Part4-Notes-431-2011-F

Part4-Notes-431-2011-F - Review Notes for Loss Models 1 -...

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Review Notes for Loss Models 1 - ACTSC 431/831, FALL 2011 Part 4 – Frequency and Severity with Coverage Modifications Let X be the ground-up loss for an insurance policy or an insurer and assume that X is a continuous r.v. with cdf F ( x ), sf S ( x ), and pdf f ( x ). 1. An ordinary deductible of d : With the deductible, the policy pays nothing if X d and pays X - d if X > d . (a) The per-payment random variable Y P is given by Y P = undefined , X d X - d, X > d = X - d | X > d. (b) The per-loss random variable Y L is given by Y L = 0 , X d X - d, X > d = ( X - d ) + = X - X d. (c) E ( Y L ) and E ( Y P ) are also called expected cost per loss and expected cost per payment , respectively, for the ordinary deductible. 2. A franchise deductible of d : With the deductible, the policy pays nothing if X d and pays X if X > d . (a) The per-payment random variable Y P is given by Y P = undefined , X d X, X > d = X | X > d. (b) The per-loss random variable Y L is given by Y L = 0 , X d, X, X > d. (c) The cdf of Y P is given by F Y P ( y ) = Pr { Y P y } = Pr { X y | X > d } = 0 , y d, F ( y ) - F ( d ) 1 - F ( d ) , y > d. (d) The sf of Y P is given by S Y P ( y ) = 1 - F Y P ( y ) = Pr { Y P > y } = Pr { X > y | X > d } = 1 , y d, S ( y ) S ( d ) , y > d. 1
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(e) If X has a pdf f ( x ) , then Y P is a continuous r.v. with pdf f Y P ( y ) = F 0 Y P ( y ) = 0 , y d, f ( y ) 1 - F ( d ) , y > d. (f) The mean of Y P = X | X > d is given by E ( Y P ) = E ( X | X > d ) = Z 0 S Y P ( y ) dy = d + R d S ( y ) dy S ( d ) = d + e ( d ) . or E ( Y P ) = Z 0 yf Y P ( y ) dy = R d yf ( y ) dy 1 - F ( d ) . (g)
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This note was uploaded on 02/08/2012 for the course ACTSC 431 taught by Professor Laundriualt during the Fall '09 term at Waterloo.

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Part4-Notes-431-2011-F - Review Notes for Loss Models 1 -...

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