<p>Not 100% sure of what you need help with, but ill throw some things here and if it doesn't help tell me and ill delete this answer. Since you asked to &quot;mathematically show why this is the case&quot;, i will try to show why <span class="math-container">$\beta = 1$</span> in your example.</p>
<p>If <span class="math-container">$\varepsilon_t$</span> is the part of the nominal return not explained by inflation, than it should respect some kind of exogeneity. Lets take strict exogeneity: <span class="math-container">$E(\varepsilon_t \mid \pi_t) = 0$</span>. Now consider the following regressions for period 1 and 2</p>
<p><span class="math-container">\begin{align*}
    R_{i1} = \alpha + \beta \pi_1 + \varepsilon_1 \\
    R_{i2} = \alpha + \beta \pi_2 + \varepsilon_2
\end{align*}</span></p>
<p>Taking the expected value in both of them</p>
<p><span class="math-container">\begin{align*}
    E(R_{i1} \mid \pi_1) &amp;= E(\alpha \mid \pi_1) + E(\beta \pi_1 \mid \pi_1) + E(\varepsilon_1 \mid \pi_1) \\
&amp; = E(\alpha \mid \pi_1) + E(\beta \pi_1 \mid \pi_1) \\
    E(R_{i2} \mid \pi_2) &amp;= E(\alpha \mid \pi_2) + E(\beta \pi_2 \mid \pi_2) + E(\varepsilon_2 \mid \pi_2) \\
&amp; = E(\alpha \mid \pi_2) + E(\beta \pi_2 \mid \pi_2)
\end{align*}</span></p>
<p>Now let us consider your example. When <span class="math-container">$\pi_1 = 1\%$</span>, we should have (with certainty) <span class="math-container">$R_{i1} = 7\%$</span>. Similarly, when <span class="math-container">$\pi_2 = 2\%$</span>, we should have <span class="math-container">$R_{i2} = 8\%$</span>. This means that <span class="math-container">$E(R_{i1} \mid \pi_1 = 1\%) = 7\%$</span> and <span class="math-container">$E(R_{i2} \mid \pi_2 = 2\%) = 8\%$</span>. Let <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> be arbitrary constants, then</p>
<p><span class="math-container">\begin{align*}
    7\% = E(R_{i1} \mid \pi_1 = 1\%) &amp; = E(\alpha \mid \pi_1 = 1\%) + E(\beta \pi_1 \mid \pi_1 = 1\%) \\
7\% &amp;= \alpha + \beta 1\% \\
\end{align*}</span></p>
<p>and</p>
<p><span class="math-container">\begin{align*}
    8\% = E(R_{i2} \mid \pi_2 = 2\%) &amp; = E(\alpha \mid \pi_2 = 2\%) + E(\beta \pi_2 \mid \pi_2 = 2\%) \\
8\% &amp;= \alpha + \beta 2\% \\
\end{align*}</span></p>
<p>Since these two equations are satisfied simultaniously, we have the following system of equations</p>
<p><span class="math-container">\begin{align*}
7\% &amp;= \alpha + \beta 1\% \\
8\% &amp;= \alpha + \beta 2\%
\end{align*}</span></p>
<p>subtracting the first equation in the second one</p>
<p><span class="math-container">\begin{align*}
1\% = \beta 1\% \iff \beta = 1
\end{align*}</span></p>

