# Personal Taxes

# Personal Taxes

Brennan [1973] examined the effect of taxes on the Sharpe-Lintner CAPM. Before the Tax Reform Act of 1986, capital gains were taxed in the US at a lower rate than dividends. This implied that the equilibrium expected return on an equity should be a function of both its *β* and its dividend yield. Moreover, since marginal tax rates differ between individuals, two-fund separation no longer holds. Even if homogeneous expectations of expected before-tax returns are assumed, expected after-tax returns will differ and the optimal portfolio of risky assets will depend on an individual's (marginal) tax rate.

Assuming end-of-period dividends across securities (as represented by the vector *δ*) are known with certainty, and that preferences are defined on moments after-tax, Brennan shows that the following equilibrium relationship holds:

where *δ _{m}* is the dividend yield on the market portfolio and

*τ*is a non-negative coefficient reflecting a weighted average of individuals' tax rates on ordinary income and capitalgains.

This model has been generalised by Litzenberger & Ramaswamy [1979] to account for a progressive tax scheme and income related constraints on borrowing (in order to rule out the possibility of negative taxes). Their equilibrium relationship is:

where the term *a* is related to the existence of constraints on borrowing, the term *b* is a measure of global market risk-aversion, and *τ* reflects a weighted average of individuals' marginal tax rates.

Both equations (5.8) and (5.9) hold assuming that dividends are exogenous. Miller & Modigliani [1961] and Black & Scholes [1974], *inter alios*, have argued that companies will adjust their dividend policies until they are in equilibrium. The spectrum of policies offered would be such that any one company would be unable to affect the price of its shares by (marginal) changes in its policy. In this case, the standard CAPM relationship would hold.