# What does stochastic premium mean

## Calculation of premiums

### Bachelor thesis, 2012

### 46 pages, grade: 1.3

Reading sample

Karlsruher Institute for Technology

Properties of premium principles. . . . . . . . . . . . . . . . . . . .

The net premium principle. . . . . . . . . . . . . . . . . . . . . . . . . .

The percentile principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Explicit reward principles

The expected value principle. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The semivariance principle. . . . . . . . . . . . . . . . . . . . . . . . . . .

The standard deviation principle. . . . . . . . . . . . . . . . . . . . . .

The semi-standard deviation principle. . . . . . . . . . . . . . . . . . . .

The mean value principle. . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Esscher principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Karlsruhe principle. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bonuses and loss functions

The net premium principle. . . . . . . . . . . . . . . . . . . . . . . . . .

The expected value principle. . . . . . . . . . . . . . . . . . . . . . . . .

The mean value principle. . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Esscher principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Experience pricing. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bonuses and utility functions

Arrow Pratt measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The division of the premium

Balance in the collective. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Additive bonus principles. . . . . . . . . . . . . . . . . . . . . . . . .

Subadditive premium principles. . . . . . . . . . . . . . . . . . . . . . . .

The principle of covariance. . . . . . . . . . . . . . . . . . . . . . . . . . . .

This thesis deals with the calculation of premiums related to the insurance

essence - a topic of great importance. On the one hand, it must be ensured that

the insurance companies are in a position to cover their

Performance promise to the policyholders as part of the insurance

Compliance with contracts with a probability bordering on certainty. On the other hand

the calculation of premiums should be "fair" so that every policyholder

assessed according to the risk of its accidental future loss amounts to be insured

and the premium is not set inappropriately or unjustifiably high.

In the following, we will assume an insurance company that

ches one or more risks insured and for each risk as well as a stock of

Wants to determine a premium for risks.

Since the premiums in life insurance are based on financial mathematics and

biometric calculation bases (see [10, Chapter 5]), which are not

are the subject of this work, we will focus on general insurance

to a certain shape. Here we assume that the risks to be insured are through

the random future loss amounts are determined.

There are many different concepts for calculating premiums in property insurance.

to lean. One of these concepts is the use of so-called premium principles,

what this thesis is about.

An important requirement here is that the

The accumulated premium does not depend on the distribution of the damage levels of the risk

however, from the joint distribution of the loss amounts of several risks.

When we talk about the premium, we always mean the risk premium, which is derived from the gross premium

mie is to be distinguished. In contrast to the risk premium, the gross premium

Surcharges for profit and operating costs as well as a discount for income from capital

The premium actually required in practice depends on other parameters

such as the size of the insurer, which means that it is better or worse

Can be achieved collectively (this becomes clearer in Chapter 6), the market data

such as the interest rate, the demand or the prices of the competition, or

your own strategic goals (see also [5, page 89]).

Before we get into the matter, I would now like to give a brief overview of

the individual chapters give:

First, in Chapter 2, we will discuss the Reward Principles and some other important terms

mathematically precise and then desired properties of premium

get to know principles. Finally, we will introduce two basic principles of rewards

present and examine with regard to their properties.

In Chapter 3 we consider explicit premium principles, which are characterized by

that for each risk they explicitly calculate the premium through a function of the expected values

assess certain functions of risk. We're going to be many different

Get to know the premium principles and their characteristics.

In Chapter 4 we deal with the connection between the principle of bonuses and

pleasure functions. We will see how some of the bonuses presented up to then

justify principles by minimizing a loss function.

In Chapter 5 we deal with utility functions and their application in the zero-

Benefit principle to set premiums. Using some definitions and sentences will be

we examine the zero-utility principle in terms of a desired property and

consider the Arrow-Pratt measure in connection with utility functions.

Finally, Chapter 6 deals with the division of the premium, it now becomes a stock

viewed from risks. To this end, we will examine further properties of the premium principle

get to know pien, whereby the balance in the collective is extremely desirable. We

will see an example of how the premium for a portfolio relates to the individual risks

ken and generalize this procedure in the course of the covariance principle.

This work is essentially based on [10, Chapter 10]. The places that are in terms of content

are not taken from [10, chapter 10], but from another source, I have as

such marked.

Before I start with the basic concepts of the premium principle, risk premium, net premium and

To make the safety margin more precise, you should first consider some basic quantities

to be introduced. This leads to the following definitions:

(R) is called the set of all discrete random variables X: - R.

Definition 2.2. A figure H: L

Bonus principle if it fulfills the following three conditions:

= H (X) = H (Y) for all X, Y L

({X> H (X)})> 0 for all X L

is the name of the class of risks that can be insured under the H premium principle.

Definition 2.3. H (X) is the risk premium for a risk X L

In the following it will be referred to briefly as the premium.

[X] is the net premium for a risk X L

(X) - E [X] is called the safety margin with regard to a risk X L

Definition 2.4. Let c be the safety margin and

[X] + c the premium for risk X.

{X> E [X] + c} Ruin under the safety margin c.

({X> E [X] + c}] denotes the probability of ruin.

The interpretation in the

The first condition means that for risks with an equal distribution of the damage

the same premium is set at the same level.

The second condition also says that for every risk the premium is at least as high

large as the net premium is; a positive safety surcharge is levied. These

The requirement is important because in the case of H

(X)

of ruin in the individual model for a homogeneous stock (X

identically distributed) due to the strong law of large numbers almost certainly against 1

The third condition requires that the amount of damage of each risk with strictly positive

Probability is greater than the premium and thus ensures that the insurance

insurance company cannot make a sure profit. Hence they are called

Requirement also no-arbitrage condition.

Theorem 2.5. A constant risk cannot be insured under any premium principle.

