[R-SIG-Finance] PCA in Risk Control with R

Wed Feb 17 13:34:22 CET 2010

hello Julien,

Thank you for your response.  I get Point 1.

On point 2, stating your comment in another way (to make sure I
understand it): since we do not have a distribution of the normal
modes, no scale factor exists by which to shock them.

Here is my thinking:
1. Work with returns
2 .Calculate the sigma of these returns (assuming a normal for ease)
3. Scale by sigma (which explains why it appears as a mult factor in step 5)
4. Calculate the normal modes
5. Return to original coordinate system by  (this might be somewhat
incorrect but, if not, then it provides a way to stress at a given
confidence level):
 PCA| i,m = exp( FI(0.995) * N|i,m * S|m * SQRT(L|i) )

i = i-th Principal component
m = forward month
FI = inv normal at a CL
N|i,m = eigenvectors
S|mm = std dev
L|i = eigenvalue

So my way of stressing is scaling up that S by the "2.57" (the
FI(0.995)).  Is this wrong?  Am I missing something?

Benji

On Wed, Feb 17, 2010 at 3:38 AM, julien cuisinier
<j_cuisinier at hotmail.com> wrote:
> Hi Benji,
>
>
> Welcome to the list & good luck with the posting guide ;-)
>
> I am not a PCA expert, but in my opinion there are 2 items in your question:
>
> 1. run PCA on original data set, choose relevant factors/components, do
> something with these factors, back to "original" data set. check
> http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf
> there is a small description of getting back to old axis system
>
> 2. Do something to the factors (risk focus): it seems you do factor push
> stress testing (i.e. each pushing factor right/left by an arbitrary amount,
> pick the side hurting the port the most, repeat this for each factors and
> apply each factor "worst" move to the portfolio. I therefore do not see
> where/why the last computation comes in (trying to get a risk measure at a
> certain confidence level). The components moves are arbitrary and the risk
> measure resulting on the portfolio level will also be, no probabilities
> associated hence no confidence level.
>
> One can decide to model each component with a parametric distribution then
> derive what risk measure is relevant - the big advatages of PCA being that
> you do not have to care about components comovement, as long as the
> assumption of linearity is acceptable. But this is simply another approach,
> not something on top of the factor push you are trying
>
>
> Hope this helps
>
>
> Rgds,
> Julien
>
> PS: the "parametric approach" used in your first post on top of the factor
> push stuff seems to use simple normal distrib VaR, if you go that route I
> would probably start by trying to model volatility clustering (GARCH)
> instead of relying on simple historical estimates...hope I understood your
> post well, I do not mean to lecture you of course...
>
>> From: benjifamel at gmail.com
>> To: cmdr_rogue at hotmail.com
>> Date: Tue, 16 Feb 2010 20:26:47 -0500
>> CC: r-sig-finance at stat.math.ethz.ch
>> Subject: Re: [R-SIG-Finance] PCA in Risk Control with R
>>
>>
>> Sarbo,
>>
>> Thank you. This answers how you rebuild the curve.
>>
>> Now this may be a silly question but how do you shock the resulting
>> curve to a given conf level?
>>
>> Thanks again,
>>
>> Benji.
>> Sent from my mobile device.
>>
>> On Feb 16, 2010, at 8:02 PM, Sarbo <cmdr_rogue at hotmail.com> wrote:
>>
>> > Hi Benji- you're in luck. I've done exactly this sort of thing in the
>> > past. Here is the code that I wrote to do the job:
>> >
>> > BuildPCACurves <- function(data, ncomps = 3){
>> > if (class(data) != 'matrix'){A <- as.matrix(data)} else A <- data
>> > M <- t(A) %*% A
>> > eigens <- eigen(M)$vectors
>> > eigens[,1] <- -eigens[,1]
>> > selectors <- matrix(0, nrow = ncol(A), ncol = ncol(A))
>> > diagentries <- c(rep(1, ncomps), rep(0, ncol(A) - ncomps))
>> > diag(selectors) <- diagentries
>> > coefficients <- A %*% eigens
>> > newcurves <- coefficients %*% selectors %*% t(eigens)
>> > coef2 <- apply(coefficients, 2, diff)
>> > means <- apply(coef2, 2, mean)
>> > stdevs <- apply(coef2, 2, sd)
>> > RMSE <- sqrt(sum((apply(newcurves - A, 2, sum)) ^ 2))
>> > output = list(original = A, rebuilt = newcurves, means =
>> > means[1:ncomps], stds = stdevs[1:ncomps], RMSE = RMSE)
>> > return(output)
>> > }
>> >
>> > This doesn't use the actual "princomp" function in R, but it does
>> > exactly the same thing; it just uses the underlying matrix theory
>> > behind
>> > PCA itself.
>> >
>> > On Tue, 2010-02-16 at 19:15 -0500, Benji Famel wrote:
>> >
>> >> I like the idea, and I have attached a sample of data. I think this
>> >> should work as the file is not a binary one. The data represents
>> >> daily NYMEX data for Natural gas.
>> >> The columns are flat prices (not returns) and represent:
