Delta Normal VaR - simultaneous long and short

[afterworkguinness]

Hi David,
Thanks for all your help leading up to the exam. I came across this scenario and was not sure how to solve it:

What is the 99% 1 day VaR for the portfolio:

long 200,000 USD 1 day volatility 15%
short 200,000 USD 1 day volatility 25%

Being long and short the same notional thew me off.

Thanks

 

[shady007]

Hi David,
Thanks for all your help leading up to the exam. I came across this scenario and was not sure how to solve it:

What is the 99% 1 day VaR for the portfolio:

long 200,000 USD 1 day volatility 15%
short 200,000 USD 1 day volatility 25%

Being long and short the same notional thew me off.

Thanks

I remember similar question in Part2 actual exam...You just need to find the individual VAR at 99% C.I. If I am not wrong, the correlation between long positions was also given. But the actual position assumed was long and short. So the VAR of portfolio = Sqrt(VarA^2 + VarB^2 - 2.VarA.VarB.Correlation coefficient).

The logic is that even if they are positively correlated, there will be diversification benefit since the positions are long and short. The diversification benefit will be maximum when the correlation is 1.

 

[ShaktiRathore]

correlation=-1 as they are long and short positions on the same asset so they move in opposite directions
sd(portfolio)=sqrt(.15^2+.25^2-2*.15*.25*1)
sd(portfolio)=sqrt(.0225+.0625-.075)
sd(portfolio)=sqrt(.01)=.1=10% is the volatility of portfolio which is less than the individual positions due to diversification !!!
VaR(portfolio)=sd(portfolio)*z(99%)*Value of portfolio
VaR(portfolio)=10%*2.33*400000= 93200 USD

thanks

 

[shady007]

correlation=-1 as they are long and short positions on the same asset so they move in opposite directions
sd(portfolio)=sqrt(.15^2+.25^2-2*.15*.25*1)
sd(portfolio)=sqrt(.0225+.0625-.075)
sd(portfolio)=sqrt(.01)=.1=10% is the volatility of portfolio which is less than the individual positions due to diversification !!!
VaR(portfolio)=sd(portfolio)*z(99%)*Value of portfolio
VaR(portfolio)=10%*2.33*400000= 93200 USD

thanks

If it is the same asset, wont the volatility be same? I thought they are different assets....Btw ur approach is also correct....

 

[ShaktiRathore]

@shady007;
the above positions can be call/put options with different maturities on the same underlying asset , hence different volatility .

thanks

 

[David Harper CFA FRM]

ShaktiRathore i agree with 10% portfolio volatility but is the $400,000 value correct?

because I get:
individual VaR (long position) = $200*15%*2.33 = $69.79,
individual VaR (short position) = abs(-$200)*25%*2.33 = $116.32
diversified VaR = SQRT($69.79^2 + $116.32^2 + 2*$69.79*$116.32*(-1.0 correlation) = $46.53

similar to operating on portfolio volatility:
portfolio volatility = sqrt[$30^2 + (-50)^2 + 2*30*(-50)*(+1.0 correlation) = $20.00.
Then 20*2.33 = 46.53

btw, 15% and 25% are almost certainly not daily volatilities, that is absurdly high for daily ... I assume the question gave these as per annum, so they would need to be scaled down to daily.

 

[ABFRM]

hello david. i dont see any problem @ Shakti's approach because if u short 200,000 and long 200,000. The total portfolio value shud be 400,000. Is this correct if not then wat will be the value of the portfolio?

portfolio volatility = sqrt[$30^2 + (-50)^2 + 2*30*(-50)*(+1.0 correlation) = $20.00.
Then 20*2.33 = 46.53
In this solution portfolio value is missing. Are u deducing it as 1?

 

[David Harper CFA FRM]

Hi abhishek, I think the portfolio value is zero (short 200 funds long 200). I actually think you could define any initial portfolio value. For example, assume P = $400,000; if so, then the weights are +50% and -50%, such that:

• if P = 400K, w1 = 50% and w2 = -50% (i.e., in order to correctly invest $200K and short 200K), then P volatility = SQRT[50%^2*15%^2 + -50%^2*25%^2 + 2*50%*-50%*15%*25%*1.0 correlation = 5%, such that P VaR($) = 5%*2.33*400,000 = $46,600
• Or, assume P = $1,000,000, then weights must be +20% and -20% with P volatility (%) = 2% and P VaR ($) = 2%*2.33*$1,000,000 = $46,600

my point above was that, if portfolio volatility = 10%, then we must be assuming weights are 100% and -100%, such that portfolio value must be $200,000 (not $400,000) in which case the correct VaR = 10%*2.33*$200,000 = $46,600.

So, I do agree with 46.53, I just think it's a little confusing when trying to get VaR from the portfolio % in a long/short ...
... basically because the weights cannot sum to 1.0, they sum to 0.0. If a portfolio is long (w) and short the same (w), we cannot find a w% - w% = 1.0

Thanks,

 

[ABFRM]

Thanks for great explanation as usual it is clear to me.... But in practice how this can be achieved? i understood that he is shorting the portfolio of 200,000 dollars but at the same time he is long on 200,000 since he is short there is no initial cash outlay so that he can use the same cash for buying another portfolio. Is my understanding correct?

