# Question in Q-Bank

Can anyone enlighten me on this one ? An analyst is trying to determine whether stock market returns are related to size and the market-to-book ratio, through the use of multiple regression. However, the analyst uses returns of portfolios of stocks instead of individual stocks in the regression. Which of the following is a valid reason why the analyst uses portfolios? The use of portfolios: A) will increase the power of the test by giving the test statistic more degrees of freedom. B) will remove the existence of multicollinearity from the data, reducing the likelihood of type II error. C) reduces the standard deviation of the residual, which will increase the power of the test. Your answer: B was incorrect. The correct answer was C) reduces the standard deviation of the residual, which will increase the power of the test.

Portfolio residuals would be lower via diversification where an individual stocks risk can have a wide st dev.

This question looks to mesh both Quant and PM; If you think about how the risk of a portfolio of assets differs from the risk of a single asset you’ll see the answer. Cov(asset1,asset1) = Var(asset) = (B of asset^2)(std dev of market^2) + Std dev of error^2 Cov(asset1,asset2) = (B of asset1 * B of asset 2)(std dev of market^2) As you increase the number of assets in your portfolio the std dev of error (nonsystematic risk) decreases and your returns become more reliable increasing the power of your test.

Makes sense guys, now that I think of it, it only sounds logical, thanx !

The multi regression that they are talking about here is very closly related to ‘Fama French model’ in Equity, which also takes size effect and growth effect. I opted C, by choice of elimination: A) will increase the power of the test by giving the test statistic more degrees of freedom. - If we choose portfolio of assets, then DOF will be less not more. B) will remove the existence of multicollinearity from the data, reducing the likelihood of type II error. - Irrespecitve of single asset or portfolio of assets in the regression, there could be possible to have correlation btw the two independent variable (ie Size and Mkt/BV). C) reduces the standard deviation of the residual, which will increase the power of the test.