RISK INS / 4.1 & 4.2

Premium principles depend exclusively on

Assumptions

Uninsurable Risk

We will assume that X is a bounded non-negative random variable.

Most premium principles can also be applied to unbounded and possibly negative claims.

When the result is an infinite premium, the risk at hand is uninsurable.

Premium Principles

1. Net Premium

2. Expected Value Principle

3. Variance Principle

4. Standard Deviation Principle

5. Exponential Principle

Premium Principles

1. Zero Utility Premium

2. Mean Value Principle

In these two premium principles, the ‘parameter’ is a function

Premium Principles

1. Percentile Premium

2. Maximal Loss Principle

3. Esscher Principle

These premium principles are chiefly of theoretical importance

Properties of Premium Principles

If π is additive as well as iterative, and S is a compound r.v. with N terms distributed as X

Premium Principle (Properties) Summary

Premium Reduction by Coinsurance

Bonus-Malus Systems

Introduction

A Generic Bonus-Malus System

Bonusmalus Scale

A Generic Bonus-Malus System

Introduction

A Generic Bonus-Malus System

How to determine Premium?

& Bonusmalus Scale

A Generic Bonus-Malus System

Ex post factor
Fixed ex ante

A Generic Bonus-Malus System

In the investigation, the following was found:

Note on bonus-malus system

Bonus-Malus Systems

Markov Analysis

Bonus-Malus Systems: Markov Analysis

Transition Matrix

Bonus-Malus Systems: Markov Analysis

Steady State Distribution

Loimaranta Efficiency

Interpreting the Efficiency Curve

Finding Steady State Premiums & Loimaranta efficiency

π = …

when dealing with Bonus-Malus systems modeled as Markov chains

Solving for the Stationary Distribution (π)

Calculating the Long-Run Expected Premium