# PCA for Risk bucketing

I'm working on a project to justify the use the certain tenors (2y, 5y, 10y, 30y) for risk bucketing. I'm a little stuck after calculating the principal components. Just to describe my approach-

a) Got the GBP swap rates for 1y - 60y for 2 years of historical data (Let's say X matrix).

b) Calculated the daily differences and then the correlation matrix

c) Calculated the eigen values and eigen vector matrix (say A)

d) To get back the PC rates Y (say, PC01, PC02,...PC60), I did a matrix multiplication of demeaned X matrix and eigen vectors (say, X_dm and A)

e) Plotted PC01, PC02 and PC03 and the shape confirmed what is expected for the level, slope and curvature

f) From what I had read in an article, if I plotted PC01 against 5y or 10y swap rate, they would mostly be following the same pattern. However, this is something I couldn't confirm. (Providing a link of the article)

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?

Unfortunately, since I'm working on this from my office, I can't paste any figures/charts I have generated.

Articles referred - https://www.garp.org/#!/risk-intelligence/all/all/a1Z1W000003rQUYUA2

https://www.clarusft.com/principal-component-analysis-of-the-swap-curve-an-introduction/

One more question - if anyone goes through the GARP article, how is the figure 2 actually generated? Any ideas?

• Is your question a technical one, or on the interpretation of your results? Can help with both, but cant tell from your question which one it is. For the figure you asked about, it's probably generated using ggcorrplot in R – Bram Feb 20 at 7:24
• Hi Bram, i guess its both ,i.e. after step e , how do I proceed to show that 2y, 5y, 10y, 30y move independently but other tenors may not. I plotted PC01, PC02, PC03 along with the four rates , but didnt see any pattern. Also ran a correlation among these to see if it would give any meaningful results, but it didnt prove much. – access_nash Feb 20 at 8:16
• 3. Relative Value Analysis – richness/cheapness of the curve can be monitored by PCA residuals, What do they mean by pca residuals here? @Bram – Permian Feb 21 at 10:25
• @ Permian - residuals refer to the rest of the principal components that account for a very small percentage of the total variance. i.e. if your first three PCs explain 97% of your data, then the rest of the 3% are the residuals – access_nash Feb 21 at 12:27

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% and 4th pc:5%) and if the 2Y/5Y/10Y/30Y swap rates respectively account for most of the variance in each principal component, then the knowledge of those variables gives a good grasp of the total information contained in the swap rate curve.

You could also do clustering (4 clusters for example) and examine which variable define its class the most.

However, IMO the use the tenors (2y, 5y, 10y, 30y) for risk bucketing is simply due to the fact they are traded the most.

• " if the 2Y/5Y/10Y/30Y swap rates respectively account for most of the variance in each principal component" - and how do I get this information? Isn't the matrix multiplication of the demeaned variables and the factor loadings equal to the factor score matrix? How do I interpret in the way as you have mentioned? – access_nash Feb 20 at 13:22
• PCA results provide factor coefficients for every variable on each component which identify the relative weight of each variable in the component in a factor analysis. Those coefficients are used to compute the factor scores I mentionned. The larger the absolute value of the coefficient, the more important the corresponding variable is in defining the component. I believe that those coefficients are comprised between -1 and 1 (I look at them like some sort of correlation measure but I am not sos ure about this). – Wane Mamadou Feb 20 at 14:33
• And yes, the factor score matrix is computed as you say. You can also create a variable for each principal component and estimate a linear model to see which variable is relevant in the explanation of one factor – Wane Mamadou Feb 20 at 14:36
• Thanks for your input. I will also consider the clustering approach – access_nash Feb 21 at 12:28

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 portfolio $$Q$$ such that the PnL of $$P + Q$$ is minimised over some metric, i.e. final PnL or norm of daily pnl changes.

The question you are then asking is if you restrict yourself to having only swaps in $$Q$$ from 2Y, 5Y, 10Y, 30Y buckets, is this effective, and is this more effective than some other combination of buckets.

It seems like a computationally intensive statistical analysis or machine learning task. Generate lots of random portfolios $$P_i$$, calculate the hedges $$Q_i$$ with bucket restrictions on $$Q_i$$ and see what you get. Note I suggest this since you might also be able to factor convexity/cross gamma in this scenario which is likely to be a concern in terms of accrued PnL.

Alternatively (for a linear analysis) you could use your covariance matrix and assume a multi-variate Guassian distribution of rates and possibly come up with the proof mathematically on paper. I might try that tomorrow and re post.

# Modelling Under MultiVariate Normal

If you make the assumption that market movements are multivariate-normally distributed with mean zero and covariance estimator as above then, $$\mathbf{\Delta r} \sim \mathcal{N}(\mathbf{0}, \mathbf{\Sigma})$$.

