Tail Conditional Expectations for Extended Exponential Dispersion Models

Transcription

1 American Researc Journal of Matematics Original Article ISSN Volume 1 Issue Tail Conditional Expectations for Extended Exponential Dispersion Models Ye (Zoe) Ye Qiang Wu and Don Hong 1 Program of Actuarial Science Department of Matematical Sciences Middle Tennessee State University Murfreesboro Tennessee USA Abstract: Te measurement of te tail conditional expectation also called tail value-at-risk elps insurance companies determine te amount of capital needed to pay claims resulting from catastropic events wen premium revenues are insufficient In tis paper we extend te exponential dispersion models and derive tail conditional expectation forms of te new models tat generalize results and discussions of paper by Landsman and Valdez (005) Keywords: Exponential dispersion model tail conditional expectations tail value-at-risk I INTRODUCTION Insurance companies often need to determine te amount of capital to pay claims resulting from catastropic events wen premium revenues are insufficient Determining tese amounts is a difficult process We need to model te probability distribution of te incurred and occurring losses More importantly we sould determine te best risk measurement [1] to find te amount of loss to cover te claims Tail conditional expectation (TCE) as become increasingly popular for measuring tis kind of risk especially te adequacy of existing capital and te possibility of financial ruin [3 4 9] It is one of te newest risk modeling tecniques adopted by te insurance industry Suppose tat for some period of time an insurance company faces te risk of losing an amount of Its distribution function and tail function are denoted by F Pr( x) S ( ) Pr( ) and x x Te function S is also called te survival function in life contingency literature Te tail conditional expectation of is defined as TCE E( x) (11) Wic is also called te tail value at risk (TVaR) We can interpret tis risk measure as te mean of worst losses beyond a certain level Te formula used to evaluate TCE is Provided tat S 0 Te value of x is usually set based on te potential value of te loss S ( x ) 1 q q 1 TCE tdf( t) (1) S x xq of te distribution wit te property were 0q 1 Wit a large value of q te value x q is considered to be te amount of loss unlikely to appen in a normal situation but could potentially ruin te business It is terefore important for a company to monitor and prepare for suc an extreme situation TCE provides a good measure for tis purpose Te class of exponential dispersion models as served as error distributions for generalized linear models in te sense developed by [13] wic includes many well known distributions suc as: Poisson Binomial Normal Gamma etc Also credibility formulas of exponential dispersion models given by [ ] preserve te property of a 1 Corresponding Autor: donong@mtsuedu wwwarjonlineorg 8

2 ISSN predictive mean A toroug and systematic investigation of exponential dispersion models was done by Jorgensen [5 6 7] In [1] te reproductive and additive models [see (1) () ] and teir tail conditional expectations are studied In tis paper we furter extend te models to an even more general form [see (31) ] and investigate te tail conditional expectation we provide a brief review of te exponential dispersion family (EDF) We study te generalized exponential dispersion family investigate its properties and derive te formula for its tail conditional expectation Finally we summarize conclusions II EPONENTIAL DISPERSION MODELS A random variable is said to belong to te EDF of distributions if its probability measure P is absolutely continuous wit respect to some measure Q and can be represented as follows dp e dq x (1) [ x ( )] ( ) for some function κ(θ) Te parameter is named te canonical parameter belonging to te set { R ( ) } Te parameter is called te index parameter belonging to te set of positive real numbers { 0} R Te representation in (1) is called te reproductive form of EDF and is denoted by ~ ED( ) for a random variable belonging to tis family Wit te transformation of Y a so-called additive form can be obtained Its probability measure P is absolutely continuous wit respect to some measure Q wic can be represented as [ y ( )] dp e dq ( y) () If te measure Q in (1) is absolutely continuous wit respect to a Lebesgue measure ten te density of as te form Te same can be said about additive model f x e q x [ x ( )] ( ) ( ) (3) ED ( ) and Y as te density f y e q y [ y ( )] Y ( ) ( ) (4) Consider te loss random variable belonging to te family of exponential dispersion models in reproductive or additive form We will denote te tail probability function by S( ) Tis simplifies te notation by dropping te subscript wen no confusion can appen and empasizes its dependence on te parameters and In [1] te TCEs are given for bot reproductive form and additive form of exponential dispersion models Suppose tat te random variable belongs to EDF If te survival function as partial derivative wit respect to ( ) is a differentiable function and one can differentiate te survival function S( ) in under te integral sign ten For ~ ED( ) te reproductive form of EDF Were For ~ ED ( ) te additive form of EDF TCE (5) log S( x ) wwwarjonlineorg 9

