BT IS A GREAT BUY!
27 Aug 2008
Feb
18
FRM 2008 Early Bird Episode #6 (volatility)
by David Harper, CFA, FRM, CIPM
Contents
- This week's episode: How to estimate volatility
- Video tutorial (episode #6)
- Practice questions (5) and a note about EditGrid (working spreadsheet answers!)
- Previous Early Bird emails
Learning Objectives for Episode #6: Volatility
About a month ago, a journalist interviewed me as background for his article on "the death of VaR." He asked if VaR was broken because asset returns were more volatile than a normal distribution. He makes a understandable mistake by linking VaR reflexively to normality. However, VaR isn't necessarily "normal" and neither is volatility.
This week is an introduction to volatility estimates. My 30-minute tutorial reviews three volatility estimate approaches (that will surely repeat in the 2008 FRM): moving average (MA), exponentially weighted moving average (EWMA), and GARCH(1,1). These Early Birds are quantitative primers, so at the end of the video tutorial, we work three practice problems. They are roughly the level of difficulty you can expect on the FRM exam.
VaR is just a quantile
Value at risk (VaR) is just a quantile: what's the expected loss (over a specified horizon) at the 9Xth percentile? This VaR question, albeit technically flawed, is indisputably useful. The thornier issues are, how do actually we figure it and how do we complement its weaknesses? There are three broad approaches:
- Historical simulation
- Parametric
- Monte Carlo simulation
Further, any two of the above can be blended into a hybrid approach; e.g., historical can be informed with parameters, Monte Carlo can be parametric or not.
The parametric approach is not data-intensive because the range of outcomes are neatly summarized by a probability distribution. The probability distribution may be normal or may be non-normal. To summarize, value at risk (VaR) may or may not be parametric; if VaR is parametric, the parameters may or may not belong to a normal (gaussian) distribution. If we use a normal distribution, we only need the mean and volatility, other distributions are likely to have other/additional parameters.
So volatility (standard deviation) is a common parameter that, in addition to the mean, fully describes a normal distribution. The normal may be used in a parametric approach to VaR, but within parametric approaches there are non-normal distributions. And, there are non-parametric VaR approaches.
Volatility is an input (a parameter) in the parametric approach
As a risk metric, volatility (standard deviation) is very popular and flawed. Consider three hypothetical series of five-day period returns:
- -20%, -20%, -20%, -20%, -20% (i.e., price dropping 20% per period)
- +10%, +10%, +10%, +10%, +10% (i.e., price increasing 10% per period)
- +10%, +21%, +33%, +46%, +59% (i.e., price increasing exponentially)
The first two series both have zero volatility because they have no dispersion, although one grows at +10% and the other drops at -20%. The third series, which grows exponentially, has a volatility of about 17%. This is why many prefer downside risk metrics like semivariance.
Historical volatility forecast
This week's episode is informed by Chapter 19 of John Hull's Options, Futures, and Other Derivatives. This chapter will probably repeat as an assigned reading in the 2008 FRM. We review the following ideas:
- The basic definition of historical volatility: a sample standard deviation taken over some historical window (e.g., 30 days, 250 days)
- For short periods, we can make two simplifying assumptions. This reduces variance (the square of volatility) to the really simply moving average (MA). The moving average (MA) is the average of the series' squared returns.
- The moving average is equal-weighted: every return (observation) in the historical series is assigned the same weight. This is unrealistic. We'd prefer to assign greater weight to more recent returns.
- The exponentially weighted moving average (EWMA) achieves this goal. EWMA assigns weights (to squared returns) that decline exponentially as the series goes back in time.
- GARCH(1,1) improves on EWMA by adding a term for mean reversion. Where EWMA is perfectly "persistent" in adjusting to new information, GARCH(1,1) exhibits "decay" toward a long-run average variance.
- All three above (MA, EWMA, and GARCH) are particular cases of the ARCH(m) model.
Video Tutorial
The 30-minute video tutorial is located here (with a table of contents).
If you are a paid member, you can also access this in the member section (where you will also find the downloadable slides, if you would like to view those. As well as an ipod format file.)
Practice Questions
I've got five practice questions for you this week! If you don't think you need the practice, can you meet my grab bag challenge (see question 5 below)?
Question #1 (coffee index)
Setup:
For the week ending February 8th, 2008, the Coffee sub-index of the Dow Jones AIG Commodity Index closed at the following prices: 65.967 (Mon), 65.925 (Tue), 66.183 (Wed), 66.256 (Thr), 68.052 (Fr i).
