# Tag Info

16

Yes, the weights of the first eigenvector of a covariance matrix represent the market factor and also the largest source of systematic risk (variation of returns). Why PCA? Well, PCA simply identifies the eigenvector that maximally explains the variance of the system. It turns out that this is the "market factor" - i.e. the tendency of securities to rise ...

7

PCA(Principal Component Analysis) is the most interesting topic in QF. PCA is at the heart of quantitative data analysis. It is used in factor analysis, factor loadings, finding principal component of interest rate term structure for derivative and option pricing, data compression, eigenfaces( find the best match from a set of pictures with a , say, fuzzy ...

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About a year ago I saw a presentation by Attilio Meucci in London. The twist of his work is a little bit different compared to yours but the general approach is similar and there is lot to be learned from his accompanying paper: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1358533 Here he is also using PCA for dimensionality reduction constructing ...

6

Here is a structured list of your bullet points: covariance, correlation, PCA, factor analysis, Are similar. They are based on Gaussian assumptions (i.e. correlations means dependencies) and try to identify common factors (i.e. a variable in small dimension) explaining the observed relationships. co-integration is more specific in the sense that you ...

6

1) Eigenvector times minus one is also an eigenvector (with the same eigenvalue). 2) Distinct eigenvectors of a symmetrical matrix (i.e. covariance) are orthogonal. 1 and 2 imply that you can multiply a subset of all the eigenvectors of a symmetrical matrix by minus one an you still get a full set of eigenvectors Which means, just impose that the first ...

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PCA gives you a decomposition of the covariance matrix of the form $$\Sigma = V \Lambda V^T$$ where $\Lambda$ is diagonal with the eigenvalues in the diagonal. Your portfolio variance is $$w^T \Sigma w = (V^T w )^T \Lambda (V^T w)$$ On the other hand if you take your return matrix $R$ and define $$F = V^T R$$ then the covariance matrix of these so ...

5

Apart from numerical stability errors, Cholesky and PCA (without dim reduction) shall produce exactly the same distribution, they are two symmetric decomposition of the same covariance matrix and thus are equivalent for transforming a standard normal vector. Of course when doing different things with PCA components, such as in dim reduction or quasi Monte ...

4

You can compute the PCA on overlapping windows, and try to match the eigenvectors: you may need to change not only their sign (since only the eigenspaces are well-defined, the sign of the eigenvectors is arbitrary) but also their order. Here is some (untested) R code to do this. # Sample data k <- 7 n <- 50 found <- FALSE while(!found) { x <- ...

4

They are not mutually exclusive. PCA and clustering are similar but used for different purposes. You could use PCA to whittle down 10 risk factors to say 4 uncorrelated factors, and you could combine securities with different FACTORS into different clusters with offsetting returns and variance characteristics. However, when you say you want to derive risk ...

4

The first principal component of a large covariance matrix is extremely expensive to replicate in a real portfolio. While it is true principal components provide true (ex post) orthogonal factors, this is not necessarily relevant to the business of risk management. The market index is what most investors are benchmarked by, and is furthermore often ...

4

a) because it does not matter how you weigh each constituents as long as the methodology is publicly accessible and as long as it more or less reflects the original intent. That is why there are market cap weighted indexes but also why there are indexes that apply different weighting methodologies. b) because PCA is computationally way more expensive. Why ...

3

To make things really clear, you have an original matrix $X$ of size $300 \times 10$ with all your returns. Now what you do is that you choose the first $k=5$ eigenvectors (i.e. enough to get 80% of the variation given your data) and you form a vector $U$ of size $10 \times 5$. Each of the columns of $U$ represents a portfolio of the original dataset, and ...

3

Regarding the second part of your question - You are running into the classic N>T problem (N=# assets; T=# of observations). Therefore the number of parameters you must estimate grows geometrically with each N, but only arithmetically for each day of observation. Because you are estimating the diagonal portion of the covariance matrix you must estimate ...

3

I've played around with both schemes, but not for portfolio optimization. I used PCA on some interest rate models. That turned into a Partial Least Squares scheme, then into some non-linear thing. I wasn't impressed with the results. My Cluster Analysis scheme morphed into a classification scheme, and it turned out that the K-Nearest-Neighbor method ...

