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# Stress Test Covariance and Correlation Matrix

#### hsuwang

##### Member
Hello David,
I've seen the terms "covariance matrix" and "correlation matrix" a couple of times now, and I think I roughly know what they are and how they work, but I'm not sure as to how they apply and are being used in scenario analysis (stress testing). Also I am getting a bit overwhelmed by the covariance matrix concept used in Cholesky factorization. Can I ask for your suggestion on the readings that might help me get a solid understanding of the matrixes and how they are being used in both scenario analysis and cholesky factorization? I only read your 08' note on "stress testing" and haven't gone through the assigned readings for Jorion's "stress testing" yet, so maybe that would be a start?

Thank you!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Jack,

These are each big topics. Please note I uploaded a structured monte carlo @

This may helpful because I used a correlation matrix as input into the covariance matrix
(just as covariance embeds correlation per, COV = COR*VOL*VOL, it is true in matrix form too)
Then on the 2nd tab, I actually spent the time to hand-build a 5-factor Cholesky; the Cholesky is just matrix math; complex maybe but it's role is straightforward: it uses information in the covariance matrix to convert a vector of random, independent returns into a vector of random, but correlated returns. (it is colored red b/c you don't need to know for exam)

I attached two pdfs from my elibrary: the PRM matrix chapter is best I've got on Cholesky - really accessible!
And then a (free) article from 2007 Garp risk review that I found very helpful...Please see attached below

In regard to stressing the covariance matrix, this is a whole topic and can be a simple as: use an old crisis period, extract the covariance matrix from that period, and run of test of today's portfolio through it. Many variations and details ensue...

Best i have on this, unsurprisingly, is Carol Alexander's Vol IV on Value at Risk
http://www.amazon.com/Market-Risk-Analysis-Value-Models/dp/0470997885/ref=sr_1_3?ie=UTF8&s=books&qid=1246647336&sr=8-3

However, IMO, for purposes of sitting the full FRM, you probably don't have time to take a deep dive on this...
e.g., Jorion's Ch 12 on stress testing is *very* introductory, just a laundry list...

You raise a great point, though: it occurs to me only now that GARP has not "introduced" the covariance matrix in the readings (?!)
Since it is fashionable to beat up on it nowadays, maybe good idea to know why it's getting dumped on

I will add, since you raised this point, a simple VaR XLS that uses convariance matrix maybe with a simple stress example...(I have an old version i can update)...I hope this is a helpful start...David

#### hsuwang

##### Member
Hello David,
Thank you so much for the reply, I feel that I'm always getting more than I ask for from your resourceful replies! THANKS!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Sure thing Jack, in all sincerity, as i said before, you've asked several questions that are relevant and often tend to hit common obstacles - so I do try to pull in answers that may be helpful (including to others) generally for the exam...(it keeps me fresh too, on topics, so i am benefitting too!) David

#### fxlprasetyo

##### New Member
Hi Jack,

In regard to stressing the covariance matrix, this is a whole topic and can be a simple as: use an old crisis period, extract the covariance matrix from that period, and run of test of today's portfolio through it. Many variations and details ensue...

I will add, since you raised this point, a simple VaR XLS that uses convariance matrix maybe with a simple stress example...(I have an old version i can update)...I hope this is a helpful start...David

Hi David, regarding stressing the covariance matrix, have you ever come across where the stress is no from historical crisis period? For example, if we are looking more into parametric stress testing, ie. S&P 500 down (x) % and retrieve the covariance matrix of SP500 with other correlated factors. Thank you.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @fxlprasetyo

I think (in the FRM literature) we tend to sub-divide stress testing into either (i) scenario analyses or (ii) factor push models. The former, scenario analysis, suggests either historical or imagined (aka, hypothetical) scenarios that can be conveyed, or at least connected coherently, by a narrative; the most obvious example being Dodd-Frank's supervisory scenarios which stress 28 variables (eg, GDP) as a set of shocks that are either adverse or severely adverse, but in a plausible, coherent narrative.