If we round the daily periodic returns to the nearest 0.1%, the daily returns for the week were:
| Day | Period Return |
| Mon | 2.6% |
| Tue | (0.1%) |
| Wed | 0.4% |
| Thr | 0.1% |
| Fri | 2.7% |
Questions:
(i) What is the five-day moving average (MA) estimate of volatility (i.e., this is "simple volatility" where we make a simplifying assumption that the average return is zero due to the short interval)?
(ii) The actual average daily return, for the five days, is actually 1.14% (i.e., the mean return is non-zero). How will the population volatility (population standard deviation) compare to the moving average? Will it be higher, lower, or similar?
(iii) How will the sample volatility (sample standard deviation) compare to the population standard deviation?
(iv) Assume the daily volatility is the volatility we computed in part (i) under the moving average approach. If there are 250 trading days in a year, what is the annualized volatility?
Question #2 (EWMA)
We are going to use the exponentially weighted moving average (EWMA) model. Assume lambda (i.e., the decay factor) is 90%, that yesterday’s (day n-1) volatility was 2.0% and that yesterday’s daily return was +3.0% (periodic return for day n-1).
(i) What is today’s volatility estimate (day n) under the exponentially weighted moving average (EWMA)?
(ii) If the decay factor is 90% (lambda = 0.9), what weight is effectively assigned to the most recent return (return-squared)?
(iii) If the decay factor is 90% (lambda = 0.9), what weight is effectively assigned to day n-5?
(iv) What is effectively the persistence parameter for the EWMA model?
Question #3 (GARCH)
Yesterday, the Euro/Yen exchange rate closed at 155 (155 Yen per 1 Euro); today the exchange rate closed at 158 Yen/Euro. Further, yesterday's volatility estimate (the n-1 estimate) was 1.0%. Our GARCH(1,1) parameters are the following:
omega = 0.000012
alpha = 0.1
beta = 0.78
(i) What is today's updated volatility estimate?
(ii) What is the implied long-run volatility?
Question #4 (GARCH)
Assume current annualized volatility is 30%. Under a daily model, the GARCH(1,1) parameter estimates are:
omega = 0.00018
alpha = 0.10
beta = 0.70
(i) What is the implied long-run average daily volatility?
(ii) What is the persistence of the specification?
(iii) What is the daily volatility forecast in four days (n+4)?
Question #5 (Volatility Grab Bag)
(i) In regard to volatility estimate approaches, what is the single key difference between moving average (MA) and exponentially weighted moving average (EWMA)?
(ii) What is the single key difference between exponentially weighted moving average (EWMA) and generalized auto regressive conditional heteroscedasticity (GARCH)?
(iii) What is the significance of lambda in EWMA?
(iv) Empirically, volatility tends to cluster (i.e., is "sticky" to itself). How would this be reflected in the GARCH(1,1) parameters? How would its opposite be reflected?
(v) GARCH(1,1) is generally considered superior to EWMA, but when would EWMA be better?
Answers to practice questions
Please: for some answers, I provide a link to an fully functioning EditGrid spreadsheet. These are working solutions, I hope they help you follow the math. You can browse them without software and without charge! You can also easily open them into MS Excel (File > Export As)!
Previous Early Bird Episodes
In case you missed them, the previous early bird emails (with the practice questions) are found here:
- Episode #1 (notation, compounding, matrix)
- Episode #2 (distributions: binomial, Poisson, normal)
- Episode #3 (linear regression)
- Episode #4 (functions and factor models)
- Episode #5 (derivatives)
- And the answers to the practice questions are in this forum area
That's all for this week. Good luck and see you next week!
David Harper, CFA, FRM, CIPM
Founder
www.bionicturtle.com
Comments
Hey David,
Where is tutorial No. 6. It seems you have loaded Tutorial no. 5 again.
Please link it correctly.
Thanks.
Neha
Hi Neha, I fixed the link above. Apologies
Please also note: as a member, you can always the most recent screencast at the top of the member page @ http://www.bionicturtle.com/frm
- David
Hi David,
Where can I download the ipod versions of early bird video 5,6? Also, ipod version of tutorial 4 is not working for me. Is anyone facing the same problem? If not, which software is to be used to view these file formats on PC? Any info on this will be very helpful.
Thanks,
Dipranil
Dipranil,
The ipod versions are on the member page. Upper left, below “Blog Member Updates:” the 2nd entry.
(the new website locates the ipod version next to the movie so that will be more convenient)
We can load the .mv4 here on our test machine, but on the forum, somebody else had a problem with ipod. I don’t think it’s a *file problem* but we aren’t sure yet why some seem to have no problem but other do. The ipod version is new this year, so any feedback help on this would be appreciated…
David
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