SEEM 3580 Risk Analysis for Financial Engineering YANG Chen, Spring 2022
Course Project
(Due on April 10, 2022, 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:
• 1,000 shares of Amazon.com, Inc.;
• 1,500 shares of Booking Holdings Inc;
• 15,000 shares of Biogen 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
The daily data of these three factors can be downloaded from Yahoo Finance at
https://finance.yahoo.com, with the tickers AMZN, BKNG, and BIIB, respectively.
There are three periods of interest: (I) 2007-07-03 to 2009-06-30, (II) 2019-03-01 to
2021-02-26, and (III) 2021-03-01 to 2022-03-01.
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SEEM 3580 Risk Analysis for Financial Engineering YANG Chen, Spring 2022
(a) Define the trading days as those dates with values for all three factors. There
are a total of NI , NII and NIII trading days for Period I, II and III,
respectively. What are the values of NI , NII and NIII?
(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 . For Period I, for each factor, find its daily values, its daily ratio
changes, and its daily absolute changes, on every trading days. 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? Repeat these steps and answer the questions for Period II.
Part II: VaR Estimation
(c) What is the total portfolio value as of 2021-02-26 (at day end)?
(d) What are the 1-day 99% Market VaRs on 2021-02-26 estimated 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 the daily portfolio losses during Period III, and compare them to
the RiskMetrics VaR calculated in (d). On how many trading days does the
daily loss exceed the RiskMetrics VaR? Does this suggest that RiskMetrics
VaR underestimate the risk?
Part III: Stress Testing
With the above preparations and VaR estimations, you now perform a stress test
and study the impact of the stress scenarios on VaR estimations. You choose Period
I as the stress period. Since this period includes a financial crisis, intuitively the
stress tests are expected to produce larger values of VaR compared to Part II.
(f) For RiskMetrics, you redo the VaR estimation in Part II using “stressed”
model parameters σ∆yi and ρi,j, i, j = 1, 2, 3, i ̸= j, where ρi,j and σ∆yi are
estimated from data in Period I rather than Period II. What is the 1-day 99%
VaR on 2021-02-26 under the “stressed” parameters?
(g) For Historical Simulation, you redo the VaR estimation in Part II using
“stressed” scenarios. That is, you use the observed daily ratio changes of the
risk factors during Period I to generate risk factor scenarios one day after
2021-02-26. What is the resulted 1-day 99% VaR on 2021-02-26?
(h) Compare the two VaR values in (f) and (g) with those calculated in (d). Can
you explain your observation? Based on this, can you suggest and describe an
improved method? (Implementation is not required.)
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SEEM 3580 Risk Analysis for Financial Engineering YANG Chen, Spring 2022
Remarks:
• For Data Processing:
– To calculate the absolute/ratio changes on the first trading day in each
period, the values for y0i are to be taken as the factor values on the last
trading day immediately before this day. For example, you need to use
the Adjusted Close Prices on 2019-02-28 in order to calculate the
absolute/ratio changes in the first trading day of Period II.
– If the downloaded factor values have misaligned dates, use the
intersection of the dates available to all factors as trading days. That is,
if the value of at least one factor is missing on a given day, remove this
day from the sample.
– The estimation of sample standard deviation and correlation can be done
using the STDEV and CORREL function in Excel, respectively.
• For the RiskMetrics Approach:
– For 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.
– Part III is similar but uses Period I data for parameter estimation.
– Note that in Part III, only the parameter estimations are changed. The
VaR is still calculated for the portfolio on 2021-02-26.
• For the Historical Simulation Approach:
– The Historical Simulation has NII scenarios for Part II and NI scenarios
for Part III (not necessarily 500 scenarios).
– Note that in Part III, only the factor scenarios are changed. The VaR is
still calculated for the portfolio on 2021-02-26.
– To obtain the α-th percentile in Excel, you may use PERCENTILE.INC()
function.
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