The Dividend Puzzle Unpuzzled

Abstract

A claim of solving the dividend puzzle, as formulated by F. Black, is made. Various stock pricing formulas are established, extending substantially the classical Miller-Modigliani theory of valuation of shares in a way to contradict their main conclusion that dividends are irrelevant. Indeed, the main stock pricing formula obtained implies that if the company's costs depend on the available cash, then dividends affect the stock pricing function in a substantial manner. On the other hand, if the company's costs are independent of the available cash, if the dividend tax rate is constant, and under a further technical condition, the established formula is consistent with the Miller-Modigliani's conclusion of dividend irrelevancy. The presented theory is in the context of Miller-Modigliani's stream of dividends approach to valuation of shares, facilitated by the latest developments in the Black-Scholes type pricing theory for incomplete multivariable markets. Various cash flows are modeled, some using Ito SDEs, so that the value of the company is not assumed to be tradable, yielding market incompleteness. In spite of market incompleteness, under some simplifying assumptions, referred to as the Basic Equity Model, the established stock pricing formulas yield unique prices, i.e., prices independent of the investor's risk aversion. Some more general models when this property of prices does not hold are considered as well. The intrinsic non-uniqueness of fair stock prices in such cases is settled by choosing the value of the investor's HARA risk aversion parameter, while if the CARA wealth utility function is used, fair prices remain unique.