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Dokko, Yoon

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Dividend smoothing, the present value model, and negative autocorrelations of stock / 91-151

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Faculty Working Paper 91-0151330B3851991:151 COPY 2STXDividend Smoothing, the Present Value Model,and Negative Autocorrelations of StockYoon DokkoDepartment of FinanceBureau of Economic and Business ResearchCollege of Commerce and Business AdministrationUniversity of Illinois at Urbana-ChampaignBEBRFACULTY WORKING PAPER NO. 91-0151College of Commerce and Business AdministrationUniversity of Illinois at Urbana-ChampaignJune 1991Dividend Smoothing, the Present Value Model,and Negative Autocorrelations of StockYoon Dokko, Assistant ProfessorDepartment of FinanceI would like to thank Hyuk Choe and Robert Edelstein for their suggestions on thispaper. Of course, I am responsible for any remaining errors.Digitized by the Internet Archivein 2011 with funding fromUniversity of Illinois Urbana-Champaignhttp://www.archive.org/details/dividendsmoothin91151dokkDividend Smoothing, the Present Value Model, and NegativeAutocorrelations of Stock Price ChangesAbstractA consequence of partial dividend smoothing is that dividends revert (slowly)to their targets and the stock price reverts to the present value of expectedfuture target dividends. This target reverting can cause stock price changesto be negatively autocorrelated. As dividend smoothing increases, the neg-ative autocorrelation becomes less significant. The negative autocorrelationappears to be a "V" shaped function of the length of holding periods. Asstock price volatility increases relative to dividend volatility, the negativeautocorrelation becomes more significant.IntroductionRecent research contributions by DeBondt and Thaler (1985), Fama andFrench (19S7), and Poterba and Summers (1988), among others, find thatstock price changes (especially in the long run) are NEGATIVELY AUTO-CORRELATED. This finding contradicts the long-standing hypothesis in theFinance literature that the stock price is a random walk.The objective of this paper is to analyze the observed negative autocor-relations of stock price changes. Our analysis is developed upon the presentvalue model and motivated by the well-known fact that corporate managerssmooth their dividend payments. The dividend depends in part on a targetdividend and in part on previous years' dividends. 1 The degree of dividendsmoothing is inversely related to the speed of dividend adjustment to thetarget. Hence, under partial dividend smoothing, the dividend reverts par-tially to the target. Holding the discount rate constant, the stock price willalso revert to the present value of expected future target dividends. Wewill show below how the negative autocorrelations of stock price changesare created by the "target reverting" process of stock prices, and how theyare affected by the length of holding periods and the degree of dividendsmoothing.I. AnalysisFollowing Lintner (1956) and Fama and Babiak (1968), we consider a simplemodel for dividend smoothing:Dt = 7(A*-A-i) + A-i (i)'See Lintner (1956), Brittain (1966), Fama and Babiak (1968), Marsh and Merton(1987), and Choe (1990), among others.where Dt is the dividend paid for period r, D\ is the target dividend forperiod t, and 7(0 < 7 < 1) is the speed of the dividend adjustment towardthe target. (1 — 7) is the degree of dividend smoothing. When 7 = 1,the dividend immediately reverts to the target level and is not smoothed.If 7 = 0, the dividend will never revert to the target and is completelysmoothed.The present value model iswhere Pt is the stock price at the beginning of period t + 1 (or at time t),Dt+i is the dividend paid during