The seminal study by Fama and MacBeth (1973) initiated a stream of papers testing for the cross-sectional relation between return and risk. The debate wether beta is a valid measure of risk has been renimated by Fama and French (1992) and subsequent studies. Rather than focusing on exogenous variables that have a larger explanatory power than an asset's beta in cross sectional tests, we assume the matrix of variances-covariances to follow a time varying ARCH process. Using monthly data from the UK market from February 1975 to December 1996, we compare the cross sectional return-risk relations obtained with an unconditional specification for asset's betas to those obtained when the estimated betas are based on an ARCH model. We also investigate the Pettengill, Sundaram and Mathure (1995) approach, which allows a negative cross sectional return-risk relation in periods in which the market portfolio yields a negative return relative to the risk free rate. These tests are also carried out on samples pertaining to a specific month and on samples from which a particular month is removed. Our result suggest that CAPM holds in downward moving markets than in upward moving markets hence beta is a more appropriate measure of risk in bear markets.