DEPARTMENT OF STATISTICS
HSTS418 RISK THEORY 2
TUTORIAL AND ASSIGNMENT
Assignment Question: 3 and 4: Due Friday 10 March 2023 @ 1000 hours
1.
Define each of the following terms:
(a) Ruin,
(b) Surplus at time t,
(c) Adjustment coefficient, and
(d) Reinsurance.
2.
State and prove the Lundberg’s inequality. State its importance in Ruin Theory.
3.
𝑆(𝑡) is a compound Poisson process with parameter 40 and claim size distribution
is log 𝑁(5, 4).
(a) Find 𝐸[𝑆(10)].
(b) The initial surplus is 400 000 and the rate of premium income is 41 000 units per unit
time. Assuming that 𝑈(𝑡) can be approximated by a normal distribution, find
probability that 𝑈(10) < 0.
4.
Claims arrive as a Poisson process with rate 𝞴. Individual claim sizes are exponentially
distributed with mean 100. The insurer uses a premium loading of 0.2. a proportional
reinsurance arrangement has been proposed, with a retained proportion of 𝛼. The
reinsurer uses a security loading of 0.4.
(a) State the range of values of 𝛼 such that the probability of ruin is less than 1.
2𝛼−1
(b) Show that the adjustment coefficient,𝑅, is given by 𝑅 =
.
100𝛼(7𝛼−1)
(c) Hence find the value of 𝛼 that maximises 𝑅.
(d) Determine the expected profit per unit time and calculate the upper bound for the
probability of ultimate ruin for the value of 𝛼 in (c) above.
5.
An insurer knows from experience that the number of claims received per month has a
Poisson distribution with mean 15 and claim amounts have an exponential distribution
with mean 500. The insurer uses a security loading of 30%.
(a) Calculate the insurer’s adjustment coefficient.
(b) Determine the upper bound for the insurer’s probability of ruin, if the insurer sets an
initial surplus of 12000.
6.
The aggregate daily claims ($000) on a certain portfolio follow a Poisson process with
mean 30. Individual claim sizes ($000) follow a gamma distribution with parameter 40 and
800. The insurer adds a premium loading of 50%.
(a) Calculate the adjustment coefficient.
(b) Hence calculate the minimum amount of capital needed to ensure that the probability
of ultimate ruin is less than 0.1%.
(c) Calculate the expected daily profit on this portfolio of policies.
7.
A Poisson claims process has a security loading
3
7
function 𝑓(𝑥) = 2 𝑒 −3𝑥 + 2 𝑒 −7𝑥 , 𝑥 > 0.
𝜃 = 0.4 and claims density
(a) Derive the moment generating function for the claim size distribution and state the
values of 𝑡 for which it is valid.
(b) Calculate the value of the adjustment coefficient.
(c) Find 𝜓(𝑢).
8.
𝑆(𝑡) is a compound Poisson process with parameter 40 and claim size distribution
is log 𝑁(5, 4).
(c) Find 𝐸[𝑆(10)].
(d) The initial surplus is 400 000 and the rate of premium income is 41 000 units per unit
time. Assuming that 𝑈(𝑡) can be approximated by a normal distribution, find
probability that 𝑈(10) < 0.
9.
An insurer knows from experience that the number of claims received per month has a
Poisson distribution with mean 15 and claim amounts have an exponential distribution
with mean 500. The insurer uses a security loading of 30%.
(c) Calculate the insurer’s adjustment coefficient.
(d) Determine the upper bound for the insurer’s probability of ruin, if the insurer sets an
initial surplus of 12000.
10. The aggregate daily claims ($000) on a certain portfolio follow a Poisson process with
mean 30. Individual claim sizes ($000) follow a gamma distribution with parameter 40 and
800. The insurer adds a premium loading of 50%.
(d) Calculate the adjustment coefficient.
(e) Hence calculate the minimum amount of capital needed to ensure that the probability
of ultimate ruin is less than 0.1%.
(f) Calculate the expected daily profit on this portfolio of policies.
11. A Poisson claims process has a security loading
3
7
function 𝑓(𝑥) = 𝑒 −3𝑥 + 𝑒 −7𝑥 , 𝑥 > 0.
2
𝜃 = 0.4 and claims density
2
(d) Derive the moment generating function for the claim size distribution and state the
values of 𝑡 for which it is valid.
(e) Calculate the value of the adjustment coefficient.
Find 𝜓(𝑢).
12. A specialist insurer that provides insurance against breakdown of photocopying
equipment calculates its premiums using a credibility formula. Based on the
company recent experience of all copiers, the premium for this year should be $100
per machine. The company’s experience for a new model of copier, which is
considered to be more reliable, indicates that the premium should be $60 per
machine. If the credibility factor is 0.75, what premium should the insurer charge
for insuring the new model?
13. Claims for a particular risk arrive according to the Poisson process with rate 𝞴. The
claim sizes are independent and identically with density 𝑓(𝑥) and are independent
of the claims arrival process. Assume that, there is a constant 𝛾 (0 < 𝛾 < ∞) such
that lim 𝑀(𝑟) = ∞ where 𝑀(𝑟) is the moment generating function of a claim.
𝑟→𝛾
Premiums are received at a constant rate with premium loading 𝜃 > 0.
(a) (i) Define the adjustment coefficient 𝑅.
(ii) Define the surplus process and the probability of ruin, 𝜓(𝑢), with initial
surplus 𝑢 > 0.
(iii) Write the Lundeberg’s inequality.
1
(b) Derive the adjustment coefficient if 𝑓(𝑥) = 𝜇 𝑒 −𝑥/𝜇 , 𝑥 > 0 𝑎𝑛𝑑 𝜃 > 0.25.
(c) Consider the case where 𝑓(𝑥) = 0.5𝑒 −𝑥 (1 + 2𝑒 −𝑥 ) 𝑎𝑛𝑑 𝜃 = 0.25.
(i)
Calculate the expected claim size.
(ii)
Calculate the corresponding adjustment coefficient and determine an
upper bound for 𝜓(15).
END OF WORKSHEET