period t+ 1 (or from time t + i' — 1 throughtime r+t), k is the discount rate, which is assumed to be constant, and Et isthe investor's expectations operator conditional upon information availableat time r.Since equation (2) means that EtPt+i = (1 + k)Pt — EtDt+\, from equa-tion (1) we haveEtPt+l = (i + *)Pt - 7£tAVi-(i-7)A. (3)Rational investors recognize corporate dividend smoothing and incorporatethe dividend smoothing behavior (equation 1) into the present value model(equation 2). This generatesWe define P* as the present value of expected future target dividends (here-after referred to as the target price); P; = ££1 j^jfi. Solving for (l-y)Dtin equation (4) yields(1- 7)A = (7 + W-7(l +W. (5)In equation (3), we substitute the right hand side of equation (5) for (1 —y)Dt and then substitute £t(Pt*+1 + A*+i) for C 1 + k )Pt- Tnis ^eldsEtPt+1 = 7£tp;+1 + (i- 7)pt . (6)Equation (6) shows that the speed of the stock price adjustment towardthe target price is the same as that of the dividend adjustment.Since Pt+i = EtPt+i H-fy+i, where rjt+ i is assumed to be a rational stockprice forecast error such that cov( rjt1 rjt+i) — for all i ^ 0, from equation(6) we have (time subscripts are reduced by 1)Pt = 7^-iP; + (1 - 7)Pt-i + %• (7)If dividends are completely smoothed (i.e., A = 1 or 7 = 0), the stock priceis a random walk and cov(Pt+2T — Pt+n Pt+r — Pt) = for all r > 1.Let r be the length of holding periods, and A = 1 — 7 (A measures thedegree of dividend smoothing). The change in stock prices over r periodsisPt+r-Pt = p-AL)-l{7(^Hr-ll?W-ft-l/T) + ^T-*} (8)where L is the backward shift operator.We assume that <7 2 (»7t) = tf for all t. To compute the first-order au-tocorrelation of Pt+T — Pt for < A < 1, we need to assume a stochasticprocess for D* (and thus for P(*). We consider two cases: (i) D* is a whitenoise around some mean; (ii) Dj is a random walk.A. Case 1: when D\ is a white noise around some mean.We assume thatDmt= D + et (9)where D is the mean of D*, and e t is a white noise. It follows that P* =D/k, and £t+T_ 1P;+T - Et. x P; = for all t.Changes in stock prices over r periods arefl+r-A = £^Wi-(l-AT)i>Vi, (10-a)j=0 3=0andpt+2T -p<+T = $;:r+i-(i-AT)E Ai^-ij=0where $1+?+! denotes the terms with T7t+2T , • • • ,tyt+T+1 ,For < A < 1, we have2(1 -A')var(Pt+r -P,) = „ l*> (11-a)1-A2and-(i-y) 21-A2cov(Pt+2T -Pt+T ,Pt+T -P( ) =-1 —J-a* (11-b)The first-order autocorrelation of stock price changes over r periods, /(r, 7),is/(nA) = ^(l-A')- (12)Holding r constant and for < A < 1, we find thatdJW1 > °- <13 >This implies that increased dividend smoothing reduces the magnitude ofnegative autocorrelations of stock price changes. Fama and French (1987)find that the negative autocorrelations of stock price changes for the 1941-1985 time period are less significant than those for the 1926-1985 timeperiod. Similarly, Kim, Nelson, and Startz (1989) find that negative au-tocorrelations of stock price changes may not exist during the post-WorldWar II period. These findings could be attributed to temporal shifts individend smoothing. In fact, A during the pre-war period is smaller thanthat during the post-war period. In particular, after the corporate incometax reform in 1952, 2 the dividend appears to depend mostly on the pre-vious year's dividend. It is found, using S & P annual data, that A's areabout 0.25 to 0.30 and 0.75 to 0.85, respectively, for the 1932-1951 timeperiod and the 1952-1986 time period.3 We may conjecture that increaseddividend smoothing in recent years has reduced negative autocorrelationsof stock price changes.Holding A constant between and 1, we find that*%2 < o. (u)This result would be consistent with Poterba and Summers' finding thatas the length of holding periods increases, the magnitude of negative auto-correlations of stock returns tends to increase. However, Fama and FrenchIn 1952, the statutory corporate income tax rate was raised from 15 percent to 52percent.3A similar result is found in Fama and French (1988) and Choe (1990).