Why we are solving for (weight*covariance)^2 to calculate portfolio variance

Hi Ryan!

Great to have you in the course and thanks for reaching out!

Can you please confirm that you are referring to the notation that appears in, for example, minute 1:22 of the video? If that's the one, then portfolio variance is exactly how you say it is; we are just using matrix notation to represent it. Please feel free to refer to an explanation/definition of matrix multiplication or reply to this message should you need a clarification on this subject matter. Thank you!

Looking forward to your answer.
Best,
Martin

Hi Martin,

I have been really enjoying your style of teaching in the python courses so far and thanks for taking the time to answer! 

If i understand you correctly, you're saying:

(w·Cov)^2 = the minute 1:22 econometric matrix notation = single number = portfolio variance = (w1σ1 + w2σ2)^2

I understand the mechanics of it, that squaring (w·Cov)^2 in matrix form is represented by WT * (Covariance Matrix) * W and other online sources have confirmed that Expected portfolio variance = WT * (Covariance Matrix) * W

What I don't quite understand is why this equation, (w·Cov)^2, represents portfolio variance. What does using the weight multiplied by covariance and then squaring it have to do with portfolio variance?

Hi Ryan!
Please excuse me for the delayed response. 

Yes, that's what I meant. 

Regarding your question. The Covariance matrix contains the relationship between each pair of stocks. This is, indeed, all portfolio variance is! Please note that we are then multiplying each variance by the relevant weight with which each stock participates in the entire portfolio.

If we didn't use the weights, we would have had an equally weighted portfolio (i.e. if it was a two-stock portfolio, w1 = w2 = 0.5). If we had only the covariance matrix, we wouldn't have known how to calculate the overall variance of the portfolio, but would only have the relationships between its components. If we only had the variance of each stock, we would have ignored the relation between them (which is, indeed, the covariance between each pair of stocks).

Hope this helps.
Kind regards,
Martin