Application of the credibility principle in reinsurance pricing

Transcription

1 Appication of the credibiity principe in reinsurance pricing David Raich Angea Wünsche Bahnhofskooquium, Zurich February 203

2 Agenda. Introduction into credibiity theory 2. Some maths 3. Credibiity for reinsurance pricing 4. Appication method used for MTPL 5. Vision Page 2

3 Introduction Page 3

4 Introduction Initia situation: Portfoio of simiiar risks (coective) Individua risk within the coective Comprehensive information avaiabe for the coective (e.g. soid oss history or more) Limited data history avaiabe for individua risk GOAL: Make use of a (reevant) avaiabe information in order to get best estimate for the individua premium Page 4

5 Introduction Coective vs individua information Individua information Coective information Data stems from individua risk Contains significant random eement Different charateristics than individua risk Statistica significance Credibiity premium Page 5

6 Introduction History of credibiity theory Limited Fuctuation CT Greatest Accuracy CT Based on centra imit theorem Originay a fu or zerocredibiity method Parameters in partia mode introduced ater caibrated according to actuaria judgement Heaviy based on Bayesian statistics Best premium to charge - approach Resuts in stabe and responsive estimator Stabiity-oriented approach Precision-oriented approach Page 6

7 Some maths Page 7

8 Mathematica Formuation coective F { F H[ ] } Famiy of distributions indexed by risk profie ϑ E[ X Individua premium E[ X ] E Coective premium ] E[ X ] Individua risk Given observations Find a good estimator for theindividua premium H( ) E[ X n ] GOAL: x,, x n for an individua risk F Page 8

9 Main resuts Bayesian estimator () A priori density function of risk profie f ( x,..., ) x n Conditiona density function of osses H ( ) x f ( x ) dx Individua Premium f ( x,...,, ) x n Joint density function x Individua risk x n x 2... ( x) A posteriori pdf of risk profie P ( B) P( B A ) P( ) i i A i E X x x P A, B) P X n H ( da B ) ( P( B) Page 9

10 Main resuts Bühmann mode coective observations Z Z X n ˆ n n a Z n n n k k 2 a 2 more weight assigned to coective information more weight assigned to individua information Page 0

11 Main resuts Bühmann-Straub mode coective observations Z w n w w ˆ w w k Z Z n w n X w n a k 2 a more weight assigned to coective information 2 more weight assigned to individua information Page

12 Credibiity for reinsurance pricing Page 2

13 Genera comments What probems do we face in reinsurance pricing? Pricing XL business for motor Usuay data are ony given back for the ast 0 years Need to project osses to utimate, where deveopment can take much onger than 0 years Data are ony avaiabe excess a threshod Hence scarce data, which may be insufficient to price a cient based on experience We want to make use of a avaiabe data in market and weight a cient against the market naturay a appication fied of credibiity Page 3

14 Genera comments 35,000,000 30,000,000 25,000,000 20,000,000 5,000,000 0,000,000 Layer 5 Layer 4 Layer 3 Layer 2 Layer Retention 5,000, Page 4

15 Chaenges Chaenges: Choice of appropriate portfoio Pure niche portfoios sti require individua treatment Pricing of a ayer 3 m xs 2 m is different than pricing i xs 25 m Credibiity weight needs to be cacuated in dependency of caims size Credibiity appied to frequency / severity or rate? Parameter uncertainty? Which is the appropriate exposure measure? Page 5

16 Requirements for credibiity approach Produce reasonabe resuts i.e. increase precision Ensure stabiity and responsiveness One mode for a ayers Easy to expain Ensure consistent approach within one market Appication sti aows for underwriting judgement Any credibiity procedure requires the actuary to exercise informed judgment, using reevant information. The use of credibiity procedures is not aways a precise mathematica process (Actuaria Standards board) Page 6

17 Credibiity for reinsurance XoL pricing Initia situation: Portfoio of simiiar risks (coective) Individua risk Comprehensive information avaiabe for the coective (e.g. soid oss history or more) Limited data history avaiabe for individua risk GOAL: Make use of a (reevant) avaiabe information in order to get best estimate for the individua premium Page 7

18 Credibiity for reinsurance XoL pricing Initia situation: Net Market rate avaiabe (=Coective information) market Limited oss history avaiabe for individua portfoio GOAL: Individua cedent Make use of both information in order to get good estimate for the individua net rate Page 8

19 Appication Method used for MTPL Page 9

20 Theoretica framework Compound Gamma-Poisson Tuesday, 2 February 203 Page 20 Bahnhofskooquium - Credibiity! ) ( n e n f n N ) ( ) ( a e b f b a a The unconditiona distribution of N is negative binomia with parameter(a, b /(+b)). w w Z N Z N Z, ˆ b w w Z

21 Tuesday, 2 February 203 Page 2 Bahnhofskooquium - Credibiity m m b a / ] [ E m m b a 2 2 / Var Estimation of parameter b: m m b 2 Estimation of the parameters- Compound Gamma mode Process needs to be repeated for different threshods

22 Tuesday, 2 February 203 Page 22 Bahnhofskooquium - Credibiity m T T m T b T a ) ( ) ( ) / ( ] [ E m T T T m T b T a 2 2 ) ( ) ( ) ( ) / ( Var Estimation of parameter b(t): m T T m T T b 2 ) ( ) ( ) ( ) ( Estimation of the parameters- Compound Gamma mode

23 Estimation of the parameters- Frequency of cedents Expected Frequency of different threshods Page 23

24 Estimation of the parameters- Fit b(t) Estimation of b for different threshods inc. exponentia fit Page 24

25 Appication on cient exampe Credibiity factors for ayer: Limit xs m Page 25

26 Status quo Where are we? Credibiity weight is cacuated dependent on caim size and exposure Caibrated on frequencies Appied to the rate Underwriting jugdement is possibe, because of the range given for the weight Uncertainty of rate not expicitey taken into account, but within underwriting judgement Page 26

27 Vision Page 27

28 Vision Where to go? Appication for severity Incorporation of market rate uncertainty Expand appication towards oadings (capita intensities) Other approaches in actuaria iterature: appication on oss deveopment factors (Pinot/Gogo) making use of ower ayers for upper ayers Page 28

29 References Bühmann, H., and A. Giser, A Course in Credibiity Theory and its Appications, New York: Springer, Cockroft, M., Bayesian Credibiity for Excess-of-Loss Reinsurance, paper presented at the GIRO Conference, Parodi, P., and S. Bonche, Uncertainty-Based Credibiity and its Appication to Excess-of-Loss Reinsurance, Casuaty Actuaria Society E-Forum, Winter Mashitz, I., and G. Patrik, Credibiity for Treaty Reinsurance Excess Pricing, Casuaty Actuaria Society 990 Discussion Paper Program, pp Credibiity for a Tower of Excess Layers David R. Cark (20) An Anaysis of Excess Loss Deveopment Pinto & Gogo; Proceedings of the Casuaty Actuaria Society (PCAS) 987; Vo LXXIV.2; Page 29

30 Thank you for your attention Page 30