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# Square Root Rule with Mean Reversion & AutoCorrelation - VaR & Volatility

#### [email protected]

##### New Member
David,
I am now thoroughly confused by the Square Root Rule and scaling the VaR under the circumstance of Mean Reversion and Auto correlation. In search of an explanation, I found this thread http://www.bionicturtle.com/forum/newreply/1729/ ,

The rules for when there is mean reversion of asset returns, or mean reversion of the variance are really confusing me. I don't know how to apply either of the rules, nor when scaling up or down for either example of mean reversion. Do you have an updated thread that illustrates both cases (asset returns, and return volatility) of mean reversion, whether scaling up or down??

I have searched the internet, your notes, and even the 5th edition of the handbook for an outline or a simple explanation to help with the exam questions on this subject (& in genera). While I can recite your chart (if there is mean reversion, the SRR rule overstates the LR vol...", I still am confused! I have killed a ton of time on this, and thought I understood several times. I think that there are a few key things that I am missing to make the chart or rules make sense.

For example, does the chart assume that you are starting with the daily volatility, and if you apply the SRR by scaling it to a 1 year vol,….it overstates the long run vol? And does “overstate” this mean that it is actually too high? It is the scaled measure that is overstated right? (not the LR vol??)

Also, there is a question about calculating the one week VaR and rescaling it to a 1 day VaR, assuming that there is mean reversion. The answer is that the recalculated VaR will be less than the ORIGINAL VaR. Since the question is relative to the original VaR (not the LR), does your chart apply in this question? Is the answer different if we started with the 1 week VaR, and recalculated and scaled to a 1 day VaR, with mean reversion?

Does autocorrelation take everything in the opposite direction?

Please help! I would imagine that there may be other students who are as confused. This will be my last annoying question before the exam, I promise! And, it’s important to say that your material has really helped me to understand the material significantly. I took a course prior to the exam last November, and thought that I knew the material. I was blown away on exam day. Once I started reading your notes and material, I realized that at best, the course that I took, introduced the concepts. I am hoping to pass this time around in 2 weeks. BTW, I am not a risk management practitioner. Thanks again for your help!

Theresa

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Theresa

(my mean reversion article is here, but I think it's main purpose was to suggest that mean reversion has several definitions: http://www.bionicturtle.com/how-to/article/what_is_mean_reversion_in_financial_time_series/ )

In FRM, we focus really (only) on two mean reversions:
1. mean reversion in returns; e.g., the daily return mean reverts to LN(S1/S0)
2. mean reversion in volatility/variance; e.g., the daily variance reverts to the long-term variance (sigma)
(the are related as historical variance is typically a weighted summation of historical squared returns, so it's possible to have some of both in the series)

Especially in regard to (1), mean reversion is synonymous with negative auto- or serial-correlation; i.e., mean reversion is the "opposite" of positive autocorrelation of returns. For example, if the return today is +3%, then positive autocorrelation implies a higher probability of another high return tomorrow while mean reversion (negative serial correlation) implies a higher probability of a low/negative return tomorrow.

The only example in the FRM of (#2) is GARCH(1,1) where:
variance estimate = weight*LR variance + weight*lagged return^2 + weight*lagged variance
The key difference between GARCH(1,1) and EWMA is this mean reversion term (weight*LR variance) which is (2) not (1): the variance is reverting toward an (unconditional) long-run variance.
Either/both types of mean-reversion are violation of i.i.d. and therefore render the square root rule untrue (i.e., the key assumption of the SRR is that it dubiously assumes i.i.d., and the "independence" in i.i.d. implies autocorrelation of 0)

Okay, so with regard to the biases (here i am illustrating the same directional points as Linda Allen), let's assume the daily volatility is 1.0%.

