MTH 223: Mathematical Risk Theory
Tutorial 6
1. Assume that the ground-up loss variable X has a uniform distribution U (0, 650). Let YP = X - 150 j X > 150, YL = (X - 150)+ and Y = min{X, 150} be the per-payment variable, the per-loss variable and the limited loss variable, respectively.
(a) Find the distribution function FYP (y) for all y ∈ (-∞ , ∞).
(b) Find the distribution function FYL (y) for all y ∈ (-∞ , ∞).
(c) Find the distribution function FY (y) for all y ∈ (-∞ , ∞).
(d) Calculate the expectation of the per-loss variable.
(e) Calculate the variance of the limited loss variable.
(f) Calculate the mean excess loss function e(150).
(g) Calculate Var [(X - 150)2 j X > 150].
2. Let X be the ground-up loss for the current year for an insurer with the following pdf
(a) Suppose that a franchise deductible of 100 is applied to the loss. Let YL and YP be the per loss and per payment random variables for the current year, respectively. Find the cdfs FYL (y) and FYP (y) for all y ∈ (-∞ , ∞), and E (YL ) and E (YP ).
(b) Suppose that an ordinary deductible of 150 is applied to the loss. Calculate the loss elimination ratio for the current year.
(c) Suppose that an annual inflation rate of 5% will prevail. The insurer would like to model its ground-up loss by 1.05X for the next year meanwhile maintaining the same ordinary deductible of
150 as that in (b) for the current year.
(i) Calculate the loss elimination ratio for the next year.
(ii) To keep the same loss elimination ratio for the next year as
that for current year, what should be the ordinary deductible of d for the next year? Justify your answer.
(d) Suppose that the insurer institutes an ordinary deductible of 100 , a coinsurance of 85%, and a maximum payment of 1,700 in the current year. Let YL be the per loss random variable for the current year. Find
(i) the cdf FYL (y) for all y ∈ (-∞ , ∞);
(ii) the probability that the payment to be made by the insurer
in the current year will exceed 50 ; (iii) Var (YL ).
3. Losses have an exponential distribution with a mean of 1,000. There is a deductible of 500. Determine the amount by which the deductible would have to be raised to double the loss elimination ratio.
4. Total claims for a health plan have a Pareto distribution with α = 2 and θ = 500. The health plan implements an incentive to physicians that will pay a bonus of 50% of the amount by which total claims are less than 500 ; otherwise no bonus is paid. It is anticipated that with the incentive plan, the claim distribution will change to become Pareto with α = 2 and θ = K. With the new distribution, it turns out that expected claims plus the expected bonus is equal to expected claims prior to the bonus system. Determine the value of K.
5. Losses have a uniform. distribution from 0 to 50,000. There is a per-loss deductible of 5,000 and a policy limit of 25,000.
Determine the expected payment given that a payment has been made.
6. Losses follow a two-parameter Pareto distribution with α = 2 and θ = 5, 000. An insurance policy pays the following for each loss. There is no insurance payment for the first 1,000. For losses between 1,000 and 6,000, the insurance pays 80%. Losses above 6,000 are paid by the insured until the insured has made a total payment of 10,000. For any remaining part of the loss, the insurance pays 90%. Determine the expected insurance payment per loss.