A risk metric associated with a portfolio of risk exposures, $$P$$, is the variance-covariance VaR standard deviation: $$c = \mathbf{P^T \Sigma P} \;, \quad \text{where VaR-99pc is} \quad c_{99} = \Phi^{-1}(0.99)c$$ for $$\Phi(x)$$ the standard normal cumulative distribution function.

Suppose you want to hedge, or simulate, that portfolio with another portfolio of risk exposures, $$\mathbf{Q}$$, then the risk of the difference can be expressed as: $$c^* = \mathbf{(P^T-Q^T) \Sigma (P-Q)}$$

Obviously if you can 'trade' all exposures then the minimal risk (i.e zero) is found when $$\mathbf{P=Q}$$. However, in your case you are restricting $$\mathbf{Q}$$ so that many of its elements are zero and only those corresponding to the 2y, 5y, 10y and 30y buckets are permitted, i.e. $$\mathbf{Q} = [0,..,0,q_{2y},0,...,0,q_{5y},0,..]$$

You seek the values $$q_{2y}, q_{5y}, ..$$ such that $$c^*$$ (the residual risk) is minimal. Since this is a quadratic form there is a unique solution. I will save the matrix calculus and simply state that: $$\mathbf{\hat{Q}} = \mathbf{\hat{\hat{\Sigma}}^{-1}\hat{\Sigma}P}$$ where $$\mathbf{\hat{Q}}$$ is the non-zero elements of $$\mathbf{Q}$$, $$\mathbf{\hat{\Sigma}}$$ are the rows of $$\mathbf{\Sigma}$$ corresponding to the non-zero elements of $$\mathbf{Q}$$, and $$\mathbf{\hat{\hat{\Sigma}}}$$ are the rows and columns of $$\mathbf{\Sigma}$$ corresponding to the non-zero elements of $$\mathbf{Q}$$.

So now based on your covariance matrix you have the analytical hedge quantities, $$q_{2y}, q_{5y},..$$ which will minimise VaR for the given portfolio $$\mathbf{P}$$.

# How is this useful

1) Firstly you now have a means of assessing the quality of your hedge. $$c^*$$ is the minimum residual VaR: are you happy about its size given your known portfolio, $$\mathbf{P}$$?

2) You can choose a different combinations of hedging instruments and calculate the residual VaR again. Perhaps it is better. Obviously the more instruments you include the better smaller the residual VaR will become.

3) If you are interested in the quality of a 2y,5y,10y,30y hedge in general then you would like to examine $$E[c^*]$$, where the expectation is measured with respect to the probability space of all possible portfolios. This is quite difficult to measure since you need to define a probability space of portfolios and that is non-trivial; a balanced portfolio (offsets) is much more likely than having every bucket with large risk exposure in the same direction. However, if you assumed a uniform distribution you could probably still come up with results, probably analytic ones.

4) You can extend 3) to ascertain for your initial assumptions what is the optimal combination of allowed hedging instruments for the general case of unknown portfolio given the initial assumptions.

• Thanks....yeah I don't have to use PCA. Just that I want to replace a regression analysis that was done earlier and which seemed incorrect (regression with time series data and not accounting for serial correlation). Will have to figure out how to use this approach in practice – access_nash Feb 21 at 12:31

On the technical side of things, make sure that your actual PCA analysis is correct. A reasonable check here is to plot the loadings of all tenors (not just the few you are interested in) for the first 3 factors and see if you recover level, slope and curveature components.

Now if your results are correct, the reasoning goes something like: the first three principle components represent X% of the risk (where X probably is >= 95%). So you then can translate any exposure (from any bucket) into an exposure on the first 3 components. Finally, by picking a short, medium and long term bucket, you then should be able to minimize all risks on the 3 first components as they should be linearly independent (if you pick 4 points and three components, they aren't obviously).

Now, what the argument above ignores is whether the risk assessment that you're doing using PCA is sufficient for your purposes. For relatively simple risk management purposes, it might be, but for the purpose of managing a rates book at a bank, it probably isn't.

• Thanks for your comments. The plot for the loadings matches what I have seen in most analysis and the first 3 components do account for close to 97% of the variance. I didnt get the last part of your second paragraph though. I must say I'm not doing this for risk mgmt. More from a model review perspective where we need to document proof for bucketing used in case delta bid offer valuation adjustments. – access_nash Feb 20 at 9:33
• I will probably add a few more details when I'm back home. My firm doesn't allow me to post anything on stack exchange but only read (posting these comments on my phone right now) – access_nash Feb 20 at 9:45