Problem 1. A claim size random variable X follows a loglogistic distribution. You are given the following information.
• The 20th percentile of this claim size distribution is 350.
• The 80th percentile of this claim size distribution is 1400. Determine P(X > 700).
Problem 2. The dollar value of the total damagedone to a home due to a fire follows a Pareto distribution with α = 2.7 and θ = 45, 000. ABC Fire Insurance Company writes homeowner’s fire insurance policies. Each policy has the following coverage modifications.
• A deductible of $500.
• ABC Fire Insurance Company will pay 92% of the loss after the deductible is met.
• ABC Fire Insurance Company will pay a maximum amount of $100, 000 per fire insurance policy.
(i) (6 points) Compute the expected payment the ABC Fire Insurance Company will make per policy.
(ii) (4 points) If the dollar value of the total damage done to a home due to a fire is inflated by 15% and all coverage modifications remain unchanged, compute the expected payment the ABC Fire Insurance Company will make per policy.
Problem 3. You have been asked to build a compound frequency model of the usual form.
S = M1 + M2 + ··· + MN
where N has a negative binomial distribution with parameters r = 3 and β = 2 and M has the distribution
P(M = 1) = 0.3, P(M = 2) = 0.4, , P(M = 4) = 0.3.
Compute P(S ≤ 4).
Problem 4. You are given the following information on the losses for a particular line of business.
• For year 2002, loss sizes followed a uniform distribution between 0 and 5000.
• In year 2002 the insurer pays 100% of all losses.
• Inflation of 6% affects all losses uniformly from year 2002 to year 2003.
• In year 2003 a deductible of 200 is applied to all losses.
Compute the loss elimination ratio (LER) of the 200 deductible on year 2003 losses.
Problem 5. You are given the following distributions for independent loss random variables X1 , X2 , and X3 .
x f1(x) f2(x) f3(x)
10 0.25 0.2 0.4
20 0.75 0.2 0.6
30 – 0.6 –
Compute the net stop-loss premium to cover the aggregate loss
S = X1 + X2 + X3
for a deductible of d = 45.
Problem 6. An insurance company has decided to establish its full-credibility re- quirements for an individual state rate filing. The full-credibility standard is to be set so that the observed total amount of claims underlying the rate filing would be within 5% of the true value with probability 0.95. The claim frequency follows a Poisson distribution and the loss severity distribution has pdf
f(x) = kx, 0 ≤ x ≤ 200
for some constant k. (The density is 0 for x > 200).
Determine the expected number of claims necessary to obtain full credibility using the normal approximation.
Problem 7. In the population at large there are good and bad drivers. Good drivers make up 80% of the population and in one year have zero claims with probability 0.8, one claim with probability 0.15 and two claims with probability 0.05. Bad drivers make up the other 20% of the population and have zero, one or two claims with probabilities 0.3, 0.5 and 0.2 respectively. A certain driver has one claim in year 1 and two claims in year 2.
Compute the expected number of claims for this driver for year 3 given this informa- tion.
