Individual risk model
Introduction
For insurer there is essential to know the property of
the distribution of future claims. We are trying to
calculate the distribution of claim amount if there are
given certain distributions.
There is possible to consider this problem with two
methods:
- individual risk model (we summarize the
risks of contracts);
- complex risk model (we consider the total
amount of claims).
Insurance mathematics VII. lecture
Individual risk model
Our portfolio contains n contracts. The total claim will be as follows:
𝑆 = 𝑋1 + 𝑋2 + ⋯ + 𝑋𝑛
where 𝑋𝑖 signs the claim payment of i-th policy.
We assume that 𝑋1 ; 𝑋2 ; … ; 𝑋𝑛 are independent and its distribution is known.
Insurance mathematics VII.
lecture
Reinsurance
Modelling I.
Let X the variate of claim payment. Let P the premium of direct
insurance, F(x) the distribution function of direct insurance. Then the
reinsurance contract can be modelled with 𝑃1 reinsurance premium and
T measurable function: if the total claim payment is x then the direct
insurer will pay T(x), the reinsurance partner will pay x-T(x).
The name of T(x) is own section or own part.
In this chapter we will sign
𝐹0 𝑥 = 𝑃 𝑇𝑋 < 𝑥 , 𝐹1 𝑥 = 𝑃 𝑋 − 𝑇𝑋 < 𝑥 , 𝑃0 = 𝑃 − 𝑃1 .
The risk of direct insurer will be characterized with 𝑃0 , 𝐹0 𝑥 pair,
and the risk of reinsurer will be characterized with 𝑃1 , 𝐹1 𝑥 pair.
Insurance mathematics VII.
lecture
Reinsurance
Modelling II.
If we collect premium due exactly the risk then we get:
𝑥𝑑𝐹 𝑥 = 𝑃
𝑥𝑑𝐹0 𝑥 =
𝑇𝑥𝑑𝐹 𝑥 = 𝑃0
𝑥𝑑𝐹1 𝑥 =
(𝑥 − 𝑇𝑥)𝑑𝐹 𝑥 = 𝑃1
Rational assumption is that: 0 ≤ 𝑇 𝑥 ≤ 𝑥 ∀𝑥 ∈ 𝑅0+
It follows: 0 ≤ 𝑃0 ≤ 𝑃
Insurance mathematics VII.
lecture
Reinsurance
Special cases
If 𝑇 𝑥 = 𝑥 ∀𝑥 ∈ 𝑅0+ then there is no any reinsurance.
If 𝑇 𝑥 = 0 ∀𝑥 ∈ 𝑅0+ then the direct insurer does not keep on any risk and
of course the whole premium will be transferred to the reinsurer. The
direct insurer will get commission from the reinsurer because of treaty
administration. The name of this type is „fronting”.
The reason of fronting will be – among others – when the reinsurance
partner does not want to ground insurance company but they would like to
sell one (or more) special product. Then they are searching a company via
which they can realize it.
Insurance mathematics VII.
lecture
Reinsurance
Basic distinctions I.
The reinsurance contract will be
proportional or
non-proportional.
Under proportional reinsurance, one or more reinsurers take a stated
percentage share of each policy that an insurer produces ("writes").
This means that the reinsurer will receive that stated percentage of the
premiums and will pay the same percentage of claims. In addition,
the reinsurer will allow a "ceding commission" to the insurer to cover
the costs incurred by the insurer (marketing, underwriting, claims etc.).
Under non-proportional reinsurance the reinsurer only pays out if the
total claims suffered by the insurer in a given period exceed a stated
amount, which is called the "retention" or "priority".
Insurance mathematics VII.
lecture
Reinsurance
Basic distinctions II.
The reinsurance contract will be
treaty or
facultative.
According to treaty reinsurance the reinsurer then covers the specified share of
more than one insurance policy issued by the ceding company which come
within the scope of that contract.
Facultative reinsurance is normally purchased by ceding companies for
individual risks not covered, or insufficiently covered, by their reinsurance
treaties, for amounts in excess of the monetary limits of their reinsurance
treaties and for unusual risks.
Insurance mathematics VII.
lecture
Reinsurance
Proportional contracts I.
1. Quota share
The direct insurer transmits the 1-q part of either business line or
product, and the q-th part remains at direct insurer. Of course
reinsurance partner will reimburse the 1-q-th part of claims to the
direct insurer (and some commission for costs of direct insurer).
At first let P is the net premium of direct business.
