John A. Beekman

Interest and Mortality Randomness in Some Annuities

Abstract A model is presented which can be used when interest rates and future lifetimes are random, for certain annuities. Expressions for the mean values and the standard deviations of the present values of future payment streams are obtained. These can be used in determining contingency reserves for possible adverse interest and mortality experience for collections of life annuity contracts. Several complete examples are considered. Certain boundary crossing probabilities for the stochastic process component of the model are obtained.

Extra randomness in certain annuity models

Abstract An alternative model is presented which can be used when interest rates and future lifetimes are random, for certain annuities. The Wiener stochastic process is a major component in the model, and provides much more randomness than an earlier model. Expressions for the mean values and the standard deviations of the present values of future payment streams are obtained and illustrated. Extensive boundary crossing probabilities for the Wiener stochastic process are given. Numerical comparisons with the earlier model are provided.

Stochastic models for bond prices, function space integrals and immunization theory

Abstract This article presents several diffusion models for bond prices. By considering the spot interest rate as a state variable and invoking the no-arbitrage principle, the price of a default-free and non-callable pure discount bond is represented as a conditional expectation. The Ornstein—Uhlenbeck (O.U.) stochastic process is described, and used to model the spot rate. The O.U. process is then modified to exclude negative interest rates and the resulting bond-price partial differential equation is solved. By considering the yield rate as a state variable and using the Brownian bridge process, a simpler bond price model is obtained. Applications to immunization theory are presented.

A series for infinite time ruin probabilities

Abstract This paper presents a series method for calculating the infinite time ruin function. The terms of the series involve convolutions related to the claim size distribution. Approximations to the series are presented, with their error analyses. Three detailed examples are given, two of which involve the inverse Gaussian distribution. A discussion of that distribution is made, including the maximum likelihood estimators of its parameters. The relevance of the Poisson model for numbers of claims stochastic process is considered. Evidence from two very large studies is presented to support that model, at least for some portfolios.

Research on the collective risk stochastic process

Abstract The subject of collective risk has been extensively studied by Scandinavian actuaries. Many articles in this journal and elsewhere have treated the theory and applications of the subject. The purpose of this article is to use these existing works in applying a paper of G. Baxter and M. Donsker [4] to both the ruin and distribution branches of risk theory. In section II, the risk process is briefly described. In section III, a general theorem based on [4] is proved. This yields expressions for the ruin and distribution functions. Section IV discusses these expressions, and gives three examples, including two tables.

Refined distributions for a multi-risk stochastic process

Abstract This paper considers a collective risk model formed linearly from four stochastic processes. The first process involves random sums of random variables, and portrays the insurance claims. The other three processes are Ornstein-Uhlenbeck processes which serve as models for the random deviations in assumptions about investment performance, operating expenses, and lapse expenses. The model presented earlier (Beekman 1975b, 1976) is improved by using both calendar and operational times. Ornstein-Uhlenbeck distributions for finite time periods are derived, and tables are furnished. Probabilities of extreme deviations for the multi-risk process are discussed. The examples in (Beekman 1975b, 1976) are reconsidered, and made more realistic by an improved treatment of the time variables.

A collective risk comparative study

Abstract Infinite time ruin probabilities are computed by the convolution method, the incomplete gamma function method, and an inverse Gaussian method for a real-life example involving data from 24,000 life insurance policies. Some analysis of the methods, and computational suggestions are included.

Risk convolution calculations

Convolutions of probability distributions have long been used in collective risk theory to determine the distribution of aggregate claims. Two examples of such a distribution are given. The paper then presents results for the distribution of aggregate claims for the family of claim distributions in which claim values are equi-spaced, and equi-probable. The distribution of claims may be either the Poisson or negative binomial law. Examples and tables are included. A convolution type series for the infinite time ruin function is examined, including approximations and their error analyses.

Focusing on Short-term Achievement Gains Fails to Produce Long-term Gains

In this article, we present a conceptual framework for addressing the disproportionate representation of culturally and linguistically diverse students in special education. The cornerstone of our approach to addressing disproportionate representation is through the creation of culturally responsive educational systems. Our goal is to assist practitioners, researchers, and policy makers in coalescing 1 Writing of this article was supported by the National Center for Culturally Responsive Educational Systems (NCCRESt) under grant # H326E020003 awarded by the U. S. Department of Education’s Office of Special Education Programs. Accepted under the editorship of Sherman Dorn. Send commentary to Casey Cobb (casey.cobb@uconn.edu). Education Policy Analysis Archives Vol. 13 No. 38 2 around culturally responsive, evidence-based interventions and strategic improvements in practice and policy to improve students’ educational opportunities in general education and reduce inappropriate referrals to and placement in special education. We envision this work as cutting across three interrelated domains: policies, practices, and people. Policies include those guidelines enacted at federal, state, district, and school levels that influence funding, resource allocation, accountability, and other key aspects of schooling. We use the notion of practice in two ways, in the instrumental sense of daily practices that all cultural beings engage in to navigate and survive their worlds, and also in a technical sense to describe the procedures and strategies devised for the purpose of maximizing students’ learning outcomes. People include all those in the broad educational system: administrators, teacher educators, teachers, community members, families, and the children whose opportunities we wish to improve.

Compound Poisson processes, as modified by Ornstein-Uhlenbeck processes

Abstract Assume that {C(t), 0 ⩽ t < ∞} is a compound Poisson stochastic process, which models a collective risk situation. Let {I(t), 0 ⩽ t < ∞} be a stochastic process describing the investment performance deviations (from the expected) over time. It is assumed that the I(t) process is an Ornstein-Uhlenbeck (O.U.) process. Such a process is Gaussian (normal) and Markovian. Its conditional mean function reflects the stabilizing effects needed in a model for an economic process in which excessive movements are rare. Let (t), 0 ⩽ t < ∞ and {L(t), 0 ⩽ t < ∞} be random processes representing the deviations from the operating and lapse expense assumptions. It is assumed that they are O.U. processes, and that the four processes are independent of each other. A risk process (t), 0 ⩽ t < ∞ is formed by a linear combination of the four processes. For the risk process, probabilities of ruin are discussed. A detailed example is provided. References to the recent papers by Harald Bohman, and Olof Thorin are given.