مدل ﺳﺎزي ﺗﺎﺑﻊ ﺗﻮزﻳﻊ زﻳﺎن ﻫﺎي ﺑﻴﻤﻪ اي ﺑﺎ ﺑﻬﺮه ﮔﻴﺮي از ﺗﻮزﻳﻊ ﻫﺎي ﺗﺮﻛﻴﺒﻲ و ﻣﻔﻬﻮم ﻛﺎﭘﻴﻮﻻ

Modeling Insurance Claim Distribution via Mixture Distribution and Copula

اﻳﻦ ﺗﺤﻘﻴﻖ ﺳﻌﻲ دارد ﺑﺎ ﺑﻬﺮهﮔﻴﺮي ﻫﻤﺰﻣﺎن از ﺗﻮزﻳﻊ ﻫﺎي ﺗﺮﻛﻴﺒـﻲ و ﻣﻔﻬـﻮم ﻛـﺎﭘﻴﻮﻻ،ﺗﺎﺑﻊ ﺗﻮزﻳﻊ ﺗﻮأﻣﺎن زﻳﺎنﻫﺎي واردﺷﺪه ﺑﺮ اﻛﺴﭙﻮژرﻫﺎي ﻣﺨﺘﻠﻒ ﺗﺤﺖ ﭘﻮﺷﺶ ﻳﻚ ﺑﻴﻤﻪﻧﺎﻣﺔ ﺧـﺎص را ﻧﺴﺒﺖ ﺑﻪ ﺗﻮزﻳﻊ ﻫﺎي آﻣﺎري ﻣﻮﺟﻮد، ﺑﺎ دﻗﺖ ﺑﻴﺸﺘﺮي ﻣﺪل ﺳﺎزي ﻛﻨﺪ. در اﻳﻦ ﺗﺤﻘﻴـﻖ از ﺗﻮزﻳـﻊ ﺧﺎﺻﻲ ﻛﻪ ﺗﺮﻛﻴﺒﻲ از ﺗﻮزﻳﻊ ﺗﻲ اﺳﺘﻮدﻧﺖ ﭼﻮﻟﻪ ﻫﺎﻳﭙﺮﺑﻮﻟﻴﻚ ﺗﻌﻤﻴﻢ ﻳﺎﻓﺘـﻪ و ﻧﻈﺮﻳـﺔ ﻣﻘـﺎدﻳﺮ ﻓـﺮﻳﻦ اﺳﺖ، ﺑﺮاي ﻣﺪل ﺳﺎزي ﺗﻮاﺑﻊ زﻳـﺎن ﺣﺎﺷـﻴﻪ اي و از ﻣﻔﻬـﻮم ﻛـﺎﭘﻴﻮﻻ ﺑـﺮاي ﻣـﺪل ﺳـﺎزي ﺳـﺎﺧﺘﺎر واﺑﺴﺘﮕﻲ ﻣﻴﺎن آﻧﻬﺎ اﺳﺘﻔﺎده ﺷﺪه اﺳﺖ. ﻛﺎﭘﻴﻮﻻﻫـﺎي ﮔﻮﺳـﻲ، ﺗـﻲ، ﻓﺮاﻧـﻚ، ﮔﺎﻣﺒـﻞ و ﻛﻼﻳﺘـﻮن،ﻣﻬﻢ ﺗﺮﻳﻦ اﻧﻮاع ﻛﺎﭘﻴﻮﻻي ﺑﺮرﺳﻲ ﺷﺪه اﻧﺪ ﺗﺎ از ﺑـﻴﻦ آﻧﻬـﺎ ﺑﻬﺘـﺮﻳﻦ ﮔﺰﻳﻨـﻪ ﺑـﺮاي ﺗﺸـﺮﻳﺢ ﺳـﺎﺧﺘﺎر واﺑﺴﺘﮕﻲ زﻳﺎنﻫﺎ اﻧﺘﺨﺎب ﺷﻮد. دادهﻫﺎي ﻣﻮرد اﺳﺘﻔﺎده در اﻳﻦ ﺗﺤﻘﻴﻖ ﻣﻘﺪار ﺧﺴﺎرتﻫﺎي ﺟـﺎﻧﻲ و ﻣﺎﻟﻲ ﺑﻴﻤﻪﻧﺎﻣﻪﻫﺎي ﺷﺨﺺ ﺛﺎﻟﺚ وﺳﺎﻳﻞ ﻧﻘﻠﻴﺔ ﻣﻮﺗﻮري اﺳﺖ. ﻧﺘـﺎﻳﺞ ﺗﺤﻘﻴـﻖ ﻧﺸـﺎن ﻣـﻲ دﻫـﺪ ﺑـﺎ ﺑﻬﺮه ﮔﻴﺮي از ﺗﻮزﻳﻊ ﺗﺮﻛﻴﺒﻲ ﭘﻴﺸﻨﻬﺎدي و ﻛﺎﭘﻴﻮﻻي ﻛﻼﻳﺘﻮن ﺗﺎﺑﻊ ﺗﻮزﻳﻊ ﺗﻮأم، ﻣﻲ ﺗﻮان ﺑـﻪ ﺧـﻮﺑﻲ زﻳﺎنﻫﺎي ﻧﺸﺌﺖ ﮔﺮﻓﺘﻪ از ﺑﻴﻤﻪﻧﺎﻣﺔ ﺷﺨﺺ ﺛﺎﻟﺚ را ﻣﺪل ﺳﺎزي ﻛﺮد.

This paper analyses whether joint probability distribution function of losses due to different exposures covered under the same policy could be modeled in an appropriate manner via mixture distribution proposed and copula concept. Special type of distribution which is a mixture of Generalized Hyperbolic Skew t distribution and Extreme Value theory (EVT) has been used for modeling marginal distributions of claims and copula function has been considered as a means of modeling dependency structure among claims. Most important copula including; Gaussian, t, Frank, Gumbel and Clayton was tested from goodness of fit point of view. The data used in this study are the amount of property damage and bodily injury covered under automobile liability insurance. Results reveal that joint probability distribution of claims could be effectively modeled by Clayton copula and proposed mixture distribution.