# Math Help - (Annualised) Standard Deviation/Volatility - Standard practice and method confusion.

1. ## (Annualised) Standard Deviation/Volatility - Standard practice and method confusion.

I am trying to work out the monthly standard deviation/volatility of a portfolio to compare it relatively against a benchmark i.e. to see how much more or less volatile the client portfolio is compared to the benchmark for their returns.

I did my calculations but am having a debate with a couple of people as to what figures are used to keep things consistent and in line with standard financial practise (e.g. fund manager on their portfolio returns etc)

They are hypothetical numbers. I have done 4 calculations. My feeling is that I use either calculation 1 or 2 (and that 3 and 4 are incorrect).

I do have daily data if needed…

A couple secondary questions once I know which is the best way (on what numbers) to calculate the volatility:
1. Is it common practise to annualise the monthly standard deviation? From my experience it is… but others are saying just to show the straight figure?
2. Do/should you measure the volatility over all the months from the clients portfolio inception, or do it on just the latest e.g. 12 months, rolling, to keep it relevant?
2. When annualising (I admit my knowledge in this area is limited), say I have 13 months of data, is the formula STDEV([13 monthly values])*SQRT(12), or STDEV([13 monthly values])*SQRT(13) or other?

Thank you so much for any time and help,
Kindest Regards
KJ

2. ## Re: (Annualised) Standard Deviation/Volatility - Standard practice and method confusi

I can't answer the first two questions because they require some knowledge of the industry rather than knowledge of mathematics but I can answer the third question. For annualising you should always multiply by $\sqrt{12}$ if the variable you are measuring is being added up over twelve months. But the rates of return on money aren't being added together, the return is compounded.

If $r_1, r_2, r_3,...r_{12}$ are the rate of return on investments in each month so that r is calculated by: $\frac{\text{money at end of period}}{\text{money at start of period}}$ not by $\frac{\text{money earned over period}}{\text{money at start of period}}$

Then after 12 months the annual rate of return R is
$R=r_1 \times r_2 \times r_3 \times... r_{12}$

Taking the log (base 10) of both sides
$log(R)=log(r_1 \times r_2 \times r_3 \times... r_{12})$

$log(R)=log(r_1)+ log(r_2)+ log(r_3) +... + log(r_{12})$

Since the log of the rate of return are added together it would make sense to find their standard deviation and multiply it by $\sqrt{12}$

So for example, if you found that the standard deviation of the log of the rate of return was 0.01 then the standard deviation of the log of the annual rate of return is $0.01 \sqrt{12}=0.0346$

And the standard deviation of the annual rate of return (removing the log) is $10^{0.0346}=1.083$

This would mean that that point that is one standard deviation below the mean annual return is $\frac{\text{mean annual return}}{1.083}$

Two standard deviations below the mean would be found by
$0.01 \sqrt{12} \times 2=0.0692$

$10^{0.0692}=1.173$

$\frac{\text{mean annual return}}{1.173}$

So I would suggest that you change your analysis from looking at percentage return to looking at log return

Another thing to note is that the standard deviation of the money earned from inception doesn't produce any meaningful number because the numbers used to compute it are not measuring the same thing, the first number in column F is measuring how much is earned from 1 month of investing, the second number is measuring how much is earned from 2 months of investing, ect.

3. ## Re: (Annualised) Standard Deviation/Volatility - Standard practice and method confusi

Thank you Shakarri for the reply! working though what you wrote now.