# Fixed Income risk attribution in the historical simulation of a sovereign bond portfolio

We use historical simulation for risk analysis. I.e. for each bond there is a repricing of the form $$P_j = PV(\text{yield curve in scenario } j),$$ where the yield curve is the zero rates curve of the respective country (for a German bond I take the German curve, for an Italian bond the Italian, ...). The various scenarios of the yield curves are calculated from historical shifts along the curve applied to the present curve and for all markets simultaneously (thus modelling correlations among the bond markets).

Then for each bond we calculate returns $r_j = P_j/P_0-1$, thus the return of the price in scenario $j$ based on the current market price $P_0$. Then the portfolio return in scenario $j$ is calculated by $$r_j^P = \sum_{i=1}^N w_i r_j^i,$$ where $w_i$ is the current weight of bond $i$ in the portfolio and $r^i_j$ is its return in scenario $j$. Then portfolio volatility can be estimated as the standard deviation of $r_1^P,\ldots,r_j^K$,ie. $\sigma^P$, if we have $K$ scenarios.

Furthermore I can calculate volatiltiy contributions $\sigma_i$ for each bond by $$\sigma_i = w_i covar(r^i,r^P)/\sigma^P,$$ then $\sum_{i=1}^N \sigma_i = \sigma^P$

All this is quite standard.

In an EMU portfolio (say: Germany, France, Italy) I have various effects: overall change in interest rate levels (more or less risk free, this is more or less the German yield curve) and changes in the market valuation of more risky countries (as additional yield w.r.t the German level) - a sort of credit risk.

What methods do you apply to separate interest rate risk from credit risk in a sovereign portfolio? What are your experiences? Are there any references?

All the data I have is the curves for each country. I would like to avoid using data from credit derivatives.