Examination Paper for STAT0011 2021

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

2

0

1

8

1

0

3

5

4

0

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

1

2

3

4

5

1

2

3

2

2

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]

Question 5 [16 marks]

(a)    Suppose that you are faced with taking one of the following three actions: 1, 2, or 3 .

However, there is a loss associated with each action which depends on the state of nature. The losses corresponding to each action , = 1,2,3, and the state of nature , = 1,2,3, are represented by the following :

1

2

3

1

2

3

10

-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