代写F71AH/PT、代写Python，c++编程

日期：2022-03-04 01:39

F71AH/PT Coursework Assignment-1

Project description

The insurance business in a single period can be modeled by a so-called surplus process (S). (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

x

β+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 samples

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 language

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 holder 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 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;

? 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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