# Portfolio variance in EWMA

Discussion in 'P1.T2. Quantitative Methods (20%)' started by Ludwma, Feb 27, 2012.

1. ### LudwmaMember

Hi David,

In your "Estimating Volatilites and Correlations" practice questions, there is following question:

"If w is a column vector of portfolio weights, w(T) is the transposed row vector of the same weights and Z is a covariance matrix, which of the following is LEAST likely to suggest a violation of the consistency condition?"

One of the answers is "We compute a negative portfolio variance". What is this? Is this simply the variance of the portfolio? I'm asking this because I do not understand what is written in the solution: "w(T)Zw is the portfolio variance".

Thanks,

Fabiano
2. ### David Harper, CFA, FRM, CIPMDavid Harper

Hi FS,

w'Zw (or w^T*Z*w) is just the matrix notion for n-asset portfolio variance. We are accustomed to the 2-asset case, where variance = w1^2*variance(1) + w2^2*variance(2) + 2*w1*w2*covariance(1,2), but this is just a special case of a vector of several weights (vector w) and a covariance matrix (Z). We get the n-asset variance if we multiply: row vector of weights * covariance matrix * column vector of weights

In the Hull chapter, on which these AIMs are queried, includes a "consistency condition" which specifically is the requirement that the matrix be positive (semi) definite: http://en.wikipedia.org/wiki/Positive-definite_matrix
i.e., the matrix isn't usable if the variance is negative, a variance must be non-negative (Why?)

I hope that helps, thanks,