F71AH/PT Coursework Assignment-1
Project description
The insurance business in a single period can be modeled by a so-called surplus process (S)
defined as
S = μ n
N∑
i=0
Xi. (1)
In (1), μ is the constant premium rate and n is the total number of policies in this business;
N is a random positive integer for the number of claims and Xi is another random variables
independent from N , describing the size (amount) of i-th claim. For a S with n = 100, it is
believed that N ～ Poisson(30) distribution and Xi are i.i.d. Pareto(100, β = 3) distributed
with density
fX(x) =
β × 100β
xβ+1
, (2)
where x > 100 denotes the claim size in￡. Otherwise the density is 0. Your task is to estimate
risk measures and other quantities associated with this portfolio over a period of a single year.
(Hint: use simulation in question (1) for the estimations of other questions)
(1) Describe mathematically an algorithm which could be used transform independent sam-
ples from a U(0, 1) distribution to generate samples from the above Pareto distribution.
[2 marks]
(2) The insurer determines a premium rate μ = ￡60 in S. Use the R programming lan-
guage to estimate the Value at Risk (VaR) and the Conditional Tail Expectation (CTE)
at probability level α = 0.9. From the insurer’s point of view, use simulation to estimate
a μ such that the ruin probability is smaller than 1%, i.e.,
Pr(S < 0) ≤ 0.01.
(Hint: for the calculation of VaR & CTE, consider S as the loss of the insurer, i.e. to
estimate VaRα[ S]).
[8 marks]
(3) From a policyholder’s point of view (who is a rational investor), consider a client facing
a truncated Pareto(100, β = 3) loss Y with upper bound 1000, i.e. Pr(Y ≤ 1000) = 1.
Estimate the maximum premium μ that he or she is willing to pay to purchase this
insurance policy if the client adopts a utility function with form
u(y) = 1 exp( y
500
),
1
where y denotes the client’s wealth in ￡. The initial wealth of the policy ho
lder is
￡1000.
Comparing with μ in question (b), which premium rate would you suggest Explain
your answer. In particular, if it holds that μ < μ or μ > μ , could you explain the
motivation for this
(Hint: 1. The upper bounded Y has density fY (y) = fX(y)/0.999, for Y ∈ [100, 1000]
and fY (x) = 0 for Y > 100; 2. Consider insurer’s the pooling effect and the policy-
holder’s level of risk averse).
[5 marks]
[Total 15 marks]
Your findings should be presented in the form of a report, which should:
have a clear and logical structure;
include detail of your mathematical calculations so that your results could be reproduced
by another statistician;
include clearly labelled and correctly referenced tables and diagrams, as appropriate;
include the R code you used in an appendix (you do not need to explain individual
R commands but some comments should be included to indicate the purpose of each
section of code);
include citation and referencing for any material (books, papers, websites etc.) used.
Notes
This assignment counts for 15% of the course assessment.
You may have discussions with me or your colleagues, but your report must be your
own work. Plagiarism is a serious academic offence and carries a range of penalties,
some very serious. Copying a friend’s report or code, or copying text into your report
from another source (such as a book or website) without citing and referencing that
source, is plagiarism. Collusion is also a serious academic offence. You must not share
a copy of your report (as a hard copy or in electronic form) or your computer code with
anyone else. Penalties for plagiarism or collusion can include voiding of your mark for
the course.
Your report should be submitted through Canvas by 4th-March, 6.00 p.m., 2022.
Assignments submitted late (but within 5 working days of the deadline) will have
their mark reduced by 30%. Projects submitted more than 5 working days late
will not be marked.
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