CC BY-SA 4.0
2021-09-01T23:03:59.803Z
I think the problem with my understanding was that one needs to think of inflation and returns both as logs. That way a beta of 1 means no change in real log returns (real log return is simple log return minus log inflation). Whereas with simple returns, a beta of 1 does not mean the real return stays the same (simple return minus inflation does not accurately reflect real return, and using actual real simple return formula shows that we need beta > 1 to achieve stable real simple return. Does this sound right to you?
user3138766
CC BY-SA 4.0
2021-09-02T08:38:48.653Z
@Br.M: I don't think you need a regression. The $\beta$ is implicit in the example, once you make the assumption that asset returns that for each percentage point change in inflation returns change by the same amount (which you correctly did in your regressions).
BrsG
CC BY-SA 4.0
2021-09-03T03:40:23.887Z
I wanted someone to mathematically show how a 1% point rise in return offsets a 1% rise in inflation. To be clear we are talking about the inflation rate rising by 1% point while return rises by 1% point.
user3138766
CC BY-SA 4.0
2021-09-01T18:51:18.947Z
2021-09-01T19:12:49.107Z
2021-09-01T19:12:49.107Z
Br.M
<p><strong>Yes, if returns always change by the same percentage as inflation, it's a perfect inflation hedge.</strong></p>
<p>What is a perfect inflation hedge? It's a position whose change in returns perfectly correlates (positively or negatively) with inflation. That means you can take a (long or short) position scaled such that other investments are hedged against inflation.</p>
<p>If you knew ahead of time that whenever inflation changes by x percentage points, the return of an asset changes by the same percentage points, then that condition would be satisfied and the asset would be a perfect hedge.
This is the case in your example, provided that for <em>each</em> percentage point change in inflation the return of the asset also changes by the same amount.</p>
<p>The example is a little confusing because asset returns are explicit and relatively high (so looks like a good investment), but for hedging purposes it's only the change in returns that matters. Also, the fact that the asset is a perfect hedge doesn't tell you anything about whether you should actually hedge, that is, whether it's worth it. This decision will depend on your risk attitude, on the size of expected inflation changes, and the <em>price</em> of the asset.</p>

CC BY-SA 4.0
2021-09-02T16:46:36.600Z
Doesn’t this interpretation only work if we are using log returns and log inflation? If we use simple returns, having an inflation beta = 1 means for every 1% point rise in inflation, our real simple return actually declines a bit. Using log returns, a 1% point rise in return coinciding with a 1% point rise in inflation does yield the same real return because real log return  = nominal log return - log inflation. So in this sense I think inflation beta = 1 being a perfect inflation hedge implies we are using log returns. Correct?
user3138766
CC BY-SA 4.0
2021-09-02T08:30:22.310Z
2021-09-02T12:02:18.017Z
2021-09-02T12:02:18.017Z
BrsG
<p>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.</p>
<p>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.</p>
<p>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?”</p>
<p>Screenshot of academic paper here : <a href="https://ibb.co/y4GZXHL" rel="nofollow noreferrer">https://ibb.co/y4GZXHL</a></p>

CC BY-SA 4.0
2021-09-01T16:39:40.363Z
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
Community
CC BY-SA 4.0
2021-09-01T16:44:55.380Z
Done! Hope that helps clarify. Thank you.
user3138766
<inflation><capital-returns>
If return rises by same % as inflation rate rises, then perfect inflation hedge?
user3138766
# If return rises by same % as inflation rate rises, then perfect inflation hedge?

(Asked by: username_0 on 2021-09-01)

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?”

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

**Tags:** inflation, capital-returns
> **Comments:**
> **username_1**: Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.  
> **username_0**: Done! Hope that helps clarify. Thank you.  

---

## 2 answers

### Answer by username_2 on 2021-09-01. Score: 4

Not 100% sure of what you need help with, but ill throw some things here and if it doesn't help tell me and ill delete this answer. Since you asked to "mathematically show why this is the case", i will try to show why $\beta = 1$ in your example.

If $\varepsilon\_t$ is the part of the nominal return not explained by inflation, than it should respect some kind of exogeneity. Lets take strict exogeneity: $E(\varepsilon\_t \mid \pi\_t) = 0$. Now consider the following regressions for period 1 and 2

\begin{align\*}
R\_{i1} = \alpha + \beta \pi\_1 + \varepsilon\_1 \\
R\_{i2} = \alpha + \beta \pi\_2 + \varepsilon\_2
\end{align\*}

