Annual Residual Risk (w) from Quarterly

CFA Level II Portfolio Management
#1

I am working through a question and I have to annualize the residual error which is given as:

w = 1% quarterly

then the answer annualilizes this by doing 1 * SQRT (4) = Annualized w of 2…

I cant not for the life of me understand how this annulizes the residual error… anyone?

#2

Variances are additive, so,

W1² + W2² + W3² + W4² = Wannual²

If all of the quarterly residuals are (assumed to be) the same (=W), then,

Wannual² = W² + W² + W² + W²

=4W²

So,

√(Wannual²) = √(4W²)

Wannual = 2W

#3

Awesome thanks bud

#4

My pleasure.

#5

OK I have a follow up question then. In the same question it gives the annual residual returns of 0.5 quarterly. And then they solve for the annual buy doing .5 +.5 +.5 +.5 = 2

Shouldnt it be 0.5 squared + 0.5 squared + 0.5 squared + 0.5 squared = 2 squared = SQRT 2 = 1?

#6

Nope.

It’s akin to calculating total returns: technically they should be compounding, but they’re making it simpler by summing.