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Part4-Notes - Review Notes for Loss Models 1 ACTSC 431/831...

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Review Notes for Loss Models 1 - ACTSC 431/831, FALL 2008 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) The mean of Y P is given by E ( Y P ) = E ( X - d | X > d ) = e ( d ) = d S ( x ) dx 1 - F ( d ) . (d) The mean of Y L is given by E ( Y L ) = E [( X - d ) + ] = E ( X ) - E ( X d ) = d S ( x ) dx. (e) The relationship between the moments of Y P and Y L : E Y P k = E Y L k 1 - F ( d ) , k = 1 , 2 , .... (f) The per-payment random variable Y P is also called excess loss variable or left truncated and shifted variable and the per-loss random variable Y L is also called left censored and shifted variable. They have been discussed in detail in Part 1. (g) E ( Y L ) and E ( Y P ) are also called expected cost per loss and expected cost per payment , respectively, for the ordinary deductible. (h) See Part 1 for distributions and more results about Y P and Y L . 1
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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 f is given by Y P f = undefined , X d X, X > d = X | X > d. (b) The per-loss random variable Y L f is given by Y L f = 0 , X d, X, X > d. (c) The per-payment random variables Y P and Y P f are conditional positive random variables and defined only for X > d . In particular, Y P > 0 and Y P f > d hold. (d) The per-loss random variables Y L and Y L f are mixed nonnegative ran- dom variables and defined for all X . In particular, Pr { Y L = 0 } = Pr { Y L f = 0 } = Pr { X d } = F ( d ).
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