Managing Financial Risk
Problem Set 1
1 Introduction
. The goal is to measure SPY’s risk for the next day with volatility. The strategy is to calculate the conditional variance, which is the expectation of the return squared conditional on infor- mation that is available before the day. We are going to model the conditional variance with two moving average specifications, two RiskMetrics specifications, and GARCH, which needs to be estimated. That would give us five predictions. In order to evaluate which method is the most reliable, we will run a horse race among the five. That is comparing the methods in terms of their prediction accuracy track records in the history. The comparison should be conducted out-of-sample to be more fair (to avoid incorrectly favoring GARCH which is fitted in sample). Finally, the best method’s best guess is used as the final prediction. Feel
free to check whether your prediction is accurate by the end of the day. Good luck!
Students may use generative AI tools (such as ChatGpt) as an assistant when working on the problem set. The report must be written in the student’s own words. And the student should report the use of such tools in the write-up.
It is prohibited to view the course documents in previous years in finishing the homework,in cluding the solution and past student submissions.
. Please upload your responses in two files to the Canvas under folder 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 cell highlighting, number formatting etc., so that the grader’s burden is minimized.
–A summary report (pdf file) of responses to each problem, including figures and tables where required. Again, conciseness and proper formatting is required. Imagine you are presenting to clients at a real job.
. The solution with detailed grading rubric will be released after due. The grades are expected to be released in one week on Canvas. You should read the solution and learn from it,
especially for the parts that are not clear when finishing the problem set.
2 Problems
0. Please list the names of the peer students that you have discussed this problem set with, if any. 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. Download the same dataset of SPY return from 1/3/2000 that we have been working with again to get the most recent data. You want to wait for the return data for the last day before Due Date (why?). But don’t wait to the last day to work on this problem set... Start working early, and leave the last few day’s return entry empty (or some arbitrary numbers as placeholders). After the market closes on Due Date −1, update the live return data in your workbook, which should automatically update the prediction result if you have designed it well.
(Remember to save the file in .xlsx format, as .csv files will not contain the formulas.)
Calculated daily log return and daily log return squared from 1/4/2000 to Due Date −1. That is our full sample to work with.
2. On each day in the full sample, suppose you are in the morning and don’t know the day’s return yet, calculate the conditional variance by predicting the day’s return2 using
- 1-month moving average (use past 21 observations)
- 1-year moving average (use past 252 observations)
Hints:
- Some methods do not apply to some earlier days in the full sample. Please just leave them blank. It should not matter for later procedures.
- I prefer to have the prediction target and the corresponding predicted value on the same row. Always remember which variable is in what information set. For example, the prediction target and the predicted value are known on what dates, respectively?
3. Do the same task with the RiskMetrics model. Repeat the task twice with λ = 0.94 (the recommended RiskMetrics parameter) and λ = 0.96 (a supposedly less preferred parameter) respectively.
Hints: for the first day of the sample, since there is no history yet, put an arbitrary sensible number as the prediction σt(2) as a start. Write the general recursive formula from the second day of the sample onward.
4. Estimate the GARCH model in the full sample.
Hints:
- Make a sensible guess of the GARCH parameters (ω,α,β) as the initial guess.
- For the first day of the sample, since there is no history yet, put an arbitrary sensible number as the prediction σt(2) as a start (some suggests using the sample variance as an estimation of the unconditional variance). Write the general recursive formula from the second row of the sample onward.
- Calculate the log likelihood with respect to the initial guess parameters as placeholders.
- Use solver to maximize the log likelihood by changing over the choices of parameters.
- The excel solver is rather sensitive to the initial guess parameters. It only finds the local optimum, which depends on initial guess. To overcome, try a few different initial guesses, and record the local optimum with the greatest log likelihood target as the global optimum.
- To automatize this process, choose “multistart” at Solver → options → GRG Non-linear Multistart. Check both “Use Multistart” and “Require Bounds on Variables”. For this you need to set bounds for all three parameters. I simply restrict the three to be all within [0, 1]. Then solver takes a bit longer (several minutes on my machine) to automatically try a series of random initial guesses within the bounds.
5. Calculate the conditional variances with the five models. Also. report the five risk measures by converting them in terms of volatility. Check if the scale and unit of the volatility make sense. Check if assessed level of risk makes sense given the current market conditions.
How is the current market condition, and how is the current level of market risk compared to the long run average level? How is the current condition compared to the past year and past month? Given these assessments, between the two MA methods, which one tends to bias the measure of risk upward or downward and why? How about the two RiskMetrics measures? (These are more open-ended questions. The time-series visualization in question 8 might help you answer these questions.)
6. Now we need to compare the five methods. Full sample GARCH has an unfair advantage when compared with the other four directly (you will be asked why). Therefore, we are going to compare the five methods on an even ground by competing OOS. Let the first decade 1/4/2000 - 12/31/2009 be the training sample (aka in-sample), let the next portion, 1/4/2010 - the last day before Due Date be the validation sample (aka out-of-sample). Pretend now we are on 1/4/2010 morning, estimate a GARCH model in the training sample. Fix the parameters estimated IS. Use that set of parameters to predict returns2 for 1/4/2010. Then pretend it’s 1/5/2010 morning, use the same parameters estimated from the same IS (2000’s data), but use the newest 1/4/2010 return to predict for 1/5/2010. Repeat this day by day for all 2010’s onward. These are our OOS predictions.
Now we have six series of best guesses in the validation sample: 4 non-estimation based methods, GARCH full sample, and GARCH OOS. Report the prediction accuracy of the six methods respectively. Here, measure prediction accuracy in terms of mean squared error, i.e. the average of prediction errors squared in the validation sample. Which sample should be used for the comparison?
7. Which method is the best? Do the six numbers make sense to you (Are they on the same scale, does the scale make sense? Does the ordering make sense? Is RM indeed better than MA? Is the RM with the recommended λ indeed better than the other RM.
Is GARCH full sample better than GARCH OOS? Is that expected?)
When arguing for the superiority of GARCH, can you cite the small MSE in the validation sample of the GARCH estimated with the full sample? Briefly explain why not, and why the GARCH estimated IS but validated OOS is a more fair comparison with MA and
RM.
8. Draw a time-series line chart of the target series (return2 ) and the GARCH Full Sample predicted series in the validation sample to illustrate the validity of the prediction. Can you spot the most volatile period? How was the volatility model performing in that period? This is also a good visual inspection for the other methods in the previous steps. Try play with the Excel options (e.g., axis, lables, gridlines, etc.) to make the visualization clear.
9. Finally, give the one number best prediction of return squared of the Due Date, and also the corresponding volatility measure quoted as a percent. Between GARCH estimated in the full sample, and GARCH estimated IS but validated OOS, which one should we use?