2

If you're referring to this problem then there is a very complete answer on the cross validated stack exchange.

2

Have you considered using 'incremental' singular value decomposition to calculate your component scores? Each future market move (or increment) forces a recalculation of component scores given the new data. This paper outlines an algorithm to do this Fast Low-Rank Modifications of the Think Singular Value Decomposition This paper develops an identity ...

2

To answer your questions we have to take a look to what it does. PCA is mathematically defined as an orthogonal linear transformation that transforms the data to a new coordinate system, such that news vectors are orthogonals and explain the main part of the variance of the first set. It took an N x M matrice as input, N represents the differents ...

2

When I use PCA, I follow a few typical steps. First, I would apply PCA to the covariance matrix, I would then designate certain eigenvalues as dominant or significant (such as by those that contribute up to $x\%$ of variance or by RMT), and then I would identify the eigenvectors that match up with those significant eigenvalues. I think you're with me at ...

2

In the chapter that deals with NMF of the book "Programming collective intelligence" , the author did NMF on several stock trading volumes and found some comovement. I googled a little. This did NMF on 40 chinese stock close prices. This developed A variant of nonnegative matrix factorization for Stock Trend Extraction. Another google found this also did ...

2

Perhaps this paper by Hyun Woo Byun and coauthors is what you're looking for: Using a Principal Component Analysis to develop Multi-Currency Trading algorithms in the FX market They apply principal component analysis to a currency basket of 9 pairs with a 2 month rolling window. In a second step, various techniques (logistic regression, decision trees, ...

2

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 ...

2

Following @silencer's comment, your formula for variance is wrong. I would suggest that instead of trying to re-invent the wheel, you just use the formula that everyone else uses. So I'd replace your first indented line with $$w^{*}\equiv argmin\left\{ \frac{1}{2}w'\varSigma w-\lambda\left(w'\mathbf{1}-1\right)\right\}$$ which will give you $$... 2 If you look at changes of the points on the yield curve, then you probably find something stationary - right? Applying PCA on the covariance of these changes makes sense. E.g. you will find out that on PC describes a parallel shift (a change in the yield curve). Look at this question too: What do eigenvalues/eigenvectors of the yield/forward rates ... 1 The derivation is correct and given the formula you should get w^{*'} \vec{1} = 1. My guess is that the inversion of the \Omega matrix is numerically badly conditioned. Instead of implementing the formula as it is, have you tried to calculate \vec{1}^{'} \Omega^{-1} and e^{'} \Omega^{-1} only once and rewrite:$$ w^{*\prime} = \frac{1}{2}\left[ ...

1

The paper alternatives between using eigenportfolios and sector/industry ETFs for statistical arbitrage. For instance, sections 2.1-2 vs. 2.3. The trade in Section 4.1 is long some stock and short an appropriate amount of sector/industry ETFs. That being said Sections 5.3 and 5.4 discuss PCA strategies in a backtest, with relatively little additional ...

1

Q #1: I'm not sure if you have the answer quite right. The signs for the loadings are arbitrary, but you cannot take the absolute value. You can multiply by -1. Q #2: It might be helpful to think about what PCA is actually doing. This paper might be helpful: http://arxiv.org/pdf/1404.1100v1.pdf (A Tutorial on Principal Component Analysis by Jonathon ...

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Trying to answer: in the blog post that you mention the author looks at three equity funds and one REIT fund. One could say that these markets are different to FX markets (for various reasons but let's start with the question whether there is a risk premium in FX markets). what he does is the usual PCA analysis on the data. You find various questions in ...

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A PCA explains the variation in data. A slope PC is usually identified by the pattern of the signs of the loadings. If the loadings of short term contracts have the same sign which is different from the sign of the loading of longer term contracts then such a PC is identified as slope PC. It means that if this PC goes up or down it affects short term ...

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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.

1

Your approach is a good one. But before you venture too far, you should be aware of issues related to zero eigenvalues (positive semi-definiteness) of your correlation matrix $\mathbf{R}$ or covariance matrix $\mathbf{C}$. Let $p$ be the number of assets, and $t$ the number of, for example, day or bars. You probably have many more times in the time ...

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