This statement can be shown by a simple contradiction proof.

) a constant random variable and H: L

Since X is constant, X follows

), so the first moment exists and continues

According to requirement (iii) of a premium principle now applies

0

H (X)}) = P ({E [X]> H (X)}), which is obviously a contradiction in terms

to demand (ii) on a premium principle, namely E

This result is irrelevant for practice, as it makes no sense to identify risks with certain

to insure ministic known amount of damage.

The following theorem is a direct consequence of Theorem 2.5:

Theorem 2.6. Let X be the amount of damage to a

safeguardable risk. Then: E

[X]> 0 and Var (X)> 0.

any bonus principle and X

) is required, X L applies

) and thus E [X] 0. So it exists

Var (X) and this must not be equal to 0, since X is constant in this case

would not be insurable under any premium principle according to sentence 2.6. For the same

[X] must not be equal to 0, since in this case, due to X L

({X = 0}) = 1 follows and thus Var (X) = 0. All in all, it follows that E [X]> 0 and

2.1 Characteristics of premium principles

In order to then define the properties of the premium principles, we need

first the concepts of the stochastic order and the stop-loss order, which in

Actuarial science can play a significant role in comparing risks

({X> a}) P ({Y> a}) for all

is called stochastic order.

Since there are risks with the same expected value but not with a different distribution

can be compared through the stochastic order, one often also considers the stop loss

Order, which has this property.

Before we look at the different options for defining a premium principle,

employ, it should first be clarified which properties are required for a premium prin-

zip are desirable. This leads to the definition of the following four properties:

Definition 2.9. A bonus principle H: L

(i) positive homogeneous if H

(cX) = cH (X) for all X L

(ii) proportional if H

(cX) = cH (X) for all X L

(iii) isotonic with respect to the stochastic order if H

(X) H (Y) for all X, Y L

(iv) isotonic with respect to the stop-loss order if H

(X) H (Y) for all X, Y L

Note 2. Why are these properties of a premium principle of interest?

to (i): The requirement for positive homogeneity makes sense for the following reason:

the risk to premium ratio should be independent of the monetary unit used

be. For example, when changing currency, risks and rewards should be included

can be converted using the same factor.

to (i) / (ii): From the positive homogeneity of a premium principle follows the proportional

quality of a premium principle. (i) is the strongest requirement.

to (ii): The property of proportionality is in proportional reinsurance,

especially quota reinsurance, of importance, since here the reinsurance

cherer with a general quota q

(0, 1) on the inventory of risks between initial

and reinsurers are involved (see [10, Chapter 8]).

to (iii) / (iv): isotonicity of a premium principle with regard to the stochastic order as well as

the stop-loss order is desirable because both order relations are for comparison

of risks in terms of their dangerousness and for a more dangerous one

Risk should also be set at a higher premium.

to (iii) / (iv): Premium principle H is isotonic with respect to the stop-loss order

= H is isotonic with respect to the stochastic order.

This follows from remark 1

Sentence 2.10. A bonus principle H: L

a positive, homogeneous premium principle. It follows

(cX) = cH (X) for all X L

(X) H (cX). Overall, it follows that H (cX) = cH (X), so the premium principle is H

In the remainder of this chapter and in the next chapter we will discuss the principles of rewards

and study their properties.

2.2 The net premium principle

First, let's look at the simplest of all premium principles that each risk

just allocating the net premium.

) | P ({X> E [X]})> 0}

(X): = E [X] is then called the net premium principle.

The following sentence shows that we have defined such a bonus principle and that it all

desired properties met.

Theorem 2.12. The net premium principle is a positively homogeneous (and thus also pro-

portional) premium principle. It is isotonic with respect to the stop-loss order (and thus

also isotonic with respect to the stochastic order).

Proof. First, we show that the conditions for a premium principle are met:

[X] = E [Y] and thus H (X) = H (Y).

[X] = H (X) and thus in particular E [X] H (X).

({X> H (X)}) = P ({X> E [X]}> 0.

Furthermore, the positive homogeneity results from the linearity of the expected value.

Regarding isotonicity with regard to the stop-loss order:

] for all a R. If we set a = 0 it follows

Since the premium corresponds to the net premium according to the net premium principle, so

no strictly positive safety margin, which is a major disadvantage of this

2.3 The percentile principle

The following bonus principle is due to its properties, in particular with regard to the

Probability of ruin of particular concern.

) | P ({X> E [X]})>, a R

0

a})} for (0, 1).

| P ({X> a})} is then called

Percentile principle for the parameter.

The following sentence shows that we have defined such a bonus principle and that there are three

fulfills the desired four properties.

Theorem 2.14. The percentile principle is a positively homogeneous (and thus also proportionally

nales) premium principle. It is isotonic with respect to the stochastic order.

Proof. First, we show that the conditions for a premium principle are met:

| P ({X> a})} = inf {a R

(ii) It exists according to H's definition

(X) a monotonically decreasing sequence {a

}) for all n N such that H (X) = inf

({X> E [X]}). So E [X] H (X).

End of the excerpt from 46 pages

Details

- title
- Calculation of premiums
- University
- Karlsruhe Institute of Technology (KIT) (Institute for Stochastics)
- grade
- 1,3
- author
- Christoph Tiemann (Author)
- year
- 2012
- pages
- 46
- Catalog number
- V213882
- ISBN (eBook)
- 9783656422952
- ISBN (book)
- 9783656423867
- File size
- 816 KB
- language
- German
- Catchwords
- Mathematics, stochastics, premium principles, premiums, loss functions, utility functions, insurance
- Price (book)
- £ 16,99
- Price (eBook)
- £ 11,99

- Cite work
- Christoph Tiemann (author), 2012, calculation of premiums, Munich, GRIN Verlag, https://www.grin.com/document/213882

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