# Why adjust for inflation annually, as opposed to realising it after the holding period?

Why do investors offset their annual returns against inflation, when inflation isn't realised until the end of the holding period (when the investor is actually spending the money)?

An example:

Assume the following:

• Stock with a 10% annualised return.
• Stock receives 50% of its growth from dividends - these are 'immediately' re-invested.
• Holding period is 50 years.
• Inflation is 3% per year.

Most people appear to calculate the inflation-adjusted return as:

``````Inflation Adj. Return = (1 + (0.10 - 0.03))^50 = 29.457
``````

Another method, which makes more sense to me (with my current understanding), is as follows:

``````Nominal Return = (1 + 0.10)^50 = 117.391
Inflation = (1 - 0.03)^50 = 0.218
Inflation Adj. Return = 117.391 * 0.218 = 25.599
``````

The results are very different.

Method 2 only realises inflation at the end of the holding period, but still accounts for the compounding effect against the dollar over time.

Method 2 also assumes "spending" your dividends by re-investing in the same stock is not affected by inflation. The rational is since you are purchasing the same "product" as the one you've just sold, so there's no realization of any loss or gain. There is, of course, some cash drag, brokerage fees and taxes - but no/minimal inflation (to my understanding).

Why is method 1 then used instead of method 2?

• Holding period isn't necessarily a year or finishes at the end of the year. Secondly if somebody wants to modify his portfolio, he needs to understand how his portfolio is performing so he can rearrange his holdings. Dec 9, 2015 at 14:17
• In this case, we're assuming a multi-year investment: 50 years, specifically. Re. your second comment: that's fine, but my question is: why would they use method 1 to do that, rather than method 2? Assuming you're only 4 years into your investment - you can still calculate inflation-adjusted earnings using method 2 instead of method 1. My question is: why wouldn't you? Dec 9, 2015 at 14:35
• if you're using an annual inflation rate over a time period of more than a year you need to take into account that it is compounded; a 1% inflation rate is the change of prices over the last year so to cover 2 years you must either use multiple inflation rates or compound the average rate. Dec 9, 2015 at 15:17
• My question takes compounding into account in both cases - did I miss something (or did you)? Notice the `^50` in the calculations - that's for the 50 year holding period. Dec 9, 2015 at 15:48

I would use neither method. Taking a short example first, with just three compounding periods, with interest rate 10%. The start value `y0` is 1.

``````y0 = 1;
y1 = (1 + 0.1) y0;
y2 = (1 + 0.1) y1;
y3 = (1 + 0.1) y2 = 1.331
``````

So after three years the value is 1.331, the same as `y0 (1 + 0.1)^3`.

Depreciating (like inflation) by 10% (to demonstrate) gets us back to `y0 = 1`

``````y2 = y3/(1 + 0.1);
y1 = y2/(1 + 0.1);
y0 = y1/(1 + 0.1) = 1
``````

Appreciating and depreciating by 10% cancels out:

``````y0 = 1;
y1 = (1 + 0.1) y0/(1 + 0.1);
y2 = (1 + 0.1) y1/(1 + 0.1);
y3 = (1 + 0.1) y2/(1 + 0.1) = 1
``````

Appreciating by 10% interest and depreciating by 3% inflation:

``````y0 = 1;
y1 = (1 + 0.1) y0/(1 + 0.03);
y2 = (1 + 0.1) y1/(1 + 0.03);
y3 = (1 + 0.1) y2/(1 + 0.03) = 1.21805
``````

This is the same as `y0 (1 + 0.1)^3 (1 + 0.03)^-3 = 1.21805`

So for 50 years the result is `y0 (1 + 0.1)^50 (1 + 0.03)^-50 = 26.7777`

Note

You can of course use subtraction but not using the inflation figure directly. E.g.

``````x = 0.03 (1 + 0.1)/(1 + 0.03) = 0.0320388

y0 (1 + (0.1 - x))^50 = 26.7777
``````

(edit: This appears to be the Fisher equation.)

2nd Note

Further to comments, here is a chart to illustrate how much the relative performance improves when inflation is accounted for. The first fund's return is 6% and the second fund's return varies from 3% to 6%. Inflation is 3%. • Great answer, thanks Chris. Will re-read over then next few days and no doubt mark as the answer :) Dec 9, 2015 at 16:16
• Interestingly, John Bogle seems to disagree with both your method and my 2nd method. He states "deducting the same inflation rate from both figures (the CAGRs) further increases the comparative advantage of the investment with the higher return" (taken verbatim). According to Bogle, inflation has an exponential effect on the HPR that is relative to the annual investment return. This precludes you from deducing an 'inflation coefficient' for some period that can be applied to any investment of the same period (which is what both your method and my 2nd method imply). Could Bogle be wrong? Dec 9, 2015 at 16:33
• Simply put, he gives an example that makes 'equity Y' 57% better than 'equity Z' after inflation. However, before inflation, 'equity Y' is only 52% better than 'equity Z'. In both our methods (my method 2 and your method) the comparison would be unaffected by inflation. Dec 9, 2015 at 16:43
• Assuming a 50 year holding period, without inflation: 6% fund `(1 + 0.06)^50 = 18.4202`. 4% fund `(1 + 0.04)^50 = 7.1067`. That's a `38.58%` relative difference. With 3% inflation: 6% fund `(1 + 0.06)^50 * (1 + 0.03)^-50 = 4.2018`. 4% fund `(1 + 0.04)^50 * (1 + 0.03)^-50 = 1.6211`. That's still a `38.58%` relative difference. What am I missing? (The 6% fund did not perform any better, relatively speaking, with or without inflation, using the method you described.) I'm aware I must be missing something! Dec 9, 2015 at 17:18
• Without inflation, 6% fund gain divided 4% fund gain: `(((1 + 0.06)^50) - 1)/(((1 + 0.04)^50) - 1) = 2.85264` times better. With inflation: `(((1 + 0.06)^50 * (1 + 0.03)^-50) - 1)/(((1 + 0.04)^50 * (1 + 0.03)^-50) - 1) = 5.15512` times better. Dec 9, 2015 at 21:00