Test1-Solutions-S09-simplified

Test1-Solutions-S09-simplified - ACTSC 431 - Loss Models 1...

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Unformatted text preview: ACTSC 431 - Loss Models 1 TEST #1 1. (20 marks) Suppose that the ground-up loss random variable X has density function f X ( x ) = &e & &x & a + 2 (1 & a ) e & &x , x , for a 2 (0 ; 1) and & > . (a) (1 mark) Determine the survival distribution of the random variable X . Solution : By integrating the density from x to 1 , one &nds that F X ( x ) = Z 1 x f X ( y ) dy = Z 1 x &e & &y & a + 2 (1 & a ) e & &y dy = ae & &x + (1 & a ) e & 2 &x , for x . (b) (3 marks) Using (a), &nd the hazard rate of the random variable X . Is the random variable X DFR, IFR or neither? Solution : By de&nition, the hazard rate of the random variable X is given by h X ( x ) = f X ( x ) F X ( x ) = &e & &x & a + 2 (1 & a ) e & &x ae & &x + (1 & a ) e & 2 &x = & & a + 2 (1 & a ) e & &x a + (1 & a ) e & &x = & & a + (1 & a ) e & &x + (1 & a ) e & &x a + (1 & a ) e & &x = & + & (1 & a ) e & &x a + (1 & a ) e & &x = & + & (1 & a ) ae &x + (1 & a ) . (1) Given that a 2 (0 ; 1) , the hazard rate is a decreasing function in x . To convince yourself, one can di/erentiate (1) with respect to x to show that d dx h X ( x ) = & & (1 & a ) a&e &x ( ae &x + (1 & a )) 2 < . One concludes that X is DFR . (c) (4 marks) Show that the Value-at-Risk at level 100 p % , namely V ar p ( X ) , is given by V aR p ( X ) = & 1 & ln p a 2 + 4 (1 & a ) (1 & p ) & a 2 (1 & a...
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This note was uploaded on 11/23/2010 for the course ACTSC act431 taught by Professor Na during the Spring '09 term at Waterloo.

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Test1-Solutions-S09-simplified - ACTSC 431 - Loss Models 1...

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