Value at Risk for Equal Long and Short Positions in One Stock
Summary
The document examines how to interpret Value at Risk for equal long and short holdings in the same stock. It calculates separate position risks using the stated share count, price, daily volatility, and confidence multiplier, then asks how to combine them while handling the signs and correlation. The central portfolio insight is that identical opposing exposures offset: their return sensitivities have opposite signs, even though the underlying stock's return is perfectly correlated with itself.
The responses distinguish the positive standalone risk of each leg from the net risk of the combined position and conclude that an exactly matched long and short is fully hedged, giving zero portfolio VaR under the stated assumptions. This illustration is limited to the same asset and equal quantities; it does not address basis risk, mismatched positions, financing, transaction costs, or changing exposures. The discussion also shows why correlation of asset returns and the signed exposures in a portfolio calculation must be treated separately.
Key ideas
- Each standalone long or short position can have positive VaR.
- Equal and opposite positions in the same asset cancel their market exposure.
- The asset return correlation remains positive; the short exposure contributes the opposite sign.
- The zero-risk conclusion assumes exactly matched positions in the same stock.
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Full text
# Value at Risk - Long/Short position # Value at Risk - Long/Short position 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 by user3748386 (score 1) https://quant.stackexchange.com/a/37102 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 by i_love_rain (score 0) https://quant.stackexchange.com/a/37021 I think VaR for the short position should be VaR = 2%*(-2000)*2.33 = -93.2 ## Answer by Wojciech Szajnar (score 0) https://quant.stackexchange.com/a/47289 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 :)
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