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MATH3066/6130 Actuarial Mathematics

创建日期：2024-05-28 17:47:01

所属分类：数学mathematics

标签： MATH3066

Actuarial Mathematics

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MATH3066/6130 Actuarial Mathematics II

Practical Assignment
This assignment is worth 20% of the overall mark for the course.
Completed work should be submitted via Blackboard before 23:59 pm on,
Monday 15 May 2023. The deadline is strict and penalties for late work will
be applied in accordance with the University’s late work policy.
Your submission should include a written report and the Excel spreadsheet
of your calculations. Note that all the answers must be presented in
your written report. Therefore please avoid expressions such as ”Please see
the spreadsheet” in the report. The Excel spreadsheet is submitted to prove
that the work is done using Excel and also to check the accuracy of the answers
presented in the report.
There is a strict limit of four A4 pages for the written report, which is easily
sufficient to receive full credit. Font sizes of at least 11pt must be used. Care-
ful explanation and clear presentation are important. All coursework must be
carried out and written up independently (see University’s Academic Integrity
Guidance)
To submit your report and Excel spreadsheet, go to the Blackboard page
of MATH3066/6130, under the Assignments tab there is an assignment called
”Practical Assignment Submission”. In your submission please attach the fol-
lowing two files:
(1) The report in a file called report-ID.pdf, where ID is your student ID
number;
(2) The Excel spreadsheet called spreadsheet-ID.xls, where ID is your student
ID number.
• Question 1 Let S ∼ CP (λ = 2, Gamma(3, 2)).
(a) Let c = 3.2. Calculate ψ1(u, 1) for u = 1, 5, 10, 20, using
(i) the normal approximation, [1]
(ii) the translated Gamma approximation. [1]
(b) Explain carefully how we can obtain ψ1(u, 1) using an exact expression.
[2]
(c) Let u = 1. Calculate the safety loading parameter θ using the translated
Gamma and normal approximation such ψ1(u, 1) = 0.05 and the premium
rate is calculated using
(i) c = (1 + θ)E[S], [1]
(ii) c = E[S] + θ
√
V [S]. [1]
• Question 2 Let U be the surplus process given by
Ut = u+ ct− St,
where S ∼ CP (λ = 2, FX) with X ∼ Gamma(3, 2), c is the premium
income per year, received continuously at constant rate and u is the initial
capital. Let c = (1 + θ)E[S] and θ = 0.2.
(a) Calculate the adjustment coefficient R. [1]
(b) Use a translated Gamma approximation for S to simulate 2, 000
realisations of U (for each period). Plot the probability of ruin
ψh=0.35(u, t) for different time horizons t = 0.7, 1.05, 1.4, ..., 7 and
u = 0.5. Comment on your result. [2]
(c) Estimate the probability that the lowest negative surplus is less than
−1 for all t = 0.7, 1.05, 1.4, ..., 7 . Discuss the effect of c and u on this
probability. [2]
The insurer has purchased a reinsurance from a reinsurer. The insurer
will pay a proportion β ∈ (0, 1) of every claim and the rest is paid by the
reinsurer. Let θβ be the loading in its premiums. Let cβ be the reinsurance
premium. The surplus process is now
Uβt = uβ + (c− cβ) t− β
Nt∑
i=1
Xi,
where cβ = (1 + θβ)(1− β)E[S].
(d) Write down the conditions needed to be satisfied by c− cβ and β. [1]
(e) Let β = 0.3, θβ = 0.25 and θ = 0.2. Repeat parts (b) and (c) for U
β
t .
Comment on your findings. [2]
(f) Calculate the adjustment coefficient Rβ . Comment on your result.[2]
(g) Using normal approximation, show that, if the one-year probabilities
of ruin for Uβt and Ut are equal, we have
uβ = βu+ (1− β)(θβ − θ)E[S]
[2]
(h) Discuss the effect of θβ on u− uβ . Comment on your findings. [2]

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