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STAT0020 Risk and Insurance Analytics

创建日期：2024-05-06 14:05:39

所属分类：统计学statistics

标签： STAT0020

Risk and Insurance Analytics

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STAT0020 - Quantitative Modelling of Operational

Risk and Insurance Analytics
LEVEL: : Undergraduate
Undergraduate (Masters Level)
Postgraduate
TIME : 14:30
This paper is suitable for candidates who attended classes for this
module in the following academic year(s):
2020/21
TURN OVER
STAT0020- Examination Paper 2020/2021 Page 1
STAT0020 : Quantitative Modelling of Operational Risk
and Insurance Analytics
2020/2021
• Answer ALL questions.
• You have three hours to complete this paper.
• After the three hours has elapsed, you have one additional hour to upload your solutions.
• You may submit only one answer to each question.
• The relative weights attached to each question are Question 1 (15 marks), Question 2 (15
marks), Question 3 (20 marks), Question 4 (30 marks), Question 5 (20 marks).
• The numbers in square brackets indicate the relative weights attached to each part question.
• Marks are awarded not only for the final result but also for the clarity of your answer.
• Show your full working for all questions. Do not write formulas alone without any comment
about what you are calculating.
Administrative details
• This is an open-book exam. You may use your course materials to answer questions.
• Some questions may ask you to solve, or not to solve, a problem in a particular way; please
take note of this. Failure to do so may result in marks being deducted.
• You may not contact the course lecturer with any questions, even if you want to clarify
something or report an error on the paper. If you have any doubts about a question, make a note
in your answer explaining the assumptions that you are making in answering it.
Formatting your solution for submission
• You should submit ONE pdf document that contains your solutions for all questions/ part-
questions. Please follow UCL’s guidance on combining text and photographed/ scanned work.
• Make sure that your handwritten solutions are clear and are readable in the document you
submit.
TURN OVER
STAT0020- Examination Paper 2020/2021 Page 2
Plagiarism and collusion
• You must work alone. In particular, any discussion of the paper with anyone else is not
acceptable. You are encouraged to read the Department of Statistical Science’s advice on
collusion and plagiarism
• Parts of your submission will be screened via Turnitin to check for plagiarism and collusion.
• If there is any doubt as to whether the solutions you submit are entirely your own work you
may be required to participate in an investigatory viva to establish authorship.
CONTINUED
STAT0020- Examination Paper 2020/2021 Page 3
Q 1 (a) Describe Pillar I and Pillar II of the Basel II banking regulation. What is the role
that each of these pillars plays in the regulation? In forming your answer, describe
the different risk types considered and the modelling approaches allowed for in
each framework according to the Basel II banking regulation. Answers should be
approximately one paragraph per pillar. [8]
(b) In 2008 Capital charges were modified Under the Basel III banking regulation.
What was the motivation for these changes? Explain in detail the Countercyclical
Capital Buffers. [7]
Q 2 (a) A financial institution has the following gross income (GI) for the Basel II stan-
dard business lines, reported in units of 1, 000, 000: Corporate Finance (CF);
Trading and Sales (TS); Retail Banking (RB); Commercial Banking (CB); Pay-
ment and settlement (PS); Agency Services (AS); Asset Management (AM); and
Retail Brokerage (RB).
Year CF TS RB CB PS AS AM RB
2015 100 50 60 2000 100 100 50 200
2016 120 -50 60 0 120 20 20 100
2017 30 500 40 2200 300 900 10 90
2018 -10 -10 78 100 600 100 90 50
2019 140 20 10 2000 100 400 80 63
Based on the reported GI’s find the capital requirement for 2017 for the opera-
tional risk model comprised of the Standardised approach (TSA) using regulatory
weighting for the following Basel lines of Business (LOB).
