Pricing of MAXL treaties
Conference R in Insurance – 29/06/2015
Indra LOLJEEH – Technical Advisor Analytics at QBE Re – li@qbere.com
Agenda
•
Reminder about reinsurance treaties
•
MAXL treaties
•
Pricing of MAXL treaties
•
Structure of the pricing program
•
Conclusion
2
Agenda
•
Reminder about reinsurance treaties
–
Reinsurance basic treaties
– Aggregate clauses in reinsurance
•
MAXL treaties
•
Pricing of MAXL treaties
•
Structure of the pricing program
•
Conclusion
3
Traditional reinsurance
Quota
share
(QS)
Stop
loss
(SL)
Surplus
(SP)
Excess
of loss
(XL)
4
Quota-Share
•
Quota
share
(QS)
Stop
loss
(SL)
Surplus
(SP)
Excess
of loss
(XL)
Cession rate: 𝛼𝑖 = 𝛼 for all risks 𝑖
5
Surplus
•
Cession rate: 𝛼𝑖 ≠ 𝛼 for all risks 𝑖
•
𝛼𝑖 = max 0,1 −
𝑅
𝑆𝐼𝑖
where 𝑅 is called the surplus line
Surplus 3 lines of 2
10
8
6
Reinsured
4
Retention
2
0
1
2
3
4
5
6
7
8
Risk number
6
Excess of loss
•
•
•
Notation: C xs D
𝑋 original claim amount
𝑋 𝑅𝑒𝑖𝑛𝑠 = min 𝐶, max 0, 𝑋 − 𝐷
𝑋 𝑅𝑒𝑡 = 𝑋 − 𝑋 𝑅𝑒𝑖𝑛𝑠
Excess of loss 5xs5 and 10xs10
45
40
Claim amount
•
35
30
25
Retention
20
15
Layer 2
10
Layer 1
5
Retention
-
1
2
3
4
5
6
7
8
9 10
Claim
7
Stop loss
•
•
•
•
Notation: C xs D
𝑆 = 𝑋1 + … + 𝑋𝑁
𝑆 𝑅𝑒𝑖𝑛𝑠 = min 𝐶, max 0, 𝑆 − 𝐷
𝑆 𝑅𝑒𝑡 = 𝑆 − 𝑆 𝑅𝑒𝑖𝑛𝑠
Quota
share
(QS)
Stop
loss
(SL)
Surplus
(SP)
Excess
of loss
(XL)
8
Aggregate clauses in reinsurance
•
AAD: Aggregate Annual Deductible
•
AAL: Aggregate Annual Limit
•
Link with Stop-Loss treaties
•
 These elements are the basis of most currently existing reinsurance
structures
9
Agenda
•
Reminder about reinsurance treaties
•
MAXL treaties
–
Concept of MAXL treaties
– Illustration/Example
•
Pricing of MAXL treaties
•
Structure of the pricing program
•
Conclusion
10
Concept of MAXL treaties
•
Multiline:
–
Reinsurance treaty covering several lines of business at the same time
– Common protection for several portfolios
•
Aggregate
–
Structure that contains aggregate clauses (MAAD/MAAL)
– These clauses are applied on a global level
•
eXcess of Loss
–
Non-proportional structure
– Protection against medium/high claims
11
MAXL Example
12
MAXL Example
Eligible
Claim
Activity
Interior
Priority
13
Advantages of such structures
•
Improved protection against medium sized claims
–
Towards frequency
– Towards changes in legislation
•
Protection of insurer’s global results
•
Can cover several lines of business at the same time
•
Cheaper than lower XL cover
•
Optimization of the reinsurance structure (Solvency II)
14
Agenda
•
Reminder about reinsurance treaties
•
MAXL treaties
•
Pricing of MAXL treaties
–
Collective risk model / Monte-Carlo methods
– Problems inherent to MAXL treaties
 Correlations
 Lines of business (ST/LT)
 Time
– What R brought to us
•
Structure of the pricing program
•
Conclusion
15
Theory behind the pricing of MAXL
•
Collective Risk model: Independency between frequency and severity
–
Determination of the loss distribution of the eligible layers
– Determination of the frequency distributions
•
Aggregation of the loss distributions
–
Use of copulas
– Dependency structure between the lines of business ?
– Dependency above thresholds ?
– Long tail vs. Short tail businesses ?
•
Use of Monte-Carlo simulations
–
Simulation for each line of business
– Aggregation to obtain aggregate loss distribution
16
Problems inherent to such methods
•
Time-consuming process
 Need for an efficient software
•
Many underlying structures possible
 Need for program flexibility
•
Still some developments needed for the copula part
 Need for a modular program
•
Difficult to take account of some distributions (CAT Softwares)
17
Problems inherent to such methods
•
Time-consuming process
 Need for an efficient software
•
Many underlying structures possible
 Need for program flexibility
•
Still some developments needed for the copula part
 Need for a modular program
•
Difficult to take account of some distributions (CAT Softwares)
R is our savior !!!
18
Agenda
•
Reminder about reinsurance treaties
•
MAXL treaties
•
Pricing of MAXL treaties
•
Structure of the pricing program
•
Conclusion
19
Structure of the pricing program
1 - Inputs
Excel
Template
1/2 - Margin
distribution by LoB
1 - Inuring layers
1 - Dependency
structure between
risks
1 - MAXL Program
3 - Monte-Carlo
Simulations
4 - Calculation of
eligible losses
R tool
5 - Aggregation per
LoB
6 -Global
aggregation
Outputs
Excel Template
PDF Report
7 - Results by
alternative
Global report
20
Global distribution
Average
STD
CoV
VaR 99,5%
12.511.322
7.147.174
57,13%
39.272.602
RP
3
14.220.596
5
17.613.749
10
21.887.814
21
Agenda
•
Reminder about reinsurance treaties
•
MAXL treaties
•
Pricing of MAXL treaties
•
Structure of the pricing program
•
Conclusion
22
Advantages of R in our case
•
Ideal for simulation based tools
•
Simplification of the audit process
•
Program easy to modify and improve
•
Packages already existing
•
Possibility for further analysis on data
–
Graphs
– Sensitivity analysis
– Reporting
23
Questions?
24