It means that 𝑇 𝑥 = 𝑞 ∙ 𝑥
Then:
𝑃1 =
𝑋𝑑𝐹1 𝑥 =
𝑋 − 𝑇𝑋 𝑑𝐹 𝑥 = (1 − 𝑞)
Insurance mathematics VII.
lecture
𝑋𝑑𝐹 𝑥 = 1 − 𝑞 ∙ 𝑃
Reinsurance
Proportional contracts II.
It means that regarding net premium the ceded premium is fair.
But there are costs (and profit rates) also.
Let the cost need of direct insurer is 𝐶1 , the cost need of
reinsurer is 𝐶2 (𝐶1 >𝐶2 ).
Then the gross premium of direct insurance will be as follows:
𝑃
Π=
1 − 𝐶1
The ceded premium will be the next:
𝑃
Π1 = (1 − 𝑞) ∙
1 − 𝐶1
Insurance mathematics VII.
lecture
Reinsurance
Proportional contracts III.
But the ceded premium with cost need of reinsurer would be as follows:
𝑃
′
Π1 = (1 − 𝑞) ∙
1 − 𝐶2
It means that the fair reinsurance commission would be as next:
Π1′
1 − 𝐶1
𝐶3 = 1 −
=1−
Π1
1 − 𝐶2
Advantages of quota share treaty:
- decreasing the fluctuation of profit/loss
- decreasing the capital need (direct insurer)
- simple administration
Disadvantage of quota share treaty:
- direct insurer can not select between
„good” and „bad” risks
Insurance mathematics VII.
lecture
Reinsurance
Proportional contracts IV.
2. Surplus
Surplus treaty is similar to quota share, but in this case direct insurer can
decide q own part per risk. There are two limits about own part: the
reinsurer takes over maximum the c-fold of own part (take over c layer), i.e.
1
𝑞≥
1+𝑐
In the other side
𝑅
𝑞≤𝑆
where R the maximum of own part, S the sum insured (probable maximum
loss).
It follows:
𝑅
1
≥
𝑆 1+𝑐
𝑆 ≤ (1 + 𝑐) ∙ 𝑅
Insurance mathematics VII.
lecture
Reinsurance
Proportional contracts V.
Example:
We suppose that there are 9 layers and the maximum own part is 100.
1000
reinsurers
part
free choice
200 100 10
100
200
own part
Insurance mathematics VII.
lecture
1000
Reinsurance
Proportional contracts VI.
Advantage of surplus treaty:
direct insurer can choose more q related to good
risks – reinsurer can give back premium refund
Disadvantage of quota share treaty:
- more complicated administration (because of
individual register of q-s)
Insurance mathematics VII.
lecture
Reinsurance
Proportional contracts VII.
Statement:
Let X is a risk and 𝐷12 > 0 for which 0 ≤ 𝐷12 ≤ 𝐷 2 (𝑋). We suppose
that the variance of ceded portfolio is 𝐷12 , i.e. 𝐷 2 𝑋 − 𝑇(𝑋) = 𝐷12 .
Then there is a q for which 𝑇 𝑥 = 𝑞 ∙ 𝑥 is optimal, i.e. 𝐷 2 𝑇(𝑋 ) is
minimal.
Proof:
Let 𝑞 = 1 −
𝐷12 1
( 2 )2 .
𝐷 (𝑋)
Then 𝐷 2 𝑋 − 𝑇(𝑋) = 𝐷12 = (1 − 𝑞)2 ∙ 𝐷 2 (𝑋)
We sign 𝑇1 𝑋 = 𝑇 𝑋 − 𝑞 ∙ 𝑋
If 𝐷12 = 𝐷 2 𝑋 then with q=0 the statement is true.
Insurance mathematics VII.
lecture
Reinsurance
Proportional contracts VIII.
Proof (continued):
If 𝐷12 < 𝐷 2 𝑋 then 𝑋 − 𝑇 𝑋 = 𝑋 − 𝑞 ∙ 𝑋 − 𝑇1 (𝑋)
It follows:
𝐷2 𝑋 − 𝑇(𝑋) = 𝐷2 𝑋 − 𝑞 ∙ 𝑋 − 𝑇1 (𝑋) =
((𝑥 − 𝑞𝑥 − 𝑇1 𝑥 − 𝐸 𝑋 − 𝑞𝑋 − 𝑇1 𝑋 )2 𝑑𝐹 𝑥 =
=
=
=
((1 − 𝑞)2 𝑥 − 𝐸 𝑋
(((𝑥 − 𝐸 𝑋 ) ∙ (1 − 𝑞) − (𝑇1 𝑥 − 𝐸 𝑇1 𝑋 )2 𝑑𝐹 𝑥 =
2
− 2(1 − 𝑞) 𝑥 − 𝐸 𝑋 (𝑇1 𝑥 − 𝐸 𝑇1 𝑋
= 1 − 𝑞 2 𝐷2 𝑋 − 2 1 − 𝑞
𝑥 − 𝐸 𝑋 (𝑇1 𝑥 − 𝐸 𝑇1 𝑋
Insurance mathematics VII.
