Some Analytical and Empirical Results on the Relationship between CAPM Beta and Downside Beta

Using a data generating process in the mean-variance framework a relationship between CAPM beta and downside beta is derived. The derived relationship reveals that when the target rate exceeds (is lower than) the risk-free rate downside beta is higher (lower) than CAPM beta for low CAPM beta portfolios and the difference between the two betas is large in less volatile markets. The converse holds for high CAPM beta portfolios. Generally, for portfolios with abnormal returns, the standard error in the estimated downside beta could be considerably higher than the standard error in the estimated CAPM beta