(1987) observe that the negative autocorrelations reach a maximum for 3-to 5-year stock returns and then decrease toward zero as the length of hold-ing periods increases. The negative sign of ^''may not be the case forallr.B. Case 2: when D\ is a random walk.We assume thatd; = A-i+« (is)where et is a white noise. It follows thatDt+1 -EtDt+1 = 7€t+1 (16)andp; = *»;_,+<* (n)where ut = t t /k. Equation (17) generatesEt+T-iPf+r — Et-\P? = P?+T-i—Pt-v= ut + u>t+ i -I- • • • + w<+T_i.(18)Equation (8), the stock price change over r periods, becomesPt+r-Pt = (l-AI)-1 {7W.+r-l+-+1Wl + I|t+T -^}. (19)For computing the first-order autocorrelation of Pt+T — Pti we need tounderstand the relationship between the stock price forecast error (rjt+i) ^dthe dividend forecast error (7£ t+i ). This is seen by substituting £^i ^{i+ffl*'6for Pt+1 and ££, %%$ for Pt in the present value model, Pt+l = (l+*)Pt-EtDt+ i -f »7t+i- It follows thatj=i (I+iy• (20)By the law of iterative conditional expectations, we have^i+iA+i+i — EtA+i+i = a» (A+i — ^tA+i) + &,«+i= at-7et+1 + £,t+1 (21)where a, is a regression coefficient, and £,-,*+ 1 is a regression error such thatcov(et+1 ,f, tf+1 ) = for all i. It is convenient to approximate a, as41 - A,+1a. =l-T^r . (22)Equation (20) becomes•_" l - aw " &w**_,+,£(TW £?uWwhere c<+1 is replaced by fcu>t+1 (see equation 17), and £t+i = HSi Tr+fcT7 'Equation (23) shows that the stock price forecast error is in principle de-termined by the dividend forecast error (wt+i = et /k) and "other" forecasterrors (&+i)- Hereafter,4Ftom equation (1), we have E, +1 D,+i+l - E,Dt+1 +i = 7(£t+iA"+i+i - ^i£f+i+f) +\(Et+lDt 4.i - EtDt+i) = T€l+1 (l + A + .. + A1) = *=^p(A+i - EtDt+1 ). For the lastequality, see equation (16).Substituting equation (23) into equation (19) generates (see AppendixA)P,+r-P, = (l-AI)-WT-l«^t+T + 7E ^t+T-j + (7 - C)Ut + 6+J= lT" 61T-l= E {(1 - Xi ) + «tf} wk+r-i + (1 - <0(1 - ^)E A^-ij=0 j=0T-l 00and+ £A'6+r-i -(i-A*)£A'6_.ilj=0 >=0-(i-at)i;a^+t-;i=o00(24-a)T (l-c)(l-V)A^AVii=oT-l OO- (i - y)E Ai^+T-i - (i - a')ate *&-;j=0 j=0(24-b)where $!+?+! denotes the terms with r/t+2T," *,»7«+t+i,&+2t,- ",6+t+i-We assume that cov(u>t+i ,(t+J ) = for all t and ji, and var((t ) = a\ forall t. Since & also is a rational forecast error, cov(ft , &+,) = for all t ^ 0.We express o^ as q<7*, where q is a positive (but unknown) constant, and,8without loss of generality, a^ = 1. For < A < 1, we have5«(*, - Pt) = r + 2(l-V)fa-_(lr cXc +A)}t (25.a)and,(1-A^{g -(1- C)(c + A)}1-A 2(25-b)cov(Pt+2T - Pt+T , Pt+T - Pt ) =The first-order autocorrelation of stock price changes over r periods is/l ' '*' " t(1-A>) + 2(1-A'){9 -(1-c)(c + A)}- l26;This autocorrelation can be positive if {q — (1 — c)(c + A)} is negative. Thishappens for < q < 1, because —(1 — c)(c + A) decreases from to —1 asA increases from to 1.When A = (i.e., if dividends revert immediately to the target), wehave/(t,A = 0,<?) = -^L < o. (27)T + ZqThis illustrates that stock price forecast errors in the target reverting pro-cess cause stock price changes to be negatively autocorrelated.The sign of gj is not clear. However, unless the dividend is extremelysmoothed, it is likely thata/(r,A,g)drand< for r < some r*, (28-a)> for r > some r*. (28-b)5It can be shown that dvar(Pl+T - Pt)/dr > 0, and var(/><+ i — Pt ) is positive for anyq > 0. Hence, var(Pl+r — Pt ) is positive for any r.9This result implies that the pattern of /(r, A, q) is a "V" shape with respectto the length of holding periods. To prove this, let x = AT(0 < x < A) sothat r = gj, A = q — (1 — c)(c + A), which is assumed to be positive forthe autocorrelation to be negative,6 and B = ncj-, which is negative. Weexpress equation (26) as** = jJX^r (29)The sign of g£ is the opposite of the sign of §£; as r increases from 1 to oo,x decreases from A to 0. The sign of J£ is the same as the sign of g(x):7g(x) = 2Bx\nx + 2Ax(l-x) + B(l-x). (30)It follows thatg(x = 0) = B < (31-a)g(x = \) = A(l-A)|(l + A)^2+^+2>l} > 0(?) (31-b)^ = 2J51ni + B + 2^(l-2i)= 2(1 -A2)r + if^+ 2^(1 -2AT ) > 0(?) (31-c)In AUnless the dividend is extremely smoothed (i.e., if A is not close to 1),g(x = A) and fj are likely to be positive. Then, f£ < (i.e., |f > 0) forsmall r (i.e., large x), and |£ > (i.e., |f < 0) for large r (i.e., small x).Assuming that k = 0.08 and q = 1, Table 1 computes /(r, A) forr = 1,2, •••,10 and for A = 0.25, 0.50, and 0.75. These A values wouldcorrespond to those of the pre-war period, the 1920s-1980s period, and theWe assume that q > 1, which implies that o\> at*g(x) — is the first-order condition for the maximum negative autocorrelation.10post-war period, respectively. For A = 0.25, 2-year stock price changeshave the largest negative autocorrelation. For A = 0.75, 6-year stock pricechanges have the largest negative autocorrelation. The patterns of thesecomputed negative autocorrelations appear to resemble those observed byFama and French (1987).Finally, holding r and A constant, we find thatdJ^l3l < 0. (32)dqAs q increases, the magnitude of the negative autocorrelation of stock pricechanges increases. Equation (23) shows that the variability of stock prices(<j*) is determined by the variability of dividends (a* ) and the variability ofother variables (<r|). Since <7* = qa^ from equation (23) we have c2 + q =oi/<7*. Hence, an increase in q means that the variability of stock pricesincreases relative to the variability of dividends. To the extent that firm sizeis inversely related to the magnitude of g, our result corroborates Zarowin's(1990) finding that negative autocorrelations of stock price changes areobserved mostly for small firms.8II. Summary and Concluding RemarksWhen the dividend paid reverts to a target dividend, the stock price alsoreverts to the present value of expected future target dividends. Rationalforecast errors of this target reverting process can create negative autocor-relations of stock price changes. We further find that (1) the significance ofthe negative autocorrelation is inversely related to the degree of dividendsmoothing; (2) the negative autocorrelation appears to be a "V" shaped'See also Chopra, Lakonishok and Ritter (1991).11function of the length of holding periods; (3) as the variability of stockprices increases relative to the variability of dividends, the negative auto-correlation becomes more significant. Future empirical studies can, usingcross-section data, test for the characteristics of the behavior of stock pricechanges which are predicted by our present value model analysis.12ReferencesBrittain, John A., Corporate Dividend Policy, Washington, D.C.: The Brook-ings Institution, 1966.Choe, Hyuk, Intertemporal and Cross-Sectional Variation of Corporate Div-idend Policy, Unpublished Ph.D. Dissertation, University of Chicago,December 1990.Chopra, Navin, Josef Lakonishok, and Jay R. Ritter, "Performance Mea-surement Methodology and the Question of Whether Stocks Overreact,"Working Paper, University of Illinois, 1991.DeBondt, Warner F., and Richard M. Thaler, "Does the Stock Market Over-react?" Journal of Finance vol. 40, no. 3 (July 1985), pp. 793-805.Fama, Eugene F., and Harvey Babiak, "Dividend Policy: An EmpiricalAnalysis," Journal of American Statistical Association vol. 63, no. 324(December 1968), pp. 1132-61.Fama, Eugene F., and Kenneth R. French, "Permanent and TemporaryComponents of Stock