How do you get the market beta for each stock given a variance covariance matrix?

Question

How do you get the market beta for each stock given a variance covariance matrix?

Answer 1

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RE:

How do you get the market beta for each stock given a variance covariance matrix?

The question is "Assume the market consists of only the securities in the variance-covariance matrix below. The matrix displays the variance of each of the seven stocks along the diagonal and each of the covariance in the off-diagonal cells. For each stock, what is the market beta?"



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Answer 2

You need to know the weights of each of the stocks in the market portfolio. Then to get the betas you:

a) Calculate the variance of the market portfolio using the weights and the variance covariance matrix

b) Divide each covariance by the variance of the market portfolio for the beta



Note that a is a fairly lengthy process for a test.

Answer 3

Re: Add'l details - Yes, safe to assume the weights of each are 1/7.



The LONG WAY....

Here's a link for calc'g the Portfolio Variance (click on #3 Mathematical model) at the bottom of the box you'll see the formula for a 3 asset portfolio - you have 7 assets so each asset is weighted 1/7 = 0.142857 - lots of math.

http://en.wikipedia.org/wiki/Modern_portfolio_theory#Mathematical_model



To calc that Port var. you will need the correlation coefficients p(x,y) ,

p(x,y) = covar(X,Y) / (std devX * std devY)...you have the covars in your matrix, and you can easily calc each std dev by taking the square root of the variance

you can find that formula here:

http://en.wikipedia.org/wiki/Correlation



Beta for each asset is..example Asset A: Covar(returnA,returnMkt) / Var(returnMkt)...where, in your case, the "Market" is the Portfolio.

see here: http://en.wikipedia.org/wiki/Beta_(finance)



The SHORT WAY...

the variance of an equally weighted portfolio (in your case, that represents the entire market) reduces to: [(1/n)*mean variance] + { [(n-1)/n] * mean covariance }

for a 7 asset port...Variance = [ 0.142857*(sum variances/7) ] + [ 0.857143 *(sum covariances/7) ]

see here: http://blog.5m10y.com/2010/04/06/deriving-the-variance-of-equally-weighted-n-asset-portfolio/



then for Beta of each asset "a" you need: Covar(Ra,Rmkt) / Variance(Rmkt)...remember your portfolio IS the market.



FYI - if they were to ask you what the Portfolio Beta is, you know it's 1, since the portfolio IS the market...(by definition the market Beta is 1, the covariance of an asset with itself is it's variance, so Mkt (i.e. HERE Port) Beta = Covar(mkt,mkt) / variance(mkt) = var(mkt)/var(mkt) = 1...) that would be a trick question that, if you know this answer, would save you lots of time.