Question about Bionic Turtle's 2009 FRM Program
07 Jan 2009
Mar
24
FRM 2008 (Regular Season) Episode #1
by David Harper, CFA, FRM, CIPM
FRM |
Contents
- 2008 FRM Screencast (movie) Tutorial Schedule
- About Episode #1
- New spreadsheets added
- Questions associated with the tutorial
2008 Screencast schedule
Hello! We published today the first episode for the 2008 FRM regular season. A new screencast tutorial will publish every other Monday. The next (Gujarati's Econometrics on distributions & inferences) will publish on April 7th, then April 21st (Gujarati on regression), and so on. Please bookmark this page which contains the schedule.
| Date | Topic |
| 3/24 | Quant A (Allen's Quantifying Volatility and Wilmott's Value at Risk) |
| 4/7 | Quant B (Gujarati's Econometrics: Distribution and inference) |
| 4/21 | Quant C (Gujarati's linear regression) |
| TBD | Live Webinar |
| 5/5 | Market Risk A |
| 5/19 | Market Risk B |
| 6/2 | Market Risk C |
| 6/16 | Value at Risk A |
| 6/30 | Credit Risk A |
| 7/14 | Credit Risk B |
| 7/28 | Credit Risk C |
| 8/11 | Operational A |
| 8/25 | Operational B |
| 9/8 | Operational C |
| 9/22 | Basel II |
| 10/6 | Investment A |
| 10/20 | Invesment B |
| 11/3 | Cram Session Recorded |
| TBD | Cram Session Live Webinar |
About Episode #1
This episode reviews two densely-packed assigned readings. Linda Allen's chapter (Quantifying Volatility in VaR Models) repeats from prior years, but Wilmott's chapter is new. Key themes include:
The difference between convenient math assumptions and actual asset returns
In particular, asset returns are not normal and i.i.d. (independent and identically distributed random variables) is not realistic. About non-normal distributions, this not only refers the shape of the distributions (as reflected in the moments) but that a distribution changes over time. It is "time-varying." If a distribution is conditional or regime-switching, note that it must be time-varying.
In regard to i.i.d., we "abuse" this assumption routinely. If you look at Wilmott's VaR calculation, it embeds the square root of time rule; variance scales directly with time. But this requires i.i.d. and we cannot depend on i.i.d. In reference to volatility, the violation in is sort of subtle. It's not really independence (it turns out we can safely sort of assume returns are independent period to period; i.e., random walk). Rather, variance/volatility is not identical from one period to the next . This happens because, as Allen says, volatility is "sticky to itself" or mean-reverting. GARCH(1,1) explicitly models this realism: autoregressive (the "A" in GARCH) refers to a variance sticky to its past, and heteroskedastic (the "H") means the variance is not constant (not identical in i.i.d).
Volatility estimate approaches
We need to know three parametric approaches to estimating volatility: historical standard deviation (moving average. MA), EWMA, and GARCH(1,1). I already devoted an entire early bird to this topic. You are doing great when you see all of them as special cases of ARCH(m): where MA assigns equal weights to squared returns, EWMA assigns exponentially declining weights, and GARCH does the same but adds mean reversion.
Nonparametric approaches (volatility or VaR)
First, keep in mind the distinction between volatility and VaR. Volatility may be an input into a parametric VaR but VaR does not require a volatility parameter if it uses a simulation. Between these two assigned readings, we have the following non parametric approaches:
- Historical simulation. Sort returns worst-to-best and look down (lookup) the list.
- Multivariate density estimation (MDE). Like the ARCH(m) models but instead of weighting (squared) returns by time, assign more weight to returns that belong to an economic state that is similar to today's (where economic state = vector of variables we define).
- Allen's hybrid: Like historical simulation, sort returns from worst to best, but also weight them according to their proximity in time. I think this is a hard idea, and frankly I did not fully grasp this idea until I created a spreadsheet for it (see below).
- Wilmott's Monte Carlo simulation: specify an algorithm that contains a placeholder for a random factor (e.g, GMB for stocks). Run algorithm many times. Then look down (lookup) the results. Historical simulation and Monte Carlo simulation share this trait: the last step is to sort the returns (actual or simulated) and find the 95th or 99th percentile ranked outcome.
- Wilmott's bootstrap (also a simulation). Don't need an algorithm. To step forward in time, retrieve an actual set of returns from the past and use those indexed returns.
Different correlation types
Finally, I'd like to point out these readings tease out three different types of correlations. Think about a portfolio as a matrix where rows are asset returns (row 1 = asset #1, row 2 = asset #2) and columns are days (column 1 = 3/3/2008, column 2 = 3/4/2008). Each cell then contains the asset return on a given day.
| Portfolio Asset | Today | n+1 | n+2 | etc |
| Asset A | return (A,n) | return (A,n+1) | return (A,n+2) | return (A,...) |
| Asset B | return (B,n) | return (B,n+1) | return (B,n+2) | return (B,...) |
| Asset C | return (C,n) | return (C,n+1) | return (C,n+2) | return (C,...) |
We can then reference three different types of correlations:
- Correlation between assets on a given day; correlation within a matrix column (Wilmott calls “correlation between assets” and technically can be called spatial or cross-sectional correlation)
- For a given asset (a given row), correlation in returns from one day to the next (autocorrelation or serial correlation)
- For a given asset, correlation in return volatility from one day to the next; this requires us to transform the matrix above, but this is also correlation within the (transformed matrix) row. This is also auto or serial correlation, but in regard to return volatility instead of returns.
