Current vs. Permanent Earnings for Estimating
Alternative Dividend Payment Behavioral Model:
Theory, Methods and Applications*
Cheng-Few Lee
Hong-Yi Chen
Alice Lee
Tzu Tai
Rutgers University, USA
lee@business.rutgers.edu
National Central University, Taiwan
State Street, USA
Rutgers University, USA
* Speech to be delivered to Stockholm University on April 29-30, 2013.
Flow Chart of Dividend Policy and Dividend Behavior
Dividend
Policy
Dividend
Irrelevance Theory
Dividend
Relevance Theory
without Tax Effect
Empirical Work
Theory
Time-Series
Dividend
Relevance Theory
with Tax Effect
CAPM
Approach
Non-CAPM
Approach
Cross-Sectional
2
Flow Chart of Dividend Policy and Dividend Behavior
Dividend
Relevance Theory
without Tax Effect
Theory
Empirical Work
Time-Series
Dividend
Behavior
Cross-Sectional
Signaling
Hypothesis
Free Cash Flow
Hypothesis
Flexibility
Hypothesis
Life-Cycle
Theory
Price Multiplier
Model
Cov. btw
ROE and g
Partial Adjustment
Process
Information
Content or
Adaptive
Expectations
Myopic Dividend
Policy
Residual Dividend
Policy
3
Empirical Analyses in Dividend Policy Research
Descriptive Data
Analysis
Time Series
Cross-Sectional
Regression
Analysis
Time Series
Probit / Logit
Cross-Sectional
Fama-MacBeth
Procedure
Panel Data
Analysis
Seemly
Uncorrelated
Regression
(SUR)
Fixed Effect
Model
4
Outline
1. Introduction
2. Theoretical determination of firm’s permanent and transitory earnings and
dividends
3. Alternative methods for decomposing current EPS into permanent- and
transitory-EPS components
3.1 Darby’s (1974) method
3.2 Lee and Primeaux’s (1991) method
3.3 Garrett and Priestley’s (2000) Kalman filter method
3.4 Lambrecht and Myer’s (2012) method
4. Empirical results in estimating two alternative dividend behavior models
4.1 Darby’s method and Lee and Primeaux’s method
4.1.1 Results from 608 Individual Regressions
4.1.2 Results from Pooled Regression
4.2 Lambrecht and Myer’s method
4.3 Combined model
4.3.1 Results from 605 Individual Regressions
4.3.2 Results from Pooled Regression
5 Summary and Concluding Remarks
5
Introduction
1. Earnings of a firm are allocated to retained earnings or dividend payments by
a financial decision.
2. The earnings of a firm can be classified into either permanent or transitory
components; permanent earning power creates the permanent component, and
the transitory component is composed of income of a temporary nature.
3. Forecasts of dividends are important to both security analysts and financial
managers, and either conditional or unconditional methods are generally used
to forecast dividend payments.
4. Lintner (1956) developed his well-known dividend payment behavior model
to describe how managers determine their dividend payment. This model has
been widely used in finance research (see Marsh and Merton, 1987; Lee and
Primeaux,1991; Garrett and Priestley, 2000; Lambrecht and Myer, 2012).
5. However, using current EPS to estimate Lintner’s dividend behavior model
might be also subject to measurement error problem (see Miller and
Modigliani, 1966)) .
6
The main purposes of this paper are…
1. To theoretically explain why firms generally allocate permanent
earnings and transitory earnings between dividends payments and
retained earnings
2. To develop alternative methods for decomposing current earnings
and dividends into permanent and transitory components
3. To empirically estimate alternative dividend payment behavior models
by using two alternative permanent EPS estimates for both individual
firms and pooled data
4. To test Lambrecht and Myer’s theoretically results related to
