MAST90051 Mathematics of RiskProblem Sheet 6/2024
Assignment 1
Posted on LMS on Tuesday 27 August 2024 (announced via LMS on the same day).
Submission deadline:23:59 AEST on Friday 13 September 2024.
Yoursolutions must be submitted via Canvas/Gradescope (see below). Late submissions
will receive no mark unless you have a valid reason for extension, in which case you should
contact the lecturerpriorto the submission deadline.
Print your name, student ID, the subject name & codein the top right corner on
thefirst pageof your solutions. Your solutions must be written on blank A4 size paper.
There is no need to typeset your solutions, but you may wish to do so. Material on different
sized paper will not be marked.
All problems should be attempted.Working and/or reasoning must be givento obtain
full credit. Wherever possible, give final answers with all numerical components thereof (e.g.,
coefficients) as decimal numbers (using “scientific notation” is OK). The form and neatness
of work can be considered in marking.
Do not include tables of data sets in your solutions!Reasonable data summaries
would be enough. Please do not exceed 15 pages, OK?
If you typeset your solutions, produce asingle PDF filewith your submission. If you wrote
them, scan your submission to asingle PDF filewith a mobile phone
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or a scanner. Scan
from directly above to avoid any excessive keystone effect. Ensure your scans are of good
quality. Check that all pages are in correct order, are clearly readable and cropped to the A4
borders of the original page. Poorly scanned submissions may be impossible to mark.
Upload the PDF filevia the Canvas Assignments menu. In the process, you will have to
submit your PDF to the Gradescope tool by first selecting your PDF file and then clicking
on Upload PDF. Gradescope will then ask you to identify on which of the uploaded pages
your answers to each question are located. You must do that!
1. LetLbe a random loss with an absolutely continuous DFFand densityf. All
we know about this density is thatf(x) = 4x
−4
lnxforx >1.5.
(a) Compute the DF tail
F(x), x≥1.5,forLand hence find the values of
VaR
α
(F) and ES
α
(F) at the following four confidence levels:α= 0.95; 0.99;
0.995; 0.999 (with four digits after the decimal point each). Calculate the ratios
ES
α
(L)/VaR
α
(L) for the aboveα-values. Show your work. Present your results
in the form of a nice table. Briefly comment on your findings.[5 marks]
(b) Model assumptions are never perfect. Suppose that thetrueloss DFF
L
is
within 0.007 of the assumed DFF:
|F
L
(x)−F(x)|≤0.007, x∈R.
Under that assumption, in what range can the true value of VaR
α
(F
L
) be for the
confidence levelsα= 0.95; 0.99; 0.995? Comment on your findings.[3 marks]
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You may wish to useCamScanner.
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Hints: It may happen that you will need to use numerical procedures to do some of the tasks
from this problem. Note that, when using real maths to solve real world problems, one can
rarely avoid using such procedures. You can use any suitable software package you wish.
For instance,Mathematicawould be fine, butMATLAB,Ror evenMS Excelcould also
be used. For a list of commercial software available for downloading to our students, go to
https://studentit.unimelb.edu.au/software
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. Or use any other suitable software of your
choice. And do not forget the golden rule for using software packages:when all else fails, read
the manual.
2. On 4 January 2022, one of our former MSc students (we will refer to her as
Ms X) purchasedλ
1
= 10 Lockheed Martin Corporation shares (LMT, traded on
New York Stock Excahnge, in USD),λ
2
= 2,000 Whitehaven Coal Ltd shares
(WHC, traded on the Australian Stock Exchange, in AUD) and alsoλ
3
= 200
Huyndai Corporation shares (traded on Korea Exchange, in KRW). Being a shy
risk-averse person, Ms X kept the composition of this portfolio unchanged until
28 December 2023.
Your lecturer kindly downloaded the stock prices
3
and the relevant currency
exchange rates daily data from the WWW
4
, “cleaned”
5
the data and put the
relevant to this question data in one Excel file,Data_for_asst_1_2024.xlsx.
Download that file from LMS (you will find a link to it in “module”Week 06),
figure out what columns in it contain what (NB: the prices of LMT and Huyndai
Corporation are given in USD and KRW, resp.) and use the data from that file
when doing this question. We labelled the days for which the prices are presented
in the Excel file byt= 0,1,2,...,T= 516.
(a) What risk factorsZ
t
= (Z
t,1
,...,Z
t,d
)
′
are appropriate for this portfolio?
