Premium principles
Introduction
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.
Insurance mathematics IV. lecture
Premium principles
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Premium principles
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Premium principles
Classical principles I.
Definition:
Π is expected value premium principle with λ parameter, if
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Premium principles
Classical principles II.
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Premium principles
Classical principles III.
Proof:
At first we suppose that Π is expected value principle. Then
Π ξ𝑛 = (1 + λ) ∙ 𝐸(𝜉𝑛 ) and Π ξ = 1 + λ ∙ 𝐸 ξ . Therefore
lim Π ξ𝑛 − ξ = lim (1 + λ) ∙ 𝐸 ξ𝑛 − ξ = 0, which proves (i)
𝑛→∞
𝑛→∞
ii) Π δ𝑐 = 1 + λ ∙ 𝐸 δ𝑐 = 1 + λ ∙ 𝑐
iii) Π 𝑡𝑄 + 1 − 𝑡 𝑆 = 1 + λ ∙ 𝐸 𝑡𝑄 + 1 − 𝑡 𝑆 = 𝑡 ∙ 1 + λ ∙ E Q +
+ 1−𝑡 ∙ 1+λ ∙𝐸 𝑆 =𝑡 ∙Π Q + 1−𝑡 ∙Π 𝑆
Insurance mathematics IV. lecture
Premium principles
Classical principles IV.
Proof (continued):
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Premium principles
Classical 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.
Insurance mathematics IV. lecture
Premium principles
Classical 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
Insurance mathematics IV. lecture
Premium principles
Classical principles VII.
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Premium principles
Classical principles VIII.
Proof of lemma:
(E ξ 𝐴 − 𝐸 ξ )2 = (
Ω
≤(
Ω
χ𝐴
∙ ξ𝑑𝑃 −
𝑃(𝐴)
χ𝐴
∙ 𝜉 − 𝐸 ξ 𝑑𝑃 −
𝑃 𝐴
2
ξ𝑑𝑃) ≤
Ω
2
2
(𝜉 − 𝐸 ξ )𝑑𝑃) = (
Ω
Ω
Insurance mathematics IV. lecture
χ𝐴
(
− 1) ∙ (𝜉 − 𝐸 ξ )𝑑𝑃) ≤
𝑃 𝐴
Premium principles
Classical principles IX.
Proof of statement:
Let
Π ξ = 𝑝𝐸 ξ + 1 − 𝑝 𝑟𝜀 ≥ 𝑝𝐸 ξ + 1 − 𝑝 𝐸 ξ ξ < 𝑟𝜀 = 𝐸 ξ
, because
Insurance mathematics IV. lecture
Premium principles
Classical principles X.
Q.e.d.
Definition:
Π is variance premium principle with β parameter, if
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Premium principles
Classical 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.
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Premium principles
Classical principles XII.
Definition: Let semi-variance of ξ the next formula:
Definition:
Π is semi-variance premium principle with β parameter, if
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Premium principles
Classical principles XIII.
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Premium principles
Classical principles XIV.
Proof (continued):
We use the earlier inequalities for these variate:
It follows:
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Premium principles
Classical principles XV.
Let ξ1 the next variate:
𝜎2
𝜎2
𝜇2
𝑃 ξ1 = 0 = 2
; 𝑃 ξ1 = 𝜇 +
= 2
𝜎 + 𝜇2
𝜇
𝜎 + 𝜇2
Insurance mathematics IV. lecture
Premium principles
Mathematical properties of premium principles I.
Definition:  ( ) is risk-loading, if ( )  E( )   H
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.
Insurance mathematics IV. lecture
Premium principles
Mathematical properties of premium principles II.
Definition:  ( ) is no unjustified risk-loading, if a risk  is equal c  0
almost everywhere then  ( )  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 ( )  sup( x : P(  x)  0)   H 
Insured will not pay more than its maximum risk (with positive
probability).
Definition:  ( ) has translation invariance property, if
(  a)  ( )  a   H , a  0
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 fixed
amount a.
Insurance mathematics IV. lecture
Premium principles
Mathematical properties of premium principles III.
Definition:  ( ) has scale invariance property, if
 (b   )  b   ( )   H  , b  0
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 the
premium 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 ξ
to be greater than twice the premium of ξ.
Insurance mathematics IV. lecture
Premium principles
Mathematical properties of premium principles IV.
Statement: If Π(ξ) has scale invariance and translation invariance properties
then Π(ξ) is no unjustified risk-loading.
Proof:
We assume that ξ ≡ 𝑐. Then (because of scale invariance property):
Π 0 =Π 0∙ξ = 0∙Π ξ =0
Finally we get:
Π ξ =Π c =Π 0+c =Π 0 +𝑐 =0+𝑐 =𝑐
Insurance mathematics IV. lecture
Premium principles
Mathematical properties of premium principles V.
Definition: Π(ξ) 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 noarbitrage argument to justify the additivity property.
Definition: Π(ξ) 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 noarbitrage 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 Π ξ + η cannot be less than
Π ξ + Π η fails because it is generally not possible for the buyer of
insurance to sell insurance for the two risks separately.
Insurance mathematics IV. lecture
Premium principles
Mathematical properties of premium principles VI.
Definition: Π(ξ) 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 Π(2 ∙ ξ) ≥ 2 ∙ Π(ξ) because of such surplus constraints. Note
that both sub-additivity and super-additivity properties can be weakened by
requiring only Π(𝑏 ∙ ξ) ≤ 𝑏 ∙ Π(ξ) or Π(𝑏 ∙ ξ) ≥ 𝑏 ∙ Π(ξ) for b > 0, respectively.
Next, we weaken the additivity property by requiring additivity only for
certain insurance risks.
Definition: Π(ξ) is additive for independent risks, if
Π ξ+η =Π ξ +Π η
∀ξ, η 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.
Insurance mathematics IV. lecture
Premium principles
Mathematical properties of premium principles VII.
Definition: Π(ξ) is monoton, if
ξ 𝜔 ≥ η 𝜔 ∀𝜔 ∈ Ω → Π(ξ) ≥ Π(η)
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:
Insurance mathematics IV. lecture
Premium principles
Mathematical 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
Insurance mathematics IV. lecture
Premium principles
Mathematical 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.
Insurance mathematics IV. lecture
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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