alternative dividend payment behavior models.
7
Theoretical determination of firm’s permanent
and transitory earnings and dividends
• The permanent-income hypothesis explains that consumption is not a
function of current income but a function of permanent income (Milton
Friedman,1957).Current income is divided into two components:
E = EP + ET
(1)
where E is the current income per share of the firm, EP is the permanent
income per share of the firm, and ET is the transitory income per share of
the firm.
• Dividends can also be divided into two similar components:
D = D P + DT
(2)
where D is the current dividends per share paid by the firm, DP is the
permanent dividends per share paid by the firm, and DT is the transitory
dividends per share paid by the firm.
8
Theoretical determination of firm’s permanent
and transitory earnings and dividends
• Permanent dividends are dividends that the business firm
systematically pays based on its permanent earnings, dividends
paid out of transitory earnings would constitute extra
dividends.
• All income is either paid out in dividends or retained by the
business in the form of retained earnings:
E =EP+ ET, E − (DP + DT) − R = 0
(3)
where R is the retained earnings per share of the firm, the other
terms are as defined in Equations (1) and (2), and ET and DT are
‘‘random” or “chance” variations in income and dividends per
share.
9
Theoretical determination of firm’s permanent
and transitory earnings and dividends
• Different sources of dividend payment (i.e., permanent or
current income) may have different implications in
determining a firm’s dividend payment behavior.
• This condition provides the motivation for examining both
permanent and current earnings per share for describing a
firm’s dividend payment behavior in the empirical section of
this work.
• In the next section, we will discuss alternative methods for
decomposing current EPS into permanent and transitory EPS
components.
10
Alternative methods for decomposing current EPS into
permanent- and transitory-EPS components
• Darby’s (1974) method
• The relationship between current dividend and permanent earning can be
defined as
Di ,t     EiP,t  i ,t
(4)
where Di ,t and EiP,t are current DPS and permanent EPS for ith firm in period t
respectively.
• Since permanent earnings are not directly observable, we use Eq. (5) to
estimate EiP,t as:
EiP,t  (1  i ) Ei ,t  i (1  C) E Pi ,t 1,
0  i  1
(5)
where λi represents the weight used to calculate the permanent earnings per share
and C represents the trend rate of EPS growth.
• Initial value of permanent EPS EPi,0 and trend rate C can be derived from Eqs.
(6) and (7):
Log Ei,t = a1 + a2t + ut
(6)
(7) 11
E Pi ,0  eaˆ1 ,
log(1  C)  aˆ2
Alternative methods for decomposing current EPS into
permanent- and transitory-EPS components
• Lee and Primeaux’s (1991) method
By using the adaptive-expectation model, the permanent EPS, EPi,t, can be
determined as
EiP,t  (1  i ) Ei ,t  i E Pi ,t 1 ,
0  i  1
(9)
By Koyck transformation, we can use Eq. (10) to obtain λi for firm i:
Di ,t  0  0 Ei ,t   Di ,t 1
(10)
where 0   (1  i ), 0   (1  i ), and   i
• Then, by using estimated λi , current EPS, and initial permanent EPS which
was described in Darby’s method, we can estimate permanent EPS in
period t.
12
Alternative methods for decomposing current EPS into
permanent- and transitory-EPS components
• Garrett and Priestley’s (2000) Kalman filter method
• Garrett and Priestley use Kalman filter to deal with measurement Eq. (11)
and transition Eqs. (12) and (13) and to estimate unobservable permanent
