Differential taxation and security market lines–a clarification

The literature on differential taxation of asset specific payoffs presents different representations of security market lines (SMLs), with puzzling features such as parallel lines and lines with the same intercept but different slopes - a security market fan. We show that these results can be traced back to using different combinations of pre-tax and after-tax figures for betas and expected returns. Above all, for betas based on returns and a stochastic discount factor (SDF), there is a single SML in after-tax expected return and after-tax beta space. Using pre-tax values for betas and/or expected returns leads to different representations in the respective beta and expected return space. With betas that use the market return instead of the SDF, it is important to distinguish whether tax payments are redistributed back to the agents. If they are, a single SML can only be obtained through a beta using the pre-tax market return and the after-tax return of single assets. In addition to SML representations, we also discuss differential taxation with respect to mean-variance frontiers in expected return and standard deviation spaces.