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Question_Set_1_-_Solutions Spring 2011

Question_Set_1_-_Solutions Spring 2011 - ACTSC 431 Loss...

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Unformatted text preview: ACTSC 431 - Loss Models 1 Exercises 1. Suppose that the ground-up loss r.v. X has the distribution function F X ( x ) = 1 & & 1 & x & ¡ & , ¡ x ¡ & and ¡ > . An ordinary deductible of d < & is applied to each loss. (a) Show that the distribution of the amount paid per payment Y p is the same as the distribution of X with & replaced by & & d . Solution: When losses are subject to a deductible d , Y p is de&ned as Y p = X & d j X > d . Therefore, F Y p ( y ) = Pr( X & d ¡ y j X > d ) = Pr( d < X ¡ y + d ) Pr( X > d ) = F X ( y + d ) & F X ( d ) 1 & F X ( d ) , where F X ( x ) = ¢ 1 & £ 1 & x ¡ ¤ & , ¡ x ¡ & 1 , x > & . Therefore, F Y p ( y ) = F X ( y + d ) & & 1 & £ 1 & d ¡ ¤ & ¡ £ 1 & d ¡ ¤ & = ( 1 & ( 1 & y + d & ) ¡ & ( 1 & ( 1 & d & ) ¡ ) ( 1 & d & ) ¡ , ¡ y ¡ & & d 1 , y > & & d = ( 1 & ( 1 & y + d & ) ¡ ( 1 & d & ) ¡ , ¡ y ¡ & & d 1 , y > & & d = ( 1 & & 1 & y ¡ & d ¡ & , ¡ y ¡ & & d 1 , y > & & d ; which is of the same form as the c.d.f. of X with & replaced by & & d . (b) Does X have an IMRL or DMRL distribution? Justify. Solution: Given that Y p is the residual lifetime distribution, its mean is given by E [ Y p ] = e X ( d ) = Z ¡ & d F Y p ( y ) dy = Z ¡ & d ¥ 1 & y & & d ¦ & dy = & & d ¡ + 1 , 1 which is clearly a non-increasing function in d . Thus, X is DMRL. Alternatively, one can prove that X is IFR given that the hazard rate is de&ned as h X ( x ) = f X ( x ) F X ( x ) = & ¡ & x , for ¡ x ¡ ¡ . Since X is IFR, X is also DMRL. (c) Determine the loss elimination ratio (LER). Solution: By de&nition, LER = 1 & E [ Y L ] E [ X ] = 1 & E [ Y P ](1 & F Y L (0)) E [ X ] = 1 & e X ( d )(1 & F X ( d )) E [ X ] , where F X ( d ) = 1 & & 1 & d ¡ ¡ & , E [ X ] = ¡ & & 1 , and e X ( d ) = ¡ & d & & 1 . Thus, LER = 1 & ¡ & d ¡ & 1 & d ¡ ¡ & = 1 & & 1 & d ¡ ¡ & +1 . 2 2. The density function of the ground-up loss r.v. X is given by f X ( x ) = 1 100 & q + 1 100 (1 & q ) x ¡ e & x 100 , x ¡ , where < q < 1 . With an ordinary deductible of 100 , the expected value of the per payment r.v. is known to be 125. Determine the expected value of the per payment r.v. for a deductible of 200 ....
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  • Spring '09
  • laundriualt
  • Probability theory, Exponential distribution, Trigraph, probability density function, Survival analysis

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