Dividend smoothing, the present value model, and negative autocorrelations of stock

Faculty Working Paper 91-0151

330

B385

1991:151 COPY 2

STX

Dividend Smoothing, the Present Value Model,
and Negative Autocorrelations of Stock

Yoon Dokko

Department of Finance

Bureau of Economic and Business Research

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO. 91-0151

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

June 1991

Dividend Smoothing, the Present Value Model,
and Negative Autocorrelations of Stock

Yoon Dokko, Assistant Professor
Department of Finance

I would like to thank Hyuk Choe and Robert Edelstein for their suggestions on this
paper. Of course, I am responsible for any remaining errors.

Digitized by the Internet Archive

in 2011 with funding from

University of Illinois Urbana-Champaign

http://www.archive.org/details/dividendsmoothin91151dokk

Dividend Smoothing, the Present Value Model, and Negative
Autocorrelations of Stock Price Changes

Abstract

A consequence of partial dividend smoothing is that dividends revert (slowly)
to their targets and the stock price reverts to the present value of expected
future target dividends. This target reverting can cause stock price changes
to be negatively autocorrelated. As dividend smoothing increases, the neg-
ative autocorrelation becomes less significant. The negative autocorrelation
appears to be a "V" shaped function of the length of holding periods. As
stock price volatility increases relative to dividend volatility, the negative
autocorrelation becomes more significant.

Introduction

Recent research contributions by DeBondt and Thaler (1985), Fama and
French (19S7), and Poterba and Summers (1988), among others, find that
stock price changes (especially in the long run) are NEGATIVELY AUTO-
CORRELATED. This finding contradicts the long-standing hypothesis in the
Finance 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 present
value model and motivated by the well-known fact that corporate managers
smooth their dividend payments. The dividend depends in part on a target
dividend and in part on previous years' dividends.1 The degree of dividend
smoothing is inversely related to the speed of dividend adjustment to the
target. Hence, under partial dividend smoothing, the dividend reverts par-
tially to the target. Holding the discount rate constant, the stock price will
also revert to the present value of expected future target dividends. We
will show below how the negative autocorrelations of stock price changes
are created by the "target reverting" process of stock prices, and how they
are affected by the length of holding periods and the degree of dividend
smoothing.

I. Analysis

Following Lintner (1956) and Fama and Babiak (1968), we consider a simple
model 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 for
period t, and 7(0 < 7 < 1) is the speed of the dividend adjustment toward
the 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 completely
smoothed.

The present value model is

where 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 through
time r+t), k is the discount rate, which is assumed to be constant, and Et is
the investor's expectations operator conditional upon information available
at time r.

Since equation (2) means that EtPt+i = (1 + k)Pt — EtDt+\, from equa-
tion (1) we have

EtPt+l = (i + *)Pt-7£tAVi-(i-7)A. (3)

Rational investors recognize corporate dividend smoothing and incorporate
the dividend smoothing behavior (equation 1) into the present value model
(equation 2). This generates

We 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)Dt
in 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 C1 + k)Pt- Tnis ^elds

EtPt+1 = 7£tp;+1 + (i-7)pt. (6)

Equation (6) shows that the speed of the stock price adjustment toward
the 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 stock
price forecast error such that cov( rjt1rjt+i) — 0 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 price
is a random walk and cov(Pt+2T — Pt+n Pt+r — Pt) = 0 for all r > 1.

Let r be the length of holding periods, and A = 1 — 7 (A measures the
degree of dividend smoothing). The change in stock prices over r periods
is

Pt+r-Pt = p-AL)-l{7(^Hr-ll?W-ft-l/T) + ^T-*} (8)

where L is the backward shift operator.

We assume that <72(»7t) = tf for all t. To compute the first-order au-
tocorrelation of Pt+T — Pt for 0 < A < 1, we need to assume a stochastic
process for D* (and thus for P(*). We consider two cases: (i) D* is a white
noise around some mean; (ii) Dj is a random walk.

