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I am aware of an abundant literature on Principal Component Analysis (PCA) application for yield curves. All of these papers to me look merely a statistics-oriented results. Most of the papers argue that by "decoding" yield curve by applying PCA, we are able to explain yield curve movements. And the result of application are just 3 PCs, i.e., level, slope and curvature, which do not have any economic interpretation (to the best of my knowledge some authors try to give some economic interpretation, but those do not appear convincing to me).

Question What is the applied use of PCA for yield curve? Is there any real-life example of this?

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    $\begingroup$ It makes a user more comfortable with pricing derivatives assuming 1 factor models - as most curve moments are parallel. $\endgroup$ Commented Apr 19, 2024 at 21:58
  • $\begingroup$ Most of the papers argue that its use is explaining curve movements though. $\endgroup$ Commented Apr 22, 2024 at 8:20
  • $\begingroup$ there are not tradeoffs in it's use and one can use it anywhere one feels it is useful. Ultimately wherever you want to know the joint distribution of rates is where PCA will be useful. $\endgroup$ Commented Apr 22, 2024 at 9:24

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My view on this is summarised in my book "Pricing and Trading Interest Rate Derivatives". In it I give a chapter on PCA calculation and demonstrate calculations and applications. But my overall frank assessment as publicised is that I am not a fan of using PCA.

That is not to say it doesn't have merit, I just have the opinion that there are better methods, which supersede it, for various different reasons.

A couple of examples:

Using PCA for risk management?: No thanks, I prefer using VaR related tools under an assumed multivariate normal distribution. The distribution uses exactly the same covariance matrix from which the PCA is calculated but you can do much more (analytically) with VaR analysis which is more relatable to the actual instruments you can trade in a market. You can never, practically trade a PC in isolation so its property of independence to another PC (which is basically one of the main proponents of the theory) is made moot.

Using PCA to generate simulated market movements: No thanks, since the PCA is generated from a covariance matrix, which is a symmetric positive definitive matrix, just use the Cholesky decomposition and a series of normal random variables instead. Easier to do mathematically with the same result.

I should add that I really tried to get into PCA a number of times over the years and try to find a way for it to really work for me. And I tried hard! But couldn't.

One thing I probably would use it for though is if I wanted to create a smooth (denoised) correlation matrix. The poor man's approach sees one selecting the first few PCs and reconstituting a correlation matrix from those. But possibly, if I looked hard enough at research there might be a better way of doing that than using PCA as well. My ignorance in this problem here probably lends PCA a hand.

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  • $\begingroup$ What I feel from your answer, you are also skeptical (don't see much value) about usefulness of PCA application in the context of yield curves, right? $\endgroup$ Commented Apr 22, 2024 at 8:13
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    $\begingroup$ It's better than nothing. Would I cultivate a field with horse and plough? Absolutely. Would I if I had access to a tractor and a cultivator? No. $\endgroup$ Commented Apr 22, 2024 at 15:36
  • $\begingroup$ I see, thank you! But how would you describe uses of PCA in the framework of yield curve? $\endgroup$ Commented Apr 22, 2024 at 17:03
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    $\begingroup$ I work with yield curves every day. I know about PCA and how to apply it. I never use PCA. Surely, that is telling enough for you, along with my answer, as to my opinion on the use of PCA for yield curves? $\endgroup$ Commented Apr 22, 2024 at 17:58
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Let me, as it often happens, disagree with everyone ;)

There is a flame taking place now between LeCun and Musk about what science is. Keeping it simple, science is, among other things, about finding efficient ways of recording the observed quantities. Compressing them, so to speak. PCA, is a wonderful example of it.

  1. As ws already mentioned, denoising correlation matrix is already a good thing. However using such matrix to justify variance/covariance "market risk" under the jointly Gaussian assumption is not good. Variance/covariance is bad, even though FRTB SA attempts to legitimise it in the most brutal way.

  2. Doing nonparametric VAR/ES on the proncipal factors is perhaps better than on the factors themselves, if PCA indicated lower dimensionality. Practically speaking, the confidence interval of VAR (which, being a statistic does have a confidence interval) will be less. That said, in the world where AR(n) is routinely estimated by regressions, who cares?

The true value of lower dimensionality is in the long-term risk.

  1. For the nonlinear pricing problems up to and including XVA gives you much better control of the model specification. You will be usually using a low-dimensional HJM/Cheyette, so you need both sensible historical correlation matrix and just the count of the factors for dynamics.

