When to skip weights in volatility?

[Newkirk]

Hi,
I've seen portfolio volatility calculated with weights ie: 2 asset portfolio with 60% in A and 40% in B volatility - sqrt(0.6^2*volatility_A^2 +0.4^2*Volatility_b+2*0.6*0.4*Covariance)

I've also at times seen it without any weights.

Can anyone give me some pointers why we would skip the weights ?

Tx

 

[ami44]

Hi Newkirk,

The mathematically answer is:
If you have two random variables A and B with standard deviation s(A) and s(B), than the standard deciation of the sum A+B is
s(A+B) = sqrt(s(A)^2 + s(B)^2 + 2 * s(A) * s(B) * cov(A,B) )

For example consider two portfolios with A and B being the portfolio value in a year. Lets say A has a mean 1000 EUR and s(A) = 200 EUR, while B has a mean of 400 EUR with s(B) = 20 EUR. Both portfolios are assumed independent i.e. Cov(A,B) = 0.
It follows, that A + B has a mean of 1400 EUR and a standard deviation of s(A+B) = 201

Now consider two random variables that represent the return of the portfolio in a year
Ra = A / Va - 1 and Rb = B / Vb - 1
with Va and Vb being the current values of the portfolios and A and B still being the value in a year. The return of the combined portfolio is the weighted sum of the single returns:
Ra+b = (A+B) / (Va + Vb) - 1 = Ra * Va/(Va+Vb) + Rb * Vb/(Va+Vb)
Put this in the formula for the standard deviation above and you get:
s(Ra+b) = sqrt(Va/(Va+Vb)^2 * s(Ra)^2 + Vb/(Va+Vb)^2 * s(Rb)^2 )
Actually you also need to know that s(x * A)^2 = x^2 * s(A)^2 for any number x and any random variable A.

Or to summarize: if you simply add two random variables you use the formula without weights. If you have the weighted sum, as for example the return of a combined portfolio you also use the formula with weights for the standard derivation.

I hope that was understandable, otherwise feel free keep asking.

 

[Newkirk]

Thanks for answering! Your answer helped.