QuestionSet1 - ACTSC 431/831 QUESTION SET 1 1(a Suppose the...

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ACTSC 431/831 – QUESTION SET 1 1. (a) Suppose the random variable X LOGN( μ, σ ). Show that for any d > 0, Z d 0 xf X ( x ) dx = E ( X ln( d ) - μ - σ 2 σ where Φ( x ) is the standard normal cdf. (b) The cost of repairing a type of property following storm damage is known to have a LOGN(5,2.5) distribution. An insurer sells policies to cover these losses subject to a policy deductible of 600. i. Calculate the expected amount paid per payment. ii. Calculate the LER. iii. Calculate the expected amount the insurer will pay in respect of a loss which exceeds 600. iv. If, as a result of inflation, all losses increase by 10% but the policy excess remains at 600, calculate the revised amount the insurer will pay in respect of a loss which exceeds 600. 2. The pdf for the amount of a single loss is given by f ( x ) = 0 . 01(1 - q + 0 . 01 qx ) e - 0 . 01 x for x > 0 where q is an unknown parameter. With an ordinary deductible of 100, however, the ex- pected value of the per payment random variable is known to be 125. Using this information, determine the expected value of the per payment random variable for a deductible of 200. 3. Suppose that the cdf of a loss random variable X is given by F ( x ) = 1 - 1 - x θ · α for 0 x θ and α > 0 . An ordinary deductible of d < θ is applied to each loss. (a) Consider Y p , the amount of payment given that a non-zero payment has been made. Show that Y p has the same distribution as X , but with θ replaced by θ - d . (b) Does X have an IMRL or DMRL distribution? Justify. (c) Determine E ( Y p ). (d) Determine the expected value E ( X d ) of the losses eliminated by the deductible d . (e) The loss elimination ratio (LER) indicates the proportion of loss eliminated as a result of having a deductible in the policy. If α = 4 and θ = 1200, what deductible is needed to eliminate 20% of the losses?
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