A. Case 1: when D\ is a white noise around some mean.

We assume that

Dmt = D + et (9)

where D is the mean of D*, and et is a white noise. It follows that P* =
D/k, and £t+T_1P;+T - Et.xP; = 0 for all t.
Changes in stock prices over r periods are

fl+r-A = £^Wi-(l-AT)i>Vi, (10-a)

j=0 3=0

and

pt+2T-p<+T = $;:r+i-(i-AT)EAi^-i

j=0

where $1+?+! denotes the terms with T7t+2T, • • • ,tyt+T+1,
For 0 < A < 1, we have

2(1 -A')

var(Pt+r-P,) = „ l*> (11-a)

1-A2
and

-(i-y)2

1-A2

cov(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 0 < A < 1, we find that

dJW1 > °- <13>

This implies that increased dividend smoothing reduces the magnitude of
negative 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 time
period. Similarly, Kim, Nelson, and Startz (1989) find that negative au-
tocorrelations of stock price changes may not exist during the post- World
War II period. These findings could be attributed to temporal shifts in
dividend smoothing. In fact, A during the pre-war period is smaller than
that during the post-war period. In particular, after the corporate income
tax 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 are
about 0.25 to 0.30 and 0.75 to 0.85, respectively, for the 1932-1951 time
period and the 1952-1986 time period.3 We may conjecture that increased
dividend smoothing in recent years has reduced negative autocorrelations
of stock price changes.

Holding A constant between 0 and 1, we find that

*%2 < o. (u)

This result would be consistent with Poterba and Summers' finding that
as the length of holding periods increases, the magnitude of negative auto-
correlations of stock returns tends to increase. However, Fama and French

In 1952, the statutory corporate income tax rate was raised from 15 percent to 52
percent.

3 A 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 for
allr.

B. Case 2: when D\ is a random walk.

We assume that

d; = A-i+« (is)

where et is a white noise. It follows that

Dt+1-EtDt+1 = 7€t+1 (16)

and

p; = *»;_,+<* (n)

where ut = tt/k. Equation (17) generates

Et+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, becomes

Pt+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 to
understand the relationship between the stock price forecast error (rjt+i) ^d
the dividend forecast error (7£t+i ). This is seen by substituting £^i ^{i+ffl*'

6

for Pt+1 and ££, %%$■ for Pt in the present value model, Pt+l = (l+*)Pt-
EtDt+i -f »7t+i- It follows that

j=i (I+iy • (20)

By the law of iterative conditional expectations, we have

^i+iA+i+i — Et A+i+i = a» ( A+i — ^t A+i) + &,«+i

= at-7et+1 + £,t+1 (21)

where a, is a regression coefficient, and £,-,*+ 1 is a regression error such that
cov(et+1,f,tf+1) = 0 for all i. It is convenient to approximate a, as4

1 - A,+1
a. = l-T^r. (22)

Equation (20) becomes

• _ " l - aw " &w

** _ ,+,£(TW £?uW

where 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" forecast
errors (&+i)- Hereafter,

4Ftom equation (1), we have E,+1D,+i+l - E,Dt+1+i = 7(£t+iA"+i+i - ^i£f+i+f) +
\(Et+lDt4.i - EtDt+i) = T€l+1(l + A + .. + A1) = *=^p(A+i - EtDt+1). For the last
equality, see equation (16).

