How Outlier Returns Distort Annualized Risk Statistics Across Frequencies

# Answer by Enrico Schumann (score 3, accepted)


# Single outsized daily return value creates substantive discrepancy between annualized variance calculated from daily vs monthly returns












I am new here, and to the field. I hope my clunkiness in expressing myself can be forgiven.

My situation is as follows: I have around three years of daily return data for some financial asset. Out of these three years there is a single day on which the return is outsized compared to all the other days. (Or well, it is actually more than a single day, but for the sake of the argument let's assume we have one clear outlier.)

This single day of outsized return causes the variance (and stdev) calculation to be very sensitive to the return interval that I choose to use.

If I would lower the resolution and translate the daily return data into monthly return data, the annualized variance of the asset would shoot up (by a lot!) due to the simple fact that we still have one data point with an outsized return, but now this data point represents a month instead of a day. In other words, its relative impact on the annual variance has gone up.

The end result is that I end up with completely different variances (and as a result covariances, beta measures, standard deviations, Sharpe ratios, etc.) depending on the return interval that I choose to use. In my specific case: when changing from daily to monthly returns, the resulting Sharpe Ratio more than halves and the resulting beta more than doubles(!)

Two questions:

- I suppose the lowest resolution available return interval (in my case: daily) gives me the most accurate approximations of variance (and all the indicators that follow from it)?

- Is this discrepancy common? When I read a Sharpe Ratio or an alpha measure on some fund manager's performance sheet, am I supposed to assume that these numbers can easily double (or halve) just by changing the return interval from which they are calculated??

## Answer by Enrico Schumann (score 3, accepted)

https://quant.stackexchange.com/a/76009

Regarding 2:

It is common that single (or a few) return-observations have disproportional influence on summary statistics. See, for instance, this paper:

```
@ARTICLE{,
  author       = {Daniella Acker and Nigel W. Duck},
  title        = {Reference-Day Risk and the Use of Monthly Returns Data},
  journal      = {Journal of Accounting, Auditing and Finance},
  year         = 2007,
  volume       = 22,
  pages        = {527--557},
  number       = 4
}
```

I don't think this sensitivity is as "well-known" as it should be. (Richard Hamming once said that something is well-known if one can find it in the literature.)

Regarding 1:

In a single sentence, instead of looking for a "best" or "correct" setting, a better approach is to de-emphasize or skip point estimates and always report, analyze and compare ranges of outcomes, for various settings.