3 ISSN TCE (6) In tis paper we will extend te formulas of exponential dispersion models and investigate teir properties Te first generalization we would consider is to replace te multiplier in EDF by a function of Tis is motivated by te linear exponential family (LEF) [] wic takes te form e f( x ) q p( ) r( ) x Wit tis generalization te extended model includes LEF as a special case Te second generalization is to replace te term ( ) by a general bivariate function of and Note tat for te reproductive form of EDF and (7) x ( ) log( e dq ) R (8) ( ) y log( e dq ( y)) R (9) for te additive form of EDF For general distributions Q orq te functions on te rigt of (8) or (9) sould be a general bivariate function not necessarily able to be written in te form of ( ) Terefore in EDF te coice of te distributions Q or Q consider more general distributions are restricted Te generalization of te model removes te restriction and allows us to Wat we want to do in tis paper is to combine te two generalizations and develop an extended dispersion model as well as study its properties and derive its TCE representation Suc a family of models will be referred to as te generalized exponential dispersion family (GEDF) III GENERALIZED EPONENTIAL DISPERSION FAMILY We consider te generalization of combining te reproductive form (1) and additive () as r( ) x ( ) dp e dq (31) If te measure Q is absolutely continuous wit respect to te Lebesgue measure ten te density of as te form f x e q x r( ) x ( ) ( ) ( ) (3) 31 Mean and Variance of GEDF Teorem1 Suppose tat a random variable belongs to te GEDF wose distribution is given by (31) If its probability measure P is absolutely continuous wit respect to some measure Q r( ) as second derivative and is invertible ( ) as te second partial derivative to ten te mean value of is 1 ( ) (33) r '( ) And te variance of is 1 1 ( ) Var( ) [ ]' (34) r'( ) r'( ) Proof Te generating function can be derived as follows wwwarjonlineorg 30

4 ISSN K t E e e dq x t [( t/ ) r( )] x ( ) ( ) log ( ) log{ ( )} R Let be te number suc tat r( ) t / r( ) Since r( ) is invertible we ave r 1 [ r( ) ( t / )] Ten K t r r t 1 ( ) ( [ ( ) ( / )] ( ) It follows tat its moment generating function can be written as 1 M () t e ( r ( r( ) t/ )) ( ) (35) 1 1 Knowing tat ( f )' 1/ f '( f ) for an invertible differentiable function we can find te first order derivative of te moment generating function: M () t e 1 1 r r t r r t ' ( ( ( ) / )) ( ) Wen t = 0 we obtain M (0) e 1 1 r r r r ' ( ( ( ))) ( ) wic gives te mean value 1 ( ) r '( ) For te variance of te combination function we ave ( ( ( ) / )) 1 r r ( ( ( ))) 1 r r 1 '( ( r( ))) 1 '( ( r( ) t/ )) '' 1 1 ( ) 1 ( ) r ''( ) ( ) M 3 E( ) (0) {[ ] } r '( ) [ r '( )] [ r '( )] Ten we can get te variance as 1 1 ( ) Var( ) [ ]' r'( ) r'( ) Tis completes te proof of te teorem Wen 1and ( ) ( ) log( p( )) te GEDF reduces to te LEF (7) Teorem 1 gives '( ) p'( ) ( ) r '( ) r '( ) p( ) '( ) and Var( ) Tese coincide wit te results for te linear exponential family in [9] r '( ) 3 TCE of GEDF Teorem Under te same assumptions as in Teorem 1 if one can differentiate te tail function S( ) in under te integral sign ten te tail conditional expectation is given by period TCE r '( ) (36) wwwarjonlineorg 31

