Metamodeling for Variable Annuities

Guojun Gan, Emiliano A. Valdez

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9780367779559
• eBook ISBN: 9781351166584
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

This book is devoted to the mathematical methods of metamodeling that can be used to speed up the valuation of large portfolios of variable annuities. It is suitable for advanced undergraduate students, graduate students, and practitioners. It is the goal of this book to describe the computational problems and present the metamodeling approaches in a way that can be accessible to advanced undergraduate students and practitioners. To that end, the book will not only describe the theory of these mathematical approaches, but also present the implementations.

Part III

Metamodels

7

Ordinary Kriging

A major component of a metamodeling approach is the predictive model. In this chapter, we introduce the ordinary kriging method, which can serve as a predictive model. This method can be used to predict the fair market values of the guarantees embedded in variable annuities. Under the ordinary kriging method, the covariance structure of the responses is modeled and the predictions are based on the weighted average of the observed values.

7.1 Description of the Model

Ordinary kriging is a method developed in geostatistics and derives estimates that are often called best linear unbiased estimators (Isaaks and Srivastava, 1990). The estimator is linear because the estimates are weighted linear combinations of the available data. It is unbiased because the mean errors of the estimates are equal to 0. It is also the best because the variance of the errors is minimized.

The ordinary kriging method treats the unknown values as the outcome of a random process and uses a stationary random function to model the values. To describe the kriging method, let V be the stationary random function, z₁, z₂, …, z_k be locations (e.g., the representative variable annuity policies) where we a value of the random function is known, and x₁, x₂, …, x_n be locations (e.g., policies in the full inforce) where the values of the random function are unknown.

In ordinary kriging, V(z_j) and V(x_j) are random variables for all j = 1, 2, …, k and i = 1, 2, …, n. For i = 1, 2, …, n, the random variable at the unknown location x_i is estimated as

$\widehat{V}\left({\mathbf{x}}_i\right)={\displaystyle \sum _{j=1}^k{w}_{ji}V\left(z_j\right)}.$

(7.1)

Since the estimate is a weighted linear combination of random variables, it is also a random variable. The estimation error is defined as

$R\left({\mathbf{x}}_i\right)=\widehat{V}\left({\mathbf{x}}_i\right)-V\left({\mathbf{x}}_i\right).$

(7.2)

Combining Equation (7.2) and Equation (7.1), we get

$R\left({\mathbf{x}}_i\right)={\displaystyle \sum _{j=1}^k{w}_{ji}}V\left(z_j\right)-V\left({\mathbf{x}}_i\right).$

(7.3)

Taking expectation in both sides of Equation (7.3) and noting that the random function is stationary, we get

$\begin{array}{l}E\left[R\left({\mathbf{x}}_i\right)\right]={\displaystyle \sum _{j=1}^k{w}_{ji}E\left[V\left(z_j\right)\right]-E\left[V\left({\mathbf{x}}_i\right)\right]}\\ \text{...}\end{array}$

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• I Preliminaries
• II Experimental Design Methods
• III Metamodels
• A Synthetic Datasets
• Bibliography
• Index