## Pages

### Paired Samples t-test vs. CAPM Linear Regression

When I first learned that a CAPM analysis is based on a regression analysis I was both surprised and confused. It seemed like such an unusual way to test two simple hypotheses: (1) Does one portfolio (or security) outperform another (alpha), and (2) does an outperforming portfolio (or security) do so with greater risk (beta).

Within the CAPM linear regression analysis, alpha is represented by the intercept, while beta is represented by the unstandardized beta weight. However, if one wanted to know whether a portfolio, on average, outperformed another, wouldn't it be more straightforward and arguably accurate to simply test the hypothesis with a paired samples t-test? In my opinion, doing so will provide precisely the required answer. By contrast, alpha is estimated by an intercept within a CAPM regression model, which represents the estimated performance of a portfolio, when the benchmark is equal to zero. Is that what people want to know? When the S&P 500 yields a zero return, what does the competing portfolio yield? That's what alpha tells us. That's different to, 'On average, does the portfolio outperform the benchmark?'

Beta is just as problematic in my mind. Is one portfolio riskier than another? That's what beta is getting at. Why not test the variances associated with the two portfolios (or securities) directly? It can be done quite easily via the Pitman-Morgan test (chech out this). Instead, the CAPM linear regression approach considers the slope associated with a regression to be representative of the difference in variability (risk) between two portfolios. A slope in a linear regression model represents the unit change in the dependent variable for each unit increase in the independent variable. Is that what people really want to know when testing the difference in risk between two portfolios? Again, why not go straight to the variances?

In this three part series of videos, I perform a paired samples t-test within a CAPM framework. I do so on some Berkshire-Hathway data versus the S&P 500. I'd love to hear other opinions on the paired samples t-test (with Pitman-Morgan test) versus CAPM linear regression approach.