Present Values, Segmentation and Approximation Theory

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.