Risk Measures and Insurance Solvency Benchmarks

Fixed-Probability Levels in Renewal Risk Models

Vsevolod K. Malinovskii

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000411096
• Edition: 1
• Topic: Business
• Subtopic: Insurance
• Index: Business

Risk Measures and Insurance Solvency Benchmarks: Fixed-Probability Levels in Renewal Risk Models is written for academics and practitioners who are concerned about potential weaknesses of the Solvency II regulatory system. It is also intended for readers who are interested in pure and applied probability, have a taste for classical and asymptotic analysis, and are motivated to delve into rather intensive calculations.

The formal prerequisite for this book is a good background in analysis. The desired prerequisite is some degree of probability training, but someone with knowledge of the classical real-variable theory, including asymptotic methods, will also find this book interesting. For those who find the proofs too complicated, it may be reassuring that most results in this book are formulated in rather elementary terms. This book can also be used as reading material for basic courses in risk measures, insurance mathematics, and applied probability. The material of this book was partly used by the author for his courses in several universities in Moscow, Copenhagen University, and in the University of Montreal.

Features

• Requires only minimal mathematical prerequisites in analysis and probability
• Suitable for researchers and postgraduate students in related fields
• Could be used as a supplement to courses in risk measures, insurance mathematics and applied probability.

Chapter 1

Risk Measures in Finance and Insurance

This chapter is an introduction to this book and a summary of its main results. We start with quantile risk measures in finance and portfolio management and proceed to the economic capital, which is a comprehensive concept put forth for predicting the strength of financial institutions. We touch on some aspects of the capital requirements of the Solvency II regulatory system, which are identified as Value-at-Risk at the 99.5% threshold. We argue that this can create confusions when choosing an adequate risk measure. To describe the essence of this concern, we turn to the non-ruin capital calculated on the basis of probability of ruin in Lundberg's collective risk model and focus on its difference from the non-loss capital calculated on the basis of year-end financial result, both of which are Value-at-Risk quantile measures.

1.1 Risk Measures in Finance and Portfolio Management

In insurance mathematics, the losses¹ of an insurer are modeled by the random variables with continuous distributions, with possibly a small number of jumps. In order to avoid inessential technical difficulties, in the further presentation we will consider only continuous² (without jumps) distributions of losses, denote by $0<\alpha <1$ a sufficiently small number, i.e., $\alpha =0.05$ or $\alpha =0.01$, and use the shorthand notation $A:=1-\alpha$.

The following well-known definition (see, e.g., [99], § 32.2) is an integral part of the general theory of probability distributions (see, e.g., [94], [98], [103], [140], and [145]).

Definition 1.1 (Quantile of a continuous distribution)Let ξ be a random variable with continuous distribution and $A\in (0,1)$. A value of x, such

that

$$\begin{array}{l}\mathrm{P}\left\{\xi \mathrm{⩽}x\right\}=A,\hfill \end{array}$$(1.1)

denoted by x_A, is called the A-quantile of the distribution of the random variable ξ, i.e., the value below which $100A$ per cent of the distribution lies.

¹ Incidentally, we note that by return (the opposite to loss), we always mean loss or expenditure with the opposite sign. In this book, to avoid confusion, we will always adhere to the loss basis.

² That is, we exclude discrete distributions of loss X and assume that X has a probability density function (p.d.f.) f_X, such that $F_X(x)={\displaystyle {\int }_{-\infty }^x{f}_X}(z)dz$, $x\in \mathrm{R}$. Moreover, we often assume that p.d.f. f_X is bounded from above by a finite constant.

To put it in another way, the A-quantile x_A of the random va...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• 1 Risk Measures in Finance and Insurance
• 2 Fixed-probability Level in a Diffusion Model
• 3 Fixed-probability Level in An Exceptional Renewal Model
• 4 Implicit Function Defined by M-Equation
• 5 Fixed-probability Level in General Renewal Model
• 6 Case Study: Numerical Evaluation of Fixed-probability Level
• 7 Probability Mechanism of Insurance with Migration and ERS-analysis
• A Auxiliary Results from Analysis
• B Auxiliary Results from Probability
• List of Notations
• Notes and Comments
• Bibliography
• Index