5 ISSN were log S( x ) Proof Recall tat te tail function is r( ) x ( ) ( ) ( ) x S x e xdq x and ( ) log S( x ) r '( ) TCE wit TCE r( ) x ( ) dp e dq Tis formula can be rewritten as r '( ) IV CONCLUSION Tis paper examines tail conditional expectations for loss random variables tat belong to te class of generalized exponential dispersion models For te exponential dispersion models in [1] it as bot te reproductive form and additive form We extended te exponential dispersion families as GEDF based on tese two forms Te representations for its mean and variance are derived Te tail conditional expectation is caracterized By comparing our results wit tose in [1] we see tey sare great similarity but our new model covers wider scenarios we considered te two generalizations simultaneously to derive a more general and complicated distribution In tis case te general form of te distribution is r( ) x ( ) dp e dq We obtain 1 ( ) ( ) r '( ) 1 1 ( ) Var( ) [ ]' r'( ) r'( ) and TCE r '( ) Finally we remark tat te EDF contains a couple of special distributions tat are commonly seen in probability teory and used in loss modeling Tese special distributions are clearly also examples of generalized exponential dispersion models Altoug it seems tat no commonly seen special distributions belong to GEDF but not to EDF it is not difficult to teoretically construct suc distributions Te generalization allows more freedom to model loss variables wic may be beneficial wen te losses cannot be well modeled using commonly seen special distributions V ACKNOWLEDGEMENTS Te autors are grateful to te anonymous referee for elpful comments Tis researc of Don Hong was partially supported by te Beijing Hig-Caliber Overseas Talents Program and te Nort Cina University of Tecnology REFERENCES [1] Artzner P Delbaen F Eber JM and Heat D (1999) Coerent Measures of Risk Matematical Finance [] Brown LD (1987) Fundamentals of Statistical Exponential Families IMS Lecture Notes - Mono-grap Series: Vol 9 Hayward California [3] Bowers NL Gerber HU Hickman JC Jones DA and Nesbitt CJ (1997) Actuarial Mate-matics nd edition Society of Actuaries Scaumburg Illinois [4] Fundamentals of Actuarial Practice (FAP)(009) Model Selection and Solution Design Society of Actuaries wwwarjonlineorg 3

6 ISSN [5] Jorgensen B (1986) Some Properties of Exponential Dispersion Models Scandinavian Journal of Statistics B [6] Jorgensen B (1987) Exponential Dispersion Models (wit discussion) Journal of te Royal Statisti-cal Society Series B [7] Jorgensen B (1997) Te Teory of Dispersion Models Capman Hall London [8] Kaas R Danneburg D Goovaerts M (1997) Exact Credibility for Weigted Observations Astin Bulletin [9] Klugman S Panjer H and Willmot G (008) Loss Models from Data to Decisions 3rd edition [10] Landsman Z Makov U (1998) Exponential Dispersion Models and Credibility Scandinavian Actu-arial Journal 1 pp [11] Landsman Z (00) Credibility Teory: A New View from te Teory of Second Order Optimal Statistics Insurance: Matematics and Economics [1] Landsman Z Valdez E (005) Tail Conditional Expectation for Exponential Dispersion Models Astin Bulletin [13] Nelder JA Wedderburn RWM (197) Generalized Linear Models Journal of te Royal Statistical Society Series A [14] Nelder JA Verrall RJ (1997) Credibility Teory and Generalized Linear Models Astin Bulletin Vol 7 No 1 pp 71-8 wwwarjonlineorg 33