SEEM 3580 Risk Analysis for Financial Engineering YANG Chen, Spring 2024
Course Project
(Due on April 2, 2024, 23:59)
This is an individual project to be finished on your own. Please give sufficient expla-
nations to support your answers. Complete the project by answering all the questions.
Submit (1) a short report including your answers and explanations to all the ques-
tions, and (2) the procedures supporting your answers to those questions that involve
calculations (e.g. Excel files or codes in other languages, which show or reproduce
results to support your answers). Read through all the questions and remarks before
starting the project.
Market Value at Risk: Estimation and Stress Testing
Foreword: In the lecture, we have discussed various methods of VaR (Value at Risk)
estimation using recent market data. In view of the past financial crisis, it is
important for the financial institutions to estimate the level of VaR under “stressed
scenarios” (also called “Stressed VaR”), during which exceptional events happen
(e.g. the 2007-09 financial crisis were to happen again). This type of estimations
can be done via “stress tests”, which are required by the Basel Accord for banking
regulation. There are many stress testing approaches in practice, and in this project,
you will be guided through some of the simple examples.
You are a market risk manager in Financial Institution XYZ. The trading book
portfolio consists of the following positions in three stocks:
• 3,000 shares of General Electric Co.;
• 5,500 shares of Novavax Inc;
• 6,000 shares of Goldman Sachs Group Inc.
The portfolio is denominated in U.S. Dollar. In the following parts, it is assumed
that no adjustment is made to the portfolio (that is, the numbers of shares remain
the same as above). You estimate the market VaR via two approaches: the
RiskMetrics Approach and the Historical Simulation Approach. For both
approaches, the risk factors are selected as the Adjusted Close Price of these three
stocks, denoted as y1, y2, y3, respectively.
Part I: Data Collection and Processing
(a) Download the daily data of these three factors from Yahoo Finance at
https://finance.yahoo.com, with the tickers GE, NVAX, and GS, respectively.
There are three periods of interest: (I) 2007-07-03 to 2009-06-30, (II)
2020-03-13 to 2022-03-08, and (III) 2020-03-16 to 2022-03-09. Define the
trading days as those dates with values for all three factors. Note that there
should be a total of NI = 503, NII = 501 and NIII = 501 trading days for
Period I, II and III, respectively.
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SEEM 3580 Risk Analysis for Financial Engineering YANG Chen, Spring 2024
(b) Denoting the value of factor i on trading day n as yni . The daily absolute
change refers to ∆yni = y
n
i − yn−1i , while the daily ratio change refers to
yni /y
n−1
i .
(i) For Period II, for each risk factor, find its daily values, daily ratio
changes, and daily absolute changes on every trading day. What is the
sample standard deviation of the daily absolute changes for each factor,
and what is the sample correlation of the daily absolute changes for each
pair of factors?
(ii) Repeat these steps and answer the questions above in (i) for Period III.
Part II: VaR Estimation
(c) What are the total values of the trading book portfolio as of 2022-03-08 and
2022-03-09, respectively?
(d) What are the 1-day 99% Market VaRs estimated on 2022-03-08 from the data
in Period II via the RiskMetrics and the Historical Simulation approach,
respectively? Which approach produces a larger VaR? Can you explain your
observation?
(e) Calculate 1-day 99% VaRs estimated on 2022-03-09 from data in Period III
via both approaches, and compare them with the results in (d). Is the VaR
value from RiskMetrics estimated on 2023-03-09 significantly different from
that in (d)? Is the VaR value from Historical Simulation estimated on
2023-03-09 significantly different from that in (d)? Provide an explanation for
the observed difference. (Hint: examine the differences in the parameter
estimation for RiskMetrics, or in the set of factor (ratio) change scenarios for
Historical Simulation; it may also be helpful to revisit the weaknesses of these
approaches in Lecture Notes 4.)
Part III: Stress Testing
With the above preparations and VaR estimations, you now perform a stress test on
the VaR from Historical Simulation approach and study the impact of the stress
scenarios on VaR estimations. You choose Period I as the stress period, which
includes a financial crisis.
(f) With Historical Simulation approach, you re-estimate the VaR on 2022-03-08
in Part II (d) using “stressed” scenarios. That is, you use the observed daily
ratio changes of the risk factors during Period I to generate risk factor
scenarios. What is the resulted 1-day 99% Stressed VaR estimated on
2022-03-08 with Historical Simulation?
(g) Compare the VaR from (f) with that in (d). Which value is higher? Please
provide a brief explanation for your observation.
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SEEM 3580 Risk Analysis for Financial Engineering YANG Chen, Spring 2024
Remarks:
• Please provide a summary of your solution, including the numbers and
explanations, in your short report. Additionally, highlight the part in your
code corresponding to the results from (b) to (g) (e.g. highlight the
corresponding cells with yellow color in Excel, or pointing out the
corresponding variables using comments in your code).
• For Data Processing:
– There are NI − 1, NII − 1, NIII − 1 absolute/ratio changes for Period I,
II, and III. The first absolute/ratio change in each period is generated
from the risk factors on the first two days. For example, you need to use
the Adjusted Close Prices on 2020-03-13 and 2020-03-16 in order to
calculate the first absolute/ratio change in Period II.
– The estimation of sample standard deviation and correlation can be done
using the STDEV and CORREL function in Excel, respectively.
• For the RiskMetrics Approach (in Part II):
– For i, j = 1, 2, 3, the standard deviation of factor i, σ∆yi , and the
correlation between factor i and j (i ̸= j), ρi,j, are the sample standard
deviation and sample correlation you have estimated in Part I(b), using
the daily absolute changes in Period II and Period III.
• For the Historical Simulation Approach:
– The Historical Simulation has NII − 1 and NIII − 1 scenarios for Part II
and NI − 1 scenarios for Part III (not necessarily 500 scenarios).
– Note that in Part III, only the factor scenarios are changed. The VaR is
still estimated for the portfolio on 2022-03-08.
– To obtain the α-th percentile in Excel, you may use PERCENTILE.INC()
function.