Suggest and specify them. Map the risks by representing the dayt= 0,1,...,T
portfolio valueV
t
(in AUD) as a function ofZ
t
and hence give representations
for (i) the loss valuesL
t+1
(in terms of the daytrisk factors’ values and their
next day changesX
t+1
:=Z
t+1
−Z
t
), (ii) the loss operatorl
[t]
(x) and (iii) the
linearised loss operatorl
∆
[t]
(x) of the portfolio.[6 marks]
(b) Calculate the maximum difference max
1≤t≤T
|L
t
−L
∆
t
|between the true losses
and the linearised losses of the portfolio during the indicated time period. On
what day was it observed? What risk factors you think were mostly responsible
for that largest discrepancy? Explain.[2 marks]
(c) Compute the averagerelativediscrepancy between the true and linearised
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I suspect that one could also use machines in our computer labs, when there are no classes.
3
The “adjusted close” ones.
4
Stock prices came fromhttps://au.finance.yahoo.com, currency exchange rates came from the
Reserve Bank of Australia site.
5
Note that different countries may have different holidays, so there can be days where there will
be data available for some but not all assets etc. For the days for which there is data for one stock
exchange but not for some other, we just added “artificial prices” instead of the missing data (these
were just the prices from the latest day for which they were available). That’s OK for the purposes
of this exercise.
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losses and the average absoluterelativeloss/gain of the portfolio, i.e. the values
1
T
T−1
X
t=0
|L
t+1
−L
∆
t+1
|
V
t
and
1
T
T−1
X
t=0
|L
t+1
|
V
t
.
Compare the former with the latter and comment on your findings.[2 marks]
(d) Calculate the numerical values of the coefficients appearing in the expression
for the linearised loss operatorl
∆
[t
0
]
on dayt
0
:= 500, i.e., 05 Dec 2023.[1 mark]
(e) Compute the method-of-moments estimates of the mean vector and covariance
matrix for the vectorX
t
of the risk factor changes, basing on the whole data set
(from 04 Jan 2022 to 28 Dec 2023).[2 marks]
(f) Use the Variance–Covariance Method and your results from parts (d) and (e)
to calculate VaR
0.95
(L
t
0
+1
) and ES
0.95
(L
t
0
+1
). [This means you will be using
“future data” (relative to timet
0
), cf. (e). That’s OK, just do that.][3 marks]
(g) Use the historical simulation method to produce a sample of simulated losses
e
L
u
=l
[t
0
]
(X
u
), u=t
0
−n+ 1,t
0
−n+ 2,...,t
0
,forn= 400 and plot a histogram
for it. Use the sample to give a quantile estimator for VaR
0.95
(L
t
0
+1
) and an
estimator for ES
0.95
(L
t
0
+1
). Compare your estimates with the results of part (f)
and comment.[4 marks]
Hints:(g) Note that you should get
e
L
t
0
+1
=L
t
0
+1
(just in case: to check if what you are doing
is correct). To estimate ES
0.95
use an empirical quantile-based analog of that quantity (it was
given in lectures and, of course, is referred to in the text as well).
3. (a) Show that
Σ =
3−21
−23−1
1−11
is a covariance matrix.[1 mark]
(b) Compute (by hand!) the (lower triangular) Cholesky factor for the matrix Σ.
Give the final answer in matrix form, with matrix entries being decimal numbers
with four digits after the decimal point. Show your work.[2 marks]
(c) LetX= (X
1
,X
2
,X
3
)
′
be a normal random vector with mean (1,2,3)
′
and co-
variance matrix Σ.Compute and plot the densities of the subvectors [i] (X
1
,X
2
)
′
,
[ii] (X
1
,X
3
)
′
,and [iii] (X
2
,X
3
)
′
.In each of the three cases [i]–[iii], make a 3D sur-
face plot and a contour plot, both on the square [−3,5]
2
. Briefly comment on
your findings.[3 marks]
(d) Make a 3D surface plot and a contour plot for the mixture density
0.7f
(X
1
,X
2
)
′
(x
1
,x
2
) + 0.3f
(X
2
,X
3
)
′
(x
1
−1,x
2
−1),
both on the square [−3,5]
2
. You may wish to play with the mixture weights and
components’ parameters to see various shapes a two-component bivariate normal
density can have.[1 mark]
Hint:(c) This is a simple technical task that can easily be done using, say,Mathematica.
One may wish to use any other suitable software package, of course. (d) Ditto.
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