Summary Data Analytics for Non-Life Insurance (DANLI) | EOR Year 3 Tilburg University

Data Analytics For Non-Life Insurance
Max Batstra
March 29, 2026




1

,Reinsurance in the Individual Risk Model
Before starting on the next section, let us first define reinsurance:
Definition 0.1. [Reinsurance]
Reinsurance is the risk sharing/ transfer between insurers. The first-line insurer is called the cedent and buys
insurance of the second-line insurer, called the reinsurer.

1. Aim cedent: Financial stability, increase solvency. Solvency is (in the insurers case) the ability to pay
for claims, so kredietwaardigheid in dutch.
2. Aim reinsurer: Make profits.

Risk and premium transfer from cedent to reinsurer:
• Individual contracts: Yi = YiC + Yir
• Total amount: S = S C + S r

• Total premium: π = π C + π r
We assume that for fixed ξ > θ > 0:
• Total premium: π = (1 + θ) E[S]

• Premium to reinsurer: π r = (1 + ξ)E [S r ]
If we include proportional reinsurance then the total claim amounts become for some proportion α ∈ [0, 1]:
• S C = αS
• S r = (1 − α)S

And the proportional premiums become:
• π r = (1 + ξ)(1 − α) E[S]
• π c = π − π r = (1 + θ) E[S] − (1 + ξ)E [S r ] = [1 + θ − (1 + ξ)(1 − α)] E[S]




2

,    
The expected cedent’s profit: E B C = π c − E S C = [θ − ξ + ξα] E[S]

Optimization Problem: For given u and ϵ > 0:

max E[B C ] s.t. P (S c > u + π c ) ≤ ϵ
α∈[0,1]

 
Now since E B C = [θ − ξ + ξα] E[S] is strictly increasing in α we have to find the largest α
such that P (S c > u + π c ) ≤ ϵ.
Then the safety coefficient of the cedent becomes:

u + E [B c ] u + (θ − ξ + ξα)E[S] u + (θ − ξ)E[S] ξE[S]
α0c = p = p = p +p
α Var(S) α Var(S) α Var(S) Var(S)

Then we maximize the ruin probability:

P [S c > u + π c ] = 1 − Φ (α0c ) ≤ ϵ ⇔ Φ (α0c ) ≥ 1 − ϵ ⇔ α0c ≥ zϵ

There are two important cases:

• u ≥ (ξ − θ)E[S]
– α0C is decreasing in α
– P [ruin] is increasing in α
– Choose largest α ∈ [0, 1] such that α0c ≥ zϵ

• u < (ξ − θ)E[S]
– α0c is increasing in α
– P [ruin] is decreasing in α
 
– Choose α = 1 (since P [ruin] is decreasing in α and E B C increasing)




3

,Excess-of-Loss Reinsurance
We now introduce a retention level d at the individual claim level:
• Claim size retained: Yjc = min (Yi , d)
• Claim size reinsured: Yir = Yi − min (Yi , d)

This retention level makes sure the insurance company (cedent) has insurance for claims above this retention
level. This means that the reinsurer will cover the part of the claim that exceeds the predetermined threshold
d. This is also called Stop-Loss Reinsurance.

Total Claim amounts:
Pn Pn
• S c = i=1 Yic = i=1 min (Yi , d)
Pn Pn
• S r = i=1 Yir = i=1 Yi − min (Yi , d)
• S = Sc + Sr

Premiums:
• π r = (1 + ξ)E [S r ]
• πc = π − πr




4

, Chapter 2 Collective Risk Modeling
Model Assumptions 2.1. [Compound Distribution]
This is the total claim amount S over a one (accounting) year period:
N
X
S = Y1 + . . . + YN = Yi , S = 0 if N = 0
i=1

1. N is a count random variable for the number of claims, only taking values in A ⊂ N0
2. Y1 , Y2 , . . . ∼ G i.i.d. with G(0) = 0
3. N and Y1 , Y2 , . . . are independent
2. states ”Individual claim sizes Yi do not affect each other ” and
3. states ”Individual claim sizes are ot affected by the number of claims and vice versa”
Proposition 2.2. Assume S has a compound distribution then:
1. E[S] = E[N ]E [Y1 ]
2
2. Var(S) = σS2 = Var(N )E [Y1 ] + E[N ] Var (Y1 )
s
2
σS Vco (Y1 )
3. Vco(S) = = Vco(N )2 +
E[S] E[N ]

4. MS (r) = E erS = MN (ln {MY1 (r)}) , r ∈ R
 

Proofs:
hP i h hP ii hP i hP i
N N N N
1. E[S] = E i=1 Yi = E E i=1 Yi | N = E i=1 E [Yi | N ] = E i=1 E [Yi ] = E [N Y1 ] =
1
E[N ]E [Y1 ] at 1 we used independence of Y1 , Y2 , . . . and N

2.
N
! " N
#! " N
!#
X X X
Var(S) = Var Yi = Var E Yi N + E Var Yi N
LTV
i=1 i=1 i=1
N
! " N
#
X X
= Var E [Yi | N ] +E Var (Yi | N )
i=1 i=1
= Var (N E [Y1 ]) + E [N Var (Yi )]
2
= Var (N ) · E [Y1 ] + E [N ] · Var (Yi )

3. Include if necessary
4. Include if necessary



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