Insurance mathematics IV. lecture Premium principles Introduction There are a lot of principle which can be used for premium calculation of new product

Insurance mathematics IV. lecture

Premium principlesIntroduction

There are a lot of principle which can be used for premium calculation of new product (which has different advantages and disadvantages).

Actuaries need to know more principles to be well-trained to choose the most adequate principle regarding the actual environment.


Premium principles


Premium principles


Premium principlesClassical principles I.

Definition:

Π is expected value premium principle with λ parameter, if


Premium principlesClassical principles II.


Premium principlesClassical principles III.

Proof:

At first we suppose that Π is expected value principle. Then

and Therefore

, which proves (i)

ii)

iii)


Premium principlesClassical principles IV.

Proof (continued):


Premium principlesClassical principles V.

Proof (continued):

Let ξ variate which has infinite expected value. Then based on (iii) we get:

Because of monotone convergence theorem

It means that Π is expected value premium principle. Q.e.d.


Premium principlesClassical principles VI.

Definition:

Π is maximal loss premium principle with p parameter, if

Remark:

The huge risk is hazardous, this principle punishes that.

Definition:

Π is quantile premium principle with ε and p parameters, if


Premium principlesClassical principles VII.


Premium principlesClassical principles VIII.

Proof of lemma:

(E ( ξ|𝐴 )−𝐸 (ξ ))2=¿≤ ¿¿


Premium principlesClassical principles IX.

Proof of statement:

Let

, because


Premium principlesClassical principles X.

Q.e.d.

Definition:

Π is variance premium principle with β parameter, if


Premium principlesClassical principles XI.

Definition:

Π is standard deviation premium principle with β parameter, if

Remark:

The above two principles punish the difference with expected value. But there is a question why we punish if the risk less then expected value. That is why it is more useable the next principle for which it is necessary the following definition.


Premium principlesClassical principles XII.

Definition: Let semi-variance of ξ the next formula:

Definition:

Π is semi-variance premium principle with β parameter, if


Premium principlesClassical principles XIII.


Premium principlesClassical principles XIV.

Proof (continued):

We use the earlier inequalities for these variate:

It follows:


Premium principlesClassical principles XV.

Let the next variate:

𝑃 (ξ1=0 )= 𝜎 2

𝜎2+𝜇2 ;𝑃 (ξ1=𝜇+𝜎 2

𝜇 )= 𝜇2

𝜎2+𝜇2


Premium principlesMathematical properties of premium principles I.

Definition: is risk-loading, if )( HE )()(

Loading for risk is desirable because one generally requires a premium rule to charge at least the expected payout of the risk ξ, namely E(ξ), in exchange for insuring the risk. Otherwise, the insurer will lose money on average.


Premium principlesMathematical properties of premium principles II.Definition: is no unjustified risk-loading, if a risk is equal almost everywhere then

)( 0c.)( c

If we know for certain (with probability 1) that the insurance payout is c, then we have no reason to charge a risk loading because there is no uncertainty as to the payout.

Definition: is no rip-off, if)( HxPx )0)(:sup()(

Insured will not pay more than its maximum risk (with positive probability).

Definition: has translation invariance property, if)(

0, )()( aHaa

If we increase a risk ξ by a fixed amount a, then this property states that the premium for ξ + a should be the premium for ξ increased by that fixedamount a.


Premium principlesMathematical properties of premium principles III.

Definition: has scale invariance property, if)(

0, )()( bHbb

This property essentially states that the premium for doubling a risk is twice the premium of the single risk. One usually uses a no arbitrage argument to justify this rule. Indeed, if the premium for 2ξ were greater than twice thepremium of ξ, then one could buy insurance for 2 ξ by buying insurance for ξ with two different insurers, or with the same insurer under two policies.Similarly, if the premium for 2 ξ were less than twice the premium of ξ, then one could buy insurance for 2 ξ, sell insurance on ξ and ξ separately, and thereby make an arbitrage profit. Scale invariance might not be reasonable if the risk ξ is large and the insurer (or insurance market) experiences surplus constraints. In that case, we might expect the premium for 2 ξto be greater than twice the premium of ξ.


Premium principlesMathematical properties of premium principles IV.

Statement: If has scale invariance and translation invariance properties then

is no unjustified risk-loading.

Proof:

We assume that . Then (because of scale invariance property):

Finally we get:


Premium principlesMathematical properties of premium principles V.

Definition: is additive, if Π (ξ+η )=Π (ξ )+Π (η ) ∀ ξ ,η∈𝐻Π

Additivity is a stronger form of scale invariance. One can use a similar no-arbitrage argument to justify the additivity property.

Definition: is sub-additive, if Π (ξ+η )≤ Π (ξ )+Π (η ) ∀ ξ ,η∈𝐻Π

One can argue that subadditivity is a reasonable property because the no-arbitrage argument works well to ensure that the premium for the sum of two risks is not greater than the sum of the individual premiums; otherwise, the buyer of insurance would simply insure the two risks separately. However, the no-arbitrage argument that asserts that cannot be less than + fails because it is generally not possible for the buyer of insurance to sell insurance for the two risks separately.


Premium principlesMathematical properties of premium principles VI.

Definition: is super-additive, if Π (ξ+η )≥ Π (ξ )+Π (η ) ∀ ξ ,η∈𝐻Π

Super-additivity might be a reasonable property of a premium principle if there are surplus constraints that require that an insurer charge a greater risk load for insuring larger risks. For example, we might observe in the market that because of such surplus constraints. Note that both sub-additivity and super-additivity properties can be weakened by requiring only or for b > 0, respectively. Next, we weaken the additivity property by requiring additivity only for certain insurance risks.

Definition: is additive for independent risks, if

independent risks

Some actuaries might feel that additivity property is too strong and that the no-arbitrage argument only applies to risks that are independent. They, thereby, avoid the problem of surplus constraints for dependent risks.


Premium principlesMathematical properties of premium principles VII.

Definition: is monoton, if

Although the above properties are (more or less) natural the earlier reviewed principles do not satisfy these properties in each case. The next table shows a short summary which principle satisfies which property:


Premium principlesMathematical properties of premium principles VIII.

Property/principle

Expected Value Variance Standard Deviation

Independent Y Y Y

Risk loading Y Y Y

Not unjustified N Y Y

No rip-off N N N

Translation inv. N Y Y

Scale inv. Y N Y

Additivity Y N N

Sub-additivity Y N N

Super-additivity Y N N

Monotone Y N N


Premium principlesMathematical properties of premium principles IX.

As earlier it can be seen that there is no one useable method for each case, but actuaries has to calculate prudential. What is the most useful process to calculate premium?

At first we have to make a list according to the given problem which properties has to be satisfied ( it can be used two categories: „must” and „nice to have” categories).

Checking which known principle satisfies the necessary properties. If one exists we find the adequate method. If no one exists we have to define a new principle. In this case there is necessary to check each necessary requirement for the problem and the natural requirements also.


Gross premium

Premium elements:

- net premium (due to risk)

- costs (commission, maintenance cost, claims handling cost)

- safety plus

- profit rate

GP=NP+C+SP+PR

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Insurance mathematics IV. lecture Premium principles Introduction There are a lot of principle which can be used for premium calculation of new product