辅导F71AH/PT、辅导Python，c++编程
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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.
2
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.
2