EPS:
Ei ,t  EiP,t  ET i ,t
(11)
EiP,t  EiP,t 1  t 1  t
(12)
t  t 1  t
(13)
where Ei,t is current EPS, transitory EPS is ETi,t ,and the permanent EPS, EPi,t,
evolve as a random walk with a changing trend, 𝛽𝑡 .
13
Alternative methods for decomposing current EPS into
permanent- and transitory-EPS components
• 3.4 Lambrecht and Myer’s (2012) method
• Using the joint determination of manager’s rent and cash dividend payment
to equity holders, Lambrecht and Myer (2012) derive a Lintner dividend
payment behavior in terms of permanent income as:
dt  a0  a1dt 1  Yt  et
(14)
where dt and dt 1 are total dividend payout at time t and t-1 respectively; Yt is
the firm’s permanent income at time t.
• If the profit margin  t follows the autoregressive process  t   t 1  t ,
then permanent income Yt can be simplified as:
i
(15)
Y 
( K    (1     )TD )
i ,t
1  i  i
i
i ,t
i
i
i ,t 1
where Kii,t is total operating income without corporate tax for ith firm in
period t; TDi ,t 1 is the total debt for ith firm in period t-1; i is interest rate; and
is the autoregression coefficient for operating income of the firm i.
14
Alternative methods for decomposing current EPS into
permanent- and transitory-EPS components
• 3.4 Lambrecht and Myer’s (2012) method
• Lambrecht and Myer (2012) claim that Lintner model as traditionally
estimated can be defined as:
dt  b0  bTE
1
t  b2 dt 1  ut
(16)
• Their true model is:
 SOA
 2  SOA
 (1   ) SOA
dt   
TEt  SOAdt 1 
t 
TDt 1  et
1  
1  
1  
(17)
where dt  dt  dt 1 ; the current reported earnings is TEt  pt   t  TDt 1; pt and  t
are permanent and transitory components respectively. TDt 1 is the component
neither permanent nor transitory component of earnings. SOA is the speed of
adjustment;  is the constant term of dividend behavior model, it generally
used to measure the degree of reluctance to cut dividend,  defined as
percentage of earnings paid as cash dividend,  =1/(1+  ).
15
Alternative methods for decomposing current EPS into
permanent- and transitory-EPS components
• 3.4 Lambrecht and Myer’s (2012) method
• If does not approach  to 1, then equation (17) can be modified as
P
dt  b0  bTE
1
t  b2 dt 1  b3 TDt 1  ut
(18)
where TE Pt is the estimated permanent earnings.
• Equation (18) is obtained by combining Lambrecht and Myer’s (2012)
theory and estimating permanent earnings estimated by either Darby’s or
Lee and Primeaux’s method.
• This specification takes care both specification errors and the transitory
components of earnings. This new specification is most important
contribution of this research.
16
Empirical results in estimating three alternative
dividend behavior models
• Sample
• EPS and DPS data of 608 firms from Compustat which has at least
30 years data ended in 2011 to perform the empirical studies (see
Appendix C).
• The methods used to do the empirical studies include
(1) Darby’s method and Lee and Primeaux’s method
(2) Lambrecht and Myer’s method
(3) combined model as defined in Equation (18)
17
Darby’s and Lee and Primeaux’s methods
• We use current and permanent EPS measures to estimate following two
alternative dividend payment behavior models as:
Di ,t  c0  c1Ei ,t  c2 Di ,t 1  ui ,t
(19a)
Di ,t  c0  c1EiP,t  c2 Di ,t 1  ui ,t
(19b)
Di ,t  c '0  c '1 Ei ,t  c '2 Di,t 1  c3 Di,t 2  ui,t
(20a)
Di ,t  c '0  c '1 EiP,t  c '2 Di,t 1  c3 Di ,t 2  ui,t
(20b)
• Following Lee and Chen (2013), we now analyze the biased associated with
estimated c1 and c2 as follows:
Case 1: when COV ( EiP,t , Di ,t 1 )  0,
 12 ( D D  c2 D2 )
c1 12
(21a)
ˆ
plim cˆ1  c1  2
,
plim
c

