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# How do you annualise the covariance, mean, variance and standard deviation of a data set?

Hi Experts,

It is very difficult to source a reliable answer to the above question. Based on some of the limited information sourced on random sites it appears that the following is true:

Ann. Stdev = Stdev x Sqrt(Freq)
Ann. Covariance = Covariance x Freq
Ann. Variance = Variance x Freq
Ann. Mean = Mean x Freq

where, Freq is the time period frequency e.g. 52 for weekly, 12 for monthly, etc.

Can someone confirm if this is correct? Also, a little explation into the maths behind the annualisation approach would be greatly appreciate.

Thanks,
j
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1 Solution

ProfessorCommented:
Annualization can mean different things in different fields.  I'm not sure which you're referring to?

I'm not too familiar with the concept, but I do know the mathematics behind the statistics, and frankly, what you've written above doesn't make sense.  (Annualized mean = Mean x Freq) for example.  If your goal is to convert the mean for one period to another period (for example, from monthly to annual), this approach would only weight the mean by the period, which wouldn't produce a meaningful number, I don't think.

Can you say where you came up with these formulae?  And precisely what you mean by annualization (i.e. what is the context of this question)?
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Commented:
Under many definitions of annualize you could annualize X, where X is any variable which is calculated of a period of time less than a year by adding the most recent values of X together until the time period covered is one year then divide that figure by the number of time periods which make up the year.
You get into trouble if your time period does not divide up the year into integer parts.
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For example, if you had weekly averages, you would add the 52 most recent averages and divide the total by 52.
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ProfessorCommented:
Well that's the thing - most of those conversions would require a ratio, not a straight multiplication by the frequency.

I imagined he was talking about annualization in the sense of converting a one-time estimate to an annual metric (i.e. this is the mean for 1 week, what is this in terms of 1 year?)  Which would be the same number that you started with.  Which is why I asked for clarification.
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Commented:
If the time period does not go into a year an integer number of times, just use the nearest integer (both for the divisor and the number of values to be added. That will be close enough.
There is no way you can take the data for just one week and predict the corresponding result for one year unless you know beforehand that the week is typical, in which case you just let that week's value represent the year.
Anything else does indeed require clarification.
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Author Commented:
I am referring to Annulisation in a financial context. e.g. say I have 220 monthly discrete stock returns.
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Author Commented:
This is the URL where I sourced my information for covariance/variances:
http://bear.cba.ufl.edu/karceski/fin6930/pdf%20summer%202005/hw2%20handout%20fin6930.pdf

Extract:
"7. The variances and covariances that we calculated in question #6 are monthly, not annualized (if you
take the variance of monthly returns, you get a monthly variance). To annualize the variances and
covariances, we need to multiply everything by 12. (Note: to convert monthly standard deviations or
TEV to an annualized standard deviation or TEV, we multiply by the square root of 12. To convert
monthly variances or covariances to annualized variances or covariances, we multiply by 12). Compute
the annualized variance-covariance matrix of relative performance, and put this matrix in the range
W94:AH105 in the data worksheet."

This is the URL where I sourced my information for annualised mean:
http://www.stanford.edu/~wfsharpe/ws/wi_perf.htm

Extract:

"The annualized mean monthly return is simply the mean monthly return times 12. The annualized monthly standard deviation of return equals the monthly standard deviation of return times the square root of 12. The annualized geometric mean return is that return that, if earned every year, would compound to give the same cumulative value as did the investment in question. More precisely, if 1+ga raised to the number of years covered will equal one plus the cumulative return."
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Commented:
"The annualized mean monthly return is simply the mean monthly return times 12"
The only way this sentence makes sense is to assume that the first "monthly" is a typo and should have been left out.
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"More precisely, if 1+ga raised to the number of years covered will equal one plus the cumulative return."
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I am unsure about the math involved here, but I do know that the sentence will not parse in English.
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ProfessorCommented:
This seems to be closer to what I assumed.  If you are talking about returns, and you want to know a monthly return standardized to an annual metric, then you would indeed simply multiply that value by the frequency (i.e. if you have a weekly return for Week 1 and want to put it in a form to compare it with other returns that are annual, you would multiply that number by 52).  But that weekly return would be a sum, not a mean.

There's definitely something wrong with your formulas in general: the variance is always the SD^2, which doesn't hold with the formulas you gave.  So something is definitely off.

Here's a question that may help.  Let's assume a monthly return is your base piece of information.  Are you:
1) trying to estimate annual returns from a single monthly return
2) trying to estimate annual returns from daily returns in a month
3) trying to convert a year's worth of monthly returns to an annual metric
4) trying to estimate average monthly returns during a year from a monthly return
5) something else?
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Author Commented:
I can have a set of monthly or weekly or daily returns. I may have any number greather than zero of these returns. This could mean that I have 120 monthly/weekly/daily returns or even just a single monthly/weekly/daily return.
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ProfessorCommented:
If you are trying to convert whatever you have to an estimated yearly return, you'd just need to use this formula: x * (F / N)

Where:
x is the sum of all values you are referencing (i.e. if you are referencing 4 months, add the 4 months together; if you are referencing 13 weeks, add the 13 weeks together)
F is the base rate of your return (Monthly = 12, Weekly = 52, Quarterly = 4)
N is the total number of periods you are interested in (i.e. if you are referencing 13 weeks, use 13)

e.g. to annualize a 4-month running total of \$10000, you would do (10000 * (12/4)) = 30000

You should note, however, that the smaller the fraction of a year you are looking at, the less accurate this estimate will probably be.
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Author Commented:
Richdiesal - Does this work for Covariances ?
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ProfessorCommented:
Well... that's more complicated.  The covariance is essentially an index of the average amount two variables change as a function of one another, using their relative distance from their own means as a method of comparison.  Which means, when you add more pieces of data to a dataset, the covariance doesn't increase - it just gets more accurate.  When annualizing, however, you are trying to change the scale of the covariance, i.e. the number of cases you are averaging against changes.

The formula behind covariance is fairly simple...  you are summing the cross-product of all of the mean differences (Sum of (X - MeanX)(Y - MeanY)) and dividing that by the number of cases minus 1 (N-1).  So I suppose you could just multiply the covariance by (N-1), then run the formula above (x * (F/N)), then divide that value by (12-1) to change the scale.  Producing a final formula of (x*(N-1)*(F/N))/11.

But I will stress that that's a derivation, and more of an educated guess than a real answer.  My expertise is in statistics as applied to social science data, and I've never needed to convert a covariance's scale before.  I would strongly recommend seeking out a finance textbook if you're planning to use this for anything important.  There are likely standards in place there that I don't know about.  The overall return formula, I am confident in - the covariance formula, less so.
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