**Is it better to adjust a stock’s compounded return for inflation and/or taxes make it better approximate an invariant? **

An investor is interested in a stock’s return after adjustment for taxes and inflation. Thus it would be convenient to project the real after-tax compounded return to the investment horizon. Do adjustments for inflation and taxes make a compounded return a better or worse approximation to an invariant?

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In first approximation you can model inflation as a constant rate. Then the real compounded return is the nominal compounded return minus a constant.

In first approximation, you can model taxes as a constant fraction of the compounded return. Thus compounded net of taxes is the pre-tax return times a constant.

In summary, we are taking the nominal compounded return and applying an affine transformation, which does not affect (neither improves nor worsens) the invariance properties

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I agree on taxes, but would add some caveats on inflation. Inflation certainly has less volatility than the equity market, but I would hesitate to model it as a constant rate across many different markets. Even in the U.S., inflation tends to exhibit persistence and Garch effects. Brazilian inflation shows regime-switching. Further, if you’re considering a portfolio of bonds, then inflation will tend to be cointegrated or correlated with some interest rates. Hence, it may make sense to include log changes in the price level as an invariant and model it jointly with the other variables of interest. It likely does not make sense to split all your variables into real and inflation components so long as you correctly model the correlation/cointegration between the different variables with inflation.The optimization would be like a benchmark-relative optimization, except replacing a market index with the price index.

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The difference between the nominal and real compounded return on an asset is the change in the log of the CPI. That change probably is not iid. (The iid hypothesis for the change in the log of the CPI can be rejected at the .05 level using monthly US data 2001-2012 and a Chow-Denning multiple variance ratio test. An ARMA(1,3) model fits the data well.) Thus it is unlikely that the nominal and real compounded return are both iid. A more plausible possibility is that one or the other is iid. Running Chow-Denning tests (as implemented in R’s vrtest package) on the nominal and real compounded returns for 45 stocks and mutual funds based on monthly data for 2001-2012, I find that the CD1 statistic (large values of which are evidence against an iid hypothesis) is larger for nominal than for real returns for 33 of the 45 assets. The iid hypothesis can be rejected at the .05 level for 9 of the 45 assets in the case of nominal returns but for only 5 of the 45 assets in the case of real returns. My tentative conclusion is that adjustment for inflation is likely to make compounded returns better approximations to invariants.

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Another possibility is to consider the excess returns of the assets (ie nominal return less short-term funding rate). This is the true return stream investors are trying to access since they can always borrow to take this exposure. Therefore, if iid behavior is caused by speculation, then the excess returns would be the iid random variable.

As short rates are correlated with inflation, real and excess returns probably look similar to statistical tests for iid.

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