PSTAT  173 FINAL EXAM

RISK THEORY

MARCH  14,  2022

Problem 1. Let X ~ Gamma(Q = 2, θ = 2). Compute:

(1) VaR0.975 (X)

(2) eX (9.488)

(3) TVaR0.975 (X) and TVaR0.95 (X)

Problem 2.

A company insures a fleet of vehicles.  Aggregate losses have a compound Poisson distribution.  The expected number of losses is 50, and the amount of each loss is assumed to be exponential with parameter θ = 2000.

We modify this coverage by in the following ways:

(1) A deductible of 100 is imposed

(2) It can be assumed that 10% of claims will not be covered (i.e., a benefit payment will not be made to the policyholder in these cases)

What is the expected amount paid by the insurer?

Problem 3. A towing company provides all towing service to members of an Auto- mobile Club. You are given:

With the following stipulations:

(1) The automobile owner must cover 10% of the towing cost; the rest is covered by the Club

(2) The number of towings is Geometric(β = 50) (use the Appendix parameteri- zation)

(3) The number and cost of towings are independent

Using a normal approximation, what is the minimum amount will the Club need to set aside to cover all claims with a probability of at least 0.9?

Problem 4. You are given:

(1) Losses follow an exponential distribution with the same mean every year

(2) The Loss Elimination Ratio this year is 55%

(3) The ordinary deductible in the upcoming year is 3/2 the current deductible

Calculate the Loss Elimination Ratio for the upcoming year

Problem 5. The random variable for a loss X has the following characteristics:

Calculate the mean excess loss for a deductible of 25 using linear interpolation for E[X ^ 25] and F (25) (i.e. for F (25), it lies on the straight line connecting F (20) and F (50)).

Hint:  The  CDF value  at x = 150 should make E[X] easy to  compute.

Problem 6. You are given:

(i) Losses follow a single-parameter Pareto distribution with density function:

f(x) = xα+1/α,    x > 1,    0 < Q < ∞

(ii) A random sample of size five produced three losses with values 3,6 and 14 , and four losses exceeding 25 .

Calculate the maximum likelihood estimate of Q.

Problem 7. X is a discrete random variable with a probability function that is a member of the (a,b, 0) class of distributions.

You are given:

(i)   Pr(X = 0) = 2/5Pr(X = 1) = 0.25

(ii)   Pr(X = 2) = 0.03

Calculate Pr(X = 3).

Problem 8. For a lognormal distribution with parameters µ and σ you are given that the maximum likelihood estimates are µb = 2.21 and σb = 1.1.

The covariance matrix of (µ, σ) is

The mode of the lognormal distribution is g(µ, σ) = eμ-σ2 .

(a) Estimate the variance of the maximum likelihood estimate of the mode using the delta method.

(b) Estimate the 95% confidence interval for the mode.

Problem 9. You are given the following information about a general liability book of business comprised of 2500 insureds:

(i) Xi   =  Σj=1 Yij   is a random variable representing the annual loss of the ith insured.

(ii) N1 , N2 , . . . , N2500  are independent and identically distributed random variables following a Poisson distribution with parameter λ = 0.4.

(iii) Yi1, Yi2 , . . . are independent and identically distributed random variables follow- ing a Pareto distribution with α = 3.0 and θ = 1000.

(iv) The full credibility standard is to be within 5% of the expected aggregate losses 90% of the time.

Using limited fluctuation credibility theory (i.e. Chapter 16 material), determine the partial credibility factor Z of the annual loss experience for this book of business.

Problem 10. You are given the following:

• A portfolio of independent risks is divided into three classes.

• Each class contains the same number of risks.

• For all of the risks in Class 1, claim sizes follow a uniform. distribution on the interval from 0 to 400.

• For all of the risks in Class 2, claim sizes follow a uniform. distribution on the interval from 0 to 600.

• For all of the risks in Class 3, claim sizes follow a uniform. distribution on the interval from 0 to 800.

A risk is selected at random from the porfolio.  The size of the first claim observed for this risk is 340.

Determine the Bu…hlmann credibility estimate of the second claim observed for this same risk.