Inflation β = 1 meaning a “perfect inflation hedge?”

Question

I read somewhere (Ang, Brière, Signori: Inflation and Individual Equities, 2012) that in a given period if the inflation rate rises by 1% point (say from 1% to 2%), and the return of Asset A rises from 7% to 8%, then Asset A is “a perfect inflation hedge” during that period.

I am struggling to understand why this behavior means Asset A is a “perfect inflation hedge.” I’m hoping someone can mathematically show why this is the case. For example, I think of an inflation hedge in the following way: if Asset A returns 5% in period 1, and inflation was 2% in the same period, then Asset A “hedged” inflation by providing a return above and beyond the inflation rate. Where I’m getting confused is the idea that a change in the rate of inflation (from 1% in one period, to 2% in the next period), coupled with achieving a return greater than the pervious period by exactly the difference in the inflation rate (2% - 1%), means the returning asset is a “perfect inflation hedge.” I can’t seem to prove to myself that this makes sense.

Using the above example: the inflation rate is 1% in period 1, and Asset A returned 7% in period 1. The inflation rate then jumps by 1% to be 2% in period 2, and Asset A’s return jumps by 1% to be 8% in period 2. This would represent a relationship where the inflation beta is equal to 1 and can be interpreted to mean “a 1% rise in inflation means a 1% rise in returns.” How is that representative of Asset A being a “perfect inflation hedge?” I understand by increasing returns by the same increase in the rate of inflation means our (nominal return - inflation rate) stays the same, but that isn’t the actual real return. The real return would indeed change even assuming the same increases in returns and inflation rate because real return is [(1+ nominal return)/(1+ inflation rate)]-1

Screenshot of academic paper here : https://ibb.co/y4GZXHL

Answer 1

You need to think of inflation as a macro factor in the cross-section of returns.

What exactly is a good inflation hedge? $$ r_{i,t}-r_f = \alpha_i + \beta_\pi^i \epsilon_{\pi, t} + u_{i,t} $$ If $\beta_\pi^i = 1$ we have a perfect inflation hedge. Meaning that if inflation surprises $\epsilon_{\pi, t}$ go up by 1%, return on your stock goes up by 1% as well.

Regarding your second point assume everything $r_{i,t}$, $\epsilon_{\pi, t}$ is in logs.

Edit: It is super accurate:

1. Let $r^{nom}_{i,t}$ and $r_f^{nom}$ be nominal log returns.
2. To get real returns, given that we are working in logs we just subtract log inflation: $r^{nom}_{i,t} - \epsilon_{\pi, t}$ and $r^{nom}_{f} - \epsilon_{\pi, t}$
3. Not it is trivial to see that inflation cancels out.