Bruce L. Jones

A family of fractional age assumptions

Abstract This paper introduces a unifying family of fractional age assumptions (FAAs) whose members include the familiar uniform distribution of deaths, constant force, and Balducci assumptions. The family also includes a wide range of alternative assumptions that can be used when those mentioned above are inappropriate. FAAs combined with life table probabilities allow one to fully specify the age at death distribution. Traditionally, a single FAA is applied consistently across all ages. However, this frequently produces forces of mortality between integer ages that are inconsistent with the pattern of mortality rates across ages. In this paper, we introduce the idea that the FAA can be allowed to vary across ages so as to produce a more reasonable force of mortality function and more accurate actuarial present values. Our family of FAAs is also considered in estimating mortality rates. Rather than making an arbitrary assumption in order to express probabilities and densities in terms of the mortality rate, we allow the data to determine the FAA.

A critique of fractional age assumptions

Abstract Published mortality tables are usually calibrated to show the survival function of the age at death distribution at exact integer ages. Actuaries make fractional age assumptions when valuing payments that are not restricted to integer ages. A fractional age assumption is essentially an interpolation between integer age values which are accepted as given. Three fractional age assumptions have been widely used by actuaries. These are the uniform distribution of death (UDD) assumption, the constant force assumption and the hyperbolic or Balducci assumption. Under all three assumptions, the interpolated values of the survival function between two consecutive ages depend only on the survival function at those ages. While this has the advantage of simplicity, all three assumptions result in force of mortality and probability density functions with implausible discontinuities at integer ages. In this paper, we examine some families of fractional age assumptions that can be used to correct this problem. To help in choosing specific fractional age assumptions and in comparing different sets of assumptions, we present an optimality criterion based on the length of the probability density function over the range of the mortality table.

Forest Fire Risk Assessment: An Illustrative Example from Ontario, Canada

This paper presents an analysis of ignition and burn risk due to wildfire in a region of Ontario, Canada using a methodology which is applicable to the entire boreal forest region. A generalized additive model was employed to obtain ignition risk probabilities and a burn probability map using only historic ignition and fire area data. Constructing fire shapes according to an accurate physical model for fire spread, using a fuel map and realistic weather scenarios is possible with the Prometheus fire growth simulation model. Thus, we applied the Burn-P3 implementation of Prometheus to construct a more accurate burn probability map. The fuel map for the study region was verified and corrected. Burn-P3 simulations were run under the settings (related to weather) recommended in the software documentation and were found to be fairly robust to errors in the fuel map, but simulated fire sizes were substantially larger than those observed in the historic record. By adjusting the input parameters to reflect suppression effects, we obtained a model which gives more appropriate fire sizes. The resulting burn probability map suggests that risk of fire in the study area is much lower than what is predicted by Burn-P3 under its recommended settings.

A Model for Analyzing the Impact of Selective Lapsation on Mortality

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.

Asymptotic Tail Probabilities for Large Claims Reinsurance of a Portfolio of Dependent Risks

We consider a dependent portfolio of insurance contracts. Asymptotic tail probabilities of the ECOMOR and LCR reinsurance amounts are obtained under certain assumptions about the dependence structure.

Nested L-statistics and their use in comparing the riskiness of portfolios

Inspired by the problem of testing hypotheses about the equality of several risk measure values, we find that the ‘nested L-statistic’—a notion introduced herein—is natural and particularly convenient. Indeed, the test statistic that we explore in this paper is a nested L-statistic. We discuss large-sample properties of the statistic, investigate its performance using a simulation study, and consider an example involving the comparison of risk measure values where the risks of interest are those associated with tornado damage in different time periods and different regions.

Research and Reality: A Literature Review on Drawing Down Retirement Financial Savings

How DO, COULD, and SHOULD retirees draw down their financial savings? This paper reviews over one hundred papers on this topic from the perspective of individuals, families, governments and financial institutions. Three significant conceptual/methodological weaknesses in the existing literature are identified: (1) analysts have examined a limited range of self-managed drawdown strategies; (2) nearly all have ignored home ownership, pensions, debt, and government taxes and transfers when quantitatively evaluating alternative drawdown strategies and (3) there is a well-acknowledged gap between the behavior implied by economic models and that of real-life individuals, particularly when it comes to voluntary annuitization. Expanding the set of drawdown strategies evaluated (e.g. including larger payouts when life expectancy is reduced after the onset of a significant health condition, or using savings as bridge income to delay the take-up of Social Security payments), refining the income concept used, and more exact modeling of the trade-offs underlying individual decision-making will likely increase the appeal of self-managed drawdown strategies and help resolve the “annuity puzzle” that has long dominated this line of research. It may also lead to advice and financial products that will better meet the needs of retirees.

The use of phase-type models for disability insurance calculations

This paper explores the use of phase-type models in actuarial calculations for disability insurance. We demonstrate that the changes in status of disability insureds can be appropriately captured by a phase-type model. Our model represents the aging process as the passage through a number of phases of decreasing vitality. When disabled, individuals additionally pass through several stages that represent duration of disability. Recovery and mortality rates from the earlier stages are greater than those in later stages. Using such a model, explicit and easily calculable expressions are obtained for relevant probabilities and actuarial present values. This facilitates the calculation of premiums and reserves.

When Inflation Causes No Increase in Claim Amounts

It is well known that when (re)insurance coverages involve a deductible, the impact of inflation of loss amounts is distorted, and the changes in claims paid by the (re)insurer cannot be assumed to reflect the rate of inflation. A particularly interesting phenomenon occurs when losses follow a Pareto distribution. In this case, the observed loss amounts (those that exceed the deductible) are identically distributed from year to year even in the presence of inflation. Nevertheless, in this paper we succeed in estimating the inflation rate from the observations. We develop appropriate statistical inferential methods to quantify the inflation rate and illustrate them using simulated data. Our solution hinges on the recognition that the distribution of the number of observed losses changes from year to year depending on the inflation rate.

“Comparison of Future Lifetime Distribution and Its Approximations,” Esther Frostig, April 2002

BRUCE L. JONES* AND JOHN A. MEREU We found this paper to be of interest because the topic is one we recently did some work on (Jones and Mereu 2000, 2002). In our papers, we examined families of fractional age assumptions including the three traditional ones analyzed by Frostig. To help in choosing specific fractional age assumptions, we presented an optimality criterion based on the length of the probability density function over the range of the mortality table. Our findings are consistent with those obtained by Frostig. They show that UDD results are better than those found with the constant-force and the hyperbolic (Balducci) assumptions. We postulated assumptions that were better than UDD, but not as simple. These could be used if the error with UDD was not acceptable.