lecture
+ (𝑇1 𝑥 − 𝐸 𝑇1 𝑋
2
)𝑑𝐹 𝑥 =
𝑑𝐹 𝑥 + 𝐷2 (𝑇1 𝑋 )
Reinsurance
Proportional contracts IX.
Proof (continued):
We saw earlier:
𝐷 2 𝑋 − 𝑇(𝑋) = 𝐷12 = (1 − 𝑞)2 ∙ 𝐷 2 (𝑋)
1
It follows:
𝑥 − 𝐸 𝑋 (𝑇1 𝑥 − 𝐸 𝑇1 𝑋
𝑑𝐹 𝑥 = 2(1−𝑞) 𝐷2 (𝑇1 𝑋 )
For 𝑇1 𝑋 = 𝑇 𝑋 − 𝑞 ∙ 𝑋, it follows:
2
𝐷2 𝑇(𝑋) =
𝑇 𝑋 −𝐸 𝑇 𝑋
=
(𝑞 𝑥 − 𝐸 𝑋
+ (𝑇1 𝑥 − 𝐸 𝑇1 𝑋 )2 𝑑𝐹 𝑥 =
=
(𝑞 2 𝑥 − 𝐸 𝑋
2
= 𝑞 2 𝐷2 𝑋 + 2𝑞
𝑑𝐹 𝑥 =
𝑞𝑥 + 𝑇1 𝑥 − 𝐸(𝑞𝑋 + 𝑇1 (𝑋) 2 𝑑𝐹 𝑥 =
+ 2𝑞 𝑥 − 𝐸 𝑋 (𝑇1 𝑥 − 𝐸 𝑇1 𝑋
𝑥 − 𝐸 𝑋 (𝑇1 𝑥 − 𝐸 𝑇1 𝑋
Insurance mathematics VII.
lecture
+ (𝑇1 𝑥 − 𝐸 𝑇1 𝑋 )2 𝑑𝐹 𝑥 =
𝑑𝐹 𝑥 + 𝐷2 (𝑇1 𝑋 )
Reinsurance
Proportional contracts X.
Proof (continued):
From above equations we get:
𝐷2 𝑇(𝑋) = 𝑞 2 𝐷2 𝑋 +
We know that
1
𝐷2 (𝑇1 𝑋 )
1−𝑞
0 ≤ 𝑞 < 1 and 𝐷 2 (𝑇1 𝑋 ) ≥ 0, it follows:
𝐷2 (𝑇 𝑋 ) ≥ 𝑞 2 𝐷2 𝑋
If 𝑇 𝑋 ≡ 𝑞𝑋 then the equation will be valid.
Insurance mathematics VII.
lecture
Reinsurance
Non-proportional contracts I.
1. Excess of Loss (XL)
𝑋, 𝑖𝑓 𝑋 ≤ 𝑀
, 𝑀≥0
𝑀, 𝑖𝑓 𝑋 > 𝑀
Then the reinsurance premium will be as follows:
In this treaty 𝑇 𝑋 = min 𝑋, 𝑀 ,i.e. 𝑇 𝑋 =
∞
𝑀
𝑃1 =
𝑥 − 𝑇 𝑥 𝑑𝐹 𝑥 =
𝑥 − 𝑇 𝑥 𝑑𝐹 𝑥 +
0
∞
𝑀
=
𝑥 − 𝑥 𝑑𝐹 𝑥 +
0
∞
𝑥 − 𝑀 𝑑𝐹 𝑥 =
𝑀
𝑥 − 𝑇 𝑥 𝑑𝐹 𝑥 =
𝑀
∞
𝑥𝑑𝐹 𝑥 −
𝑀
𝑀𝑑𝐹 𝑥
𝑀
If F(x) is absolutely continuous then there is f(x) probability density
function and we get:
∞
𝑃1 =
∞
𝑥 ∙ 𝑓(𝑥)𝑑𝑥 − 𝑀
𝑀
𝑓 𝑥 𝑑𝑥
𝑀
Insurance mathematics VII.
lecture
Reinsurance
Non-proportional contracts II.
