Managing Financial Risk
Problem Set 2
April 28, 2024
1 Instructions
*** It is a violation of academic integrity policies to look at the solutions from previous years
when finishing this assignment.
. The end goal is to measure next week’s tail risk by calculating the VaR and ES five days into the future. To do that, we need to construct the return distribution into the future. The methods of constructing the distribution include
(a) Historical Simulation (HS)
(b) GARCH + Monte Carlo Simulation (MCS) based on conditional normality assumptions
(c) GARCH + Filtered Historical Simulation (FHS)
We are first going to compare the three methods by looking at their in-sample daily VaR performances. Based on that, we are going the conduct (b) and (c) for the 5-day cumulative VaR and ES into the future.
Besides practicing the above mentioned risk management concepts, this assignment is also designed to prepare you with several Excel “best practices” and “tricks” as we will encounter below.
. To alleviate your burden with managing the live data, let us pretend today is Friday 4/19/2024. We are in the evening and the market has closed. All data up to today is available. Let us pretend we don’t know any information beyond today. So that you don’t need to worry about updating the data, which we already practiced in PSet 1. The end goal is to measure the risk in the next week (5 business days) after today.
. Also, we are not going to worry about OOS validation this time (the concept should be familiar after PSet 1). As we will see, the conditional normality assumption already shows problems in sample. So “sample” in this assignment is the full sample from 1/4/2000 to today. Validation is done IS, but leaving out the first 200 days (since we need to wait 200 days to do the first HS with m = 200.) So the validation sample is from 10/18/2000 to today.
. To save you sometime designing your own Excel sheets, you may start from the Excel template I prepared for you “Assignment2 Template.xlsx”. This file is based on the PSet 1 solution, where I have already estimated the GARCH model in the full sample. Please take those estimates as given. You may follow the template and fill in the blanks step by step, or you may design your own which could well be better than mine.
. Again, this is an individual (not group) assignment. Please finish the assignment indepen- dently. Students are allowed to discuss with each other about the knowledge, skills, and the general strategy required to finish the problem set. Here, discussion is limited to talking, phone calls, etc. It is not allowed to discuss the results. Especially, it is not allowed to trans- mit in any form the electronic documents of the assignment before due. You are required to report all the peers with whom they have discussed the assignment to allow for cross-checking.
It is encouraged to search the Internet for technicalities.
. This document looks very long. Don’t get scared. The document is long because I wrote lots of hints to help you, not because there’s lots of work. Please read this document thoroughly.
. Due date: This problem set is due Monday May 6th (week 7) 8:30 am.
. Please upload your responses in two files to Blackboard under “HW2” submission folder in
“Assignments” before the deadline. Please include your name in the file names.
– An Excel spreadsheet with all the intermediate steps and quantitative results. Please, be concise and precise. Use proper color coding, number formatting etc., so that the grader’s burden is minimized. Think you are presenting to your boss.
– A summary report (pdf file) of responses to each problem, including figures and tables where required. You should think hard about and address every question mark. I always believe these conceptual questions are more interesting and useful than the executions in Excel. Yet, please be concise and to the point in answering each question.
. The solution and grades will be released after due.
2 Problems
0. List the names of the fellow students that you have discussed with when finishing this problem set. If none, please write “none” . Please also report any special issues here, for example requests and excuses for extension and late submission. Such issues would not be considered after homework due cutoff time. You may also report the use of generative AI tools here.
1. Draw QQ-plots of
(a) The historical distribution of full sample returns v.s. a normal distribution with the same mean and standard deviation as the the sample returns. (This was done in the Excel Hands-on in Week 1 class.)
Looking at the QQ-plot, can we conclude that the (unconditional? conditional?) distri- bution of return (is? is not?) normal. If not, in what ways it is different from normal (left?right? tail too long?short?)
(b) The historical distribution of sample zt’s v.s. a standard normal distribution. Use zt filtered by decomposition Rt = σtzt, where σt is the volatility of Rt given by GARCH. Looking at the QQ-plot, we conclude that the (unconditional? conditional?) distribution of return (is? is not?) normal. If not, in what ways it is different from normal?
(left?right? tail too long?short?)
Excel Hints:
- Maintain an Excel sheet that automatically turns a column of numbers into a QQ-plot against some Normal distribution. So that you just need to make different copies of the sheet, and only change the column of numbers, and the Normal parameters. This idea of maintaining a general purpose Excel/code/function is very useful to save time, while keeping things clear and manageable. You should always follow this idea whenever possible for any work. That is, when constructing a new thing, make it general and flexible, so that it can be easily adopted for similar situations.
- You should put the empirical distribution on the vertical axis, and normal on the hori- zontal axis. To switch the axes, right click the chart → Select Data → Legend Entries → Edit → Switch the contents in “Series X Values” and “Series Y Values” .
- The QQ-plot benchmark is the 45-degree line. So we’d better make the X and Y axis on the same scale, i.e. same interval lengths.