New Spreadsheets Added
On the member page, you will notice several new EditGrid spreadsheets. You have several ways to access them: you can simply view in the browser, you can open/save into Excel (File > Export As > Excel (.xls), or you can download the "native" Excel file (XLS) associated with the entry.
Some customers have historically found the spreadsheets useful. So this year, I am dressing them up for learning purposes (e.g., primary steps are color coded, sparse illustrative data instead of large actual datasets, I try to use the simplest possible formulas). The reason I deploy spreadsheets is to give help on the more difficult ideas. Recently, for example, a customer asked for a copy of the key rate duration calculation; a good example of where a spreadsheet might crystallize a concrete understanding. I will continue to upload spreadsheets during the year for the harder ideas (e.g., option pricing, many of the bond concepts, many of the credit risk concepts although these may require Excel, the Basel IRB, some of the hedge fund performance metrics).
For this episode #1, I uploaded the following spreadsheets:
- Allen's parametric volatilities: compares historical standard deviation (a.k.a., moving average), exponentially weighted moving average (EWMA), and GARCH(1,1)
- Allen's hybrid volatility
- Wilmott's (Taylor Series) delta/gamma approximation
- Wilmott's bootstrapping
- Wimott's Monte Carlo simulation
- Wilmott's extreme value theory (EVT) distributions: GEV & GPD
But a caveat about these spreadsheets: I do not recommend them for your "first pass" if you are just getting your feet wet in the FRM. For example, the exam treats EVT superficially and so you don't really need much familiarity with GEV & GPD distributions. I would rather see you pull down a spreadsheet when your are seeking some conceptual clarification.
However, in regard to this episode,I do recommend you take a peek at spreadsheet associated with Willmott's delta/gamma approximation. That's because the Taylor series is both thematic and difficult if you are rusty with the calculus. Don't worry: you don't need to understand the calculus that derives the Taylor Series. But you do want to understand the job of the Taylor Series and why it's useful. So, the spreadsheet illustrates Wilmott's text with a concrete example.
- First, we price an option
- Second, we adjust (shock) two of the option price model inputs, stock price and volatility. We can think of these as the risk factors. Now, the question is, what is the impact of changing (shocking) these inputs (stock price and volatility) on the option's price? Do you see how this may be the quintessential risk question; i.e., what is asset/portfolio's response to a shock in an underlying risk factor?
- Third, we perform a full-repricing: we run a second Black-Scholes on the new set of inputs. That's the long way around.
- Fourth, we perform a short cut with the (truncated) Taylor Series approximation. In this short cut method, we estimate only the change in the option price as a function of only the changed inputs (risk factors). The result is nearly the same.
In this way, the Taylor Series can be employed for many non-linear assets, where non-linear refers to the relationship between an asset's price and it's underlying inputs or risk factors. (But it cannot be used for functions which, in Linda Allen's terms, are "not well-behaved" or "exhibit extreme non-linearities").
Video Tutorial
Non-members can sample the start of the screencast tutorial here.
Paid member access the screencast in the member section (where you will also find the downloadable PPT slides, if you would like to view those. Plus an ipod format file, and a downloadable format.)
Please note: every year I get notes from very nice people warning me that the videos can be stolen because I provide downloads. We have explored security, etc, but the problem with most security methods is that they unduly inconvenience paying customers (unsecured downloads create enough issues on their own!). I decided to offer downloads and iPod because customers asked for them and my business is to serve paying customers. My business is not subsidizing unethical behavior. So, if you are a customer, please don't distribute files to other people. You may have noticed that our price is quite affordable. We can do this, in large part, only if actual consumers actually do pay. If you aren't a paying member, you are wrong to be viewing my work product without paying me. But you already know that...
Practice Questions
I wrote the questions below to provoke thinking about the ideas in the Episode. (If you doubt that I write questions freshly for each episode, consider that I do make mistakes!)
Unlike exam questions, some of the below don't have perfectly concrete, objective answers and are instead meant to elicit some thought about the similarities/differences among volatility/VaR approaches. In fact, on question #4 below, I frankly sort of stumped myself, so feel free to improve on my answer in the forum.
Question #1 (Goldman's 25-sigma loss)
Last August, after one of Goldman’s funds lost 25%+ of its value, the CFO said, ”We were seeing things that were 25-standard deviation moves, several days in a row.” .
This statement was roundly criticized on the grounds that it would take a very long time for a 25-sigma event to happen, maybe longer than the universe has existed. But the Linda Allen reading (Quantifying Volatility in VaR Models) provides more than one possible explanation for the *perception* of a 25-standard deviation loss.
