Measurement interval vs Annualized Standard Deviation

[haphols_cfa]

Hello - I am getting mixed up with regards to how shortening the measurement interval impacts annualized standard deviation. I can’t seem to understand why the info below would be incorrect.
Shortening the measurement interval for an annualized standard deviation decreases the annualized standard deviation. Here are my calculations:

If StDev is 12% annually

Semi-Annually
Are we able to say the semi-annual StDev is 6%
Annualized StrDev would be 6% * SqRt(2) = 8.48%

Monthly
Are we able to say the monthly StDev is 1%
Annualized StrDev would be 1% * SqRt(12) = 3.46%

Daily
Ex: let’s say daily standard deviation is (12%/250) = .048 StDev per day
Annualized StrDev would be 0.048 * sqRt(250) = .7589%

Conclusion: So, the shorter the measurement interval, the lower the annualized Standard Deviation. Which, when incorporated into a Sharpe Ratio would increase the Sharpe because it appears there is a decrease to volatility.

Where has my logic and/or calculations gone awry? Thank you.

 

[S2000magician]

If the annual standard deviation (of returns) is 12%, then the semiannual standard deviation (of returns) is:
12% ÷ √2 = 8.49%, not 6%.
And so on.

 

[haphols_cfa]

In searching, I found a post by cpk123 here .
His conclusion, like the others I have seen:
Lengthening measurement interval - reduces the Std Deviation - and thus Sharpe Ratio can be gamed.
But, there are no calculations. Is it possible to support his statement calculations. Perhaps that will help me see where I’m going off track.
Thank you.

 

[haphols_cfa]

S2000magician - I think the results are the same between your formula and mine.
And it appears if you calculate based on monthly StdDev and daily StdDev the StdDev will continue to decrease. So it still doesn’t seem to make sense to me. Because we are decreasing the interval and the annualized standard deviation is decreasing.
cpk123 concluded the opposite.
Thank you.

 

[cpk123]

that is not the point of the whole thing.
My understanding.
Monthly returns, monthly std deviation.
Now Annual Std Dev = Monthly Std Dev * sqrt(12).
monthly * sqrt(12) would be LESS THAN Actual Annual Std Dev based on Annual Returns. that number would be lower than an annualized standard deviation based on annual returns.
That number (monthly * sqrt(12)) would be higher than (Daily Std Deviation * sqrt(250))
due to the nature of the returns and higher smoothing when the returns are calculated on a smaller time period - this causes an issue. You would have a smaller std deviation on daily returns when compared to standard deviation on monthly returns which would be still smaller than std deviation on annual returns.
if you however used a daily std dev to calculate the annual std dev and then moved to look at the sharpe ratio - you would be gaming the process.
Now look at the impact of Sharpe ratio - and make your judgement.