Example 1.:
Let X is continuous uniform distribution on (0,100) and we suppose that
exactly 1 claim will happen. (We will calculate just the net premium.) Then
the direct premium will be as next:
0 + 100
𝑃=𝐸 𝑋 =
= 50
2
If the direct insurer want to keep maximum 80 of the claim then it has to be
transferred 20% in case of quota-share treaty. The premium of quota-share
treaty as follows:
𝑃1,𝑞 = 1 − 𝑞 ∙ 𝐸 𝑋 = 0,2 ∙ 50 = 10
But if the direct insurer will buy XL treaty then the premium will be the next:
100
𝑃1,𝑋𝐿 =
80
𝑥
𝑑𝑥 − 80
100
100
80
1
1 𝑥2
𝑑𝑥 =
− 80𝑥
100
100 2
It means that XL treaties are relative cheap.
Insurance mathematics VII.
lecture
100
=2
80
Reinsurance
Non-proportional contracts III.
Example 2.:
If there is no limit of direct insurer then it can be worthy to buy XL
treaty. Let X is exponential distribution with a parameter. (We will
calculate just the net premium.) Then the direct premium will be as
1
next:
𝑃=𝐸 𝑋 =
𝑎
We suppose that the own part is
∞
𝑎𝑥𝑒 −𝑎𝑥 𝑑𝑥 −
𝑃1,𝑋𝐿 =
𝑘
𝑎
𝑘
𝑎
𝑘
𝑎
∞
𝑎𝑒 −𝑎𝑥 𝑑𝑥 =
𝑘
𝑎
.Then the reinsurance premium will be:
1 −𝑎𝑥
𝑒
∙ (−𝑎𝑥 − 1)
𝑎
1 −𝑘
𝑘
=− 𝑒
−k − 1 − − −𝑒 −𝑘
𝑎
𝑎
Insurance mathematics VII.
lecture
∞
𝑘
𝑎
1 −𝑘
= 𝑒
𝑎
−
𝑘
−𝑒 −𝑎𝑥
𝑎
∞
𝑘
𝑎
=
Reinsurance
Non-proportional contracts IV.
Example 2. (continued):
It means that for example k=2 than the reinsurance premium will be just
about 14% of net premium.
Advantage of XL treaty:
- simple administration
Disadvantages of XL treaty:
- difficult to calculate reinsurance premium;
(usually the risk has no known distribution)
- protect just against big claims, does not
protect against more small losses.
Catastrophe XL treaty:
There is a special XL treaty: reinsurer will pay when because of one
insurance event the total claim excess a pre-defined limit.
Insurance mathematics VII.
lecture
Reinsurance
Non-proportional contracts V.
2. Stop Loss
The reinsurer will pay if the total claim of one pre-defined period
(typically one year) will excess a pre-defined limit or a predefined percentage of premium.
Let 𝑋𝑖 (1 ≤ 𝑖 ≤ 𝑁) the claim payment of i-th risk in the predefined period, than the contract can be modelled as follows:
𝑁
𝑋𝑖 , 𝑖𝑓
𝑁
𝑇(
𝑋𝑖 ) =
𝑖=1
𝑁
𝑖=1
𝑋𝑖 < 𝑀
𝑖=1
𝑁
𝑀, 𝑖𝑓
𝑋𝑖 ≥ 𝑀
𝑖=1
Insurance mathematics VII.
lecture
Reinsurance
Non-proportional contracts VI.
or
𝑁
𝑋𝑖 , 𝑖𝑓
𝑁
𝑇(
𝑋𝑖 ) =
𝑖=1
𝑁
𝑋𝑖 < 𝑞 ∙ 𝐷
𝑖=1
𝑖=1
𝑁
𝑞 ∙ 𝐷, 𝑖𝑓
𝑋𝑖 ≥ 𝑞 ∙ 𝐷
𝑖=1
q signs a claim ratio, D signs a premium income.
Statement:
If the reinsurance premium is fixed then ∃𝑀0 ≥ 0 for which
𝑋, 𝑖𝑓 𝑋 ≤ 𝑀0
𝑀, 𝑖𝑓 𝑋 > 𝑀0
is optimal, i.e. in case of any 𝑇 transformation
𝑇0 𝑋 =
𝐷2 𝑇(𝑋) ≥ 𝐷2 𝑇0 (𝑋)
Insurance mathematics VII.
lecture
Reinsurance
Non-proportional contracts VII.