c


0
2
2
( E   12 )
 D2 ( E2   12 )
i ,t i ,t 1
P
i ,t
i ,t 1
i ,t 1
P
i ,t
Case 2: when COV ( EiP,t , Di ,t 1 )  0,
plim cˆ1  c1 
E
2
P
i ,t
2


c1 12

1

, plim cˆ2  c2  c1bD E P  2
i ,t 1 i ,t
  1   E2 P (1  RE2P D ) 
 bD E P   12
i ,t 1 i ,t
i ,t
i ,t i ,t 1


(21a)
where  12 is the variance of EiT,t , bD E is the auxiliary regression coefficient of a
regressing Di,t 1 on EiP,t , and RE2 D is the correlation coefficient between EiP,t and Di ,t 1 .
P
i ,t 1 i ,t
P
i ,t i ,t 1
18
Darby’s and Lee and Primeaux’s methods
• Equations (21a) and (21b) imply that the estimated c1 and c2
are downward biased. Hence, the estimated intercept as
defined in Equation (21c) is upward biased.
(21c)
cˆ0  Di ,t  cˆ1Ei ,t  cˆ2 Di ,t 1
• Therefore, we need to deal with this kind of errors-in-variable
problem.
• First, we use Darby’s and Lee and Primeaux’s methods to
estimate permanent EPS. Then we will use DPS and both
current EPS and permanent EPS to estimate equations (19a),
(19b), (20a), and (20b) for individual firms and pooled data.
19
Darby’s and Lee and
Primeaux’s methods
Table 1 (A)
Individual Regression Results
for Equations (19a), (19b),
(20a) and (20b)
• Current EPS is used in Eqs.
(19a) and (20a).
• The permanent EPS calculated
by Darby’s method, is used in
Eqs. (19b) and (20b).
• The permanent EPS calculated
by Lee and Primeaux’s
method, is used in Eqs. (19b)*
and (20b)*.
20
Darby’s and Lee and
Primeaux’s methods
Table 1(B).
Partial Adjustment Coefficient
and Long-Term Payout Ratios
• Current EPS is used in Eqs.
(19a) and (20a).
• The permanent EPS calculated
by Darby’s method, is used in
Eqs. (19b) and (20b).
• The permanent EPS calculated
by Lee and Primeaux’s
method, is used in Eqs. (19b)*
and (20b)*.
21
Darby’s and Lee and Primeaux’s methods
• Table 2. Alternative EPS and Payout Ratios
• According to the average estimated λi by using Darby’s and Lee and
Primeaux’s methods, it implies that Lee and Primeaux’s method for
estimating permanent earnings weights more heavily on current
earnings than those from Darby’s method.
22
Darby’s and Lee and
Primeaux’s methods
Table 3
Pooled Regression Results
for Equations (19a), (19b),
(20a) and (20b)
• Current EPS is used in Eqs.
(19a) and (20a).
• The permanent EPS calculated
by Darby’s method, is used in
Eqs. (19b) and (20b).
• The permanent EPS calculated
by Lee and Primeaux’s
method, is used in Eqs. (19b)*
and (20b)*.
23
Lambrecht and Myer’s method
• Since i is not available for individual firm, therefore, we use limiting
definition of Lambrecht and Myers’ (2012) method (see equation (15)) to
estimate permanent income and apply permanent income to test dividend
payment behavior models:
di ,t  a0  a1TEi ,t  a2 d i ,t 1  ei ,t
(22a)
di ,t  a0  a1Yi ,t  a2 di ,t 1  ei ,t
(22b)
(23a)
di ,t  a0  a1TEi ,t  a2 di ,t 1  a3di ,t  2  ei ,t
(23b)
di ,t  a0  a1Yi ,t  a2 di ,t 1  a3di ,t 2  ei ,t
• In addition, Lambrecht and Myers (2012) show that the Lintner model may
be subject to the model misspecification. We therefore test the model
misspecification by using Equation (24):
(24)
di ,t  a0  a1TEi ,t  a2 di ,t 1  a3 iTDi ,t 1  ei ,t
24
Lambrecht and Myer’s method
Table 4
Individual Regression
Results for Equations (22a),
(22b), (23a), (23b) and (24)
• There are 25.45% of firms
whose dividend payouts
can be determined by their
interest expenses.
• It indicates that there exists
specification error in
Lintner’s model in terms of
current earnings.
25
Combined model
• We will modify Equation (18) in terms of EPS and DPS as follows:
Di ,t  b0  b1Ei ,t  b2 Di ,t 1  b3 Ii ,t 1  ui ,t
(25a)
Di ,t  b0  b1EiP,t  b2 Di ,t 1  b3 Ii ,t 1  ui ,t
(25b)
where Di,t and Di,t-1 are dividend per share for firm i at time t and t-1,
respectively; Ei,t and EPi,t are current and permanent EPS for firm i at time t ;
Ii,t-1 is the interest expense per share firm i at time t-1.
• Eqs. (25a) and (25b) can be used to test whether the companies’ annual
