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# Scaling VaR for IR Swap

#### rahul.goyl

##### Member
Hi David,

I wanted to know the correct measure of scaling VaR in the below case. Your help is always invaluable.

http://www.bionicturtle.com/wiki/FRM2009.L1.09/

c. In the answer above, the square-root-rule is employed to scale the daily yield volatility to one month. Why is this especially dubious here? [mine]
Scaling variance directly with time requires i.i.d., interest rates are not independent. They mean-revert.

Thanks & Regards,
Rahul

#### David Harper CFA FRM

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

The first learning XLS shows an autocorrelation adjustment:
(in order to continue service, we cannot offer everything free, of course! ... so this is for members only)

this allows you to enter a negative autocorrelation (i.e., mean reversion) which, say, given a 1% daily volatlity will produce an annual volatility that is less than (<) 1% *SQRT(250/1) per the mean reversion

... FWIW, you don't need autocorrelation adj for the exam. Rather, the important conceptual point is that scaling per square root rule (S.R.R) requires i.i.d. assumption and mean reversion (of returns) is a violation of the "independent" so S.R.R. will overstate long horizon volatility if returns mean revert (and, if returns are positively auto correlated, SRR will understate)

David

#### rahul.goyl

##### Member
Hi David,

I agree what you say but not with S.R.R, there is again an typo error on the below link. Which change the complete meaning of Positive & Negative Auto-Correlation.
Kindly have a look, I have replicated the same.

For 20g. If the Autocorrelation = .25 , implied daily vol = 2.43% not 1.47% (Which is case in N.AC of -0.25) Which mean S.R.R will Overstate if Positively Auto Correlated

20g. What if returns are mean-reverting, then how would the daily volatility compare?

e.g., if autocorrelation = -0.5, the implied daily vol = 1.09% & not 3.26% (which is case in P.AC 0.50) that means S.R.R will understate if Negative Auto Corrlated.

Below is the link of wiki, where I found this Glitch.

http://www.bionicturtle.com/wiki/FRM2009.L1.20

20g. Problem 20a above requires solving for the daily volatility under the restrictive i.i.d. assumption. Can we say anything about the daily volatility if returns are not independent; specifically, what if returns are positively auto-correlated, how would the daily volatility compare to our calculation above?
Under i.i.d., annual volatility of 30% scales to daily volatility of 1.89%; i.e., 30% * SQRT(1/252)

If returns are postively auto correlated, given annual vol of 30%, the implied daily volatilty will be less than (
<) 1.89%; e.g., if autocorrelation = 0.25, implied daily vol = 1.47%

20g. What if returns are mean-reverting, then how would the daily volatility compare?
If returns are mean reverting, given annual vol of 30%, the implied daily volatilty will be greater than (>) 1.89%;
e.g., if autocorrelation = -0.5, implied daily vol = 3.26%.

Regards,
Rahul

#### David Harper CFA FRM

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

I don't see a glitch, rather a difficult idea. The example scales "down" (from annual to daily) rather than scaling "up" (daily to annual) so the rule that SRR overstates under mean reversion refers to "scaling up" with SRR;
e.g., Annual volatility < Daily volatility * SQRT(250/1) if autocorrelation < 0; this is consistent with, under "scaling down:"
Daily volatility > Annual volatility * SQRT(1/250) if autocorrelation < 0

... I can't speak to your numbers but under n/a/c of -0.5 i got implied daily vol of 3.26% versus 1.89% under scaling down SRR. This appears to be at least directionally correct. The 1.09% looks directionally wrong b/c if scale it up = 1.09% * SQRT(250) = 17.23% and then with negative autocorrelation, we must have < 17.23% (compare to 30%)

...so, apologies I don't see the problem but it appears this may be due to the way we speak about the relative outcomes on scaling down versus up