Prices," Journal of Political Economy vol. 96, no.2 (April 1988), pp. 246-73.Fama, Eugene F., and Kenneth R. French, "Dividend Yields and ExpectedStock Returns," Journal of Financial Economics vol. 22, no. 1 (October1988), pp. 3-25.Kim, Myung Jig, Charles R. Nelson, and Richard Startz, "Mean Reversionin Stock Prices?" Working Paper, University of Washington, 1989.Lintner, John, "Distribution of Incomes among Dividends, Retained Earn-ings and Taxes," American Economic Review vol. 46, no. 2 (May 1956),pp. 97-113.Marsh, Terry A., and Robert C. Merton, "Dividend Behavior for the Aggre-gate Stock Market," Journal of Business vol. 60, no. 1 (January 1987),pp. 1-40.Poterba, James M., and Lawrence H. Summers, "Mean Reversion in StockPrices," Journal of Financial Economics vol. 22, no. 1 (October 1988),pp. 27-60.Zarowin, Paul, "Size, Seasonality, and Stock Market Overreaction," Journalof Financial and Quantitative Analysis vol. 25, no. 1 (March 1990), pp.113-25.13Appndix A: Derivation of Equations (24) and (25)Equation (24-a) : < A < 1.Pt+r-Pt = (1 - XL)' 1 {cut+T + lUt+r-i + '•' + 1(ut+i + (i - cfa + {t+T - Zt}The right hand side of this equation is rewritten asaJt+T +cXvt+r-i +c\2ut+T-2 +•• +cAT~ 1u;t+i +c\Tut +cXT+lut-i ++7u;t+T_ 1 +7Ao;<+T_ 2 +••• +7AT_2u;H. 1 +fXr~1u>t +7ATu;t_ 1 ++7u;t+T_2 + • • •+7<*>t+i +7^t +7^2a;t-i ++(7 - c)u>t +(7 - c)Au;t_! ++6+T +A^+T-lThis is rearranged as+AT- iet+i +AT6 +AT+16-i +-6 -A6-i -+ (7 + cA)u;t+T_ 1+ (7 + 7A + cA2 )wt+T_2+ (7 + 7A + 7A2 + cA3)wt+T_3+ (7 + 7A + ... + 7AT-2 + cAT" 1)u;t+1+ (-c + 7 + 7A + ..- + 7AT- 1 +cAT )a;t+ (-c + 7 + 7A+.-. + 7AT" 1+ cAT )Aa;t_iT-l OO+ E A^+r-i-(l-AT)^A^-ii=o i=oThe coefficient of o;t+T_; , for < j < r - 1, is7(l + A + ... + A'- 1 ) + cA'= 71-A'1-A+ cA' = 1 - X3 + cXj (recall 7 = 1 - A)14The coefficient of u>t-j, for j = 0, 1, • • • , is{-e + 7(l + A+- ... + AT-1) + cAT}v= {7\^ " c(l - AT )} X' = (1 - c)(l - AT )A'Equation (24-b):To compute Pt+2-r — P*+t, we replace subscript < in equation (24-a) with t+T and relegatethe terms with r]t+2Tj" •>ty+T+i>6+2T,'-'>&+r4-i to #tS+n wn^cn are unnecessary forcomputing cov(Pt+2T - P<+T,Pt+T - Pt ).Equation (25-a) : < A < 1, oj as 1, and <r| = q.var(P<+T - F«) =E {(1 - *') + cA>} 2 + (1 - c)2 (l - A')2 f) A2' + g fg A2> + (1 - A')2£ A2'}The first sum becomesT-l= £{(i-Ai)Hc2A2'+2c(i-Ai)Ai}J=0I = E{ 1 -2(l-c)A^(l-c) 2A^}>=o2(l-c)(l-AT ) (l-c)2(l-A2-)1 _ A"*"1 - A2Since £°i A2' =^ and EJ=J A2> =^ = Ez*jg+JQ it follows thatvar(P(+T -P,) = r.2d-c)(lrA^)+(l-c)»(l-^)(l + A^1-A l-A2(1 - c)2 (l - A-)2 f (l-A-)(l-hA-) + (l-AT )2 1+—r^—+ <7\ r^ /152(1 - c)(l - Xr ) (1 - c)2 (l - A*)(l + V + 1 - AT )— " : : r+= r= r +1 - A'1 - A2g(l-AT )(l + AT + l-AT )1-A22(l-c)(l-A*) 2(1 - c)2 (l - A*) 2g(l-A*)1-A 1-A2 + 1-A 22(l-AT){9 + (l-c)2 -(l-c)(l + A)}1-A2This leads to equation (24-a) in the main text.Equation (25-b): cov = cov(Pt+2T - Pt+T,Pt+T - Pt)T-lcov = (l- c)(i_AT)^{l-(l- C)A>}AJ + (l-c)2(l-AT )2AT^A2j-isj=0T-li=oooT\2\T(1-A')£A2'-(1-At )2At£A2'j=o >=on ,vi ml 1 -** (i-<0(i-ATXi + A*n (i-c)2(i-A ) AJ (1-AT)(1-AT )(1 + AT)-(1-A-)2AT )"U ^ J(1 - C)(l . \rf (1 _ c)2(1 _ Ar )2(1 ^ AT_ Ar )1-A 1-A 2g(l-AT )2 (l + AT -AT )1-A2(1 - c)(l - A-)2 (l + A) - (1 - c)2(l - AT )2- g(l - AT )21-A2-(l-AT )2 {<Z + (l-c)2 -(l-c)(l + A)}1-A2This leads to equation (25-b) in the main text.16Table 1Computing Autocorrelations of Stock Price ChangesEquation (26)Length of /(r,A. q = l,fc =: 0.08)Holding Periods A = A = A =w 0.25 0.50 0.751 -0.230 -0.140 -0.0592 -0.233* -0.182 -0.0953 -0.202 -0.186* -0.1184 -0.172 -0.175 -0.1295 -0.148 -0.159 -0.1346 -0.130 -0.144 -0.135*7 -0.116 -0.131 -0.1338 -0.105 -0.119 -0.1299 -0.095 -0.109 -0.12410 -0.087 -0.100 -0.119* denotes the largest negative autocorrelation.

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