Hi can anyone tell me how to calculate annualized standard deviation for 5 years daily prices.
If you have prices five days a week, that’s 261 prices per year. Multiply the daily σ by √261 (≈ 16.16) to get the annual σ.
If you have prices seven days a week, that’s 365 prices per year. Multiply the daily σ by √365 (≈ 19.11) to get the annual σ.
S2000 is right if you are assuming your data has no serial dependence, and therefore scales with the square root of time (i.e., raise to the one-half power). However, if your data is serially dependent and has a Hurst exponent (“H”) > 0.5, it may be preferable to scale your daily data by the estimated H of the series, and not the square root, as this would imply a true random walk in the series, which may not effectively describe the evolution of your distribution. If H > 0.5 and you assume a square root scaling factor, you may be systematically underestimating realized ex-post volatilities in your models/forecasts.
Keep in mind if working with other inputs such as a correlation matrix, it is appropriate for the matrix to be scaled in the same fashion (however, I would advise to keep in mind the effects of two potentially different H values in the joint distribution estimates). See Leon and Reveiz (2011) at https://www.bis.org/publ/bppdf/bispap58e.pdf.
Hi, thanks for the information, but i still have a doubt do i need to multiple daily returns by square root of (261*5) or should i multiply daily returns with square root(261) only??
Also please brief me about the reason also.
Thanks
Uh, well, the assumption is that the smallest instances in s series of observations (in this case, the convention is to use daily stock price movements) scale with the square root of time. The intuition for this stemmed from the physics concept of Brownian motion, named for Robert Brown but explained by Albert Einstein, to describe the path of particles through fluid. The behavior of the particles was random, hence it scales at a random walk and covers the distance/area described by a square root (i.e., a continuous process limit such that H = 0.5.).
S2000 describes 261 trading days, but some use 260, still others use 256, and so on. Truthfully, if you’re near 260, it won’t matter that much. Where that number comes into play is that you take the daily volatility and multiply it by the square root of 261 (or whatever your number of annual trading days is). If you were working with monthly data, you’d multiply by the square root of 12, etc.
Again, refer to my other post for some issues that exist with this method.
Yes i got your point
261 days are for one year only but for calculation of 5 years, shouldn’t we use 1305 days? Should not we multiply daily returns with square root of (1305) days??
If you multiply by √1,305 (≈ 36.12), you will get the 5-year σ, not the annual σ.
Good point.
I wasn’t aware of the Hurst exponent; I’ll have to study up on it.
Exactly. We use one year because we assume that the standard estimate needed is annualized vol. You are simply using the time window of data you have to come up with that (5 years in this case), but if the OP uses 36.12 as a multiplier, you don’t end up with a one-year annualized vol. If you had even more data, you’d still multiply by the square root of 261 – the only thing the increased data window does is supposedly increase the accuracy of your estimate of one-year vol. Make sense OP?
It’s an interesting body of research. Probably beyond the scope of what the OP is looking for, it seems. Of course, there remains this idea that if you do scale by something else besides the square root of time, does it lead to a Gaussian estimator like standard deviation being, itself, an inappropriate estimator for your data? Mandelbrot suggested these ideas in the 1960’s, but it seems as though Fama and the rest of the University of Chicago economics school of thought won out in defining “modern finance,” so these ideas did not end up in most textbooks. The math required for even simplistic Gaussian-based finance is difficult enough; when one adds long-term serial dependence and Cauchy distributions, the math becomes so intractable for most that people just defer to the (imperfect) convention.
However, if you ask me, it isn’t difficulty insomuch as it is an exposure problem. If we were exposed to this math in FIN101 for 60 years in addition to the limit case Gaussian random walk, it would have become rote by now.
Got your point now, thanks 