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# daily VAR 10-day trading horizon in IMA

#### ajsa

##### New Member
Hi David,

How should I understand the daily VAR here? is it just calculated daily? I think 10-day trading horizon means I need to scale the "daily" VAR (using one day volatitlty) with 10 days to get the VAR which is called the daily VAR here? so it is actually 2-week VAR right?

Thanks.

#### David Harper CFA FRM

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

(this is worth a careful look, IMO, b/c a good candiate for testing)

here is copy from Basel II, here are the first three quantitative standards:

(a) Value-at-risk” must be computed on a daily basis.
(b) In calculating the value-at-risk, a 99th percentile, one-tailed confidence interval is to be used.
(c) In calculating value-at-risk, an instantaneous price shock equivalent to a 10 day movement in prices is to be used, i.e. the minimum “holding period” will be ten trading days. Banks may use value-at-risk numbers calculated according to shorter holding periods scaled up to ten days by the square root of time

...so, yes it *must* be calculated on daily basis (this is where VaR started, short periods)
...and, yes, it's a 10-day "horizon" or "holding period"
we can scale to 10-days different ways (e.g., historical simulation, bootstrap) - although exam is likely to use SQRT(10) and SQRT(10) connotes parametric, please note that no approach is dictated, IMA can use either of our basic big three VaR approaches: HS, parametric, or MCS.
but if parametric, then *SQRT(10), you can see is *explicity* accepted in Basel... it would be 10 trading days, so it's a 2 weeks VaR although i don't think you see it expressed that way much

David

#### ajsa

##### New Member
Hi David,

Just to double-confirm, so under market risk capital, the it is 10-day VaR calculated daily, right?

if parametric, under Basel the 10-day VaR can be scaled up from 1-day VaR, or can be calculated directly (with 10-day volatility), right?

Thank you!

#### David Harper CFA FRM

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

Yes, to recap, at the advanced (internal) levels: Credit Risk and OpRisk are a 99.9% VaR and one-year holding period (we need both confidence and horizon, or we haven't defined the VaR). Basel 2 IMA market risk is "only" a 10-day 99% VaR.

Yes to either scaled or calculated directly. For example, maybe the 1-day 99% VaR is 1%, then scaling leads to 1%*SQRT(10) ... but we dangerously assumed i.i.d.
...or how might we do it "directly" instead? we could bootstrap (HS) or Monte Carlo a 10-day simulation

David

#### cqbzxk

##### Member
Hi asja,

Yes, to recap, at the advanced (internal) levels: Credit Risk and OpRisk are a 99.9% VaR and one-year holding period (we need both confidence and horizon, or we haven't defined the VaR). Basel 2 IMA market risk is "only" a 10-day 99% VaR.

Yes to either scaled or calculated directly. For example, maybe the 1-day 99% VaR is 1%, then scaling leads to 1%*SQRT(10) ... but we dangerously assumed i.i.d.
...or how might we do it "directly" instead? we could bootstrap (HS) or Monte Carlo a 10-day simulation

David
Hi David, I have a detail question about the calculation of Market Capital Charge, according to the book, it said at 99% confidence, if
Max( K* ave VaR(60days), VaR(yesterday) )
so, if exam give me ave VaR(60days)=100, VaR(yesterday)=310,
first question, which one I should Choose, 3*100 =300 compare with 310 or 3*100*sqrt(10) compare with 310,
second question, if I choose VaR(yesterday)=310, to calculate the final market risk capital, which one should I use, VaR(yesterday) or VaR(yesterday)*sqrt(10) ?
third question, how about same thing under stressed VaR, same way to calculate?
thank you!

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi David,

I am confused about the Capital requirement under the 1996 Amendment:

(1) Under the IRB approach, the VAR measure was calculated with a 10-day horizon and a 99% confidence level
(2) The Capital requirement was:

Max(VARt-1, mc * VARavg) + SRC where VARt-1 = previous day's VAR, VARavg = average VAR over past 60 days and minimum value for mc = 3
(3) Since VARt-1 < 3*VARavg, Capital requirement = 3*VARavg

(4) Since banks invariable use a one-day 99% VAR, under the 1996 Amendment, regulators explicitly stated that 10-day 99% VAR = SQRT(10)*one-day 99%VAR
(5) Hence, the capital requirement was calculated as mc* average 10-day VAR i.e. mc*SQRT(10) =mc* 3.16*average one-day 99% VAR = 3*3.16*average one-day VAR = 9.48*average one-day VAR
(6) How does the capital requirement in (5) reconcile with that stated in (2)? In (2) VARavg = average VAR over past 60 days and in (5) it is based on the average 10-day VAR at the 99% confidence level. Is this to reconcile the capital requirem
ent under (2) to be consistent with VAR calculations in (4)?
(7) I know that this set the stage for Basel II, but nevertheless!

Thanks #### Matthew Graves

##### Member
Subscriber
Hi David, I have a detail question about the calculation of Market Capital Charge, according to the book, it said at 99% confidence, if
Max( K* ave VaR(60days), VaR(yesterday) )
so, if exam give me ave VaR(60days)=100, VaR(yesterday)=310,
first question, which one I should Choose, 3*100 =300 compare with 310 or 3*100*sqrt(10) compare with 310,
second question, if I choose VaR(yesterday)=310, to calculate the final market risk capital, which one should I use, VaR(yesterday) or VaR(yesterday)*sqrt(10) ?
third question, how about same thing under stressed VaR, same way to calculate?
thank you!
I think this is a simple topic which causes significant confusion!

Think about it from the regulators perspective, they want to establish a baseline of K * your average calculated VaR for the past 60 days but do not want to ignore significant VaR spikes which might indicate distress. Hence the MCC is based on the max of either your average calculated VaR over the past 60 days and your VaR yesterday to protect against spikes.

To answer your example directly, K*VaR(60 day average) = 300, which is less than VaR(yesterday) so you must pick VaR(yesterday), in your example 310. I don't recall whether you need to use 1-Day or 10-Day VaR but I'm fairly certain it is consistent for VaR(avg) and VaR(yesterday). Happy to be corrected.

Confusion #1 - VaR10 vs. VaR1
For all VaR calcs you need a distribution of returns, the only difference is over what time period you measure the return. In 1-day VaR, the distribution is built from 1-Day returns. In 10-Day VaR it is built from 10-Day returns, usually by aggregating 1-Day returns in the case of HS.

Confusion #2 - Average VaR
If I remember correctly, the regulator specifies the time period of return data to be considered (e.g. 250 business days) for the VaR calculation. Each day, the bank must calculate it's VaR based on a sliding window from today (t) to t-250. After 60 days you will have a VaR value for each day with a slightly different window as time advances. The average VaR the regulator is referring to is the average of these VaR values that you've calculated daily for the last 60 days.

Hopefully that helps clear up some of the confusion!

#### Dr. Jayanthi Sankaran

##### Well-Known Member
Hi @Matthew Graves,

Thanks a lot for your explanation. Still not sure I understand it completely....will try a problem to understand #### FRM candidate

##### Member
Hello David,

If I am given the returns of stock for a certain period of time and I need to calculate VaR with the following formula VaR = portfolio value * (∑(R) – zσ), how do I calculate the expected return ∑(R).

Should I just take the average of the log returns, or should I calculated the cumulative return like this ∑ (((1 + day 1 return)-1)*((1 + day 2 return)-1)*((1 + day n return) -1))