EPS is following the random walk or not and also to test whether
Lambrecht and Myers’s budget constraint in Eq. (26) is held for individual
firm or not.
dt  rt   t ( K )  TDt 1  (TDt  TDt 1 )
(26)
where dt is total dividend payout at time t , TDt and TDt 1 is the total debt in
period t and t-1, respectively;  is interest rate; rt is managerial rents at time
t;  t ( K ) is gross profit at time t.
26
Combined model
Table 5
Individual Regression Results
for Equations (25a) and (25b)
• If the budget constraint does
not hold, then the term
associated with interest
expense will not necessary
exist.
• There are only 22.64%,
21.16%, or 22.15% firms
with budget constraints hold
under Lambrecht and Myers
theoretical model.
27
Combined model
Table 6.
Pooled Regression Results for
Equations (25a) and (25b)
• The insignificant coefficient
of interest expense per share
implies that the permanent
EPS not only can remove
random fluctuation of EPS
but also can remove parts of
misspecification error which
is shown by Lambrecht and
Myers.
28
Summary and Concluding Remarks
• Based upon the theories and methods developed by Marsh and Merton
(1987), Lee and Pri-meaux (1991), Garrett and Priestley (2000), and
Lambrecht and Myers (2012), we perform both theoretically analyses and
empirical studies in this paper.
The major findings are:
• The average long-term payout ratio is downward biased and the average
estimated intercept is upward biased when current instead of permanent
EPS are used in dividend behavioral model.
• We also empirically investigate Lambrecht and Myers’ misspecification
issue and find that interest expense per share might be important for
estimating dividend behavior model for some firms
Future research:
• Revise Lambrecht and Myers’ permanent earnings measurement
• Extend aggregate dividend behavior model to individual dividend behavior
model to test either signaling theory hypothesis or free cash flow
hypothesis for individual firms.
29
Reference
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Ang, J. S. (1975). “Dividend Policy: Informational Content or Partial Adjustment?” Review of Economics and
Statistics 57: 65-70.
Ando, A., and Modigliani. F. (l963). “The ‘Life Cycle’ Hypothesis of Saving.” American Economic Review 53: 55-84.
Almeida, H., M. Campello, and A. F. Galvao Jr. (2010). Measurement errors in investment equations, Review of
Financial Studies, 23, 3279-3328.
Black, F. (1976). “The Dividend Puzzle.” Journal of Portfolio Management II (Winter): 5-8.
Cochran, W. G. (1970). “Some Effects of Errors of Measurement on Multiple Correlation.” Journal of American
Statistical Association 65: 22-34.
Chen, H. Y., Gupta, M. C., Lee, A. C., and Lee, C. F. (2013). “Sustainable Growth Rate, Optimal Growth Rate,
and Optimal Payout Ratio: A Joint Optimization Approach.” Journal of Banking & Finance 37, 1205-1222.
Darby, M. R. (1972). “The Allocation of Transitory Income Among Consumers’ Assets.” American Economic
Review (September): 928-41.
Darby, M. R. (1974), “The Permanent Income Theory of Consumption—A Restatement.” Quarterly Journal of
Economic, (May): 228-50.
Dichev, I. D., and Tang, V. W. (2009) “Earnings volatility and earnings predictability.” Journal of Accounting and
Economics 47, 160–181.
Duesenberry, J. S. (1949). Income, Savings, and the Theory of Consumption Behavior. Cambridge MA: Harvard
University Press.
Eisner, R. (1967). “A Permanent Income Theory of Investment.” American Economic Review 57: 363-90.
Eisner, R. (1978). Factors in Business Investment. General Series No. 102. Washington. D.C.: National Bureau of
Economic Research.
Fama, E. F., and Babiak, H. (1968). “Dividend Policy: An Empirical Analysis.” Journal of American Statistical
Association 63: 1132-61.
Frankel, R., and Litov, L. (2009) “Earnings persistence.” Journal of Accounting and
Economics 47, 182–190.
Friedman, M. (1957). A Theory of the Consumption Function. Princeton. NJ: Princeton University Press.