Substituting equation (23) into equation (19) generates (see Appendix

A)

P,+r-P, = (l-AI)-W

T-l

«^t + T + 7 E ^t+T-j + (7 - C)Ut + 6+

J = l

T"61

T-l

= E {(1 - Xi) + «tf } wk+r-i + (1 - <0(1 - ^) E A^-i

j=0

j=0

T-l

00

and

+ £A'6+r-i-(i-A*)£A'6_.il

j=0 >=0

-(i-at)i;a^+t-;

i=o

00

(24-a)

T(l-c)(l-V)A^AVi

i=o

T-l

OO

- (i - y) E Ai^+T-i - (i - a')at e *&-;

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) = 0 for all t and ji, and var((t) = a\ for
all t. Since & also is a rational forecast error, cov(ft, &+,) = 0 for all t ^ 0.
We express o^ as q<7*, where q is a positive (but unknown) constant, and,

8

without loss of generality, a^ = 1. For 0 < A < 1, we have5

«(*, -Pt) = r + 2(l-V)fa-_(lrcXc + A)}t (25.a)

and

,(1-A^{g-(1-C)(c + A)}
1-A2

(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. This
happens for 0 < q < 1, because — (1 — c)(c + A) decreases from 0 to —1 as
A increases from 0 to 1.

When A = 0 (i.e., if dividends revert immediately to the target), we
have

/(t,A = 0,<?) = -^L < o. (27)

T + Zq

This 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 extremely
smoothed, it is likely that
a/(r,A,g)

dr

and

< 0 for r < some r*, (28-a)

> 0 for r > some r*. (28-b)

5It can be shown that dvar(Pl+T - Pt)/dr > 0, and var(/><+i — Pt) is positive for any
q > 0. Hence, var(Pl+r — Pt) is positive for any r.

9

This result implies that the pattern of /(r, A, q) is a "V" shape with respect
to the length of holding periods. To prove this, let x = AT(0 < x < A) so
that r = gj, A = q — (1 — c)(c + A), which is assumed to be positive for
the autocorrelation to be negative,6 and B = ncj-, which is negative. We
express 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):7

g(x) = 2Bx\nx + 2Ax(l-x) + B(l-x). (30)

It follows that

g(x = 0) = B < 0 (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 A

Unless 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£ < 0 (i.e., |f > 0) for
small r (i.e., large x), and |£ > 0 (i.e., |f < 0) for large r (i.e., small x).

Assuming that k = 0.08 and q = 1, Table 1 computes /(r, A) for
r = 1,2, •••,10 and for A = 0.25, 0.50, and 0.75. These A values would
correspond to those of the pre-war period, the 1920s-1980s period, and the

We assume that q > 1, which implies that o\> at*

g(x) — 0 is the first-order condition for the maximum negative autocorrelation.

10

post-war period, respectively. For A = 0.25, 2-year stock price changes
have the largest negative autocorrelation. For A = 0.75, 6-year stock price
changes have the largest negative autocorrelation. The patterns of these
computed negative autocorrelations appear to resemble those observed by
Fama and French (1987).

Finally, holding r and A constant, we find that

dJ^l3l < 0. (32)

dq

As q increases, the magnitude of the negative autocorrelation of stock price

changes increases. Equation (23) shows that the variability of stock prices

(<j*) is determined by the variability of dividends (a* ) and the variability of

other 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 prices

increases relative to the variability of dividends. To the extent that firm size

is inversely related to the magnitude of g, our result corroborates Zarowin's

(1990) finding that negative autocorrelations of stock price changes are

observed mostly for small firms.8

II. Summary and Concluding Remarks

When the dividend paid reverts to a target dividend, the stock price also
reverts to the present value of expected future target dividends. Rational
forecast errors of this target reverting process can create negative autocor-
relations of stock price changes. We further find that (1) the significance of
the negative autocorrelation is inversely related to the degree of dividend
smoothing; (2) the negative autocorrelation appears to be a "V" shaped

'See also Chopra, Lakonishok and Ritter (1991).

11

function of the length of holding periods; (3) as the variability of stock
prices increases relative to the variability of dividends, the negative auto-
correlation becomes more significant. Future empirical studies can, using
cross-section data, test for the characteristics of the behavior of stock price
changes which are predicted by our present value model analysis.

12

References

Brittain, 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 Empirical
Analysis," Journal of American Statistical Association vol. 63, no. 324
(December 1968), pp. 1132-61.

