Quantitative Modelling of Operational Risk and Insurance Analytics

UNIVERSITY COLLEGE LONDON

 

 

 

 

EXAMINATION FOR INTERNAL STUDENTS          

 

 

 

MODULE CODE

 

: STAT0020
ASSESSMENT : STAT0020A6UB  
PATTERN

 

  STAT0020A7PB
MODULE NAME

 

: Quantitative Modelling of Operational Risk and Insurance Analytics
LEVEL: : Undergraduate
 

 

  Postgraduate
DATE

 

: 17-June-2020
TIME : 12:00

 

 

 

 

This paper is suitable for candidates who attended classes for this module in the following academic year(s):

 

2019/20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TURN OVER

 

STAT0020: Quantitative Modelling of Operational Risk and Insurance Analytics 2019/2020

  • Answer ALL questions.
  • The relative weights attached to each question are Question 1 (25 marks), Question 2 (25 marks), Question 3 (25 marks), Question 4 (25 marks).
  • The numbers in square brackets indicate the relative weights attached to each part question.
  • Marks will not only be given for the final (numerical) answer but also for the accuracy and clarity of the 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.
  • UCL requires that all 24-hour online exams have a specified overall word limit. The overall word limit for this exam has been set well in excess of the expected amount of work so that you do not need to worry about exceeding it.
  • Some part-questions may indicate a word or sentence limit. You must adhere to this or risk losing marks.

Formatting your solutions for submission

  • Some part-questions require you to type your answers instead of handwriting them. These questions state [Type] at the start of the part-question. You must follow this instruction. Failure to do so may result in marks being deducted. For questions without the [Type] instruction, you may choose to type or hand-write your answer.
  • You should submit ONE 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. You are encouraged to write out solutions neatly once you are happy with them.

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, which you can find
  • 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.

1     Formulae

  • Moments

Non-central moments in terms of a characteristic function ϕX(t):

 .                                                                          (1)

Coefficient of Variation:

.                                                                          (2)

Skewness:

.                                                                          (3)

The moment generating function of a Normal random variable Y ∼N(µ,σ2) is given by:

 .                                                                          (4)

  • Distributions and Densities

Lognormal distribution and density functions for x ≥ 0, µ ∈R and σ > 0:

(5)

(6)

(7)

where

erf                                    (8)

The quantile function for a Lognormal random variable is given by  
                                               FX−1(p) = exp(µ + σΦ−1(p)),        p ∈ (0,1), (9)

where Φ(x) is the distribution function of a standard normal random variable.

One parameter Pareto distribution and density function for x ∈ [xm,∞) and α > 0

(10)

(11)

Definiton 1: A random variable is said to have a G-and-H distribution with parameters a,b,g and h if it satisfies the following transformation of the reference Normal Random variable W ∼ (0,1) according to X = a + bk(W), where

.                                                                        (12)

The distribution function of standardised G- and-H severity model is given by  
F(x) = Φ(k−1(x)). (13)

 

Q1.      (a) [Type] Describe Pillar 1 and Pillar 2 of the Basel II banking regulation and explain 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 each framework according to the Basel II banking regulation. Answer should be approximately one paragraph per pillar.         [5]

  • [Type] Explain the Top-Down approach to modelling Operational Risk capital and point out the main differences with the Bottom-Up approach. In the process provide a definition of the Standardised Approach. [5]
  • [Type] List three standard characteristics of insurable losses. Explain why some Operational Risk losses may not be directly insurable and consider one example of such loss processes. [5]
  • [Type] What is risk taxonomy? What are the major data elements that should be used to measure and manage Operational Risk? [5]
  • Provide a definition of N-decomposability of loss random variables and state the difference with N-divisible loss random variable. State which of these conditions are preserved under convolution. [3]
  • Provide a definition of the Tier 1 capital ratio under Basel II regulation. [2]

Q2.      (a) [Type] Explain the capital charges under Basel III and provide the reason for these changes. In the process explain the countercyclical capital buffer.            [4]

  • What is the difference between a spliced distribution and truncated distribution?

Explain how truncated distributions are useful in Operational Risk.                           [5]

  • For a Lognormal severity model, provide a definition of the density function when its support is restricted to [L,∞) for L > 0. [3]
  • If the loss random variable X is Lognormal then it can be expressed via transformation of a Normal random variable Y given by X = exp(Y ), where Y ∼ N(µ,σ2). Use this representation to obtain expressions for the mean and variance of a Lognormal severity model in terms of µ and σ. [3]
  • Suppose the loss sizes in 2017 follow a Lognormal distribution with parameters µ = 4.2 and σ = 2. Loss sizes grow at 6% during 2018 and 2019. Calculate f(1000), the probability density function at 1000, of the loss size distribution in
  1. [5]
  • Consider a Loss Distributional Approach model with frequency model Poisson with mean λ and Lognormal severity model . If you assume all counts and loss amounts are independent, find a closed form expression for the first order Single Loss Approximation for the Value-at-Risk capital measure. [5] Q3. (a) Consider a standard Loss Distributional Approach model for an Operational Risk loss modelling with independent frequency and severity random variables. Prove that the annual loss 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. [7]

  • The distributions for loss severity follows a single parameter Pareto distribution of the following form:

.

Determine the mean size of a loss between 10,000 and 100,000, given that the loss is between 10,000 and 100,000.        [5]

  • An insurance company has the following policyholders:

Type     Probability of claim                   Claim amount    Number of Policyholders

  • 2 k 3500
  • 6 αk 2000

Determine α and k so that the expected aggregate claims amount is 100,000 and the variance of the aggregated claim amount is minimised.         [8]

  • Consider S = X + Y to be the sum of two individual losses. Individual losses X and Y are independent. X is 0 with probability 0.1 and with probability 0.9 X has a uniform distribution on the interval (0,90). The individual loss of Y is 0 with probability 0.4 and 5 with probability 0.6.

Find

  • The cumulative distribution function and the probability density function of S.
  • The V aR0.7(S).
  • The ES0.7(S). [5]

Q4.    (a) Find an expression for the severity density of the G-and-H loss model.           [2]

  • Find an expression for the Value-at-Risk of the G-and-H loss severity model. [3]
  • Consider a Loss Distributional Approach (LDA) model with a Poisson frequency model and a G-and-H severity model. Find an expression for the asymptotic of the upper tail of the annual loss distribution for this model. [5] d) Consider a Loss Distributional Approach (LDA) model with a Poisson frequency model, with rate λ, and a G-and-H severity model. Find the Value-at-Risk Single Loss Approximation for this model for the extreme quantile level as α ↑ 1. [7] NOTE: For parts (a),(b),(c) and (d) of question B3 you may use Definition 1 provided in the Formula sheet.

 

(e) Prove that convolution is associative. First state the definition of associativity in this setting, then prove that this property holds. [8]

END OF PAPER