The beta of the CAPM measures the linear market risk. The first to assign that the beta was not linear were G.Pettengill, S.Sundaram and I.Mathur, “The Conditional Relation Between Beta and Returns”, Journal of Financial and Quantitative Analysis , 1995. They showed that the beta of a stock is different depending if the market is up or down. They did two regressions: one with positive market returns and one with negative market returns. Another approach has been developed for hedge funds by Favre and Galeano, “Hedge Funds Analysis Using Loess Fit Regression”, Journal of Alternative Investment , Spring 2002. They showed, with a powerful statistical technique, called Local Regression,, that several hedge fund indices have non-linear option payoffs.

By using polynomial regression between the stock and the market returns, it is possible to see if an asset has a higher exposure on the downside.

As many hedge funds managers use derivatives or are active managers, their returns are not linearly related to the SP500, for example. This is why the polynomial regression is appropriate. The figure below exhibits that the asset (vertical axis) is not exposed (even non-linearly) to S&P500 negative returns (left on the horizontal axis).