On the other hand, factor push is clearly a strand in the literature, although I admit that I do not have direct experience applying it in practice. Factor push should include, but hardly be limited to, (mathematical) stressing of the covariance matrix. However, in the FRM literature the most common approach seems to be using the assumption that correlations spike to perfect 1.0 (ie, worst case).

Here is Kevin Dowd (Chapter 13) on factor push:
"13.3.1 Factor Push Analysis: The simplest of these procedures is factor push analysis, in which we ‘push’ the price of each individual security or (preferably) the relevant underlying risk factor in the most disadvantageous direction and work out the combined effect of all such changes on the value of the portfolio. We have already met this type of approach in the shape of Wilson’s delta–gamma approach to VaR, which was discussed in Chapter 10. We start by specifying a level of confidence, which gives us a confidence-level parameter α. We then consider each risk factor on its own, ‘push’ it by α times its standard deviation, and revalue the portfolio at the new risk factor value; we do the same for all risk factors, and select that set of risk factor movements that has the worst effect on the portfolio value. Collecting these worst price movements for each instrument in our portfolio gives us our worst-case scenario, and the maximum loss (ML) is equal to the current value of our portfolio minus the portfolio value under this worst-case scenario …"

As I reference Carol Alexander (to answer your question), she seems to have a similarly "limited" view of factor push, akin to Jorion's approach wherein implicitly the covariance matrix spiked all the way to its worst case (i.e., all correlations to 1.0), but I see no reason to limit the factor push to this approach. She also has a good discussion of scenario analysis/stress testing in her Vol IV; e.g., emphasis mine:
"The application of worst case scenarios to stress tests may be based on hypothetical or historical events. A common hypothetical event is a six sigma event, meaning a loss that is at least six standard deviations from the expectation of a distribution. Simply put, if the historical (or hypothetical) P&L standard deviation is σ dollars, then the worst case loss is Θσ dollars. More generally, suppose we are stress-testing a portfolio that has k risk factors whose returns are denoted X1,...,Xk and whose P&L is denoted f(X1,...,Xk). Given an estimated or hypothesized value for the means µ and the standard deviations σ(i), i = 1,...,k, the six sigma loss is defined as f(μ ˆ 1 ± 6σ ˆ 1,...,μ ˆ k ± 6σ ˆk), where the + or - is chosen independently for each risk factor in order to maximize the loss.

This is an example of the factor push methodology for stress testing, in which each risk factor is ‘pushed’ by a certain amount, in a direction that will incur the greatest loss, without respecting any assumption about the risk factor correlations. More generally, a factor push method generates a P&L of the form f(μ ˆ 1 +a1σ ˆ 1,...,μ ˆ k +akσ ˆk) where the integers a1,...,ak can be positive or negative. This method is commonly used by traders for assessing the risks of their own positions, but since it takes no account of risk factor correlations the factor push methodology has limited application to firm-wide solvency assessment.

I hope that is some help ... thanks!

#### fxlprasetyo

##### New Member
Hi David,

Thank you so much for your response. Yes, I agree with you assessment, I own all four books in the series from Carol Alexander, an excellent materials on Risk Management. She covered both stressing the Covariance Matrix based on historical period as well as stressing with some standard deviation of the means (disregarding correlation). I have yet to discover resource that discussed stressing a factor by percentage, says SP 500 up/down 15% and it's effect on correlated factors as well as its dependent vars in factor model setting, have you?

Speaking of the Kevin Dowd book, is it worth purchasing, how much overlap is it with Carol's books? Would you recommend it?

Thank you.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @fxlprasetyo No, I have not encountered such a modeled approach (but the question fascinates me, and I tagged it for further research FWIW ....).

... it's a fantastic book, so strong on the quantitative aspects of certain topics (but not all; e.g., model risk is really light), but it's too old for me to recommend (published in 2005). Thanks!

#### San955

##### New Member
Hi David,

I read your comment above about Cholecky factorization. Does it mean that we decompose the covariance matrix = A* A^t and we use A to find the correlated "e"?

And is the following link: http://www.math.kent.edu/~reichel/courses/monte.carlo/alt4.7c.pdf a good/enough explanation/answer of the question: "Use Cholesky decomposition to determine the pricing process of multi-asset options. "

Thank you!