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
N
S = μ ∗ n − Xi. (1)
i=0
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
β × 100β
fX(x) = 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) Describemathematicallyanalgorithmwhichcouldbeusedtransformindependentsam- ples from a U (0, 1) distribution to generate samples from the above Pareto distribution.
[2 marks]
(2) A premium rate μ = £60 is proposed for S. Use the R programming language to estimate the Value at Risk (VaR) and the Conditional Tail Expectation (CTE) at proba- bility level α = 0.9. From the insurer’s point of view, 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 V aRp[−S]).
[8 marks]
(3) From a policyholder’s point of view (who is a rational investor), consider a client facing a Pareto(100, β = 3) loss Y with upper bound, Pr(Y ≤ 1000)=1. Estimate the maxi- mum 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
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where y denotes the client’s wealth in £. 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 (x) = fX (x)/0.999, for Y ∈ [100, 1000] and is 0 for Y > 100; 2. Consider insurer’s the pooling effect and the policyholder’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;
• includedetailofyourmathematicalcalculationssothatyourresultscouldbereproduced 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 face-to-face 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 Turnitin by 17:30, 26th, Feb, 2021. A link to the submission page is available through the ‘Assessment’ section of the course Vision page. 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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