# Interpretation of PCs

I have computed PC1 and PC2 wts on future contracts derived from cumulative log differences. How can I use them to get back the theoretical price of each contract using those 2 pcs? Thanks in advance.

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Just to be sure: you analyze log-differences - right? This is equivalent to log-returns. Then you extract the principle components and keep e.g. the "biggest" $2$. What you get back (if at all, as Freddy remarks) are the log-returns. – Richard Feb 13 '13 at 13:18
Thanks Richard. You are right, I used standardized log returns and then picked 2 biggest components. How can I use those pcs to adjust current contract’s price to theoretical? Thanks. – Lucy Feb 13 '13 at 16:19
I think for this it does not really help you. Which kind of futures are we talking about? A deviation of the traded price from the theoretical price can be a sign of lack of liquidity in the underlying. – Richard Feb 13 '13 at 16:25
Crude. Im not really sure what the deviation is going to look like since Im not sure how to get the theo price... – Lucy Feb 13 '13 at 16:43
Lucy please provide us with more info ... we can not guess what you do ... are dealing with equity index futures? Most probably a cost-of-carry approach can be applied to get the theoretical price: en.wikipedia.org/wiki/Cost_of_carry – Richard Feb 13 '13 at 20:09

Yes you can, how depends fully on your required accuracy and also whether PC1 and PC2 are sufficient in explanatory power of the log differences of your futures contract.

Also, make sure you understand the signs of the eigenvalues (sign of the PC) can be different from one experiment to the next as they are arbitrary (the values are obviously not). Here some comments on that which I found when I tried to find supporting documents:

http://stats.stackexchange.com/questions/30348/acceptable-to-reverse-score-a-principal-component

The following describes in a somewhat theoretical way how to PCA-Reverse but they also bring up couple neat examples. You are not presented here with a off-the-shelf R code toolbox that gets you to your results in the next 5 minutes but I think anyone using PCA should actually first understand the related math and stats behind it first. I am sure you will be able to easily reverse the PCA post reading this:

http://www.cs.columbia.edu/~stratos/research/pca_cca.pdf

Edit: I was actually so intrigued by the above paper's examples that I currently play with PCA and the PCA-Reverse in regards to image manipulation. Sorry that comment is not quant finance related but just wanted to share my excitement with math that sometimes overcomes me when playing with some interesting stuff.

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Thanks for your reply. The links are great. From what I understand, I can overlay the orig. log return matrix with found components and get a new matrix of log returns of contracts in terms of components. But I don’t really want to know how the contracts’ log returns looked like in the past in terms of 2 new PCs. I would like to know how to adjust current price of the future using the 2 PCs I found for each contract to get my theoretical price for each contract. In futures, pc1 is shift (the same sign) and pc2 is slope and I need to adjust current price to reflect those. Thanks again!! – Lucy Feb 13 '13 at 16:12

I found a link and I have to repeat: I don't think that PCA helps you to find a price ... it helps to model the movements of prices but not their values. You get something like a factor model ... this does not directly give you a price ... maybe you also want to have a look at this link where PCA is applied to the oil market.

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Thanks so much for taking the time to reply to me. I will read the papers and see if I'm missing something. – Lucy Feb 13 '13 at 23:58
I agree with Richard here. @Lucy, you are maybe expecting a little too much magic out of the PC box ;-) but once you setup your factor model you can certainly take a step further and model price moves. – Matt Wolf Feb 14 '13 at 18:16