# Risk Decomposition of Index linked Bonds

Do you know how to decompose the risk of index linked bonds?

To value a inflation linked bond, one plugs the real zero curve into the bond PV calculation, the real interest rate is as the fisher equation tells us: $r_{real}=r_{nom}-r_{be}$, where $r_{be}$ is the break even inflation rate.

Let's say we hava a portfolio of inflation linked bonds and I want to decompose the risk into risk of the nominal interest rate and the break-even rate, or even derive risk contributions for the indidual bonds. How can i achieve this?

Do you have any ideas or any papers covering this topic?

Thank you very much!

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You observe the nominal and real bond prices/YTMs. Transform them to zero or forward curves. Estimate the multivariate dynamics and project them to your horizon, let's call this $X$. Use the distribution of zero/forward rates to obtain the distribution of prices at the horizon (and make assumptions about how your re-invest coupons or how you use cash at maturity), call this $R(X)$. For some holdings $h$, the distribution of portfolio returns at the horizon would be $h'R(X)$.
If you followed the above approach, you can still apply the Fisher equation to obtain the breakeven inflation curve at your horizon. Alternately, you could also have the distribution of the individual bonds. Ultimately, you're gathering together some factors, call them $Z$, that you want to explain the returns of the portfolio.
You would then perform the optimization to minimize the residuals $$d_{w} \equiv argmin\left\{ E\left(\left(h'R\left(X\right)-d'Z\right)^{2}\right)\right\}$$ and solve for d, subject to whatever constraints are appropriate. Note that to compute these residuals requires the conditional distribution between X and Z, refer to the Meucci paper for more details.