# What are some advanced methods for bond risk transformations?

Consider a portfolio of bonds within a given yield curve (e.g. Gilt curve), consisting of positions in every bond in the curve. I'm looking for ways to transform the risk of the portfolio into estimated representations of the macro/on-the-run buckets.

For example, the Gilt curve (roughly 55 bonds) has a liquid 5y bond, a liquid 10y bond, and a liquid 30y bond, and the net position of the portfolio may be such that it trades similarly to some combination of the 5y, 10y, and 30y bonds.

The challenge is how to transform the risk representation of 55 bonds (a 1-dimensional vector with 55 elements) into a risk representation of $$(5y,10y,30y)$$. A few well known methods are outlined below:

Method 1: PCA

PCA (Principal Components Analysis) is a well-known method for transforming the risk of a bond portfolio into its macro components. For a bond curve with $$n$$ bonds, the portfolio risk is $$S_A\in\Re^{n\times 1}$$. After applying PCA with 3 components, we get a PCA matrix of $$P\in\Re^{n\times 3}$$. We can roughly represent the macro risk of our portfolio by doing $$S_B=P^TS_A, S_B\in\Re^{3\times 1}$$ This isn't quite what I'm after, but it will give us the outright risk (i.e. the risk of the portfolio to parallel shifts in the curve), curve risk (e.g. roughly the 5s10s position of the portfolio), and butterfly risk (e.g. roughly the 5s10s30s position of the portfolio.

Method 2: Jacobian Transformation

Consider our risk vector $$S_A$$. Now consider that we want to break our risk down into the 3 most liquid points (the 5y, 10y, and 30y points). We generate a jacobian matrix $$J\in\Re^{3\times n}$$, where $$J_{ij}$$ ($$i\in[1,3]$$ and $$j\in[1,n]$$) is a number describing how bond $$i$$ is related to bond $$j$$. This could be the $$\beta$$ from regressing the change-on-day of each bond $$j$$ against each bond $$i$$ separately. Then we can get our $$(5y,10y,30y)$$ risk vector by doing $$S_B=JS_A.$$ This will give our clear macro risk broken down into the 5y, 10y, and 30y buckets.

Other methods

My question is what other methods exist in the literature to transform the risk of a full bond portfolio into more macro components to get a simplified view of the risk.

I'm generally looking for more advanced methods that may not typically be implemented in practice due to impracticalities. Happy to receive both explanations of the methods as well as references to literature.

• >After applying PCA with 3 components. Isn't this number rather arbitrary? I could see how higher order PCs can be difficult to interpret, but still why not choose more or less components to explain a certain threshold in variance. – Sergei Rodionov Mar 23 at 8:19
• I understand that with risk representation you mean something like $DV01=PV(y+1bp)-PV(y)$ or $CS01=PV(r,cs+1bp)-PV(r,cs)$ or something alike? – Kermittfrog Mar 23 at 10:20
• @SergeiRodionov The number is rather arbitrary (except for PC1 since this quite explicitly explains the outright movements) - hence why I'm in search for other methods... – quanty Mar 23 at 21:39
• @Kermittfrog By risk representation I mean a vector of $n$ numbers where each number is the dv01 risk in the bond, for example, risk vevtor $S=(r_1,r_2,r_3,r_4,r_5,r_6,r_7,r_8,r_9,r_{10},r_{12},r_{14},r_{15},r_{16},r_{18},r_{20},r_{22},r_{25},r_{27},r_{28},r_{30},r_{35},r_{37},r_{39},r_{40})$ where $r_i$ is the dv01 position in bond $i$. I'm looking for transformation methods to reduce this down to broad macro exposure such as $(r_5,r_{10},r_{30})$ which is much more useful for managing overall risk – quanty Mar 23 at 21:44
• The transformations that I commonly see in banks' risk teams is to calculate bonds' zero rate sensitivities w.r.t. various 'pillar' nodes on a 'base/pure' interest rate curve (let's say swap for simplicity) and another 'stacked' curve of credit spreads. The idea here is to more or less manage the bond portfolio's risk using liquid reference instruments for interest rate risk (swaps) and for the credit spread risk - As you have stated in your method 2, more or less. This way, risk/pnl is linearly attributed to reference instruments' performance. – Kermittfrog Mar 24 at 7:20