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# GARP.2010.PQ.P2Questions about Long Short VaR (garp10-p2-17)

#### cqbzxk

##### Member
17. The bank’s trading book consists of the following two assets:\ Correlation (A, B) = 0.2

How would the daily VaR at 99% level change if the bank sells 50 worth of asset A and buys 50 worth of asset B?
Assume there are 250 trading days in a year.
a. 0.2286
b. 0.4581
c. 0.7705
d. 0.7798

Hi, I found this questions from practice exam, I am not sure my understanding is right, can anyone verify if my solution is correct? thanks

First Step: Calculate one day portfolio sigma and portfolio return from original position:
Sigma P = sqrt( wA^2*SigmaA^2 + wB^2*SigmaB^2 + 2*wA*wB*SigmaA*SigmaB*correlation)
Sigma P (1day)=Sigma P/sqrt(250)
E(Rp 1day)= wA*rA+wB*rB/250
VaR(P 1day)=(Sigma P(1day)*2.33-E(Rp1day) ) * (100+50)

Second Step: Calculate one day portfolio sigma and return after position change, which wA=0.33 wB=0.67, other things are same => new VaR(P 1day)
Finally, Use VaR(P 1day)-new VaR(P 1day) =>my answer is about 0.4866 similar to B but not exactly same, so this is why I want to make sure if it's correct. thanks

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#### cqbzxk

##### Member
And also, for another question is, assume we have a portfolio, long asset A $100, short asset B$50, how do we judge the weight? still wA=67% wB=33%?

so in calculation, because of short B, set wA positive and wB negative, in equation I assume wB is positive, so I use - to minus,
Sigma P = sqrt( wA^2*SigmaA^2 + wB^2*SigmaB^2 - 2*wA*wB*SigmaA*SigmaB*correlation) ?
E(R)= wA*rA-wB*rB ?
VaR(portfolio) = (SigmaP*2.33-E(R))*V where V= 100+50=150 ?

Thanks!

#### [email protected]

##### New Member
Hi David
How do you calculate this in less than 3 mins or there is some VaR properties i need to know.
It is 2010 garp practise questions

Q)The bank's trading book consists of the following two assets:

Asset Annual return Volatility of Annual Return Value
A 10% 25% 100
B 20% 20% 50

Correlation (A,B) =0.2
How would the daily VaR at 99% level change if the bank sells 50 worth of Asset A and buys 50 worth of asset B 50? Assume there are 250 trading days in a year.

a) 0.2286
b) 0.4581
c) 0.7705
d) 07798

the answer acording to GARP is b)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @[email protected] (More detail and XLS in paid section at https://www.bionicturtle.com/forum/threads/question-17-var-valuation-garp10-p2-17.2150/ )

I don't really have a shortcut for this question. The way I would do it is:
• Individual VaR(A) = $100 * 25% * 2.33 = 58.25 • Individual VaR(B) =$50 * 20% * 2.33 = 23.30
• Portfolio VaR = sqrt(58.25^2 + 23.30^2 + 2*58.25*23.30*0.20) = $66.92 Then the trade which implies new: • Individual VaR(A) =$50 * 25% * 2.33 = 29.125
• Individual VaR(B) = $100 * 20% * 2.33 = 46.60 • Portfolio VaR = sqrt(29.125^2 + 46.60^2 + 2*29.125*46.60*0.20) =$56.69
• This is a useful (diversified) portfolio VaR where $A and$B are individual VaRs and note it has a familiar mean variance form so it's not really memorizing something new:
[VaR(A+B)]^2 = VaR($A)^2 + VaR($B)^2 + 2*VaR($A)*VaR($B)*ρ(A,B)
So Δ annual VaR = 66.92 - 56.69 = 7.24 and one-day change = 7.24/sqrt(250) = 0.457524, so looks right to me. Thanks,

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#### [email protected]

##### New Member
thanks David, it makes sense now

#### rex.w.chow

##### New Member
• Individual VaR(A) = $100 * 25% * 2.33 = 58.25 • Individual VaR(B) =$50 * 20% * 2.33 = 23.30
Hi David,

I am a bit confused. I still could not understand the solution.
I thought the formula would be VaR=+-(mean-Z_99%*Std.dev.)*S. So I put the mean of A, B into the calculation and could not work out an answer.
May I ask why the mean return here can be ignored?

Cheers,
Rex

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @rex.w.chow You are right to be confused. That question was flawed (and feedback was subsequently provided to GARP): in fact, without any specification, under an annual horizon we'd assume an absolute VaR. The question has the burden of specifying relative VaR, if that's what it wants!
• If arithmetic returns are normally distributed, absolute VaR = - µ*Δt + σ*z(α)*sqrt(Δt), and
• relative VaR = σ*z(α)*sqrt(Δt)
This is discussed further in the link to paid members at https://www.bionicturtle.com/forum/threads/question-17-var-valuation-garp10-p2-17.2150/post-7536 ie..,
Hi joeisenb,

Hi, relative/absolute are Jorion's (and Crouhy's) terms (it's actually GARP that fails to clarify in this question).
Did you see my note 71.2 above? i.e.,
"Relative VaR is loss relative to the future expected value.
Absolute VaR is the loss relative to zero; in this case, positive drift (return) offsets the loss.
Absolute VaR is the best (most encompassing) VaR and the most robust (robust to modifications like liquidity-adjustments) formula is given by Dowd"

So, to keep is simple, just taking Asset A above (1-year exp return = 10% and vol = 25%):
• 1 yr 99% relative VaR = 25%*2.33 @ 95% confidence = ~58.2%; i.e., loss relative to future expected value
• 1 yr 99% absolute VaR = -10% 25%*2.33 = ~48.2%; i.e., loss relative to zero/today's initial value
• ... where the "drift" (return) is positive, absolute value gives credit for the drift that is expected to offset the loss
The question as given by GARP (17.1), revealed by the answer, assumes a relative VaR by ignoring the expected returns but the question is flawed for not making explicit this assumption; the absolute VaR decrease will be higher than shown because Asset B also has a higher return (the question oddly has Asset B as both higher risk and higher return; often, the trade into the lower volatility would be mitigated--and reflected in absolute VaR--by a lower return asset).

David

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#### rex.w.chow

##### New Member
Hi @rex.w.chow You are right to be confused. That question was flawed (and feedback was subsequently provided to GARP): in fact, without any specification, under an annual horizon we'd assume an absolute VaR. The question has the burden of specifying relative VaR, if that's what it wants!
• If arithmetic returns are normally distributed, absolute VaR = - µ*Δt + σ*z(α)*sqrt(Δt), and
• relative VaR = σ*z(α)*sqrt(Δt)
This is discussed further in the link to paid members at https://www.bionicturtle.com/forum/threads/question-17-var-valuation-garp10-p2-17.2150/post-7536 ie..,
Thank you, David.