David

#### rahul.goyl

##### Member
Hi david,

I am still not sure, I want to share my calculation with you.
I look at this, if Vols are +ve auto correlated, it follows +ve exponents (Which is Always increasing function) (Vice-Vera for Negative)
So if we think in simple terms (Given Positive Auto-correlation) the daily vol*sqrt(252) > the Vol (Zero Correlation) (Vice-Vera for Negative)

Kindly look at the excel & plz correct me where am wrong.
Regards,
Rahul

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Rahul - sure, happy to take a look, can you attach XLS? (please note: when you reply, you have the ability to attach a file) - David

#### David Harper CFA FRM

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

I could be wrong, but I don't see the consistency in your XLS; you get daily vol = 2.4374% under AR(1) = 0.25.
But testing that, the annualized volatility = 49.9% ... so i don't agree with your last step to apply a square root rule the AR(1) annualized volatility of 38.69%. IMO, that introduces false circularity ... i don't see how you can justify your SQRT(1/252) in cell I12, as this is AR(1)=.25

I prepared this XLS:
http://sheet.zoho.com/public/btzoho/l1-20-autocorrelation-1

i.e.,
column 1: daily 1.89% scales to 30% under SRR (i.e., AR=0)
column 2: daily 1.47% scales to 30% under AR(1) = 0.25
column 3: daily 3.27% scales to 30% under AR(1) = -0.5; i.e., mean reversion in returns

Let me know, thanks! David

#### rahul.goyl

##### Member
Hi David,

If you look at the link provided by you for calculating the AR, I have replicated the same on to excel. It does takes sqrt in calculation, if you look at the formula pasted by me on my Excel sheet.

column 2: daily 1.47% scales to 30% under AR(1) = 0.25
The vol number 1.47%, am getting when I use correlation as -0.25 (Look into my excel)

column 3: daily 3.27% scales to 30% under AR(1) = -0.5; i.e., mean reversion in returns
The vol number 3.27% am getting when I use +ve Correlation of 0.50 (Look into my Excel)

http://sheet.zoho.com/public/btzoho/l1-20-autocorrelation-1
In your sheet above in the link, I am not able to see Scaling of Volatility. Could you please put formulas in row 5 ( that will clear the confusion)
It seems we agree on the scaling Factor (am comparing row 19 with my calculation & they tie back 419.11 & 84.44)

It was great Support by you David, thanks a lot for your time. I am doing my best to contribute to the forum.

Regards,
Rahul

#### David Harper CFA FRM

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

Yes, thanks for your contribution: as I do make mistakes (e.g.,, http://www.bionicturtle.com/forum/viewthread/3709/) , challenging the calcs is welcomed!

I do perceive that our calculations, including the scaling factor, agree for the most part.
In regard to calcs for row 5, I would have to reverse-engineer the scaling factor but it should not be necessary: row 5 is the daily volatility input that corresponds to annual volatility of 30% (the 30% is an output); i.e., in my XLS, daily vol is the input and 30% is solved-for annual volatility under A() = 0 | X
... actually, solving for formulas in row 5 would introduce spreadsheet circularity with formulas in row 20, so i don't see the need

.... it seems possible to me that our disagreement is "merely" in the phrasing and, at least the way i perceived my question (perhaps inaccurately), I don't support your final step: you have 38.7% scaling down to 2.44% under SRR and AR() = 0. That is a true scaling conversion, but it is not a scaling down of 30% annual volatility
so i see your final step as the answer to the question "what is the daily vol of annual 39% implied by SRR under i.i.d.?"
but not the answer to the question " what is the daily vol of annual 30% under mean reversion (non-i.i.d or A() <>0)?"

Thanks, David

#### rahul.goyl

##### Member
Hi David,
It was good learning for me to understand this concept better, obviously this make sense, for answer to your last question "what is the daily vol of annual 30% under mean reversion (non i.i.d or A()<>0), for scaling purpose I think we should be using sophisticated models such as ARCH & GARCH, that will consider mean reversion if we look from market practioner’s point of view.

Please correct me if am wrong at any point.
I understand why you were not happy with my final step calculation.

Thanks & Regards
Rahul