standard deviation in long-short portfolio?

I have a simple question though confusing.

Does it make sense that the annualized standard deviation of a long-short portfolio is higher than the long leg and the short leg (separately)?

• I will take the formula for 2 assetsas example: formula if i understand well if the correlation is negative the standard deviation of the whole portfolio decrease, am i right? – user53387 Feb 13 '17 at 9:22
• Thanks. According to your answer when taking two assets for long-short portfolio 100%(long)-100%(short) the variance will be higher because I'm taking the minus returns for the short leg? – yudyud Feb 13 '17 at 10:29

i understand well if the correlation is negative the standard deviation of the whole portfolio decrease, am i right?

Yes - if the weights are both positive, meaning you are long in both. If one goes up, the other tends to goes down, and your overall variance decreases.

The formula you reference requires that the weights add up to 1, so it won't work for a perfectly offset portfolio since the total net position is 0.

Does it make sense that the annualized standard deviation of a long-short portfolio is higher than the long leg and the short leg (separately)?

Yes, if the legs are inversely correlated. When one leg goes up, the other tends to go down, but since you are short one leg your total is higher, and vice-versa, so the overall variance is higher.

To make the formula work, let's say you hedge \$200 of A by shorting \$40 of B.

Your net total investment is \$160 (200 - 40), Then w1 would be 1.25 (200 / 160) and w2 would be -0.25 (-40 / 160). Suppose they had equal variance of 0.20 and are perfectly inversely correlated

so

s(p) = SQRT(s1^2*w1*2 + s2^2*w2^2 + 2*r*s1*s1*w1*w2)
= SQRT(.04 *1.56 + .04 *0.063+ 2*-1*.2*.2*1.25*-0.25)
= SQRT(.0625     + .0025     + .025)
= SQRT(0.09)
= 0.3

So the negative weight offsets the negative correlation and your variance is higher.

Yes, when two assets were negatively correlated (measured when both are Long).

Example: SPY and EDV.

Read Variance of a Portfolio formula that involves covariance.