Summary
This paper describes a method for computing present values for a part (segment) of a given collective with the help of the corresponding present value for the whole collective. The algorithm is given by introducing a special norm via an inner product from the present value of the whole collective. The integral is solved by approximating a function which shows the selection and the best approximation on the integrand for the present value of the whole collective. Examples demonstrate the procedure.
Zusammenfassung
Barwerte, Segmentierung und Approximationstheorie
Inhalt der Ausarbeitung ist eine Methode zur Berechnung der Barwerte für ein segmentiertes Kollektiv unter Benutzung des entsprechenden Barwertes für das ganze Kollektiv. Der Algorithmus ist definiert über eine Norm auf dem Raum der mindestens einmal stetig differenzierbaren Funktionen, induziert über ein Inneres Produkt, definiert unter Zuhilfenahme des Barwerts für das gesamte Kollektiv. Das der Barwertberechnung zugrundeliegende Integral wird durch einfache Auswertung eines geschlossenen Funktionsausdrucks – der aus der expliziten Lösung des Integrals folgt – und der besten Approximation bezüglich der so definierten Norm berechnet. Beispiele demonstrieren die Vorgehensweise.
Table of Contents
1 Introduction
2 Basics of approximation theory
2.1 Theorem
2.2 Theorem
3 Computing Present Values with Approximation Theory
4 Segmentation
5 Examples
Research Objectives and Core Topics
This paper aims to provide a computational method for determining present values within segmented insurance collectives by leveraging approximation theory in Hilbert spaces to achieve closed analytic solutions.
- Application of Hilbert space inner products to actuarial present value problems.
- Development of an algorithm for "segmented" collectives using perturbation functions.
- Approximation of insurance benefit functions using best approximation (BA) in linear spaces.
- Sensitivity analysis for insurance collectives based on death and endowment benefit structures.
- Demonstration of the computational efficiency gains compared to numerical integration.
Excerpt from the Book
3 Computing Present Values with Approximation Theory
In this chapter we give an application of theorem 2.1 (ii) on the computation problem of present values. We choose {h_1,...,h_n} like in theorem 2.2 and get with (10) v(t) h_j(t) f(t)dt = v(t) h_j(t) g_0(t)dt ; j = 1,...,n.
If the space G contains constant elements we can set for example h_i(t) = 1. This gives v(t) 1 f(t)dt = v(t) h_i(t)dt =
That means - with (11) - we are able to compute the present value by
Provided the assumptions in theorem 2.1 hold, let g_0 ∈ G. We assume R(t) and F(t) in (4) are one time continuously differentiable functions; g_0 ∈ G is the BA for the continuous function f like in (0). In (4) there is
Summary of Chapters
1 Introduction: Provides the mathematical foundation and definition of present values for death and endowment cases within a deterministic life insurance model.
2 Basics of approximation theory: Introduces Hilbert spaces and inner products, establishing the theorem for the "best approximation" (BA) of continuous functions.
3 Computing Present Values with Approximation Theory: Applies the developed approximation algorithm to the specific problem of calculating present values for insurance benefits.
4 Segmentation: Extends the methodology to segmented collectives, where the whole group is divided into subgroups with different intensities of death.
5 Examples: Demonstrates the practical implementation of the algorithm using specific mortality and interest rate parameters and validates the results through sensitivity analysis.
Keywords
Present Value, Life Insurance Mathematics, Approximation Theory, Hilbert Space, Segmentation, Inner Product, Best Approximation, Deterministic Model, Intensity of Death, Actuarial Computation, Sensitivity Analysis, Gompertz-Makeham, Stieltjes-Schärf-Integral, Numerical Integration.
Frequently Asked Questions
What is the fundamental goal of this paper?
The primary goal is to establish a method for calculating present values for segmented insurance collectives by using approximation theory in Hilbert spaces instead of relying solely on complex numerical integration.
What are the central thematic fields covered?
The work integrates life insurance mathematics, specifically mortality models and present value calculations, with functional analysis and approximation theory.
What is the core research question?
The research asks if there is a way to compute present values for segmented collectives that allows for explicit solutions and sensitivity analysis, independent of the standard integral calculations used for the whole collective.
Which scientific methodology is applied?
The paper utilizes Hilbert space theory, defining an inner product and norm to characterize the "best approximation" (BA) of insurance benefit functions within a linear space.
What topics are discussed in the main body?
The main body covers the theoretical construction of best approximations, the development of an algorithm for segmented collectives, and numerical examples based on the Gompertz-Makeham mortality rule.
How is the work characterized by its keywords?
It is characterized by terms linking actuarial mathematics to structural numerical methods, such as 'Present Value', 'Segmentation', 'Hilbert Space', and 'Best Approximation'.
How is the "segmented" collective defined?
The collective is divided into two disjoint parts, K1 and K2, based on factors like smoker status or social segmentation, each with distinct intensities of death.
What role does the "perturbation function" play in the algorithm?
The perturbation function s(t) accounts for the difference between the segmented collective and the whole collective, allowing the algorithm to compute present values for segments using the existing solution of the whole group.
Why is the proposed method advantageous for sensitivity analysis?
Once the algorithm is set up, changing the parameters for a segment does not require re-solving the entire linear system, significantly saving computational effort.
What specific role does the Gompertz-Makeham law play in the examples?
It serves as the mortality model used to define the intensity of retirement and death, providing concrete parameters for testing the approximation algorithm.
- Citation du texte
- Dr. Burkhard Disch (Auteur), 2005, Present Values, Segmentation and Approximation Theory, Munich, GRIN Verlag, https://www.grin.com/document/151731