Explain the CFO’s statement with Linda Allen’s assertions about volatility.
Question #2 (Comparing volatilities)
In regard to volatility estimate, identify:
(i) The primary advantage of EWMA over STDEV,
(ii) The primary advantage of GARCH(1,1) over EWMA,
(iii) The primary advantage of MDE over both GARCH and EWMA
(iv) The primary advantage of non-parametric over approaches to volatility
Question #3 (GARCH)
Assume you have determined the GARCH(1,1) specification for a volatility time series to be given by the following:
Variance estimate (today) = 0.02 + (0.1)(r^2) + (0.8)(v) where r = the lagged (1) return and v = the lagged (1) variance
(i) What is the implied long-run (average) variance?
(ii) If we forecast volatility forward in time based only on the “square root rule” (i.e., scale volatility by the square root of time), what is the nature of the error if the volatility is otherwise characterized by this GARCH(1,1). Admittedly, we wouldn't make this mistake under GARCH(1,1) so this isn't realistic.
(iii) What is the series’ persistence?
Question #4 (Hybrid vs. bootstrapping)
Linda Allen illustrates a "hybrid approach" to estimating value at risk (VaR) which combines, in her words, RiskMetrics and Historical Simulation (HS). Paul Wilmott illustrates a method called bootstrapping.
Are these really different, and if so, how? (Try not to peek because it makes you think about each approach).
Question #5 (EVT)
Paul Wilmott’s readings gives a very brief introduction to extreme value theory (EVT). He shows two distributions, one classical (GEV) that corresponds to a maxima/minima approach (block maxima) and one modern (GPD) corresponding to a peaks over threshold (POT) approach (or, we could say, block maxima is “fitted by” GEV and POT is fitted by GPD). What type of extreme loss dataset would tend to favor the selection of each distribution over the other?
Question #6 (Wilmott's two VaRs)
Assume portfolio value of $100 today with (return) volatility of 20%. Paul Wilmott shows two VaR formulas.
(i) If the single-period is short, what is the 95% value at risk (VaR)? (either in dollars or return)
(ii) if the single-period is long (e.g., one year), and expected return over the period is +10%, what is the 95% value at risk (VaR)? (either in dollars or return)
Question #7 (adding VaR)
Assume a portfolio of two equally-weighted assets that happen to also have equivalent VaRs (value of risk) of $100 each; i.e., VaR(A) = $100 and VaR(B) = $100.
(i) If VaR(A + B) > VaR(A) + VaR(B), what risk measure criteria is violated?
(ii) If we assume normality, and the correlation between A & B is 1.0, what is VaR (A+B)? (an unfair question at this junction, not yet covered)
(iii) If we assume normality, and the correlation between A & B is zero, what is VaR (A+B)? (also unfair)
Question #8 (three different meanings of correlation)
Think of a portfolio as a matrix where rows are asset returns (row 1 = asset #1, row 2 = asset #2) and columns are days (column 1 = 3/3/2008, column 2 = 3/4/2008). The cell contains the asset return on a given day. We can then reference three different types of correlations:
a. Correlation between assets on a given day; correlation within a matrix column (what Wilmott calls “correlation between assets” and technically can be called spatial or cross-sectional correlation)
b. For a given asset, correlation in RETURN from one day to the next; correlation within a matrix row (autocorrelation or serial correlation)
c. For a given asset, correlation in return volatility from one day to the next; this requires us to apply matrix math to the cells, but this is also correlation within the (transformed matrix) row. So this is also auto or serial correlation, but in regard to return volatility instead of returns.
(i) Which correlation is a key feature of GARCH(1,1)
(ii) Which correlation is automatically captured by Wilmott’s bootstrapping?
(iii) Which correlation(s) are captured by Allen’s MDE, if any?
(iv) The square root rule requires i.i.d. Which of the above correlations violates the i.i.d. and renders, in practical terms, the square root rule unreliable?
Answers to questions
Thanks very much. I will see you again in two weeks with another newsletter and Episode #2.
David Harper, CFA, FRM, CIPM
Founder
www.bionicturtle.com
P.S. I don't want to spam you!
Comments
Hi David. The pdf download for the slides to episode 1 doesn’t seem to work. Can you please repost. Thanks.
Hi Chih - the pointer to episode #1 looks good (i just tried it). I can send you the PDF, but might be better to figure out why it doesn’t work. On some browsers, for some reason i don’t understand, a RIGHT-CLICK and SAVE AS to local works. Does that work>
Hi David.
Was just going through the core reading Linda chapter and got confused with figure 2.4 what is the author trying to convey here....i mean he/she is not plotting many curve with different variance (conditional Normality) ...he/she is just plotting the same curve ( just normalized it with variance)......i may be wrong but is he/she using different variance for normalization so that it can be standardized ..i mean normalising each with different variance according to the period...If I am wrong can you plz describe it with a screencast tutorial
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