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# Undiversified VaR

##### Member
Hi David,

Chapter: Portfolio Risk: Analytical Methods

It says that undiversified VaR is the sum of all VaRs of the individual positions in the portfolio when none of those positions are short positions.

I couldn't understand the effect of "short positions" in the calculation of VaR. Appreciate your feedback.

Thanks

#### ShaktiRathore

##### Well-Known Member
Subscriber
David can expalin later but i think you might give a thought about my explaination also,
undiversified VaR occurs when portfolio is not diversified so that there are no benefits of diversification. When positions move in opposite directions then the overall risk of the portfolio reduces to any adverse movements. So that overall loss is less. When all the positions are affected in the same way due to market movements than we say there is no diversification benefits.
consider three long positions 1,2 and 3 than their
undiversified VaR=VaR1+VaR2+VaR3
since all the positions are affected in the same way thus there is no diversification.
Now if we introduce a short position of 2 in the portfolio,
consider also two long positions 1, and 3 than their
undiversified VaR=VaR1-VaR2+VaR3 now position of 2 is affected in opposite direction to other positions 2 and 3 so now there are benefits of diversification. Overall loss is reduced as the positions now dont move in same direction. so its not undiversified VaR but a diversified VaR. Undiversified VaR does not gain from diversification but diversified Var does.
So in the end we can not define undiversified VaR for long and short positions combinations. But it can be define when all the positions are long positions.
thanks

##### Member
Thank you ShaktiRathore! Can we calculate undiversified VaR for short positions only?

#### ShaktiRathore

##### Well-Known Member
Subscriber
As far as I can understand that undiversified VaR tell you about how bad things can go. That is during downtime all the positions are affected in the same way that is the correlation between them increases so that they almost move in same direction. The maximum loss that can occur is undiversified VaR.For short positions the I think its not required because the poistions are not affected by the downtime. Also during negative moves(during normal or boom markets) all the positions does not move in the same way as during downtime so calculating undiversified VaR for short poistions is not sensible because there can be some diversification because its very rare that all the positions move in same direction or that correlation becomes abnormally very high during normal market conditions as during downtime.
I hope u understood.
thanks

#### David Harper CFA FRM

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

Great observations! For reference, Imad refers to Jorion page 165: "Undiversified VaR: The sum of individual VaRs, or the portfolio VaR when there is no short position and all correlations are unity."

As Jorion's subsequent text indicates, I think the statement could be elaborated to:
"Undiversified VaR: The sum of individual VaRs, or the portfolio VaR when all pairwise long and pairwise short positions (i.e., long + long; short + short) have pairwise correlation of 1.0; and all pairwise long/short positions (i.e., long + short, short + long) have pairwise correlation of -1.0."
... if you have two $10 positions, worst exposure is long+long or short+short if correlation is +1.0; but if you are long/short, correlation of 1.0 is a perfect hedge and worst is correlation = -1.0 ... this formalizes Shakti's characterization "that undiversified VaR tells you how bad things can go" In analytical (mean variance) portfolio (diversified) VaR, short positions are generally handled (and easily) with negative weightings. Because a negative weight will impact the third term in the portfolio volatility formula, where w1 = weight1 and weight2 = weight2 and let w2 be negative to signify a short position in asset2: Portfolio volatility (i.e, diversified) = SQRT[ w1^2*vol1^2 + (-w2)^2*vol2^2 + 2*w1*(-w2)*vol1*vol2*correlation(1,2) ] ... a switch from long to short (-w2) will not change the (-w2)^2*vol2^2 due to squaring, but will switch the third term. This extends to matrix with n-assets; e.g., our analytical VaR learning spreadsheet mostly holds up if given negative weights However, in the two-asset VaR, we cannot (to my knowledge) adapt shorts to the general: Diversified VaR = SQRT[VaR1^2 + VaR2^2 + 2*correlation*VaR1*VaR2] ... because the individual VaRs are each absolute values. So we need to first compute portfolio volatility, then multiply that by the normal deviate (eg, 2.33 @ 99%). Here is the two-asset XLS where the 2nd column illustrates a short https://www.dropbox.com/s/n14y2yzqrekddci/0921_analytical_VaR_short.xlsx Thanks, #### gunessari1980@gmail.com ##### New Member Hi David, I am struggling with the same problem. I need to calculate VaR consisting of two currency positions ( one is negative position). EUR and GBP currency positions are 6 milyon USD and (-7) milyon USD, respectively. Daily volatilities are 1,22% and 1,09%, respectively. Corr is 66,7%. Daily portfolio VaR is 142.027 USD. When I check component VaR (63.081 USD and 78.945 USD), both are positive. That means there is no diversification because of positions, which doesn't make sense to me. At least USD component should be negative due to the fact that correlation is positive and they should be moving in opposite direction. #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @gunessari1980@gmail.com I get the same diversified VaR of$142.1 (see below). And here is the spreadsheet.
And our component VaRs (if we allow for rounding) are effectively the same.

But that's significant diversification benefits:
• The component VaRs sum to the diversified VaR; I don't think we expect either to be negative (I think my row 43 below, is not robust, somehow)
• The individual VaRs, here, are $177.50 (i.e., absolute value) and 170.29, which sum to an undiversified VaR of$347.8. For example, if you used position of +7,000 and +6,000 with correlation of 1.0, you get portfolio VaR of 347.8
• Put another way, you could also get an diversified VaR of 142.1 if you assume +7,000 and +6,000 positions, and a correlation of -2/3; i.e., hedging with a long position that is negatively correlated (-2/3) is the same as shorting a position that is positively correlated (+2/3). So, your calculations look correct to me: diversified VaR of 142.1 is substantial benefit from undiversified VaR 347.8. I hope that's helpful!

#### gunessari1980@gmail.com

##### New Member
You made me think from another angle, David. That was great help! thanks a lot. Now it makes sense to me

Gunes Sari,FRM