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Why Sharpe Ratios Can Differ Across Models and Benchmarks

Article Quant Q&A · Author: Simon Rydstedt
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Summary

The question asks whether the same portfolio or index can have different Sharpe ratios when analyzed with CAPM or Arbitrage Pricing Theory. The answers distinguish ex-post Sharpe ratios, calculated from realized excess returns and their variability, from ex-ante ratios based on expected returns. If two asset-pricing models imply different expected returns, they can produce different ex-ante Sharpe ratios even when the risk-free rate is held fixed.

The risk-free benchmark is another source of variation: using a Treasury security, deposit certificate, or bank account with a different return changes the excess-return numerator. The exchange therefore suggests that there is no contradiction in reporting different ratios when inputs or the estimation target differ. It does not set out a full calculation or address every convention, such as return frequency or annualization. To compare reported Sharpe ratios, identify whether they are realized or model-implied and check that the benchmark and return assumptions match.

Key ideas

  • Ex-post Sharpe ratios use realized returns, while ex-ante ratios depend on expected returns.
  • CAPM and APT can imply different expected returns and therefore different model-based Sharpe ratios.
  • Different risk-free benchmarks change the excess return used in the ratio.
  • A meaningful comparison requires consistent definitions, benchmarks, and return assumptions.

Tags

StatisticsRisk managementPortfolio constructionFactor investing
Full text
# Can there be different Sharpe Ratios for the same index?


# Can there be different Sharpe Ratios for the same index?












I am reviewing a fellow students paper, and it is argued in this paper that the Sharpe Ratio can differ based on which model is used to analyze the portfolio returns. Here a model based on the Arbitrage Pricing Theory which includes macroeconomic variables, and a classic CAPM model.

It is argued that the they calculate the Risk Premium based on the model, and then use this to calculate the Sharpe Ratio.

Is there any truth to this? As far as I know, from inspecting the Sharpe Ratio equation, there is only one Sharpe Ratio, period.

Thank you.

## Answer by Dave (score 2)

https://quant.stackexchange.com/a/74275

One possible explanation is that Sharpe ratio involves a comparison to a “risk-free” asset. If the two calculations use different risk-free assets that have different returns, then the Sharpe ratios would differ.

Some people might consider US Treasury bonds to be risk-free. Some might consider a certificate of deposit to be risk-free. Some might consider a bank account to be risk-free. All of these have different returns and would lead to different Sharpe ratios.

## Answer by AKdemy (score 2)

https://quant.stackexchange.com/a/74276

While differences in Risk Free rates used can lead to differences, I do not think that the difference mentioned in the paper refers to differences in risk free rates. I would say that it discusses differences in ex-ante Sharpe ratios due to differences in models.

For example, APT and CAPM will unlikely result in the same expected return of an asset. Hence, the ex-ante SR will be different, even if you use the same risk free rate.

Wikipedia's page about SR explains the difference betwenn ex-ante and ex-post SR, although I think it is quite self explanatory.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.