Ruin probabilities in models with a Markov chain dependence structure

In this paper we derive explicit expressions for the probability of ruin in a renewal risk model with dependence among the increments (Z _k ) _k>0. We study the case where the dependence structure among (Z _k ) _k>0 is driven by a Markov chain with a transition kernel that can be described via ordinary differential equations with constant coefficients.     

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 The first, believed to be new, is based on the synergy potential; this is the largest amount that can be withdrawn from the system without hurting any company in terms of expected utility.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.     

Excess Mortality In Asia Associated With Cigarette Smoking

Abstract

Cigarette smoking has reached epidemic proportions in many Asian countries, and epidemiologists predict massive increases in the number of smoking-related deaths in future decades. This information is of great interest to insurers who would like to sell coverage in these markets with smoker/nonsmoker-distinct pricing. This review examines excess mortality due to cigarette smoking in Asia as determined by a second-quarter 1998 Internet search of the world’s English-language medical literature for references published during the preceding five years. Studies to date which observed fairly low relative risks of mortality in smokers compared with nonsmokers in Asia despite a high prevalence of smoking can be explained by the fact that health outcome data represent early experience. Given similar associations between smoking and mortality in Asian and Western studies, it is likely that mortality patterns of smokers in Asia eventually will mirror those seen in the U.S. and the United Kingdom.     

Hedging Equity-Linked Life Insurance Contracts

Abstract

This paper examines a portfolio of equity-linked life insurance contracts and determines risk-minimizing hedging strategies within a discrete-time setup. As a principal example, I consider the Cox-Ross-Rubinstein model and an equity-linked pure endowment contract under which the policyholder receives max(S_T , K) at time T if he or she is then alive, where S_T is the value of a stock index at the term T of the contract and K is a guarantee stipulated by the contract. In contrast to most of the existing literature, I view the contracts as contingent claims in an incomplete model and discuss the problem of choosing an optimality criterion for hedging strategies. The subsequent analysis leads to a comparison of the risk (measured by the variance of the insurer’s loss) inherent in equity-linked contracts in the two situations where the insurer applies the risk-minimizing strategy and the insurer does not hedge. The paper includes numerical results that can be used to quantify the effect of hedging and describe how this effect varies with the size of the insurance portfolio and assumptions concerning the mortality.     

The Impact of the Inpatient Prospective Payment System and Diagnosis-Related Groups

Abstract

The outpatient prospective payment system for the Medicare program became effective Aug. 1, 2000, as mandated by the Balanced Budget Act of 1997. This outpatient program complements Medicare’s inpatient prospective payment system, which was introduced in 1983. A survey of the literature over the past 20 years is undertaken to review the effects of the inpatient prospective payment system and diagnosis-related groups (DRGs) on inpatient hospital utilization, expenditures, and outcomes. The level of the DRG payment has been questioned, as well as the process of adjusting the payment levels from one year to the next. In addition, past research has speculated that the DRG classification may not be sensitive to severity and is subject to coding ambiguities. These conclusions can be used as input to future research on the new outpatient program, as well as updating research on the inpatient program.     

Equity-Indexed Life Insurance: Pricing and Reserving Using the Principle of Equivalent Utility

Abstract

The author applies the principle of equivalent utility to price and reserve equity-indexed life insurance. Young and Zariphopoulou (2002a, b) extended this principle to price insurance products in a dynamic framework. However, in those papers, the insurance risks were independent of the risky asset in the financial market. By contrast, the death benefit for equity-indexed life insurance is a function of a risky asset; therefore, this paper further extends the principle of equivalent utility. In a second extension, the author applies the principle of equivalent utility to calculate reserves, as introduced by Gerber (1976). In a related paper, Moore and Young (2002) price equity-indexed pure endowments, the building blocks of equity-indexed life annuities.