standard deviation of annualized monthly data vs. annualized monthly standard deviation

[tushiroda]

So this is really more of a stats question but I'm a bit stuck on it and this board has some sharp minds so here goes.

I have a set of monthly data of say 100+ data points. the time conversion to annualize the standard deviation is square root of time or sqrt(12) in this case.

However, if I annualize the individual points (x12) and take the standard deviation of that, the result is ~3x of the annualized monthly standard deviation in the prior line..

I must be missing something obvious but the formulas are straightforward and look ok so I am confused as to which standard deviation to use.

thanks much in advance for any insight into this!

 

[KRice]

If you multiply the monthly data by a factor (say 12), you will scale both the mean and the standard deviation by that same factor...so I'm not sure why you got a factor of ~3 (it should be 12). The relative standard error (sd/mean) should remain the same, however. It appears that you are relying on a population standard deviation rather than a sample standard deviation. How many points are you using to create an effective annualized value?...are you taking a full year's worth of monthly results and attempting to present a single annualized result?

 

[tushiroda]

hi Kirk,

there are 94 points in this particular case so the population/sample n / n-1 shouldn't be much of a difference maker (I think).

in the case of standard deviation it is scaling by 12 as you noted just like the mean. the confusion on my end is why there is material online that in order to scale stdev by time to use the square root of time. so to convert monthly to annual you would take the monthly stdev and multiply by sqrt of 12 instead of just multiplying by 12.

so I am trying to generate annualized values (scaling by time from monthly values) from the monthly mean and stdev. for stdev is x 12 the number to use or is x SQRT(12)?

thanks!

 

[KRice]

If you are using the built in functions in Excel to compute mean and standard deviation, the standard deviation is based on whatever time period might be associated with the data points, as the formula only considers the count of data points, N (or N-1 if a sample standard deviation function was used). I believe you need to recover the square root of the sum of squared deviations and then convert that to an effective annualized amount:
standard deviation from Excel = STDEV.P(full range of 94 points) gives a monthly standard deviation

To annualize: (standard deviation from Excel) * SQRT(12/N)
This last term would be SQRT(12/94) = 0.357

 

[KRice]

Please disregard my post #4...I don't think that is correct. I'll post back if I find an answer. For clarification, does each data point represent something like a monthly return on investment?

 

[tushiroda]

hi Kirk,

the monthly data is a % delinquency. I think the expectation is for an annualization to result in a > 1 scaling for a standard deviation from monthly to annualized so is that x12 or x sqrt(12) which is x 3.46. if there is another option consider me even more confused.