Why Inflation Cancels in Cross-Sectional Excess Returns

Summary

The document addresses whether cross-sectional asset pricing models such as CAPM and the Fama-French three-factor model require nominal or inflation-adjusted returns. Its central point is that either convention can be used when comparing assets over the same period: subtracting the same period’s inflation rate from each asset’s nominal return leaves pairwise return differences unchanged. Thus, a common inflation adjustment cancels in cross-sectional comparisons.

The answer illustrates this with log returns, defining real returns as nominal returns less inflation. Subtracting one asset’s real return from another’s yields the same spread as subtracting their nominal returns, so excess-return comparisons are unaffected under this setup. The explanation is concise and addresses the cross-sectional comparison in the question, rather than offering an empirical test or a full treatment of asset-pricing model estimation. Its conclusion relies on a common inflation measure and synchronized return periods; it does not discuss cases where assets have different inflation exposures or where the choice of deflator matters.

Key ideas

• A common inflation adjustment subtracted from each asset’s return cancels in cross-sectional return spreads.
• Nominal and real pairwise return differences match when the same inflation measure and period are used.
• The answer applies this identity to excess returns in CAPM and Fama-French models.
• The explanation does not address heterogeneous inflation exposures or alternative deflators.

Tags

# Nominal vs. real (inflation-adjusted) prices/returns in cross-sectional asset pricing


# Nominal vs. real (inflation-adjusted) prices/returns in cross-sectional asset pricing












I have the impression that asset pricing models such as the CAPM or Fama & French 3 factor model typically concern nominal rather than real (inflation-adjusted) prices/returns. If this is indeed so, why is that?

Here is my guess. In cross-sectional asset pricing, there is no inherent time dimension (that is why it is called cross sectional), so the concept of inflation is irrelevant. Yet the models are estimated on data from multiple periods, so the time dimension is present in the data. Also, I suppose adjustment for inflation might not make a big difference when using daily data but it could become important when using monthly (or even lower frequency) data.

References to relevant texts would be appreciated.

Another question with a similar title but somewhat different content (more focus on continuous-time finance, risk-neutral measure and such) is this one.

## Answer by Andre (score 1, accepted)

https://quant.stackexchange.com/a/75743

You can use nominal or real returns in the CAPM or Fama-French model. Both models have expressions for excess returns. As inflation will adjust nominal returns in the same way for different assets in the cross section, inflation cancels out.

More concretely, if $r^i_t = \log(P^i_t) - \log(P^i_{t-1})$ are nominal returns of asset $i$, $\pi_t = \log(\text{CPI}_t)-\log(\text{CPI}_{t-1})$ is inflation, and $\nu^i_t = r^i_t - \pi_t$ are real returns of asset $i$, then $r^i_t - r^j_t = \nu^i_t - \nu^j_t$. Excess returns will be equal if you use nominal or real returns.

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