# What details must be considered when assessing an asset’s inflation beta?

The inflation beta of an asset is described as:

$$r_{i,t}-r_f = \alpha_i + \beta_\pi^i \epsilon_{\pi, t} + u_{i,t}$$

For shorthand, I will use “return” to mean excess return.

In academic literature it is common to define an asset as a “perfect inflation hedge” if the asset’s inflation beta = 1. If for every 1% point rise in inflation we see a 1% point rise in the return of the asset, then the asset is a “perfect inflation hedge.”

I have several concerns:

1. Does the above definition only make sense when working with log returns and log inflation? If we are working with simple returns, a 1% point rise in returns is not quite enough to combat the 1% point rise in inflation. For example, consider the following scenario: asset with simple return of 7% in period 1, and 8% in period 2. Inflation rate of 1% in period 1 and 2% in period 2. In this scenario, the inflation rate rose by 1% point over the same period returns rose by 1% point. The real simple return in period 1 = 5.94%, and in period 2 = 5.88%. Now consider if the 7% and 8% returns were log returns, and 1% and 2% were log inflation. Then, using (log return) - (log inflation) = (real log return), we have the real return in period 1 = 6%, and in period 2 = 6%. In the simple return case, an inflation beta = 1 does not yield an unchanged real return - inflation had a negative impact on our return. In the log return case, we preserve our real return. Does this not mean an inflation beta = 1 = “perfect inflation hedge” rely on our using log returns?

2. Could we use simple returns and actual inflation for inflation beta?

3. In addition to an inflation beta = 1, do we not also need to have Pearson’s r = 1 for the asset to be considered a “perfect inflation hedge?”

4. Assuming we do need to use log returns and log inflation, how does that change the interpretation? If using simple returns, if I’m right you wouldn’t be able to use inflation beta = 1 to describe a perfect hedge, but you could say “for every 1% point rise in actual inflation (not log inflation), then we expect a say, 5% point, rise in the nominal simple return of the asset in question. If we use log returns and log inflation, doesn’t this interpretation lose tons of meaning since we are interpreting an inflation beta = 1 as meaning “for every 1% point rise in log inflation, we expect a, say, 5% point rise in the nominal log return of the asset in question?

I’m hoping someone will take the time to hit all the above concerns in a very convincing way, leaving no doubt whatsoever. I’m grateful and thankful for your time.