30
Garrett, I. and Priestley, R. (2000). “Dividend Behavior and Dividend Signaling.” The Journal of Financial and
Quantitative Analysis (June) 35: 173-189.
Reference
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Higgins, R. (1977). “How Much Growth Can a Firm Afford?” Financial Management 6, 7-16.
Higgins R. C. (1981) “Sustainable growth under Inflation.” Financial Management 10, 36-40.
Higgins R. C. (2008) Analysis for financial management, 9th ed. (McGraw-Hill, Inc, New York, NY).
Johnston. J. (1972). Econometric Methods, 2nd ed. New York; McGraw-Hill.
Kmenta, J. (1986). Elements of Econometrics, second edition. New York: Macmillan.
Lambrecht, B. M. and Myers, S. C. (2012). “A Lintner Model of Payout and Managerial Rents.” Journal of Finance
(October) 67: 1761-1810
Latane, H. A., and Jones, C. P. (1979). “Standardized Unexpected Earnings-1971-77.” Journal of Finance 34: 71724,
Lee, C. F., and Chen, H.Y. (2013). “Alternative Errors-in-Variables Models and Their Applications in Finance
Research,” working paper.
Lee, C. F., M. C. Gupta, H. Y. Chen, and Lee, A. C. (2011) “Optimal payout ratio under uncertainty and the
flexibility hypothesis: Theory and empirical evidence.” Journal of Corporate Finance, 17: 483-501.
Lee, C. F., and Primeaux, W.J.(1991). “ Current- Versus Permanent- Dividend Payments Behavioral Model:
Methods and Applications,” Advances in Quantitative Analysis of Finance and Accounting. Vol 1 (Part A) 109- 130.
Lee, C. F., M. Djarraya, and C. Wu. (1987) “A further empirical investigation of the dividend adjustment process.”
Journal of Econometrics,267-285.
Leibenstein, H. (1950). “Bandwagon, Snob, and Veblen Effects in the Theory of Consumers’ Demand,” Quarterly
Journal of Economics (May): 183-207.
Lintner, J. (1956). “Distribution of Income of Corporations Among Dividends, Retained Earnings, and Taxes.”
American Economic Review (May): 97-113.
Marsh, T.A. and Merton, R.C. (1987). “Dividend Behavior for the Aggregate Stock Market.” The Journal of
Business (Jan) 60: 1-40.
Miller, M. H., and Modigliani, F. (1961). “Dividend Policy, Growth and Valuation of Shares.” Journal of Business 34
(October): 411-33.
Miller, M. H., and Modigliani, F. (1966). “Some Estimates of the Coast of Capital to the Electric Utility Industry.”
31
American Economic Review 56: 334-91.
Reference
•
•
•
•
•
•
Miller, M. H., and Scholes, M. S. (1982). “Dividends and Taxes: Some Empirical Evidence.” Journal of Political
Economy.
Modigliani, F., and Miller, M. H. (1958). “The Cost of Capital, Corporation Finance and The Theory of Investment.”
American Economic Review 4B: 261-97.
Modigliani, F., and Miller, M. H. (1963). “Corporate Income Tax and the Cost of Capital: A Correction.” American
Economic Review 53: 433-43.
Peterson, W. C. (1978). Income. Employment and Growth. 4th ed. New York: W. W. Norton.
Wang, N., (2003). “Caballero Meets Bewley: The Permanent-Income Hypothesis in General Equilibrium.”
American Economic Review 93, 927-936.
Weston, F.J., Brigham, E., and Besley,S., (2004) “Essentials of Managerial Finance,” Cengage South-Western
32
Optimal payout ratio under uncertainty and the
flexibility hypothesis: Theory and empirical evidence
The Model for Optimal Dividend Policy
- Let r (t ) represent the initial assets of the firm and h(t )
represent the growth rate. Then, the earnings of this firm are
given by Eq. (1), which is
x(t )  r (t ) A(o)eht
- The retained earnings of the firm, y  t , can be expressed as
y (t )  x(t )  m(t )d (t )
where m(t ) is the number of shares outstanding, and
d (t ) is dividend per share at time t.
33
The Model for Optimal Dividend Policy
- The stock price should equal the present value of this certainty
equivalent dividend stream discounted at the cost of capital (k)
of the firm.
p(o)   dˆ (t )e kt dt
T
0
( A  bI  h) A(o)eth   m(t ) p(t )   A(o) 2  (t ) 2  (t ) 2 e 2th  kt
 [