Taking the expected value in both of them

\begin{align\*}
E(R\_{i1} \mid \pi\_1) &= E(\alpha \mid \pi\_1) + E(\beta \pi\_1 \mid \pi\_1) + E(\varepsilon\_1 \mid \pi\_1) \\
& = E(\alpha \mid \pi\_1) + E(\beta \pi\_1 \mid \pi\_1) \\
E(R\_{i2} \mid \pi\_2) &= E(\alpha \mid \pi\_2) + E(\beta \pi\_2 \mid \pi\_2) + E(\varepsilon\_2 \mid \pi\_2) \\
& = E(\alpha \mid \pi\_2) + E(\beta \pi\_2 \mid \pi\_2)
\end{align\*}

Now let us consider your example. When $\pi\_1 = 1\%$, we should have (with certainty) $R\_{i1} = 7\%$. Similarly, when $\pi\_2 = 2\%$, we should have $R\_{i2} = 8\%$. This means that $E(R\_{i1} \mid \pi\_1 = 1\%) = 7\%$ and $E(R\_{i2} \mid \pi\_2 = 2\%) = 8\%$. Let $\alpha$ and $\beta$ be arbitrary constants, then

\begin{align\*}
7\% = E(R\_{i1} \mid \pi\_1 = 1\%) & = E(\alpha \mid \pi\_1 = 1\%) + E(\beta \pi\_1 \mid \pi\_1 = 1\%) \\
7\% &= \alpha + \beta 1\% \\
\end{align\*}

and

\begin{align\*}
8\% = E(R\_{i2} \mid \pi\_2 = 2\%) & = E(\alpha \mid \pi\_2 = 2\%) + E(\beta \pi\_2 \mid \pi\_2 = 2\%) \\
8\% &= \alpha + \beta 2\% \\
\end{align\*}

Since these two equations are satisfied simultaniously, we have the following system of equations

\begin{align\*}
7\% &= \alpha + \beta 1\% \\
8\% &= \alpha + \beta 2\%
\end{align\*}

subtracting the first equation in the second one

\begin{align\*}
1\% = \beta 1\% \iff \beta = 1
\end{align\*}

> **Comments:**
> **username_0**: I think the problem with my understanding was that one needs to think of inflation and returns both as logs. That way a beta of 1 means no change in real log returns (real log return is simple log return minus log inflation). Whereas with simple returns, a beta of 1 does not mean the real return stays the same (simple return minus inflation does not accurately reflect real return, and using actual real simple return formula shows that we need beta > 1 to achieve stable real simple return. Does this sound right to you?  
> **username_3**: @Br.M: I don't think you need a regression. The $\beta$ is implicit in the example, once you make the assumption that asset returns that for each percentage point change in inflation returns change by the same amount (which you correctly did in your regressions).  
> **username_0**: I wanted someone to mathematically show how a 1% point rise in return offsets a 1% rise in inflation. To be clear we are talking about the inflation rate rising by 1% point while return rises by 1% point.  

---

### Answer by username_3 on 2021-09-02. Score: 1

**Yes, if returns always change by the same percentage as inflation, it's a perfect inflation hedge.**

What is a perfect inflation hedge? It's a position whose change in returns perfectly correlates (positively or negatively) with inflation. That means you can take a (long or short) position scaled such that other investments are hedged against inflation.

If you knew ahead of time that whenever inflation changes by x percentage points, the return of an asset changes by the same percentage points, then that condition would be satisfied and the asset would be a perfect hedge.
This is the case in your example, provided that for *each* percentage point change in inflation the return of the asset also changes by the same amount.

The example is a little confusing because asset returns are explicit and relatively high (so looks like a good investment), but for hedging purposes it's only the change in returns that matters. Also, the fact that the asset is a perfect hedge doesn't tell you anything about whether you should actually hedge, that is, whether it's worth it. This decision will depend on your risk attitude, on the size of expected inflation changes, and the *price* of the asset.

> **Comments:**
> **username_0**: Doesn’t this interpretation only work if we are using log returns and log inflation? If we use simple returns, having an inflation beta = 1 means for every 1% point rise in inflation, our real simple return actually declines a bit. Using log returns, a 1% point rise in return coinciding with a 1% point rise in inflation does yield the same real return because real log return  = nominal log return - log inflation. So in this sense I think inflation beta = 1 being a perfect inflation hedge implies we are using log returns. Correct?  

---

