Negative Standard Deviation? about, question, possible, excel

smartpants

Ok heres a question i ws thinking about when doing some work in excel...

The variance of a 2 asset port = (Wa^2)(Var a) + (Wb^2)(Var b) + 2(Wa)(Wb)(Std a)(Std b)(Correl (a,b))

My question is this...lets say the correlation is -1. Wouldnt it be possible to get a variance of the portfolio that is negative? What would that mean? or am i thinking a bout this wrong?

tikfed

Variance is squared, removes the negative, and standard dev is obviously the square root of variance which makes it impossible to have a negative standard deviation.

Standard deviation can be zero, but never negative. Both sides of the distribution result in positive value for standard deviation.

I think you're thinking about this one a little too hard.



Edited 2 time(s). Last edit at Thursday, June 23, 2011 at 11:39AM by Chuckrox8.

hw0799

Spanishesk Wrote:
-------------------------------------------------------
> Ok heres a question i ws thinking about when doing
> some work in excel...
>
> The variance of a 2 asset port = (Wa^2)(Var a) +
> (Wb^2)(Var b) + 2(Wa)(Wb)(Std a)(Std b)(Correl
> (a,b))
>
> My question is this...lets say the correlation is
> -1. Wouldnt it be possible to get a variance of
> the portfolio that is negative? What would that
> mean? or am i thinking a bout this wrong?

Yes. If the third part of the equation is negative enough to make the summation negative, variance become negative and standard deviation become meaning less as there is no square root for negatives.

However, it is a mathematical truth that third part of the equation can't make the summation negative. The most it can do is making the variance zero (I don't know how to prove it, but know as a fact).

kingstongal

The least you can get is when the variances are equal, the weight is equal and correlation is -1. In this case the formula reduces [by letting Wa = Wb, and var(a) = var(b)] to:

2*(Wa^2)(Var(a)) + (-1)*2(Wa^2)(Var(a)) = 0

You will never get a negative variance.

canadiananalyst

Spanishesk Wrote:
-------------------------------------------------------
> Ok heres a question i ws thinking about when doing
> some work in excel...
>
> The variance of a 2 asset port = (Wa^2)(Var a) +
> (Wb^2)(Var b) + 2(Wa)(Wb)(Std a)(Std b)(Correl
> (a,b))
>
> My question is this...lets say the correlation is
> -1. Wouldnt it be possible to get a variance of
> the portfolio that is negative? What would that
> mean? or am i thinking a bout this wrong?

The variance of a 2 asset port = (Wa^2)(Var a) +(Wb^2)(Var b)
+2(Wa)(Wb)(Std a)(Std b)(Correl(a,b)) = (wa*Std a-wb*Std b)^2+2*Wa*Wb*Std a*Std b*(1+cor(a,b)). Each term is non-negative. Therefore, the sum is non-negative.

anish

Forget about formulas. Just think for a second...what is standard deviation? And how can it be negative?

Sunshine4ever

Iginla2010 Wrote:
-------------------------------------------------------
> Forget about formulas. Just think for a
> second...what is standard deviation? And how can
> it be negative?


my thoughts exactly....

IAmNeil

obviously it cant be negative. But mathematically i looked at the equation and started thinking if the third part of the equation was negative enough, it could lead to a negative result. Was just trying to see if it was possible in the equation. I am fully aware that negative std dev wouldnt make sense.

sabaruch

In a two asset portfolio assuming perfect negative correlation your standard deviation would be zero. The third part of the variance equation would most certainly be negative, but the first two expressions would be equal to that amount.

Part of the equation can certainly be less than zero, but it will always be offset by the first two expressions.

bolligerallstar

Chuckrox8 Wrote:
> In a two asset portfolio assuming perfect negative
> correlation your standard deviation would be zero.

Given a careful selection of the weights obviously.

There are some neat things about this question though...if you look at the three asset equation (and more assets as well), if you choose the correlations carelessly, it is actually possible to have the formula produce a seemingly negative variance. This is because as you add assets to the portfolio, the correlations start to become constrained based on their mutual correlations with the other assets.

e.g. if a and b are correlated, and b and c are correlated, then the correlation between a and c is bounded to something tighter than [-1,1]. If you want a tangible example of this, assume corr(a,b) = -1, and corr(b,c) = -1. What do you think the corr(a,c) has to be? (It’s 1) It can no longer be the entire range [-1,1]. If you assume it's less than 1, than weird things can happen, like you can produce a negative variance using the multi asset version of the above equation.

More technically, this is related to the covariance matrix and its property of being positive semi definite. If the matrix doesn't have this property then it could not have been obtained using real numbers. I actually busted a major investment consulting firm using an imaginary covariance matrix for portfolio construction, for ALL their clients, a few years ago.

But like everyone has already said, it's variance; the definition of variance is axiomatic....and it must be positive in the real numbers.