Scaling DV01 by a Principal Component Yield-Curve Shock
Summary
The document asks how to express a swap’s DV01 as exposure to the first principal component of daily U.S. Treasury yield changes. The proposed method normalizes the PC1 loading vector to a one-basis-point move at a chosen anchor tenor, then uses the resulting shifts across maturities to represent a curve-wide shock. The example anchors the fourth tenor, identified as 5Y, and reports smaller moves at the short end and larger moves at the long end.
This is a conceptual question rather than a validated risk framework. The text does not establish that the anchor tenor is the most likely to move by one basis point, or that the resulting PC-scaled measure is inherently more realistic than DV01. PCA loadings describe historical covariance patterns, while a position’s risk also depends on its tenor-specific sensitivities and the sample and scaling choices. The calculation is therefore best read as a proposed way to define a PC1 shock scenario, not as a general replacement for DV01.
Key ideas
- A principal component can describe a correlated yield-curve move across multiple maturities.
- The proposed calculation rescales PC1 so the chosen anchor tenor moves by one basis point.
- The example’s normalized PC1 shock implies different changes at the other tenors.
- A PC-based risk measure depends on the historical sample, scaling convention, and chosen anchor.
- The document raises, but does not resolve, whether this measure is superior to DV01.
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Full text
# How to interpret DV01 in terms of PCA equivalent?
# How to interpret DV01 in terms of PCA equivalent?
I have computed the PCs for daily changes in the UST yield curve over the last 5 years using the covariance matrix and can explain 94% of the variance using the first two components, which is good enough for what I'm doing.
The first component is level with each tenor implying a move in the same direction. What I'm struggling with is how to interpret this in terms of a 'DV01 equivalent' move. Let's say that I have a swap position that has a DV01 of $100k. My goal is to determine the PC1 equivalent of this number, which I think will be a more realistic measure that as each point will move in the same direction, but not necessarily by 1 basis point.
I do the following: a) determine an anchor point (e.g. 5Y) and reformulate the PC1 in terms of a 5Y equivalent, as I think that the tenor that will most likely move by 1bp is the 5Y tenor; b) I then determine the shift applied to the rest of the yield curve using the following formulation below,
$ \textbf{PC1} = \Bigg[ \textbf{SD} \cdot \bigg[ \textbf{P} \cdot \big[ \textbf{Sc} \cdot \textbf{SD} \cdot \textbf{P} \big] ^ {-1} \bigg] \Bigg] \cdot \big[1\big]$
where,
- $\textbf{SD}$ is the standard deviation matrix of size $n \times n$ where $n$ is the number of tenors
- $\textbf{P}$ is the first Principal Component vector, which is simply the eigenvalues of the covariance matrix of the daily returns of size $n \times 1$
- $\textbf{Sc}$ is a scalar vector that represents a 1bp shock at the 5Y point, so is a $n \times 1$ vector with 0 at each element apart from 1 at the corresponding 5Y tenor
The output of the above expansion does indeed lead to an $n \times 1$ vector that has a shift for every tenor and exactly 1 at the 5Y point, so the result does appear to be correct.
However, I just wanted to confirm that I was thinking of this correctly. The following vectors are the first two PCs,
```
PC2 PC1
-0.15028 0.05154
-0.32138 0.13310
-0.41856 0.20852
-0.41465 0.26116
-0.33315 0.29555
-0.19980 0.31919
-0.04874 0.33416
0.08760 0.34173
0.19759 0.34324
0.27930 0.34002
0.33455 0.33347
0.37023 0.32502
```
I have assumed the vector $\textbf{Sc}$ is $[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]$.
So if the 5Y tenor corresponds to the 4th element in the vector, the equivalent 5Y PC1 measure result (in basis points),
```
PC1 5Y equivalent (bps)
0.071660453
0.336473214
0.707324248
1
1.204548309
1.340536029
1.429582409
1.489268877
1.517832566
1.518270303
1.496125159
1.462623964
```
So am I right in saying that the result above is more realistic as a risk measure than DV01 in that if the 5Y point moves by 1bps, the long end will move by 1.4-1.5bps and the short end by 0.7bps?
Thank you.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.