Fama, Eugene F., and Kenneth R. French, "Permanent and Temporary
Components 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 Expected
Stock Returns," Journal of Financial Economics vol. 22, no. 1 (October
1988), pp. 3-25.

Kim, Myung Jig, Charles R. Nelson, and Richard Startz, "Mean Reversion
in 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 Stock
Prices," Journal of Financial Economics vol. 22, no. 1 (October 1988),
pp. 27-60.

Zarowin, Paul, "Size, Seasonality, and Stock Market Overreaction," Journal
of Financial and Quantitative Analysis vol. 25, no. 1 (March 1990), pp.
113-25.

13

Appndix A: Derivation of Equations (24) and (25)
Equation (24-a): 0 < 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 as

aJt+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-l

This is rearranged as

+AT-ie

t+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_i

T-l

OO

+ EA^+r-i-(l-AT)^A^-i

i=o i=o

The coefficient of o;t+T_;, for 0 < j < r - 1, is

7(l + A + ... + A'-1) + cA'

= 7

1-A'
1- A

+ cA' = 1 - X3 + cXj (recall 7 = 1 - A)

14

The 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 relegate
the terms with r]t+2Tj" •>ty+T+i>6+2T,'-'>&+r4-i to #tS+n wn^cn are unnecessary for
computing cov(Pt+2T - P<+T,Pt+T - Pt).

Equation (25- a): 0 < 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 f g A2> + (1 - A')2 £ A2' }
The first sum becomes

T-l

= £{(i-Ai)Hc2A2'+2c(i-Ai)Ai}

J=0

I = E{1-2(l-c)A^(l-c)2A^}
>=o

2(l-c)(l-AT) (l-c)2(l-A2-)

1 _ A "*" 1 - A2

Since £°i0 A2' = ^ and EJ=J A2> = ^ = Ez*jg+JQ it follows that
var(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)21

+ — r^ — + <7\ r^ /

15

2(1 - c)(l - Xr) (1 - c)2(l - A*)(l + V + 1 - AT)

— " : : r

+

= r

= r +

1 - A ' 1 - A2

g(l-AT)(l + AT + l-AT)
1-A2
2(l-c)(l-A*) 2(1 - c)2(l - A*) 2g(l-A*)

1-A 1-A2 + 1-A2

2(l-AT){9 + (l-c)2-(l-c)(l + A)}

1-A2

This leads to equation (24-a) in the main text.

Equation (25-b): cov = cov(Pt+2T - Pt+T,Pt+T - Pt)

T-l

cov = (l-c)(i_AT)^{l-(l-C)A>}AJ + (l-c)2(l-AT)2AT^A2j

-is

j=0

T-l

i=o

oo

T\2\T

(1-A')£A2'-(1-At)2At£A2'
j=o >=o

n ,vi ml1-** (i-<0(i-ATXi + A*n (i-c)2(i-AT)2A

J(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-A2

g(l-AT)2(l + AT-AT)
1-A2
(1 - c)(l - A-)2(l + A) - (1 - c)2(l - AT)2 - g(l - AT)2

1-A2
-(l-AT)2{<Z + (l-c)2-(l-c)(l + A)}
1-A2

This leads to equation (25-b) in the main text.

16

Table 1
Computing Autocorrelations of Stock Price Changes

Equation (26)

Length of

/(r,A.

q = l,fc =

: 0.08)

Holding Periods

A =

A =

A =

w

0.25

0.50

0.75

1

-0.230

-0.140

-0.059

2

-0.233*

-0.182

-0.095

3

-0.202

-0.186*

-0.118

4

-0.172

-0.175

-0.129

5

-0.148

-0.159

-0.134

6

-0.130

-0.144

-0.135*

7

-0.116

-0.131

-0.133

8

-0.105

-0.119

-0.129

9

-0.095

-0.109

-0.124

10

-0.087

-0.100

-0.119

* denotes the largest negative autocorrelation.