Using PCA to Simulate Bond Yield Curves for PRIIP Category 3
Summary
The answer explains how to model a bond for PRIIP Category 3 by representing its value as dependent on a yield curve across maturities. Rather than simulating every tenor’s rate independently, it recommends applying principal component analysis to the covariance matrix of curve returns, retaining the three factors associated with the largest eigenvalues, simulating those factors, and reconstructing curve return paths from the PCA transformation. The bond can then be repriced from the simulated curve, with interpolation used to obtain the yield at its maturity.
The response clarifies that the relevant curve may include multiple tenors, even when the target is one bond; the inputs therefore describe curve movements rather than just that bond’s price returns. It does not resolve the question’s detailed uncertainties about bootstrap sampling, combining bond and equity results, or calculating portfolio VaR. It offers a conceptual outline, without worked calculations or evidence comparing this procedure with alternatives.
Key ideas
- Model a bond as priced from a yield curve spanning multiple maturities.
- Apply PCA to the covariance of returns across the curve’s tenors.
- Simulate the three principal factors with the largest eigenvalues and reconstruct curve return paths.
- Reprice the bond using its yield interpolated from each simulated curve.
- The answer does not specify how to combine bond and equity bootstrap results for portfolio VaR.
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# PRIIP Category 3 Curves # PRIIP Category 3 Curves Good evening, I've tried searching similar posts, but most are unanswered or in a more advanced step than what I'm trying to achieve. I've managed to do the boostrap method for spot prices underlyings. I've confirmed the results with the EIOPA slides and it's working as intended. What I don't understand however, is what to do when my underlying is a bond. The legislation states that I must do a PCA to my residual maturity matrix. Is this matrix for all my bonds (ex: each column is the maturity of a bond?) or just for one underlying? Why do I need a matrix in this case instead of a vector? The rest of the steps seem clear except for what they mean about the return matrix. Again, if I have one underlying that is a bond, is this just a vector of returns (calculated from the market prices of my bond) like I have for spot price underlyings? In step 23.b) I also have questions: It seems I will create a NxT matrix similar to the other bootstrap method, where N is 10k (simulation number) and T is the number of days in the RHP, what I don't understand is what value I'm going to sample to fulfil my matrix. The steps don't seem very clear to me. And in the end, how shall I join both boostrap methodologies results to value my PRIIP? If I have 1 bond and 1 equity, I'm going to obtain 10k results for each, do I take the 9750th value of both vectors and sum them to obtain my total VaR ? Thanks for the help, and please ask for more details if I wasn't clear explaining a point. ## Answer by Antoine Conze (score 0, accepted) https://quant.stackexchange.com/a/37078 Think of the specific bond you are interested in as being priced off a curve (e.g. a govt bonds par rates curve). The curve is actually represented by a vector of rates $(R_1,...,R_I)$ indexed by tenors $(T_1,...,T_I)$, and when repricing a specific bond you first interpolate its yield in the curve. Now if you want to generate sample random paths for your bond price, you first need to generate sample random paths for your curve $(R_1,...,R_I)$. Now if $I$ is large that would be time consuming, so what the legislator says is - do a PCA on the variance-covariance matrix for the returns of $(R_1,...,R_I)$; - select the 3 principal factors $(x_1, x_2, x_3)$ (those that correspond to the 3 largest eigenvalues); - generate sample random paths for $(x_1, x_2, x_3)$, then reconstruct the sample paths for the returns of $(R_1,...,R_I)$ from $(x_1, x_2, x_3)$ and the PCA transition matrix.
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