•      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 (14 marks), Question 2 (24 marks), Question 3 (21 marks), Question 4 (6 marks), Question 5 (16 marks), Question 6 (19 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.

Administrative details

•      This is an open-book exam. You may use your course materials to answer questions.

• 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. You should also fill out the exam paper query form online.

•      Some part-questions require text-based answers; many of these will indicate the maximum number of words you may write. You must adhere to this or you may lose marks.

•      Some questions may ask you to approach a problem in a particular way; please take note of this. Failure to do so may result in marks being deducted.

Formatting your solutions for submission

x    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.

x    You should submit ONE pdf document that contains your solutions for all questions/ part-  questions. Please followUCL's guidance on combining text and photographed/ scanned work.

x     Make sure that your handwritten solutions are clear and are readable in the document you submit.

Plagiarism and collusion

x    You must work alone. In particular, any discussion of the paper with anyone else is not acceptable. You are encouraged to read theDepartment of Statistical Science's advice on collusion and plagiarism.

x     Parts of your submission will be screened via Turnitin to check for plagiarism and collusion.

x    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.

Question 1 [14 marks]

Suppose that a parameter can take only three values, = 0, = 1 or = 2. Your prior on before observing any data is ( = 0) = 0.3, ( = 1) = 0.5 and ( = 2) = 0.2. The distribution of is as follows:

(| = 0) = (2,3)

(| = 1) = (1,2)

(| = 2) = (3,3)

Your task is to estimate the value of . Let action 0   corresponds to claiming that = 0, 1 corresponds to claiming that = 1,  and 2   corresponds to claiming that = 2. The  losses corresponding to each action and values of are represented by the following lossmatrix:

= 0 = 1 = 2

0

1

For example, the loss associated with action 0  when the true value is = 1 is ( = 1, 0 ) = 1. You obtain one observation = 2.

(a) Compute the posterior distribution for .

(b) Compute the Bayesian expected loss associated with all three actions given this observation,

and decide which action to take.

[7]

Question 2 [24 marks]

Consider the following historical record for the daily log-returns of a financial stock: 1  = 0.4, 2  = −0.61, 3  = −0.53, 4  = 0.22, 5  = −0.018, 6  = −0.39

The log-returns are assumed to be independent and identically distributed draws from a Normal distribution with known mean 0 and unknown variance 2 .

(a)    Using the - (, ) distribution as a conjugate prior for 2, derive its posterior

distribution  given the  data. You  do  not  need to  evaluate  any  integrals  or  normalising constants.                                                                                                                              [5]

(b)   Let denote the  log-return on a  particular day in the future. Show that the  posterior predictive distribution is of the following form:

1 ̃ (̃ + )

√2 (̃) ( + ) ̃+

for some values of ̃  and . Using the numeric values of 1, … , 6   provided above, give expressions for ̃ and in terms of and . State the range of possible values that can take.                                                                                                                                       [7]

(c) [Type] State the Pickands − Balkema −de Haan theorem. Explain using your own words how this theorem can be used in risk analysis. Use no more than 100 words.                             [4]

(d)   Find the probability of log-return, 7 , on the next day being between −0.75 and −0.7 using a  Generalised  Pareto  Distribution  (GPD)  with  threshold = −0.38.  Use  the  method  of moments to estimate the GPD parameters.                                                                          [8]

Question 3 [21 marks]

A local meteorological office located on an island in the Pacific Ocean would like to model the number of hurricanes that occur in a year. Let be a discrete random variable which represents the number of hurricanes that occurred in year . The company considers a Poisson model to estimate the frequency of hurricanes, i.e. . . (). The following historical record shows the number of hurricanes over the past 5 years:

Year, Number of hurricanes

The meteorological office suspects that during this period there might have been a structural change in the hurricane frequency due to climate change. In particular, the meteorological office believes that the change point has occurred in year = 3. Thus, observations 4  and 5  still come from a Poisson distribution but with a different parameter , i.e.:

(1 )  if ≤ 3

Define the two models:

0 : There has been no change point in the hurricane frequency.

1 : There has been a change point at = 3.

(a)    Compute the marginal likelihood for Model 0 without a change point. Note that you should

use the Gamma(1, 1) prior in Model 0 .                                                                               [6]

(b)   Compute the marginal likelihood for Model 1 with a single change point in year = 3. Note

that you should use the Gamma(1, 1) prior for both segments in Model 1 .                        [11]

(c)    Compute the posterior distribution for both models, and decide whether there has been a structural change in the hurricane frequency in year = 3. Both models 0 and 1 are equally likely a priori.                                                                                                                          [4]

Question 4 [6 marks]

Let = () − (−1) denote the daily log-return on a financial asset, where is the opening daily price at time . The following model has been proposed to describe the behaviour of log- returns:

=

= . . (0,1)

= + −1 + 1

where , , > 0 and + < 1.

Derive the unconditional variance of , Var(). Clearly show all the steps leading to your answer.

[6]

-3

-6

8

-3

-5

-8

-2

6

For example, if you take action 1, and the state of nature is 1, then the incurred loss is 10.

(i)     State for each action 1, 2  and 3, whether it is admissible or inadmissible.                      [3]

(ii)    Find the minimax nonrandomized action. Clearly show all the steps leading to your answer. [5]

(b)   Let . . ()  where > 0  is unknown. It  is of interest to estimate using the following loss function:

(, ) = log () + + log ()

where , > 0  are constants. The prior density that reflects prior information about is () = − , where > 0. Let = (0, ∞) be the action space, so that ∈ .

Assuming that no data are available, compute the Bayesian point estimate of under the

loss function (, ).                                                                                                              [8]

Question 6 [19 marks]

(a)    Consider the following information on the hypothetical portfolio of £7,000 invested in two

assets. The information on each daily asset return is provided in the table below. It is assumed that these returns are jointly normally distributed.

Mean

Standard deviation

Portfolio weights

Portfolio value

Correlation coefficient

0.008

0.3

0.2

£7,000

0.2

Standard normal distribution table.

()

-2.326

0.01

-2.054

0.02

-1.881

0.03

-1.751

0.04

-1.645

0.05

-1.555

0.06

Compute the 99% 1-day Value-at-Risk (VaR) of the portfolio in value terms. Interpret your

findings.

[5]

(b) [Type] Suppose that you have been hired as a risk analyst by a financial company to assess

its  current  internal  approach  to  computing  the  99%  10-day  Value-at-Risk  (VaR)  for  a particular financial asset. Let denote the daily log-return on that asset which evolves over time as follows:

= +

where

= + ∑ − =1

= . . (0,1)

= + ∑ − + ∑ − [ −<0] + ∑ =1 =1 =1