Value at Risk - Long/Short position

Question

I have a simple question on the VaR for a portfolio that consists of a long and short position. Say I have a portfolio consisting of the following positions:

• long 1000 shares of stock X

• short 1000 shares of stock X

Let's assume the daily volatility for Apple stock is 2% and that the stock trades at 2 dollar/share so my investment is 4000 dollar in total. I want to calculate the VaR for each of the positions separately and for the full portfolio, but I am a little bit confused with the signs associated with long and short position. Intuitively, the portfolio should have VaR = 0 as the long and short position in the same stock with equal investment have a neutralizing effect.

(1) VaR for the long position VaR = 2%*2000*2.33 = 93.2

(2) VaR for the short position VaR = 2%*abs(-2000)*2.33 = 93.2

Now, for the combined position, my question is which signs to use? Do we say:

VaR = 2.33*sqrt((40)^2+(-40)^2+2*40*(-40)*corr) = 0 since corr = 1

VaR = 2.33*sqrt((40)^2+40^2+2*40*40*corr) = 0 since corr = -1

Since you have the same stock X, the correlation is simply equal to 1 or do we say it is equal to -1 because we have a short and long position in it? So generally my question is, do we take the absolut value if the position is short or?

Thanks!

Answer 1

i_love_rain: I don´t think it should be negative. A negative Value at Risk should be a potential profit as stated here (or not exist at all): Why is Value at Risk non-negative?.

Seperately both positions have a positive VaR as stated by the OP. Both positions could lose money seperately, one when the share price rises, one when it falls.

The negative correlation as second option presented by the OP is right. If both positions were long (so it would be the same position anyway) the correlation would be 1. When one stock is long, the other short the correlation is -1.

Answer 2

I think VaR for the short position should be VaR = 2%*(-2000)*2.33 = -93.2

Answer 3

Since you have (exactly) the same long and short position in (exactly) the same asset, your portfolio is perfectly hedged, what means VaR is equal to zero. There is actually no room for VaR consideration :)