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26

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 ...

11

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 ...

10

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 ...

8

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 ...

8

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 <- ...

7

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 ...

7

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 ...

6

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 ...

6

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 ...

6

What you basically do here is a Principal Component Analysis (PCA). A good starting point in the financial sphere is Managing Diversification by Attilio Meucci (2010) Page 3: "The most natural choice of uncorrelated risk sources is provided by the principal component decomposition of the returns covariance [...] The eigenvectors define a set of N ...

6

I would do as follows: A) First do PCA on an arbitrage-free monthly curve (assuming the most granular contract you will use is individual months). To ensure no arbitrages, you will need to drop out certain contracts, I would drop the most illiquid ones. To give you an example, if you are in Dec, you might see Jan, Feb and Mar quoted, but also Q1. In this ...

5

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 ...

5

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 ...

4

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  w^{*}=\...

4

You can see my remark above for some more words on PCA for the yield curve and an interesting paper. About the question whether it helps us to creat a risk model: PCA on the yield curve changes (!) tells us: what are dominant moves (it turn out it is a pralell-shift, steepening and curvature change)? This gives us a picture and language to think and speak ...

3

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 ...

3

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, ...

3

The first principle component of interest rates will not help you capture the term structure better at all. It will basically remove all term structure affects you are going to see. When we decompose the returns on interest rates you are going to get 3 PC's which explain 99.9% of the variance. PC1 - Level of the interest rates (~90% of variance) PC2 - ...

3

I am also interested in resolving this problem, although, decided not to create separate thread for it yet. This is kind of continuation of previous question below. https://stats.stackexchange.com/questions/34396/im-getting-jumpy-loadings-in-rollapply-pca-in-r-can-i-fix-it In factor analysis, specifically PCA, sign of the loadings does not mean anything, ...

3

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

3

I just want to mention that it's highly prevalent to apply PCA to rate levels in rich/cheap analyses. Personally I prefer that... There's an old MS publication that discusses this very topic and the recommendation is to use level PCA for rich/cheap, and to use change PCA for risk management. There's a really good Salomon paper (Principles of Principal ...

3

Let's use the following returns matrix, X 2Y 5Y 10Y -------------------------- 0.0143 0.0910 0.1451 0.1791 0.3505 0.4588 0.0572 0.1358 0.0120 0.0357 0.1809 0.2884 -0.0571 -0.1096 -0.0719 0.0286 0.0710 0.1319 0.0429 0.1806 0.2754 -0.0357 -0.0579 -0.1075 0.0714 0.2513 0.4304 -0.0214 -0....

3

Broadly speaking, as you probably already know, there are 2 approaches to estimating large covariance matrices: 1) Shrinkage Methods like Ledoit-Wolf that try to reduce the noise in a large matrix (N by N) that has been estimated using the conventional method. 2) Factor Models of Covariance as described in for example Connor Korajczik 2007 that assume that ...

3

The point of PCA is that your components are supposed to represent axes of principal variation. I.e. if you just use one principal component you can describe the most variation of true market movements with that, than you can with any other relative combinations of instruments. So if your component (eigenvector) is: [2y,5y,7y,10y] = ~[25,33,24,23], where ...

3

It depends on the intended end-use of your model, but generally-speaking, if you were solely trying to measure and forecast inflation levels or the GDP deflator over the course of a year (including the use of, say, the GDP deflator percentage change in March, as a factor that somehow goes into your April forecast), you would need to consider seasonal ...

3

To justify the use of tenors 2Y, 5Y, 10Y, 30Y for risk bucketing, you could analyse up to the first four principal components and examine which variables summarize better the information displayed on each axis using the factor score. For example, if the first four pc contains 90% of the available information (let's say 1st pc: 40%, 2nd pc: 30%, 3rd pc: 15% ...

3

So my question is how do I prove that the use of 2y, 5y, 10y and 30y is justified for risk bucketing and not other alternate buckets? Ok so just to pose a second viewpoint but why do you have to necessarily use PCA to do this? You are basically trying to show that given any underlying swap portfolio $P$ you can find a set of trades / risk exposures in ...

3

What you are describing is not mathematically plausible. Firstly, but less important, a PC is a normalised vector (an eigenvector) meaning if it has more than one non-zero element they will always be less than one. Of course you can scale the PC but technically any feature will never be worth one in the direction of the PC unless every other feature has ...

3

The first observation I make is that the proportion of variance is not very high for the first PCs, with the implication that I would hypothesise that the PCs are not very stable, nor reliable. (You can test this by varying the sample period and analysing the consistency of the PCs) If the PCs are not stable from period to period then information you can ...

2

If you are asking which of the 10 variables is contributing most to the principle component, then look at your first eigenvector; each value reflects a single variable, so the largest value (by magnitude) in that eigenvector should give the variable with the largest contribution. Note that a large negative number means anticorrelation. The matrix you have ...

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