Real-Rate Sensitivity of Inflation-Linked Bonds

# Interest rate risk of inflation-linked bonds


# Interest rate risk of inflation-linked bonds












According to the Fisher equation, the nominal interest rate $i$ is (approximately) equal to the sum of the real interest rate $r(i)$ and the inflation rate $\pi(i)$. Both the real interest rate and the inflation rate depend on the nominal interest rate in what follows. The present value $PV$ of an inflation linked bond is: \begin{equation} PV=\sum_{t=1}^T{\frac{CF_t\cdot(1+\pi)^t}{(1+r+\pi)^t}} \end{equation} In my opinion, the application of the multivariate chain rule gives: \begin{align} \frac{\partial PV}{\partial i} &=\frac{\partial PV}{\partial r} \cdot \frac{\partial r}{\partial i} + \frac{\partial PV}{\partial \pi} \cdot \frac{\partial \pi}{\partial i}\\ &=\frac{\partial PV}{\partial r} \cdot \Bigg(\frac{\partial r}{\partial i}-\frac{r}{1+\pi} \cdot \frac{\partial \pi}{\partial i} \Bigg) \end{align} because $\frac{\partial PV}{\partial \pi}=-\frac{r}{1+\pi} \cdot \frac{\partial PV}{\partial r}$. However, page two of https://corporate.nordea.com/api/research/attachment/2801 says that $\frac{\partial PV}{\partial \pi}=0$. I do understand that the absolute value of $-\frac{r}{1+\pi} \cdot \frac{\partial PV}{\partial r}$ is rather small, but is it really equal to zero?

Furthermore, is my interpretation correct that inflation risk is part of interest rate risk?

Given the first answer, I would like to clarify my question. To this end, I assume that there is a linear relationship between the real interest rate $r$ and the nominal interest rate $i$: $r=\beta_1\cdot i + \beta_0$. This assumption is in line with the figure on page 3 of the above link. Then, we can continue as follows: \begin{align} \frac{\partial PV}{\partial r} \cdot \Bigg(\frac{\partial r}{\partial i}-\frac{r}{1+\pi} \cdot \frac{\partial \pi}{\partial i} \Bigg) &= \frac{\partial PV}{\partial r} \cdot \Bigg(\frac{\partial r}{\partial i}-\frac{r}{1+\pi} \cdot \frac{\partial (i-r)}{\partial i} \Bigg)\\ &= \frac{\partial PV}{\partial r} \cdot \Bigg(\frac{\partial r}{\partial i}-\frac{r}{1+\pi} \cdot \Big(1 - \frac{\partial r}{\partial i}\Big) \Bigg)\\ &= \frac{\partial PV}{\partial r} \cdot \Bigg(\beta_1-\frac{r}{1+\pi} \cdot \Big(1 - \beta_1\Big) \Bigg)\\ &\geq \frac{\partial PV}{\partial r}\\ \end{align} where I assumed in the last step that $\frac{r}{1+r+\pi}<\beta_1\leq1\Leftrightarrow0<\beta_1-\frac{r}{1+\pi}\cdot(1-\beta_1)\leq1$. (Please note that $\frac{\partial PV}{\partial i}<0$ and $\frac{\partial PV}{\partial r}<0$.) So, in summary, $0>\frac{\partial PV}{\partial i}\geq\frac{\partial PV}{\partial r}$ if $\frac{r}{1+r+\pi}<\beta_1\leq1$, right? This implies that replacing the interest rate sensitivity $\frac{\partial PV}{\partial i}$ by the real interest rate sensitivity $\frac{\partial PV}{\partial r}$ of an inflation linked bond is conservative as long as the condition $\frac{r}{1+r+\pi}<\beta_1\leq1$ holds, correct? If the condition does not hold, we have to multiply the real interest rate sensitivity by the adjustment factor $\Bigg(\beta_1-\frac{r}{1+\pi} \cdot \Big(1 - \beta_1\Big) \Bigg)$. In any case, there is no need to deal explicitly with inflation risk. Do I have an error in thinking?

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/81985

The Fisher equation is more correctly written as $$1+n = (1+r)(1+\pi)$$, where n is the nominal rate , r is the real rate and pi is inflation. Your version approximates the RHS but is not exact. If you plug the proper version into your equation you will find out that the value of an inflation linked bond is indeed just a function of the real interest rate.

## Answer by Jo&#227;o (score 1)

https://quant.stackexchange.com/a/81960

If you over simplify yes:

$$dPV/dπ = 0$$

But this either assumes that:

- Bond values don´t depend on real rates.

- $r=0$, which is also unlikely, since real rates are typically positive.

But of course that this doesn´t happen in real life since inflation affects both discounting and cash flows (for inflation-linked bonds). Even with a small impact assuming that $dPV/dπ = 0$ would just fall under specific conditions.

Conditions which, from the link that you provided:

This is true because the bond structure neutralizes inflation risk.

But for academic questions yes I don´t see the problem if you note it is as an assumption.

" Furthermore, is my interpretation correct that inflation risk is part of interest rate risk? "

For a nominal bond, inflation risk and interest rate risk are connected:

- If inflation rises, nominal rates often increase, causing bond prices to fall.

For an inflation-linked bond, cash flows adjust with inflation, reducing direct inflation risk. but, real rates still matter, so interest rate risk remains.

## Answer by Jan (score 0)

https://quant.stackexchange.com/a/81987

Thank you very much, dm63! I think now I got it:

This implies that the interest rate risk of an inflation linked bond reduces to the risk of changes in the real interest rate:

Is this true now?