>> >> 1. Date
>> >> 2. Prompt contract
>> >> 3. Back contract (2nd month out)
>> >> 4. Far contract (3d month out)
>> >> 5. etc.
>> >> After transferring the data to R through RExcel as MktData, I
>> >> execute
>> >> the following code:
>> >>
>> >> MktReturns.d <- MktData
>> >>
>> >> # ----------- Data Preparation ------------------
>> >> for (i in 1:ncol(MktData) ) {
>> >> MktReturns.d[,i] <- Fin.Calcs.logreturns(x=MktData[,i], deltaT=
>> >> 1, pad=T)
>> >> }
>> >> MktReturns.d <- na.omit(MktReturns.d)
>> >>
>> >> #PERFORM PCA ON DAILIES (good for 1 day risk... if I wanted the
>> >> weekly risk, I would work with weekly returns)
>> >> pcdat.d <- princomp(MktReturns.d, cor=TRUE) # - It will use
>> >> correlation matrix so NO need to scale
>> >> the.summary.d <- summary(pcdat.d) # - It will print standard
>> >> deviation and proportion of variances for each component
>> >> the.loadings.d <- loadings(pcdat.d) # - it will give information
>> >> how
>> >> much each variable contribute to each component.
>> >> the.scores.d <- pcdat.d$scores # - It will plot scores of each
>> >> observation for each variable
>> >> whichQuantile <- quantile(rnorm(1000000),probs=c(0.95))
>> >>
>> >> PC <- exp(whichQuantile*t(pcdat.d[[2]])*sqrt(pcdat.d[[1]])*sd
>> >> (MktReturns.d))
>> >> # note that if I wanted to work with daily returns but calculate
>> >> the 1
>> >> week risk, I woudl be multiplying above with sqrt(5)
>> >>
>> >> Hope this helps.
>> >>
>> >> Benji
>> >>
>> >> On Tue, Feb 16, 2010 at 6:54 PM, Brian G. Peterson <brian at braverock.com
>> >> > wrote:
>> >>> Why don't you disguise a subset of your data and provide a working
>> >>> example?
>> >>>
>> >>> Both you and the list will get more out of it if we can all work on
>> >>> something that is actually executable in R, per the posting guide.
>> >>>
>> >>> Your problem is interesting and relevant, so put a little more
>> >>> effort into
>> >>> it, and I'm sure you'll get collaborators in working through it.
>> >>>
>> >>> Regards,
>> >>>
>> >>> - Brian
>> >>>
>> >>> Benji Famel wrote:
>> >>>>
>> >>>> Hello,
>> >>>>
>> >>>> my apologies if I do something wrong - first posting for me.
>> >>>>
>> >>>> I am trying to apply PCA on the daily history of a bunch of forward
>> >>>> curves and run into my depths of ignorance. I would appreciate
>> >>>> some
>> >>>> help...
>> >>>>
>> >>>> My aim is to use PCA for risk control. I.e. estimate the
>> >>>> eigenverctors and eigenvalues and build the principal components at
>> >>>> some confidence level, e.g. 95%. If, for example, we were
>> >>>> looking at
>> >>>> the first 3 components only, I would
>> >>>> - estimate PC1up, PC1dn, PC2up, PC2dn, PC3up and PC3dn.
>> >>>>
>> >>>> Let's assume that
>> >>>> - PC1up is worse for my position than PC1dn,
>> >>>> - PC2up is worse than PC2dn and
>> >>>> - PC3dn is worse than PC3up
>> >>>> I would then 'add' these worse for me components (PC1up, PC2up and
>> >>>> PC3dn) and run my position through them to get a measure of risk at
>> >>>> that confidence level.
>> >>>>
>> >>>> To do the PCA, I first foundthe log returns, let's call them
>> >>>> Returns.
>> >>>> I then do:
>> >>>> pcdat <-princomp(Returns, cor=TRUE)
>> >>>> and calculate the principal components like this (this is where I
>> >>>> am
>> >>>> very foggy...):
>> >>>>
>> >>>> PC <- exp(someQuantile*t(pcdat[[2]])*sqrt(pcdat[[1]])*sd
>> >>>> (Returns)) #
>> >>>> somQuantile = 1.64 for a 95% CL
>> >>>>
>> >>>>
>> >>>> As much as I looked around, people discuss the benefits of PC but
>> >>>> not
>> >>>> how to recombine the principal components at some confidence
>> >>>> interval
>> >>>> to get a shocked curve.
>> >>>>
>> >>>> Could anyone help?
>> >>>>
>> >>>> Thank you,
>> >>>> Benji
>> >>>>
>> >>>> _______________________________________________
>> >>>> R-SIG-Finance at stat.math.ethz.ch mailing list
>> >>>> https://stat.ethz.ch/mailman/listinfo/r-sig-finance
>> >>>> -- Subscriber-posting only. If you want to post, subscribe first.
>> >>>> -- Also note that this is not the r-help list where general R
>> >>>> questions
>> >>>> should go.
>> >>>
>> >>>
>> >>> --
>> >>> Brian G. Peterson
>> >>> http://braverock.com/brian/
>> >>> Ph: 773-459-4973
>> >>> IM: bgpbraverock
>> >>>
>> >> _______________________________________________
>> >> R-SIG-Finance at stat.math.ethz.ch mailing list
>> >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance
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>> >> -- Also note that this is not the r-help list where general R
>> >> questions should go.
>> >
>> >
>> >
>> > [[alternative HTML version deleted]]
>> >
>> > _______________________________________________
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>>
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