Pettengill, Sundaram, and Mathur 1( by beta. Tests of this assertion examine portfolios of securities to reduce both estimation error and nonsystematic risk. The relationship tested is represented as (1) E(RP)=R,+B *(E(Rm)-R) where E(Rp) is the expected return for the risky portfolio P, R is the current risk-free rate, Bp is the covariance between the portfolios return and the markets return divided by the variance of the market, and E(Rm)is the expected return to The interrelationship between these variables provides crucial implica for testing the relationship between beta and returns. On the assumption of a positive risk-return tradeoff, the expected return to the market must be greater than the risk-free return(or all investors would hold the risk-free security ) Since the term(E(Rm)-Ry)must be positive, the expected return to any risky portfolio is a positive function of beta. This relationship has prompted researchers to examine the validity of the SLB by testing for a positive relationship between returns and beta. Since these tests use realized returns instead of expected returns, we argue that the validity of the SLB model is not directly examined. Indeed, recognition of a second critical relationship between the predicted market returns and the risk free return suggests that previous tests of the relationship between beta and returns must be modified The need to modify previous tests results from the model's requirement that a portion of the market return distribution be below the risk-free rate. In addition to the expectation that, on average, the market return be greater than the risk-free rate, investors must perceive a nonzero probability that the realized market return will be less than the risk-free return. If investors were certain that the market eturn would always be greater than the risk-free rate, no investor would hold the risk-free security. This second requirement suggests that the relationship between beta and realized returns varies from the relationship between beta and expected return required by Equation(1). However, the model does not provide a direct indication of the relationship between portfolio beta and portfolio returns when the realized market return is less than the risk-free return. a further examination as detailed below, shows that an inverse relationship between beta and returns can be reasonably inferred during such periods In order to draw this inference, it is necessary to provide an analysis of the portfolio return distribution implied by the SLB model. This model shows that the expected return for each portfolio is a function of the risk-free return, the portfolio beta, and the expected return to the market. The expected return for the portfolio is the mean of the distribution for all possible returns for that portfolio in the appropriate return period. Identical with the market return, for all portfolios with a positive beta, the expected value for the return distribution must be greater than the risk-free rate and the return distribution must contain a non-zero probability of realizing a return below the risk-free rate. To arrive at testable implications, we must extend this analysis to examine the differences in the return distributions of portfolios with different betas Portfolios with higher betas have higher expected returns because of higher risks. For high beta portfolios to have higher risk, there must be some level of realized return for which the probability of exceeding that particular return is