Summary
Subject of this paper is a method for computing present values with the Arithmetic-Geometric-Mean iteration (AGM). The classical integral-fomula for present values – as for-mulated by Reichel [1] – is solved explicitly by approximating the integral using exponential or trigonometric sums. Using the Elliptic-Integral Presentation (first kind), sequences are con-structed which converge with second order accuracy in the case of exponential, sinus and co-sine functions. Thus, the present value is defined as the limit of a rapidly converging sequence. Examples demonstrate the procedures.
Zusammenfassung
Die Berechnung von Barwerten mit dem AGM
Inhalt der Ausarbeitung ist die Berechnung von Barwerten mit dem Arithmetisch-Geometrischen-Mittel. Die bekannte Integraldarstellung von Reichel [1] wird explizit gelöst unter Zu-hilfenahme einer Approximation des Integranden mit einer speziellen Form von Exponentialsummen oder trigonometrischen Polynomen. Durch die Darstellung des Grenzwertes des AGM mit dem Elliptischen Integral erster Art werden Folgen konstruiert, die quadratisch ge-gen die Exponentialfunktion bzw. Sinus und Cosinus konvergieren. Dadurch wird der Barwert durch eine rasch konvergente Folge dargestellt.
Beispiele demonstrieren die Vorgehensweise.
Table of Contents
1. Introduction
2. Problem and Basic Assumptions
3. Basics on the AGM
4. Computing Sinus and Cosine by AGM
5. Describing the Algorithems with the AGM
5.1. The Exponential Function
5.2. The Trigonometric Functions
Research Objectives & Core Themes
This paper presents an efficient methodology for calculating present values in life insurance mathematics by utilizing the Arithmetic-Geometric Mean (AGM) iteration. The research aims to replace standard numerical approximation techniques by transforming integrals into rapidly converging sequences, thereby providing a more precise and computationally efficient alternative for evaluating actuarial formulas involving exponential and trigonometric functions.
- Application of the Arithmetic-Geometric Mean (AGM) iteration for actuarial computations.
- Approximation of present value integrals through exponential and trigonometric sums.
- Development of second-order convergence algorithms for transcendental functions.
- Synthesis of classical mathematical theory with modern insurance computation problems.
- Efficiency validation through numerical examples and error analysis.
Excerpt from the Book
3. Basics on the AGM
The AGM iteration is defined as an iterative scheme starting with arbitrary positive real numbers z0 and w0. For simplification, we assume 0 < z0 < w0. The iteration is defined for all n ∈ IN by
w_{n+1} = (z_n + w_n) / 2 and z_{n+1} = √(z_n · w_n) . (15)
The following Lemma provides basic facts: Lemma 3.1. (i) The iteration (15) converges to a real limit, i.e. lim_{n→∞} z_n = lim_{n→∞} w_n = AG(z_0, w_0). (ii) z_n and w_n are always nonzero.
Summary of Chapters
1. Introduction: Outlines the mathematical foundation of present value calculation in life insurance and introduces the motivation for using alternative approximation methods.
2. Problem and Basic Assumptions: Rephrases the primary insurance integrals into forms suitable for approximation via exponential or trigonometric sums.
3. Basics on the AGM: Defines the Arithmetic-Geometric Mean iteration and proves its convergence properties as a prerequisite for the developed algorithm.
4. Computing Sinus and Cosine by AGM: Extends the AGM theory to trigonometric functions using Landen's transformation and elliptic integrals.
5. Describing the Algorithems with the AGM: Provides the practical implementation of the theoretical models for exponential and trigonometric functions, supported by numerical examples.
Key Words
Arithmetic-Geometric Mean, AGM, present value, life insurance mathematics, actuarial science, elliptic integrals, second-order convergence, exponential function, trigonometric functions, numerical approximation, Gauss-Legendre, iterative scheme, transcendental functions, integration, Newton's method.
Frequently Asked Questions
What is the primary focus of this research paper?
The paper focuses on an efficient computational method for determining present values in life insurance mathematics using the Arithmetic-Geometric Mean (AGM) iteration.
What are the main subject areas explored in the document?
The core subjects include actuarial mathematics, the theory of elliptic integrals, numerical analysis, and algorithms for the rapid computation of elementary functions.
What is the core objective or research question?
The objective is to replace traditional, computationally heavy integration methods with a faster, second-order convergent algorithm that maintains high precision for insurance calculations.
Which scientific methodology is primarily applied?
The work utilizes the AGM iteration method to construct sequences that converge to a limit, effectively approximating complex integrals without losing critical information.
What topics are covered in the main body?
The main body treats the formulation of present value integrals, the theory of the AGM iteration, the adaptation of this iteration for trigonometric functions, and the practical implementation for exponential and periodic functions.
Which terms best characterize the research?
Key characterizations include Arithmetic-Geometric Mean, present value, second-order convergence, and actuarial computation.
Why is the AGM method considered superior for this specific actuarial problem?
The AGM method allows for computing present values with second-order accuracy, meaning the number of correct digits roughly doubles with each iteration, leading to significant computational efficiency.
How does the author handle the potential for computational errors in the AGM?
The author highlights that the AGM is not inherently self-correcting and suggests combining it with Newton's Inverse Iteration to ensure the precision and stability of the result.
What is the significance of the numerical examples provided?
The examples serve to validate that the presented algorithms provide accurate results for real-world insurance durations (e.g., age 30) with very few computational steps compared to traditional summations.
- Quote paper
- Dr. Burkhard Disch (Author), 2002, Computing present values by the AGM, Munich, GRIN Verlag, https://www.grin.com/document/151733
