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From my understanding, the annualized standard deviation of daily returns is generally higher than of annualized standard deviation of weekly returns is generally higher than.... monthly...quarterly... year standard deviation.

My question is... when is this general rule not true? What would need to happen? or what would the portfolio look like where the longer time frame annualized has a higher standard deviation than the shorter term annualized standard deviation?

Could people give some examples and explanations please

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  • $\begingroup$ I think this is a very good question but the title could be clearer ... could you find one? $\endgroup$ – Richard Jan 11 '16 at 12:22
  • $\begingroup$ that's the best i can come up with, if anybody wants to edit it to make it more clear, then go ahead $\endgroup$ – jason Jan 11 '16 at 12:42
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    $\begingroup$ I tried .. ;) just adding characters for the remark to be able to post. $\endgroup$ – Richard Jan 11 '16 at 12:44
  • $\begingroup$ why not "Historical volatility as a function of return period lengths" ? $\endgroup$ – will May 11 '16 at 10:33
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In the set of an index where all insturments are traded in the same time zone I would agree that vola pa from say weekly returns is lower than from daily returns. Besides this, the distribution of weekly returns should look "more" Gaussian than the one of daily returns. This is called aggregational Gaussianity e.g. in the paper by Rogers and Zhang. The term "more Gaussian" can be concretized that excess-kurtosis tends to decrease.

However if you can assume that the assets traded have closing prices taken at some quite different point in time (e.g. US and JP) then correlations will be underestimated. Thus in a long-only portfolio (or an index) volatility will increase if you use weekly returns as this mismatch does not matter that much with weekly returns. On the other hand a short position could hedge more effectively observing weekly returns. Finally if you have to convert foreign stocks to your home currency you have to take the "closing price" of the currency. Again the choice of the point in time matters less with weekly returns.

Looking at ADR/GDRs and the local stocks also show weak correlations on a daily basis whereas on a weekly basis they show a correlation close to one.

These phenomena lead to auto-correlation in portfolio/index return time series. These auto-correlations are reduced if one passes to lower-frequency time series (e.g. weekly) or they should be adressed as we do it in our paper.

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  • $\begingroup$ You see something related to what you say here when you look at the correlation between assets closing at different times. as the return period is increased (but not too much) it approaches a more meaningful correlation when the time zones are different. $\endgroup$ – will May 11 '16 at 10:36
  • $\begingroup$ I am not sure whether I correctly understand your remark. But yes: increasing the return period reduced the effect of different closing times. $\endgroup$ – Richard May 11 '16 at 11:35
  • $\begingroup$ if you look at a correlation swap between an american asset and an american asset, the period of the returns (i.e. $R_i = \frac{S_i}{S_{i-p}}$) makes little to no difference (in the long term). If you take an american and a japanese asset though, the correlation is lower for 1 day returns and then will approach the true correlation as you increase $p$ (while becoming less responsive to changes in correlation). $\endgroup$ – will May 11 '16 at 13:23
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It is most common to use the "square root of time" method to scale volatility (i.e. standard deviation of returns) to a year (annualize it) if needed, i.e. if the estimate is based on a sample with higher frequency (daily, weekly,..).

Mathematically this requires the underlying stochastic process $(X_t)_{t\in T}$ (I've omitted some technical prerequisites here) to be i.i.d., meaning the random variables $X_t$ have the same distribution and are independent.

So far for the theory but in practice this is quite often not the case meaning stochastic independence cannot be assumed. For example the empirical time series exhibit autocorrelation which is a hint for some kind of dependence.

I would suggest the paper of Diebold as a starting point for further reading:

Converting 1-Day Volatility to h-Day Volatility: Scaling by $\sqrt t$ is Worse than You Think. Wharton Financial Institutions Center, Working Paper 97-34.

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