1    Formulae

1.1    Moments

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

dn                『

Coefficient of Variation:

CoV [X] =                                                                    (2)

Skewness:

γ =    E[(X o µ)3]                                                           (3)

(E[(X o µ)2])3/2 .

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

E [exp(tY)] = exp ╱µt + σ 2t2 、.                                         (4)

1.2    Distributions and Densities

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

FX (x; µ, σ)   =    + erf ┌ ln^(x)2σ(o)µ ┐             fx (x; µ, σ)   =    exp ╱ o  (ln 2(x)σ2(o) µ)2 、

where

erf(x) =  2     x e −t2 dt.

The quantile function for a Lognormal random variable is given by

FX(−)1 (p) = exp(µ + σΦ− 1 (p)),    p e (0, 1),

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

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

FX (x) = 1 o ╱、α

fX (x) =

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

k(w) = ╱ exp(gg(w)) o 1 、exp ╱ 、 .

The distribution function of standardised G- and-H severity model is given by

F (x) = Φ(k − 1 (x)).

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] (b) [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]

(c) [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]

(d) [Type] What is risk taxonomy? What are the major data elements that should be used to measure and manage Operational Risk?                                              [5]

(e) 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

(f) 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]

(b) What is the difference between a spliced distribution and truncated distribution?

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

(c) For a Lognormal severity model, provide a definition of the density function when its support is restricted to [L, o) for L > 0.           [3]

(d) If the loss random variable X is Lognormal then it can be expressed via trans- formation 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]

(e) 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 2019.                            [5]

(f) 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]

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

fX (x) = ╱ 、 . ╱ 、−4 ,    x > 1000.

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]

(c) An insurance company has the following policyholders:                                  Type   Probability of claim   Claim amount   Number of Policyholders

1                   0.2                          k                            3500

2                   0.6                        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]

(d) 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

(i) The cumulative distribution function and the probability density function of S.

(ii) The VaR6.7 (S).

(b) Find an expression for the Value-at-Risk of the G-and-H loss severity model.  [3] (c) 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 1.     [7]

NOTE: For parts (a),(b),(c) and (d) of question B3 you may use Definition 1 pro- vided in the Formula sheet.