  2. For long-term risk problems, e.g. counterparty risk, model risk, composite stohastic/deterministic scenarios and trading strategy design, you need to balance the richness of the model so as not to miss out the essential components, with keeping as few factors as possible for performance reasons.

Finally, I'd say modelling rate curves is, perhaps, the easiest multifactor problem. This is because processes are more or less stationary, so PCA is more of a nice additional tool.

PCA as a percieved panacea for the dimensionality reduction in modelling the families of the asset-like quantities results in researches rushing into modelling dependency between returs (with the goal of PCA'ing their correlation matrices) and possibly missing out on stuff like cointegratiok or Garch. Now this is myopic and bad.

Shapes of the USD Libor curves in rhe past few years. Source: BBG

enter image description here

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A few thoughts - sorry, too long for a comment.

Different risk measures are more useful for different uses and different users. For example, imagine a "first line" trader who takes views like "I bet that the 10Y UST - 2Y UST spread will widen!", knows that his book doesn't have much sensitivity to a parallel shift of the curve, doesn't care about other sensitivities that he believes to be immaterial, doesn't even care about the sensitivity to the historical 2nd PC because he thinks he sees the future better than the history predicts and than the other market participants, which is fine. In a large enough organization, there will be a "second line" risk person, whose pay does not improve if the first line wins his bet, who might want to monitor the sensitivities to some numbers of standard deviations of the 2nd PC, providing a little more color than both the VaR, which mashes everything into one number, and the sensitivities to 1bp bump, which don't use historical volatilities.

For example, consider the interest rate risk of an fx forward, where in 1 year you will pay some amount of USD and will receive some amount of emerging market local currency whose present value right now has the same order of magnitude as the USD leg. Suppose you see a risk report showing that the market to market of the USD leg will change by some amount of the USD interest rate moves 1bp (i.e. the dv01); and likewise a much smaller amount by which the local currency leg's dollar pv will change if the local currency interest rate moves 1bp. That looks confusing and useless, until you recall that the local currency interest rate is about 50% per annum, and historically fluctuates hundreds of basis points every day. The VaR is more informative, but it's one number. Few people have a VaR tool that would show component VaR or margin VaR down to the level of one side of an fx forward. Whereas, expressing the interest rate sensitivities as a dollar impact s of 1 standard deviation move of the first PC, this combining the sensitivity with historical volatility, shows more than the dv01.

Or, imagine a junior "first line" trader whose job is to flatten the unwanted rate risk arising as a side effect of more senior traders taking bets on credit spreads or commodities. The only risk measures he cares about are the ones he's tasked with flattening, calculated by the various pricing models. His second line market risk guy might check that he's flattening adequately and ponder whether more sensitivities should be calculated. His second line product control guy might be looking at the sensitivities to curve fitting instruments, ie futures, to minimize unexplained P&L, rather than to the 18m, 1y, 2y.. par swap rates that the market risk guys might monitor. His second line model risk guy might be monitoring some cross-gammas between rates and time in the context of ongoing performance monitoring of pricing models. Etc

Factor sensitivities often need to be aggregable. If you're the only one looking at your risk, then you can choose your tenor buckets to be just 0-2Y and 2Y-$\infty$, and look to sensitivities to forward rates, rather than par rates. But if a second line person needs to add up your sensitivities with someone else's, then you need to agree on what market factors are perturbed and how. Eg you don't want to add up one books sensitivity to forward rate down 1bp to another books sensitivity to par rate up 1bp. If you do want PC sensitivities, then you want to ensure that everyone uses the same loadings.

VaR/ES/SVaR is not enough. Since everyone remember LTCM, let's recall what killed them, a much simpler trade than the complicated stuff on which they earned lots of money:

  • receive US Treasury + Fixed

  • pay USD swap rate

Their view was simply that the swap rate - treasury spread would usually be less than Fixed. But Sandy Weill decided to unwind Salomon's Relative Value book, and the swap rate widened relative to treasury like it never did historically, and LTCM couldn't meet margin calls. You wouldn't see this possibility in VaR. You'd need to look at the spread sensitivity ro decide if it is within your risk appetite. Or, consider what happened to the cash-ftures basis in March 2020 - it widened like it never did before.

MC for interest rate VaR is tricky. If you simulate each tenor with volatilities and correlations, and look at the scenarios in the tail, you will feel that these scenarios don't seem realistic - despite having the specified volatilities and correlations, the curve just doesn't move like this. If you use only the first 3 PCs for MC, then the simulations won't be unusual enough. You need to useat least 5-6 PCs.

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