Hi Imad, Shakti,

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,

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