Homework 1
April 25, 2023
1 Problems
Use the provided data on the SP500 prices from January 1st 2000 to December 31st 2013.
1 Using a qq-plot answer the following two questions:
• Are the log-return unconditionally normally distributed (i.e. are Rt normally dis-
tributed)?
• Are the log-return conditionally normally distributed (i.e. are zt = Rtσt normally dis-
tributed)?
– if σ2t follows the Risk Metrics model?
– if σ2t follows a GARCH model?
– if σ2t is measured by the Realized Variance?
2 Compute the V aR0.01t+1 and the ES
0.01
t+1 from January 1st 2011 to December 31st 2013 using:
(a) Historical Simulation with m=200
(b) a normal distribution for the return innovation and the GARCH model for the variance.
(c) a standardized t-student distribution for the return innovation (only for VaR) and the
GARCH model for the variance.
(d) Filtered Historical Simulation with m=500 and the GARCH model for the variance.
3 Plot the daily losses along with the VaR and ES measures just computed.
4 Which of the above methods for VaR do you think is better?
5 At the end of the day on December 31st 2010, compute V aR0.01t,t+1, V aR
0.01
t,t+2 , ...,V aR
0.01
t,t+5 and
the same for ES using:
• Monte Carlo simulation with normally distributed innovations and GARCH variance
(simulate 500 paths)
• Filtered Historical Simulation with GARCH variance (simulate 500 paths and use
the last 500 observations)
1
2 Upload on Blackboard
Please upload only one Homework per group. Make sure that your submission includes:
• Excel spreadsheet in which you show all the work you have done. You can put comments
in order to make more clear your procedure. Please, be clean and precise.
• A short report (pdf file) with figures, tables and a concise description of the results. Think
as if you had to present your results to your boss.
3 Deadline and Solution
• Deadline: before the next class.
• The solution and the Rubric will be posted on Blackboard after the next class:
Look for the file Var_ES.xlsx