∞
Proof:
We know that 𝑃1 = 𝑥 − 𝑀 𝑑𝐹 𝑥
in case of XL treaty.
𝑀
Generally 𝑃1 is continuous in M and strictly monotone decreasing.
If M=0 then 𝑃1 = 𝑃 and lim 𝑃1 = 0. Then because of Bolzano theorem
𝑀→∞
∃𝑀0 that for pre-defined 𝑃1
𝑃1 =
Than we will get as follows:
∞
2
𝐷 𝑇(𝑋) =
∞
𝑇 𝑥 −𝐸 𝑇 𝑋
0
2
∞
𝑀0
𝑥 − 𝑀0 𝑑𝐹 𝑥
∞
𝑑𝐹 𝑥 =
( 𝑇 𝑥 − 𝑀0 ) + (𝑀0 −𝐸 𝑇 𝑋
𝑑𝐹 𝑥 =
0
( 𝑇 𝑥 − 𝑀0 2 +2 𝑇 𝑥 − 𝑀0 (𝑀0 −𝐸 𝑇 𝑋 ) + (𝑀0 −𝐸 𝑇 𝑋
=
2
0
Insurance mathematics VII.
lecture
2
)𝑑𝐹 𝑥 =
Reinsurance
Non-proportional contracts VIII.
Proof (continued):
∞
∞
𝑇 𝑥 − 𝑀0 2 𝑑𝐹𝑥 + 2(
=
0
∞
𝑇 𝑥 𝑑𝐹 𝑥 − 𝑀0 ) ∙ (𝑀0 −𝐸 𝑇 𝑋
0
2
𝑇 𝑥 − 𝑀0 2 𝑑𝐹𝑥 − 2(𝑀0 −𝐸 𝑇 𝑋
=
+ (𝑀0 −𝐸 𝑇 𝑋
0
∞
0
2
=
𝑇 𝑥 − 𝑀0 2 𝑑𝐹𝑥 +
=
0
∞
𝑇 𝑥 − 𝑀0 2 𝑑𝐹𝑥 − (𝑀0 −𝐸 𝑇 𝑋
+
2
𝑀0
𝑇 𝑥 − 𝑀0 2 𝑑𝐹𝑥 − (𝑀0 −𝐸 𝑇 𝑋
=
+ (𝑀0 −𝐸 𝑇 𝑋
𝑀0
In the first integral 𝑇 𝑥 < 𝑥 < 𝑀0
2
(𝑇 𝑥 − 𝑀0 )2 ≥ (𝑥 − 𝑀0 )2
The second integral is non-negative that is why:
𝑀0
𝐷 2 𝑇(𝑋) ≥
𝑥 − 𝑀0 2 𝑑𝐹𝑥 − (𝑀0 −𝐸 𝑇 𝑋
0
Insurance mathematics VII.
lecture
2
2
=
Reinsurance
Non-proportional contracts IX.
Proof (continued):
Whereas because of definition 𝑇0 𝑥 ≡ 𝑥, 𝑖𝑓 𝑥 ≤ 𝑀0 we will get the
next:
𝑀0
𝑀0
𝑇0 𝑥 − 𝑀0 2 𝑑𝐹𝑥 =
0
𝑥 − 𝑀0 2 𝑑𝐹𝑥
0
and 𝑇0 𝑥 ≡ 𝑀0 , 𝑖𝑓 𝑥 > 𝑀0 , it means:
∞
𝑇0 𝑥 − 𝑀0 2 𝑑𝐹𝑥 = 0
It follows: 𝐷2 𝑇0 (𝑋) =
𝑀0
𝑀0
0
𝑥 − 𝑀0 2 𝑑𝐹𝑥 − (𝑀0 −𝐸 𝑇 𝑋
It means: 𝐷2 𝑇(𝑋) ≥ 𝐷2 𝑇0 (𝑋)
Insurance mathematics VII.
lecture
2
Reinsurance
Comparison
QS
Surplus
XL
Cat XL
Stop Loss
proportional
proportional
nonproportional
nonproportional
nonproportional
Administ.
easy
difficult
easy
depend on
definition
easy
Selection
no
possible
no
no
no
Is it a part of
reinsurer in
each claim
yes
yes
no
no
no
Prem. calc.
easy
easy
difficult
subjective
difficult
proportional
proportional
cheap
relative cheap
relative
expensive
Saving against
big claim freq.
no
no
no
no
yes
Saving against
cumulated cl.
no
no
no
yes
uninterested
Saving against
bad loss ratio
no
no
no
no
yes
Type
Premium
Insurance mathematics VII.
lecture