mean reversion (#1) in returns: when scaling up, mean reversion in returns implies an "actual" long horizon volatility that is less than volatility implied by SRR
If daily vol = 1.0%, then SRR says annual vol = 1%*SQRT(250) = 15.8% but only if returns are i.i.d.
Now assume returns are mean reverting (#1 above not #2), which is the same as saying auto-correlation is negative (between 0 and -1)
With regard to this MEAN REVERSION (negative autocorrelation) in RETURNS:
"True" Annual volatility will be less than 15.8% (just as true annual volatility would be greater than 15.8% if there were positive return serial correlation)
... so we can say "if returns mean revert, the SRR [15.8%] will overstate the true long horizon mean reversion"

mean reversion (#1) in returns, but scaling down; in this case, scaling down gives an actual short-horizon volatility that is greater than the volatility implied by SRR
Recall that if daily vol = 1%, then SRR implies annual vol = 15.8%. Now assume instead mean reversion such that 1% daily would scale accurately (with mean reversion) to true annual vol of 11.0%
What if i just gave you 11.0% as the annual volatility?
SRR says daily = 11% * SQRT(1/250) = 0.7%
but the daily is really 1%
such that "scaling down" under mean reversion implies: SRR [giving 0.7%] will understate the true daily volatility [1%]

#2 in variance/volatility: it depends on whether we are currently above/below the long-run variance:
this could be, for example, the long-run daily volatility is 0.5%
Now our current daily vol (1%) is greater than average long run VOLATILTY (0.5%) and the SRR will also overstate.
This is just more like a "gravitational pull;" if the average long run daily vol were 2%, then SRR (15.8%) would understate as the actual would be nearer to 2%*SQRT(250)

I hope that helps and that your studies are going well, thanks for the kind words! (GARP is unlikely to query deeply on this, btw)

David

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#### [email protected]

##### New Member
Thank you so much, David!
I will be running through this carefully tomorrow.
Theresa

#### southeuro

##### Member
David, I am confused. In the FRM 2009 sample exam bootcamp answers, you write "if returns are mean reverting given annual volatility of 30% the implied daily volatility will be greater than 1.89% eg if autocorrelation is -0.5 implied daily vol will be 3.26%." You also mention that with positive autocorrelation implied daily vol will be 1.47%. I thought mean reversion acts as a "negative force" and pulls the implied daily vol down and autocorrelation acts as a "positive force" as you stated above in your post. Thanks for clarifying.

#### southeuro

##### Member
addendum: also how do you find 3.26 and 1.47. what is the formula to find the implied vol? many thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi southeuro,

I tried to explain exactly this directly above here. It is probably easier to see the calcs side by side, i attached a snapshot of the learning XLS, but where i just added some columns, see below (here is the same spreadsheet)

What you say is true, but i think the confusion arises when "scaling down" (eg) from one year to one day. Consider the following, from below:
• 1.89% daily vol with correlation = 0 scales to 30% annual due to SRR: 1.89%*sqrt(252) = 30%
• 1.466% daily vol with correlation = 0 scales to a lessor 23.26%
• 3.265% daily vol with correlation = 0 scales to a greater 51.83%
• but 1.466% daily requires positive A/C of 0.25 to scale to 30.0%; i.e., +A/C increases annual vol from 23.26 to 30%, autocorrelation as a "positive force"
• but 3.265% daily vol with neg A/C of -0.5 to scale to a 30%; i.e., negative A/C as a "negative force"
but due to this, when scaling down from annual 30% vol and positive correlation, our daily vol is HIGHER than the 1.890%, but this is just because we are going in the other direction. positive autocorrelation remains a "positive" force. With respect to " also how do you find 3.26 and 1.47. what is the formula to find the implied [not implied: you mean autocorrelated] vol?", please see the XLS if you want to explore, it's beyond my current scope, our immediate concern is just the directional impact of autocorrelation. thanks,

#### nit

##### New Member
Hi David,

The VaR conversion formula
VaR= standard deviation * z value * portfolio value * squared root of n (1)

I do not understand why we times squared root of n?
Is the value "squared root of n" comes from formula SE= Standard deviation divided by squared root of n.
Is the standard deviation (1) * squared root of n equal to the standard deviation of population in n next days?

