# Assignment-1

```SEMESTER 1 - ASSIGNMENT 1
BAS 310- Risk Theory for Actuarial Science
Lecturer: Mr. Saviour Chibeti
February 7, 2022
INSTRUCTIONS TO STUDENTS:
(1)
(2)
Use the appropriate student's cover page to complete your details.
(3)
Your assignment must be submitted in soft copy (typed or scanned) of which
bad handwriting will result in loss of marks.
(4)
Answer ALL the questions in this paper, showing all the necessary workings
where applicable.
(5)
the name of the Lecturer and the Course details on your cover page
(6)
Plagiarism is a serious oense so do not copy from your classmates, same
assignments will automatically attract a fail on the assignments and a serious
punishment.
(7) There are ve questions, attempt all the questions
(8) Deadline: 4th March,2022 NOTE THAT LATE SUBMISSIONS SHOW NOT BE ATTENDED TO.
1
QUESTION ONE
(a) Loss amounts under a class of insurance policies follow an exponential distribution with
mean 100. The insurance company wishes to enter into an individual excess of loss
reinsurance arrangement with retention level M set such that 8 out of 10 claims will not
involve the reinsurer.
[4 marks]
(i) Find the retention M
For a given claim, let XI denote the amount paid by the insurer and XR the amount
paid by the reinsurer.
(ii) Calculate E(XI ) and E(XR ).
[6 marks]
(b) Losses on a portfolio of insurance policies in 2021 are assumed to have an exponential
distribution with parameter λ. In 2022 loss amounts have increased by a factor k (so that
a loss incurred in 2022 is k times an equivalent loss incurred in 2021). Show that the
distribution of loss amounts in 2022 is also exponential and determine the parameter of
the distribution
[10 marks]
.
QUESTION TWO
(a) The loss function for a decision problem is given below.
θ1
D1
D2
D3
D4
30
20
17
35
θ2
20
25
19
20
θ3
15
10
24
17
(i) Explain which strategies, if any, are dominated.
(ii) Find the minimax solution to this decision problem.
(b)
[4 marks]
[2 marks]
(i) For the Pareto distribution with parameters α and λ, derive the median
.
[4 marks]
(ii) For α = 2 and λ = 3, comment on this pareto distribution. [3 marks]
(c) Claim amounts on a portfolio of insurance policies follow a Weibull distribution. The
median claim amount is K1, 000 and 90% of claims are less than K5,000.
Estimate the parameters of the Weibull distribution, using the method of moments
.
[7 marks]
QUESTION THREE
(a) A portfolio of insurance policies contains two types of risk. Type I risks make up 80% of
claims and give rise to loss amounts which follow a normal distribution with mean 100
2
and variance 400. Type II risks give rise to loss amounts which are normally distributed
with mean 115 and variance 900.
(i) Calculate the mean and variance of the loss amount for a randomly chosen claim
.
[6 marks]
(ii) Explain whether the loss amount for a randomly chosen claim follows a normal distribution
[2 marks]
The insurance company has in place an excess of loss reinsurance arrangement with
retention 130.
(iii) Calculate the probability that a randomly chosen claim from the portfolio results
in a payment by the reinsurer.
[4 marks]
(iv) Calculate the proportion of claims involving the reinsurer that arise from Type
II risks.
[3 marks]
(b) Claims on a certain type of insurance policy are believed to follow an exponential distribution. The upper quartile claim size is 240.
[5
Calculate the mean claim size
marks]
QUESTION FOUR
(a) An insurer believes that claims from a particular type of policy follow a Pareto distribution
with parameters α = 2.5 and γ = 300. The insurer wishes to introduce a deductible such
that 25% of losses result in no claim on the insurer.
(i) Calculate the size of the deductible
[4
(ii) Calculate the average claim amount net of the deductible
marks]
[6 marks]
(b) X and Y are independent Poisson random variables with mean λ. Derive the moment
generating function of X , and hence show that X + Y also has a Poisson distribution
.
[10 marks]
QUESTION FIVE
(a) The number of claims on a portfolio of washing machine insurance policies follows a Poisson distribution with parameter 50. Individual claim amounts for repairs are a random
variable 100X where X has a distribution with probability density function

3

2

 32 (6x − x − 5)
f (x) =



0
(1, 5)
elsewhere
In addition, for each claim (independently of the cost of the repair) there is a 30% chance
that an additional xed amount of K200 will be payable in respect of water damage
3
(i) Calculate the mean and variance of the total individual claim amounts
.
[7 marks]
(ii) Calculate the mean and variance of the aggregate claims on the portfolio
.
[3 marks]
(b) A random variable X follows a gamma distribution with parameters α and β .
(i) Derive the moment generating function (MGF) of X .
(ii) Derive the coecient of skewness of X .
END OF AN ASSIGNMENT
4
[3 marks]
[7 marks]
```