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8

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 ...


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 ...


7

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

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 ...


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

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

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

The literature on cointegration in large datasets or panels is really the only place where I've seen this sort of issue discussed. Breitung and Pesaran, among other places, talks about it. I would recommend applying the PCA to the rate changes (perhaps with some kind of zero lower bound adjustment). Then, take the cumulative sum of each of the factors. ...


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

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

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

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

Let $S$ be your risk sarray, expressed in pv01, for each of your (implied) 10 instruments. You restrict the array to all zeroes except those corresponding to the 5Y, 7Y and 10Y risks, e.g. if 1Y:10Y were your instruments you would have: $$S = [0, 0, 0, 0, w_1, 0, -1, 0, 0, w_2]^T $$ You seek the solution of $w_1$, $w_2$ such that your risk expressed in PCs ...


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 ...


3

Simple Directionality Spread Trade Hedge If the sum of the risks of the trade $t$ are zero (as in the case of the 2Y5Y10Y spread trade) that immediately gives a starting point from which to make a simple calculation for an adjustment. For example if one assumes that the first principal component is the outright market driver and that the factor loadings ...


3

ML is a very broad term. Do you mean linear regression? To you mean random forests? People use all of these approaches with various degrees of success. Bloomberg will have a story every few months about a big quant/ML fund starting or being shut down. PCA specifically is used quite a bit in fixed income to model the underlying characteristics of fixed ...


3

A classic problem, been there, done that, didn't buy the T-shirt ;-) PCA and clustering (K-means, or hierarchical) are similar but different. They're both "unsupervised learning" methods; but one is essentially descriptive, while the other is essentially pragmatic and expedient. People want both, but they need to prioritise one first! Your PC1/PC2 ...


2

A way to go would be to linearly build indepedant interest rates to eliminate correlation effects. How do you do that ? You linearly build orthogonal interest rates from your starting ones. This is totaly equivalent to diagonalising correlation matrix, which is the principle of PCA. Using information criteria you can then choose to remove lowest components,...


2

Nelson Siegel seems to be pretty standard too


2

When estimating covariance matrices, you run into problems as the number of assets/risk factors approaches or exceeds the number of observations. Some eigenvalues will go to zero, or be very small. This will mean that the covariance matrix is positive semi-definite instead of positive definite. Since the Cholesky decomposition requires a positive definite ...


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 for ...


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 ...


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