LOB CF TS RB CB PS As AM RB
Weighting 18% 18% 12% 15% 18% 15% 12% 12%
Show all working and provide a definition of the capital requirement formula be-
fore performing calculations. [6]
(b) Consider the Bottom-Up approach to modelling Operational Risk under the Loss
Distributional Approach modelling framework. Provide two qualitative and two
quantitative requirements that should be satisfied under Basel II regulations for
a sophisticated financial institution to be allowed to report capital under the Ad-
vance Measurement Approach. [5]
(c) Provide the definition of the Tier 1 Capital ratio and calculate this for the example
in Table 1 [4]
TURN OVER
STAT0020- Examination Paper 2020/2021 Page 4
Risk Weight Asset Amount RWA
0% Cash(*) $10 $0
Treasury Bills(*) $50 $0
Long-Term Securities $100 $0
20% Municipal Bonds(*) $20 $4
Items in collection $20 $4
50% Residential Mortgages(*) $300 $150
100% AA+ rated loan(*) $20 $20
AAA- rated(*) $55 $55
BB-rated commercial loans $200 $200
Sovereign loans B-rated $50 $50
Reserve for loan losses $10 $10
Total $835 $493
Table 1: Example of Risk Weighted assets calculation under Basel II.
(*) in the table, denotes eligible Tier 1 Capital.
Q 3 (a) Claims frequency follows a distribution in the (a, b, 0) class. You are given that
(i) The probability of 4 claims is 0.066116.
(ii) The probability of 5 claims is 0.068761.
(iii) The probability of 6 claims is 0.068761.
Calculate the probability of no claims. [5]
(b) You sell a single auto insurance policy. The number of claims on this policy are
Poisson distributed with a mean of 2. The size of each claim follows a zero-
truncated Poisson distribution with λ = 0.8. S is the aggregate claim amount for
this policy.
Calculate
(i) P (S = 0) [3]
(ii) P (S = 3) [7]
(c) Provide a detailed definition of big-O notation and hence prove that 4n5+3n3+2
is O(n5) and it is not O(n4). [5]
Q 4 Consider a Poisson-Stable family LDA model, where the severity distribution for inde-
pendent and identically distributed losses Xi is α-stable with α = 1, β = 0, γ > 0 and
δ = 0 and the support on [0,∞).
(a) Conditional on the number of annual losses N to be equal to n, find a closed form
expression for the density function of the annual loss for Zn =
∑n
i=1Xi. [5]
(b) Consider a standard LDA model for an operational risk loss modelling with in-
dependent frequency and severity random variables. Prove that the annual loss
CONTINUED
STAT0020- Examination Paper 2020/2021 Page 5
distribution has a characteristic function χ(t) given by
χ(t) = [g ∗ φ](t),
where g is the probability generating function of the frequency model and φ is the
characteristic function of the severity model. [5]
(c) Find a closed form expression for the characteristic function χ(t) of the Poisson-
Stable LDA model when the parameters satisfy α = 1 β = 0, γ > 0 and δ = 0.[4]
d) By using the results in part (c), show that the resulting LDA model produces a
heavy tailed annual loss model and define what you mean by a heavy tailed annual
loss model in your answer. [6]
(e) Find a closed form asymptotic expression for this Poisson-Stable family LDA
model density function as the loss amount grows to infinity. [5]
(f) In the case that γ = 1, find a closed form Single Loss Approximation of the Value
at Risk measure at level α for the Poisson-Stable LDA model. [5]
Q 5 (a) The loss X follows a continuous spliced distribution which is proportional to an
exponential distribution with mean 3 from 0 to 4 and it is proportional to a uniform
distribution from 4 to 6. What is the Pr(3 < X < 5)? [5]
(b) Let N(t) be the number of losses for the time interval [0, t]. We assume that N(t)
follows a Poisson distribution P(λ). We recall that:
ex =
∞∑
n=0
xn
n!
(i) Calculate the first moment E[N(t)]. [3]
(ii) Show the following result:
E
[
m∏
i=0
(N(t)− i)
]
= λm+1
[3]
(iii) Then deduce the variance of N(t). [3]
(c) Let X be a random variable with the following loss distribution
k pk
0 0.65
100 0.30
200 0.03
500 0.01
1000 0.01
(i) Calculate the 90% value-at- risk. [3]
(ii) Calculate the 90% tail value-at-risk. [3]

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