 

[ShaktiRathore]

AB just to make it more clear,
If we invest 50 in risk free asset and invest 50 in portfolio then,
wB=.50, wA=.5
Now if we borrow 50 and invest it in portfolio with value 100,
wR=150% and wB=-50% so total weight is still 100%=wR+wB
similarly if the long position is 200 and then we first short sell for 200 and then invest the proceeds of 200 into the long position on same notional instruments.its similar to borrowing 200 and investing in the long position so that initially wR=wB=0 as there was no borrowing and no long position but after borrowing wB=-100% and wR=100% as all the borrowing proceeds are invested into the long position so that our overall initial portfolio weight remains unchanged i.e. at 0%(100%-100%). so our overall portfolio position is 200 and we assume weights of 1,1 for both the short and long position,
sd(portfolio)=sqrt(.15^2+.25^2-2*.15*.25*1)
sd(portfolio)=sqrt(.0225+.0625-.075)
sd(portfolio)=sqrt(.01)=.1=10% is the volatility of portfolio which is less than the individual positions due to diversification !!!
VaR(portfolio)=sd(portfolio)*z(99%)*Value of portfolio
VaR(portfolio)=10%*2.33*200000= 46600 USD

thanks

 

[ABFRM]

thanks shakti i understood.

 

[cqbzxk]

AB just to make it more clear,
If we invest 50 in risk free asset and invest 50 in portfolio then,
wB=.50, wA=.5
Now if we borrow 50 and invest it in portfolio with value 100,
wR=150% and wB=-50% so total weight is still 100%=wR+wB
similarly if the long position is 200 and then we first short sell for 200 and then invest the proceeds of 200 into the long position on same notional instruments.its similar to borrowing 200 and investing in the long position so that initially wR=wB=0 as there was no borrowing and no long position but after borrowing wB=-100% and wR=100% as all the borrowing proceeds are invested into the long position so that our overall initial portfolio weight remains unchanged i.e. at 0%(100%-100%). so our overall portfolio position is 200 and we assume weights of 1,1 for both the short and long position,
sd(portfolio)=sqrt(.15^2+.25^2-2*.15*.25*1)
sd(portfolio)=sqrt(.0225+.0625-.075)
sd(portfolio)=sqrt(.01)=.1=10% is the volatility of portfolio which is less than the individual positions due to diversification !!!
VaR(portfolio)=sd(portfolio)*z(99%)*Value of portfolio
VaR(portfolio)=10%*2.33*200000= 46600 USD

thanks

Hi ShaktiRathore, I got a question very similar to this one, but I am not sure my thought is correct, I appreciate if you can correct my thought.
suppose we have a two asset portfolio,
Asset A value is 100, annual return is 10%, annual volatility is 25%, weight is 67%,
Asset B value is 50, annual return is 20%, annual volatility is 20%, weight is 33%,
correlation is 0.2, confidence level is 99% 2.33,
If we long asset A, short asset B, what is portfolio VaR at this point?
ANS:
Sigma P = sqrt( 0.67^2*0.25^2 + 0.33^2*0.2^2 - 2*0.67*0.33*0.25*0.2*correlation)
But I am not sure the portfolio E(r): E(R)= wA*rA-wB*rB ? or E(R)=wA*rA+wB*rB?
VaR(portfolio) = (SigmaP*2.33-E(R))*V where V= 100+50=150 or 100-50=50 ?
thank you!

 

[ShaktiRathore]

hi,
long A with value 150(100+50 borrowed from B) and short B means sell B for 50 so that total portfolio position is 200 out of which wA=150/200=.75 and wB=-50/200=-.25 so that total portfolio position remains 100.[.75*200-.25*200=150-50=100]
for the short position we take a negative weight so that,
Sigma P = sqrt( .75^2*0.25^2 + 0.25^2*0.2^2 +2*.75*(-.25)*0.25*0.2*.2)
E(R)=wA*rA+(-wB) *rB=wA*rA-wB*rB
VaR(portfolio) = (SigmaP*2.33-E(R))*V where V= 150+50=200

thanks

 

[cqbzxk]

hi,
long A with value 150(100+50 borrowed from B) and short B means sell B for 50 so that total portfolio position is 200 out of which wA=150/200=.75 and wB=-50/200=-.25 so that total portfolio position remains 100.[.75*200-.25*200=150-50=100]
for the short position we take a negative weight so that,
Sigma P = sqrt( .75^2*0.25^2 + 0.25^2*0.2^2 +2*.75*(-.25)*0.25*0.2*.2)
E(R)=wA*rA+(-wB) *rB=wA*rA-wB*rB
VaR(portfolio) = (SigmaP*2.33-E(R))*V where V= 150+50=200

thanks

Hi ShaktiRathore, thanks for replying, sorry about one more question, I didn't understand why the total Value of portfolio is 200 can you explain little more? and also, the Value of asset A is 200, based on the assumption, why should I count $50 from short Asset B into long asset A instead of just leave this 50 away ? thanks