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I have a portofolio with 30 indexes and I want to calculate the annulised returns and volatility because I want to compare it with another portofolio with different number of indexes (but same time period) My data are time series with 3 month frequency from 2009-12-31 to 2020-3-31.

I know that the general formula is: $$annualised \enspace return = (1 + total \enspace returns)^N - 1$$ where $total \enspace returns$ is the last value minus the first divided by the first. $N$ is the period I want to annualised and here is my doubt.

  1. If I have 3 month frequency data, what is the best value for $N$?
  2. how to get the volatility after?
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I think all the previous answers have small mistakes:

Given that you have derived the return over the period of interest, i.e. in your case 2009-2020 we can then:

  1. Compute the return at the granularity level of your data i.e:

$r_{quarterly}=(1+r_{total_{period}})^{\frac{1}{number_{datapoints}}}-1$

This is then the return of the whole period on a quarterly basis!

  1. Now we can annualise it accordingly:

$r_{annualised}= (1+r_{quarterly})^4-1$

and this is because we have 4 times those 3 month periods in a year.

I hope this helps.

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If you compute total return as:

$$R_{2009-12-31 \rightarrow 2020-3-31} = \frac{P_{2020-3-31} + D_{2020-3-31}}{P_{2009-12-31}}-1$$

Where $D_{2020-3-31}$ are all the dividend paid in that period. Then you can do the following:

  1. First note that your sample has $12 \times 10 + 3$ months; I.e. 123 months.
  2. So the average monthly return is:

$$\bar{r}_{monthly} = (1+R_{2009-12-31 \rightarrow 2020-3-31})^{\frac{1}{123}} -1 $$

  1. The average annual return will be: $$\bar{r}_{annual} = (1+\bar{r}_{monthly})^{12} -1 $$
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  • $\begingroup$ thanks for you prompt answer. Well I have no such information as Dividend paid, so I guess that is zero in my case which turns to make your formula equal to mine (P2020/P2009 - 1) for the total return. My doubt on your approach is that you compute the monthly returns, but I have 3 monthly data so, shouldn't I calculate the tree-monthly returns first (instead of 123 months is 40 trimestral)? after this I would then follow your point 3 as you describe. How you would approach then volatility for the same purpose? Many thanks in advance $\endgroup$
    – Luigi87
    Jun 26 '20 at 13:32
  • $\begingroup$ Can you explain what you mean you only have three months of data? How does your data look like? $\endgroup$
    – phdstudent
    Jun 26 '20 at 13:39
  • $\begingroup$ sure, my bad. I mean the first sample is 2009-12-31 the second sample is 2010-03-31, my time series x-axis has 1 sample every 3 months. $\endgroup$
    – Luigi87
    Jun 26 '20 at 13:49
  • $\begingroup$ (1+𝑅2009→2020)*3/123 would be average quarterly return, but r(annual) will not change. $\endgroup$ Jan 23 at 9:30

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