At Format Chart → Axis Options → That right-most button with three bars → Units: Major: set the same number for both the vertical and horizontal axes. Then drag on the chart so that the grid lines look square (of equal widths and heights). (I don’t know how to make this exactly square, feel free to Google and let me know.)
2. Compute the VaRt(0) .05 for every t from 10/18/2000 to today using the following methods:
(a) Historical Simulation with m = 200
(b) GARCH model for the variance + Conditional Normal assumption (z follows standard normal).
(c) GARCH model for the variance + Filtered Historical Simulation with m = 200
(Can probably use a larger m for FHS. But let’s make it the same m as in HS, to make the excel a bit easier. FHS m = 200 should be already performing well.)
Hints:
- Write p = 0.05 in a cell and refer to that cell. Do not type 0.05 as a number within any formula, as we will change p later. Again, it is always a good practice to make things easily adaptable.
- 10/18/2000 is the 201th day of the sample. So, on 10/17/2000, one could calculate the first HS and FHS VaR (VaR1(0)0/(.05)18/2000). Although we don’t need to wait for 200 days to calculate the first GARCH-based VaR, let us keep it in the same sample for comparison.
- Use “Freeze Panes” under “View” menu, so that you don’t loose the column name row when looking at the 200th day below.
3. Unconditioned performances:
For the three methods respectively, construct the VaR breach indicator series. (that is, on each day t, 1 if breach happened, 0 if not.)
Calculate and report the VaR breach frequencies in the validation sample (from 10/18/2000 to today).
What is the theoretical value of the frequency? Are the three methods close to the theoretical value? If a sample frequency is higher than the theoretical value, then do you think the VaR is too high or too low in general?
4. Conditional Performances:
Bar chart (called “column” in Excel) the three breach indicator series against calendar days.
Hints:
- Since the values are either zero or one, the end result should look similar to a bar code.
- Again, the best practice: First make one chart, adjust its label, axis etc. to your liking. Then make two more copies and “Select Data” to the two other series. It would save time going back and forth the three charts, and they are standardized in terms of formats.
Eyeballing the charts, are the bars roughly “randomly” evenly spaced? Is there VaR breach clustering (meaning periods of very frequent breaches in between periods or very few breaches)?
Which method has more severe clustering? Should there be such clustering if VaR were calculated perfectly in theory? Why or why not?
Can you recognize the periods with the most clustered breaches? Is the VaR too low or too high in that period? Is this a good time to over- or under-estimate risk?
5. Change p from 0.05 to 0.01. (Supposedly, you only need to change one cell, and everything
including the charts should update automatically.)
Report breach frequencies.
Do you see that the sample breach frequency for GARCH + Normal is off the most at p = 0.01? Does that reflect an over- or under-estimation of VaR?
This implies what problem of the distributional assumption of this method? In what ways are the empirical distributions better than the normal distribution in avoiding this problem
(given the evidences from the QQ-plots)
How come the problem was not so severe when p = 0.05? (think which p is about more extreme scenarios.) Let’s say keep p = 0.05, do you think the problem could emerge again or still be hidden when we are talking about one-day ES or multi-day cumulative VaR?
6. At the end of the today compute and report VaRt(0),1 , VaRt(0),2 , ...,VaRt(0),5 and the same for 5 ES’s using:
(b) GARCH model for the variance + MCS under conditional normality (z is standard normal)
Simulate 1000 paths.
(c) GARCH model for the variance + Filtered Historical Simulation
Simulate 1000 paths. Draw historical simulation of z from the full sample.
(We are changing to full sample of historical z’s here, different from m = 200 as in Q2,
because m = 200 does not make anything easier here.)
Change to p = 0.01, and report the 10 numbers again.
Hints:
- The two methods should be all the same, except for the simulation of z’s. So make one perfectly first, then make a copy of the whole sheet and only change the z’s simulation method. Everything else should update automatically.
- Every simulated path should be the same, except for the random number generated. So make one path perfectly, then drag every column down 1000 rows. (The shortcut “Ctrl + D” can be helpful.)
- For every path, the starting point should be the same. We know today’s R and σ, which give tomorrow’s σ, according to the GARCH recursive formula just like the last PSet. So that σt+1 (already known at t) is the common starting point. Next step, simulate tomorrow’s z, that gives tomorrow’s R. Then use GARCH recursive formula to the day after tomorrow, so on so forth.
Every time when the random numbers change, do the results change? Do they change a lot? Where does the problem come from and how can you alleviate the problem?
Is VaR or ES greater for every horizon? Why?
Are VaR and ES increasing or decreasing for longer horizon? Why?
Does MCS give larger or smaller VaR than FHS? How about ES? How about when change to p = 0.01? Why?
In the end, report two numbers VaRt(0),5 and ES5 as your measure of next week’s risk.
Which method (MCS or FHS) do you prefer to use?