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.
Inhaltsverzeichnis (Table of Contents)
- Introduction
- Basics of approximation theory
- Theorem 2.1
- Theorem 2.2
- Computing Present Values with Approximation Theory
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This work aims to develop a new approach to computing present values in life insurance mathematics, specifically for segmented collectives. The method relies on the principles of approximation theory within Hilbert-Spaces, aiming to provide a more accurate and efficient alternative to traditional techniques.
- Present value calculations in life insurance
- Approximation theory using Hilbert-Spaces
- Segmented collectives and the impact on present values
- Applications of the method to actuarial computation problems
- Comparison of the new approach to traditional methods
Zusammenfassung der Kapitel (Chapter Summaries)
- Introduction: This chapter establishes the context for the work, outlining the existing methods for calculating present values in life insurance and highlighting their limitations. It introduces the concept of segmented collectives, which presents a challenge for traditional approaches. The chapter motivates the need for a new method that addresses these limitations.
- Basics of approximation theory: This chapter provides a foundation in the theory of Hilbert-Spaces, focusing on the concept of best approximation. Key theorems and definitions relevant to the subsequent application are introduced. This chapter establishes the theoretical framework for the proposed method.
- Computing Present Values with Approximation Theory: This chapter presents the application of the theoretical framework developed in the previous chapter to the problem of computing present values. It outlines a method for calculating present values for segmented collectives using best approximation within a Hilbert-Space. This chapter demonstrates the practical implementation of the new approach.
Schlüsselwörter (Keywords)
This work focuses on present value calculations in life insurance mathematics, particularly for segmented collectives. It explores the application of approximation theory within Hilbert-Spaces to provide a more accurate and efficient method for computing present values. Key concepts include best approximation, inner product, norm, and linear normed spaces. The paper analyzes the benefits of using this approach and compares it to traditional methods.
Frequently Asked Questions
What is the main objective of this paper on Present Values?
The paper develops a method for computing present values for a specific segment of a collective using the present value data of the entire collective through approximation theory.
How is the algorithm for present value calculation defined?
The algorithm uses a special norm in Hilbert-Spaces, induced by an inner product derived from the whole collective's present value.
What are segmented collectives in life insurance mathematics?
Segmented collectives refer to sub-groups within a larger insured population that require specific actuarial present value calculations.
What role does approximation theory play in this method?
It provides a framework to solve the underlying integrals by finding the best approximation of a function within a linear normed space.
How does this approach compare to traditional actuarial methods?
The paper suggests that this method can be more accurate and efficient, particularly for complex segments where traditional techniques might be limited.
- Citation du texte
- Dr. Burkhard Disch (Auteur), 2005, Present Values, Segmentation and Approximation Theory, Munich, GRIN Verlag, https://www.grin.com/document/151731
