MATH 5340MThis question paper
consists of 4 printed pages,
each of which is identified
by the Code Number MATH 5340M
©UNIVERSITY OF LEEDS
(Semester 2 2023/2024)
Assessed coursework
MATH 5340M Risk Management
This assignment counts for 30% of the total module mark
Submission deadline: 1st May 2024 at 2 pm.
RED: you must not use any content generated by Artificial Intelligence (GenAI) in any
part of this work.
You are responsible for calculating the minimal capital requirement for an investment fund.
Your firm has its own internal model and uses the following formula for calculating capital
requirements:
C = 2.5VaR0.99,
where VaR is the 1-day Value at Risk. Your fund is small and your portfolio consists of
4 stocks only: λ1 shares of Stock 1, λ2 shares of Stock 2, λ3 shares of Stock 3 and λ4
shares of Stock 4. You have access to the closing prices for a past period for the stocks.
You will find them in the file Data Student n.csv within a zip archive on MINERVA [in
Learning Resources→ Assignment→ Data.zip], where n stands for your student number.
Each student has (real) data corresponding to different stocks from different financial
sectors. The file contains prices for the stock in USD at the end of the trading day. The
number of shares of each stock in the portfolio, namely, the quantities λ1, λ2, λ3, λ4, are
provided separately in the folder. The risk factors changes for the stocks are the log-
returns
Xit+1 = log
(
Sit+1
Sit
)
, for i = 1, . . . , 4,
where Sit denotes the price of stock i at time t. Throughout this document, we will denote
by L a 1-day future loss.
You need to perform the analysis along the guidelines in Section 1 and then write a
report with the structure and format specified in Section 2.
Page 1 of 4 Please turn over
MATH 5340M
1 Analysis
Important: You are not allowed to use packages for computation of VaR, ES or any other
financial quantity unless stated otherwise.
(i) Apply the variance-covariance method to estimate VaR0.99(L) and ES0.99(L) for the
portfolio loss, and then use these to estimate C.
(ii) Apply the historical estimation method to estimate VaR0.99(L) and ES0.99(L) for the
portfolio loss, and then use these to estimate C. Compare with the results obtained via
the variance-covariance method and draw conclusions.
(iii) The purpose of this part is to understand and implement a method which has not
been seen in the lectures.
Read Chapter 14 of G. A. Holton Value-at-Risk. Theory and practice, Second Edition.
The second edition of the book is available only in the electronic form at
https://www.value-at-risk.net/.
You are allowed to use other references if you prefer but you need to include them in the
reference list.
Perform back-testing of the variance-covariance and the historical estimation meth-
ods for the computation of VaR0.99(L) using coverage tests of Section 14.3 of the book:
standard coverage test and Kupiec’s PF test with the significance level = 0.05.
1. Calculate both tests using either your own code or a package.
2. Draw conclusions about the quality of your estimators of the value at risk.
3. Explain your approach.
4. Implement the two estimators. Do not use packages/functions apart from standard
statistics functions for Chi squared and binomial distributions!
5. Find a package that implements Kupiec’s PF test, apply it to your data and verify
your results against the output of that package.
Hint: pay careful attention to how exceedances are calculated. This is explained in Sec-
tion 14.2 of the above book and succintly given in the following quotation: “We define an
exceedance as an instance of a portfolio’s single-period loss exceeding its value-at-risk
for that single period.”
(iv) Would it be reasonable to assume that the distribution of each risk factor Xi is nor-
mal? Support your argument for each risk factor Xi using a Q-Q plot and an appropriate
statistical test for normality. Interpret the outcomes of the backtesting in view of these
findings.
(v) The investment fund is aware of the deficiencies of their methodology to calculate
VaR0.99(L) using the variance-covariance method. Therefore, they return C to the regu-
lator instead of their VaR0.99 (i.e., they claim that C is their estimate of the true VaR0.99).
Report backtesting results for this reported value C (treating it as VaR0.99) using Kupiec’s
PF test with a lower (more realistic) significance level = 0.02. Does this measure pass
the test?
Page 2 of 4 Please turn over
MATH 5340M
2 Report
Your report must be written in 11pt font with 3 cm margins on each side (this document
uses such margins and font size for the main text). It must be written using professional
(formal) language. The notation and vocabulary throughout the whole report should be
consistent. Inconsistent notation and/ or terminology will be penalised. Careful proof-
reading before the submission is necessary!
Your report must follow the structure below:
Title page. Contains your student number and a title (of your choice).
Summary. This must be up to 1 page long and report the financial and statistical
aspect of the analysis with all relevant information provided in a readable and compact
form. For example, do not provide Q-Q plots and do not discuss the code. You can refer
to Section Analysis when needed. This section will account for a significant number of
marks.
Explanation: The selection of the material and its presentation is not easy and requires
a lot of planning. It should convey the message that you have done the analysis thor-
oughly and support your conclusions. The ability to do this type of writing is crucial in all
professional environments.
Analysis. Provide further details of the analysis that were not included in the Sum-
mary section. You should document your workings and explain your approach. You are
encouraged to use mathematical notation and make your arguments precise. Be concise
and do not include unnecessary information like unrelated definitions, discussions, tables
or data printouts.
Divide into steps (i)-(v) as in Section 1 of this document.
Code. Provide all the code that you used to obtain your results, including commands
for producing graphs. To answer the questions you are to use the programming language
R or Python. The code should be cleared of unnecessary content and include comments
which will allow me to understand it. You must provide an appropriate REFERENCE for
any piece of code taken from other sources including internet searches and you are not
allowed to use generative AI.
Detailed instructions
• Please ensure that you leave sufficient time to complete the online submission process,
as upload times can vary. Accessing the submission link before the deadline does NOT
constitute completion of submission. You MUST click the ‘CONFIRM’ button before the
submission deadline to be classed as submitted on time.
• The name of the report file that you upload must be your student number.
• The file you upload must be in pdf format (Portable Document Format). Make sure
that nothing has been lost in conversion (particularly from Microsoft Word) before sub-
mission.
• The Question Paper is the same for all students but each student will have to analyse
different market data (make sure you download the correct data file from Minerva).
Page 3 of 4 Please turn over
MATH 5340M
• The report must not exceed 12 pages (including all pictures, a list of references, but
excluding the code).
• You are encouraged to write your report using Latex. To this purpose, a Latex template
is provided on Minerva. Reports written in Microsoft Office Word will also be accepted
and will not incur any penalty.
• If you use Microsoft Word for your report, please to use the following format:
– Margins: top 3 cm, bottom 3 cm, left 3 cm, right 3 cm
– Font type: Arial
– Font size: 11
END
Page 4 of 4 Please turn over