#### ShaktiRathore

##### Well-Known Member
Subscriber
Square root of n.
The returns are independently and identically distributed that mean the returns of portfolio in consecutive periods are independent and have no correlation with each other(which is not a real world picture but only a theoretical assumption). So if we assume variance of returns for 1 period as Var than total variance for n periods(assuming equal variance for each period i.i.d.) is n*variance of 1 period=n*Var (variances adds up VaR(a1+a2+...an)=VaR(a1)+VaR(a2)+.....+VaR(an) when there is no correlation between returns).
Thus VaR(n-period)=n*Var
taking square root on both sides,
sqrt[VaR(n-period)]=sqrt[n*Var]
stdDev(n-period)=sqrt[n]*sqrt[Var]
stdDev(n-period)=sqrt[n]*stdDev(1-period)

thus we always scale volatility to the square root of time period over which we need to forecast volatility. Thus in VaR we always scale it to square root of time period over which we need to forecast VaR.

thanks

#### Arka Bose

##### Active Member
Hi david,
With regard to your 1st post,
Refering to the biases, #1 should be less than SRR?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @arkabose Yes, correct, I had a typo: my ">" should have been less than. I corrected the typo with language (and color coded it). Thanks!

#### AAP

##### New Member
Hi @David Harper CFA FRM ,

How you came up with individual scaling factor and VaR increase? I didnt get below.

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#### kandytsang

##### New Member
May I ask what is mean reversion?

I read the study notes but I still not understand it

More , In our study notes R25. P1.P4.Allen, P.28, there is two scenarios which illustrate “violations” in the use of the square root rule to scale volatility over time ,I don’t understand when is overstates and when is understates? Thanks a lot!

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#### Nicole Seaman

Staff member
Subscriber
May I ask what is mean reversion?

I read the study notes but I still not understand it

More , In our study notes R25. P1.P4.Allen, P.28, there is two scenarios which illustrate “violations” in the use of the square root rule to scale volatility over time ,I don’t understand when is overstates and when is understates? Thanks a lot!
Hello @kandytsang

Nicole

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
HI @kandytsang

Believe it or not there are various definitions of mean reversion, but the FRM tends to concern the two discussed in Allen (your reference).
1. The first meaning of mean reversion is negative auto-correlation (or negative auto-covariance) between successive returns in time. If yesterday's asset return is denoted r(t-1) and today's asset return is denoted r(t), then when using the square root rule to scale volatility (or VaR) we are implicitly assuming the "independence" in i.i.d. such that we are assuming today's return is not correlated to yesterday's. But if a high positive return yesterday implies that today is more likely to be negative or low, then maybe the returns in time are negatively correlated, which is negative auto-correlation and is the first type of "mean reversion." This sort implies that a 10-day volatility (or VaR) will be less than the VaR implied by simply scaling per the square root rule with *sqrt(10). Hopefully, that's a little intuitive: the returns are reversing such that they won't "diffuse" as much (in this way, positive auto-correlation has the opposite effect: scaling per the SRR will understate the n-day volatility/VaR). So, for this type of mean reversion (i.e., mean reversion in returns), the SRR overstates the "true" long horizon volatility (as indicated in Allen's Table 2.4)
2. The second meaning of mean reversion is the idea that there does exist some absolute LEVEL of volatility, like we observe in the GARCH(1,1) model. For example, say it's 1.0% per day. This is not reversion in returns (i.e., negative auto-correlation or negative auto-covariance between successive returns) but rather this is the idea that volatility (which is remind is "volatility of returns") will revert to the long-term level; this is also sometimes called "regression to the mean." The over- versus under-statement, in this case, DEPENDS on the current level. If the current volatility is 3.0%, then scaling volatility per SRR will overstate simply because we expect the volatility to get "pulled toward" the 1.0%. If the current volatility is only 50 basis points, scaling will understate. I hope that's helpful!

#### Anuja

##### New Member
Hi David,

What's the formula for calculating Scaling Factor when Auto corelation is there.

Std deviation daily is 1%
Confidence level is 99%
Target Horizon 10days
Autocorrelation is .2

How we should calculate scaling factor, std deviation & VAR over horizon..

Thanks,
Anuja

Staff member
Subscriber

Thank You David