Standard deviation (h days)
- Thread starter Ludwma
- Start date
Hi FS,
I will annotate that cell in a revision. (We definitely need to know this, the square root rule will certainly appear on the exam). Notice the first assumption is Trading days/year = 252. To scale volatility, under the i.i.d. assumption, variance scales with time; e.g., n-day variance = n*1-day variance. As variance scales with time, volatility scales (only under iid) with the square root of time, so the general form is N-day volatility = SQRT(N/Y * Y-day variance) = SQRT(N/Y) * Y-day volatility;
Given Y-day volatility, conditional on i.i.d., N-day volatility = SQRT(N/Y) * Y-day volatility. This the square root rule.
So, in this case, 10-day volatility = 12.45% * SQRT(10/252) = 2.48%
We can scale up, too. Let's scale the 10-day volatility to a 3-year volatility (not practical, the i.i.d. violation becomes acute, it's unbearable to assume both independence and same distribution over such a long time):
2.48% 10-day vol * SQRT ([252*3]/10) = 21.56%
After this, you want to be aware of the i.i.d. violation: eg., if returns either (positive) autocorrelate or mean-revert (negative auto correlation), what is the directional error?
I hope that helps, thanks,
I will annotate that cell in a revision. (We definitely need to know this, the square root rule will certainly appear on the exam). Notice the first assumption is Trading days/year = 252. To scale volatility, under the i.i.d. assumption, variance scales with time; e.g., n-day variance = n*1-day variance. As variance scales with time, volatility scales (only under iid) with the square root of time, so the general form is N-day volatility = SQRT(N/Y * Y-day variance) = SQRT(N/Y) * Y-day volatility;
Given Y-day volatility, conditional on i.i.d., N-day volatility = SQRT(N/Y) * Y-day volatility. This the square root rule.
So, in this case, 10-day volatility = 12.45% * SQRT(10/252) = 2.48%
We can scale up, too. Let's scale the 10-day volatility to a 3-year volatility (not practical, the i.i.d. violation becomes acute, it's unbearable to assume both independence and same distribution over such a long time):
2.48% 10-day vol * SQRT ([252*3]/10) = 21.56%
After this, you want to be aware of the i.i.d. violation: eg., if returns either (positive) autocorrelate or mean-revert (negative auto correlation), what is the directional error?
I hope that helps, thanks,
Hi FS,
The derivation of the AR(1) VaR is non-trivial (it would take more time than I have, and while the qualitative impact of the directional error is within scope, the actual calculation is beyond FRM scope). The formula is in the learning spreadsheet, can I refer you to that?
The reason i show it is, to give concrete illustration to the directional error (i.e., the violations to SRR noted by Linda Allen); e.g., that positive autocorrelation (an iid violation) give a higher VaR than iid.
Thanks,
The derivation of the AR(1) VaR is non-trivial (it would take more time than I have, and while the qualitative impact of the directional error is within scope, the actual calculation is beyond FRM scope). The formula is in the learning spreadsheet, can I refer you to that?
The reason i show it is, to give concrete illustration to the directional error (i.e., the violations to SRR noted by Linda Allen); e.g., that positive autocorrelation (an iid violation) give a higher VaR than iid.
Thanks,
WhizzKidd
Member
Hi David Harper CFA FRM,
I just wanted to understand something on scaling volatility. When one does readings, people plot a graph of rolling volatility. What I would like to know is that, if you have a time series of data, and the log returns of that data. Say, you want the 60-day vol today, so you take the std.dev of the log-returns for 60 points, but say you want to view this as an annual volatility. It seems that all one does is is std.dev(60 daily log-returns)*sqrt(252), to get an annualized vol, but why is it not std.dev(60 daily log-returns)*sqrt(252)/sqrt(60), so that you have a 1-day vol which you scale up to annual?
Then what is done is people plot this over time as a rolling vol, but I am concerned on the scaling? So, annual vol at a point would be std.dev(252 daily log-returns)*sqrt(252)?
I just wanted to understand something on scaling volatility. When one does readings, people plot a graph of rolling volatility. What I would like to know is that, if you have a time series of data, and the log returns of that data. Say, you want the 60-day vol today, so you take the std.dev of the log-returns for 60 points, but say you want to view this as an annual volatility. It seems that all one does is is std.dev(60 daily log-returns)*sqrt(252), to get an annualized vol, but why is it not std.dev(60 daily log-returns)*sqrt(252)/sqrt(60), so that you have a 1-day vol which you scale up to annual?
Then what is done is people plot this over time as a rolling vol, but I am concerned on the scaling? So, annual vol at a point would be std.dev(252 daily log-returns)*sqrt(252)?
Hi,
std.dev(60 daily log-returns)that you obtained is the average value of the daily volatility(not 60 day volatility,please look at formula carefully) so you scale it by sqrt(252) as there are 252 days in a year to get the annualized vol.
thanks
std.dev(60 daily log-returns)that you obtained is the average value of the daily volatility(not 60 day volatility,please look at formula carefully) so you scale it by sqrt(252) as there are 252 days in a year to get the annualized vol.
thanks
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