate tdistributions, it seems especially suitable for empirical work. Moreover, we show that the t-copula gives rise to easily computable predictive distributions that we use to generate credibility predictors. Like Bayesian methods, our copula credibility prediction methods allow us to provide an entire distribution of predicted claims, not just a point prediction.

We present an illustrative example of Massachusetts automobile claims, and compare our new credibility estimates with those currently existing in the literature.     

Utility Functions

Abstract

This article is a self-contained survey of utility functions and some of their applications. Throughout the paper the theory is illustrated by three examples: exponential utility functions, power utility functions of the first kind (such as quadratic utility functions), and power utility functions of the second kind (such as the logarithmic utility function). The postulate of equivalent expected utility can be used to replace a random gain by a fixed amount and to determine a fair premium for claims to be insured, even if the insurer’s wealth without the new contract is a random variable itself. Then n companies (or economic agents) with random wealth are considered. They are interested in exchanging wealth to improve their expected utility. The family of Pareto optimal risk exchanges is characterized by the theorem of Borch. Two specific solutions are proposed. The first, believed to be new, is based on the synergy potential; this is the largest amount that can be withdrawn from the system without hurting any company in terms of expected utility. The second is the economic equilibrium originally proposed by Borch. As by-products, the option-pricing formula of Black-Scholes can be derived and the Esscher method of option pricing can be explained.     

Summary of Results of Survey of Seminar Attendees

Abstract

The Society of Actuaries undertook a three-phase research project on mortality improvement in the three NAFTA countries: Canada, Mexico, and the U.S. Phase 1 consisted of a literature review of papers on projecting mortality levels in the future and a study of the trend in mortality improvement during this century. Phase 2 consisted of a discussion of different facets of modeling mortality rates at a seminar attended by 79 experts (actuaries, demographers, economists, and medical researchers) representing different countries. The last session of the seminar consisted of the completion of a survey by the attendees to obtain input for Phase 3, which would analyze the impact of mortality improvement on the social security system of each country. This paper summarizes the results of the survey.

The survey results illustrate the difficulty in forecasting mortality levels, because the effects of many factors that could have significant impact on mortality rates are unknown. This suggests the need for dynamic forecasting, which allows for the possibility of random shocks. A majority of the survey respondents believe that stochastic forecasting models, despite their complexity, have significant potential to add value. Respondents also believe that both historical data and cause-specific mortality forecasts are useful as input and also in validating forecasts of the aggregate levels of mortality. The challenge is to develop more sophisticated forecasting models to produce results that are relatively easy to interpret and to communicate these results to the desired audiences, including the public and policymakers.

The survey results suggest that the aggregate effect of lifestyle changes, medical advances, diseases, catastrophe, and physical environmental changes is an increase in life span. However, there is much uncertainty about the future. Respondents expect that beyond the year 2020 the mean annual rate of reduction in mortality for males age 65 and over will average about 0.58% for Canada, 0.76% for Mexico, and 0.67% for the U.S. The results for the female age 65 and over population are 0.64%, 0.83%, and 0.70%, respectively. The age 65 and over population is expected to see larger percentage reductions in mortality than the 0–14 and 15–64 populations. The reductions in male and female mortality will be ultimately the same, and the mortality levels in the three countries will ultimately converge, although differences may persist for decades.     

Growth of the Public Responsibility of Actuaries in Life Insurance

Abstract

The public responsibility of life insurance actuaries has changed from supervisory compliance with detailed state laws to certifying adherence to more general regulatory objectives complemented by actuarial standards of practice.     

Pricing Term Insurance in the Presence of a Family History of Breast or Ovarian Cancer

Abstract

We estimate the increased mortality and term life insurance costs for women who have a family history of breast or ovarian cancer. Using data from the medical literature on age-specific and family history-specific incidence rates, we develop double-decrement models to evaluate the actuarial impact of breast cancer and ovarian cancer in the family. We also calculate the increased mortality and term insurance costs for women who test positive for the BRCA1 or BRCA2 gene mutation. We find that the type of affected relative and her age at onset of the disease are key underwriting factors. We find substantial mortality increases (up to 100%) for women with two relatives with cancer and women with a first-degree relative who developed cancer at an early age. Mortality increases for women with the BRCA gene mutation reach 150%. While some females with a family history of cancer can be accepted at standard rates, others may need to be quoted substandard rates, depending on the underwriting policy of the company. Females with the gene mutation can possibly be accepted at a rate that incorporates a severe mortality surcharge.