Dividend Policy Irrelevance and Eigenvalue Location

This note gives a quali…ed a¢ rmative answer to a natural question, asked by Ohlson and motivated by some earlier work, concerning the irrelevance of dividend policy to the calculation of equity in the context of an Ohlson style general linear accounting dynamic. Does Dividend Irrelevance occur when discounting at a rate of interest R if and only if R is set equal uniquely to the dominant eigenvalue of the principal submatrix? The latter submatrix relates the accounting variables to each other in the absence of any dividend payout. The question reduces to the assertion that the maximum eigenvalue max of the following ‘bordered diagonal matrix’ 266664 1 0 0 1 0 2 1 ::: 0 0 n 1 !1 !2 !n+1 377775 : lies between the …rst largest and second largest among j 1j; :::; j nj. An a¢ rmative answer necessarily restricts the dividend policy vector (!1; :::; !n+1). The results show that an algebraic condition equivalent to dividend irrelevance derived previously is not vacuous. Key words: Dividend irrelevance, dominant eigenvalue, bordered diagonal matrix. 1. Introduction Consider the following …rst-order recurrence, which we term the Ohlson accounting dynamics. zt+1 = Azt + bdt + avt; dt+1 = wzt + dt + vt; vt+1 = + vt; =; ( ) with 0 < 1: Here zt = (z t ; :::; z n t ) 2 R; is called the accounting state vector, while dt and vt are reals de…ned for t = 0; 1; 2; ::; representing respectively the dividend and the information variable (a proxy for un-accounted value); the value of the latter is assumed to dwindle into insigni…cance over time. The vector !div = (w; ; ) is the dividend policy vector. A is a real matrix of size n n; called the reduced matrix of the system ( ). Its eigenvalues 1; 2; :::; n are listed in order of decreasing modulus, i.e. j 1j j 2j ::: j nj: The augmented matrix is given by A = A(w; ) = A b w ; regarded as a function of (w; ): Its eigenvalues will likewise be regarded as functions of (w; ) and referred to as 1; 2; :::; n+1; identi…ed through the condition i(0; :::; 0; ) = i for i = 1; :::; n; and n+1(0; :::; 0; ) = : (1.1) We will be working for the most part in an equivalent canonical setting where A is replaced by 266664 1 0 0 1 0 2 2 ::: 0 0 n n !1 !2 !n+1 377775 ; (1.2) where i = 1 for each i: Equity, de…ned as a discounted series of dividends, is regarded here as a function of the discount rate R; thus P0(R; d) = 1 X