]e dt
2
0
m(t )
m(t )
T
- A differential equation can be formulated:
p(t )  [
where
 m(t )
m(t )
 k ] p(t )  G (t )
(a  bI  h) A(o)eth   A(o)2  (t )2  (t )2 e2th
G(t ) 

m(t )
m(t )2
• Optimal Payout Ratio when   1 :
D(t ) (a  bI  h)  (h  (t ) 2  (t ) 2   (t ) 2  (t ) 2   (t ) 2  (t ) 2 )[e( h  k )(T t )  1] 

1 

x (t )
(a  bI ) 
 (t ) 2  (t ) 2 (h  k )

(a  bI  h)  [e( h  k )(T t )  1] 
 (t ) 2  (t ) 2  
=

1 
h

(a  bI ) 
(h  k ) 
 (t ) 2  (t ) 2  
34
Optimal Payout Ratio vs. Systematic Risk
  D(t ) / x (t ) 
  (t ) 2 

2

(
t
)


h
 (1 
a  bI
 e( h k )(T t )  1 
)

h

k


 High growth firms  h  a  bI  :
negative relationship between optimal payout ratio and
systematic risk.
 Low growth firms  h  a  bI  :
positive relationship between optimal payout ratio and
systematic risk
35
Multivariate Regression
 payout ratio
i ,t
ln 
 1  payout ratioi ,t




    1 Riski ,t   2 Di ,t  g  ROA  Riski ,t  3Growth _ Optioni,t   4 ln( Size)i,t  Fixed Effect Dummies  ei


Flexibility Hypothesis
FCF Hypothesis
36
36
Sustainable Growth Rate, Optimal Growth Rate, and
Optimal Payout Ratio: A Joint Optimization Approach
Joint Optimization of Growth Rate and Payout Ratio
• The new investment at time t is
t
dA  t 
g  s  ds

o
 A t   g t  A o  e
dt
 Y  t   D t   n t  p(t )  LA t 
Retained Earnings New Equity
New Debt
p  t   price per share at time t;
  degree of market perfections, 0 <   l; L = the debt to total assets ratio
n  t  P t   the proceeds of new equity issued at time t;
where D t   the total dollar dividend at time t;
37
Joint Optimization of Growth Rate and Payout Ratio
• The model defined in previous slide is for the convenience purpose. If we
want the company’s leverage ratio unchanged after the expansion of assets
then we need to modify the equation as
t
g  s ds

o
A t   g t  A  o  e
 Y  t   D t   n t  p(t )  1  D / E  Y t   D t   n t  p (t ) 
we can obtain the growth rate as
g (t ) 
Our Model
ROE 1  d 
1  ROE 1  d 

 n t  p t  / E
1  ROE 1  d  
Higgins’ sustainable g
which is the generalized version of Higgins’ (1977) sustainable growth rate
model. Our model shows that Higgins’ (1997) sustainable growth rate is
under-estimated due to the omission of the source of the growth related to
new equity issue which is the second term of our model.
38
Joint Optimization of Growth Rate and Payout Ratio
p  o    dˆ  t  e kt dt
T
Discount cash flow
0
The price per share can be expressed as PV of future dividends with a
risk adjustment.
p o 
1

n o

T
0
g  s  ds

 1
2
  2 2 0 g  s  ds   kt

2
0



r t   g t  A  o  e
 a A  o   t  n t  e
n t 
 e dt




t
t
Risk Adj.
Future Dividends
 maximize p(o) by jointly determine g(t) and n(t).
Optimal Growth Rate
g t  
*
r

go r
 rt   2

r   rt   2 go   r  go  e  
1  1   e
 go 
Logistic Equation – Verhulst (1845) => a convergence process
39
Case I: Optimal Growth Rate v.s. Time Horizon
  r   r t    2 
 e
 
*


2

g  t 
 
g* t  

2
t
 

r  r t    2
1

1

 

e
g
o 
 

When g 0  r , g*  t   0.When g 0  r , g*  t   0.When g 0  r , g*  t   0.

r
r 1 
 go
Case III: Optimal Growth Rate v.s. ROE
g *  t 
r



t
 rt    2  
 rt    2   

go  go 1  e
  g o  r 
re





2



 




 g o   r  g o  e rt    2 


2
g *  t 
When initial growth rate is lower than the target rate (ROE), r is
positive.
=> If the target rate (ROE) is higher, the adjustment process will be
faster.
40
Optimal Dividend Payout Ratio
D t 
 g * t  
 1 

Y  t  
r  t  
2
*
*
2
*
t

 2
 
kt    g *  s  ds     t     t  g  t    r  g  t      t  g  t  

0
1  e
W


3



2
*
 


2



t
r

g




t 



T
where W  
t
s
2
g  u  du  ks 2
 1
e 0
  s   r  g *  s   ds

*
• Assuming   1 and g *  t   g * ,


( g *  k )(T t )

2 
e
1 
D t  

g 

(
t
)
*
 1 
1

g




Y  t   r  t   
(g*  k ) 
 (t ) 2  


*
- Wallingford (1972), Lee et al. (2010)
41
Stochastic Growth Rate and Specification Error
dA  t 
g  s  ds

 A t   g t  A  o  e
 Y  t   D  t    n  t  p (t )  LA t 
t
o
dt
Retained Earnings New Equity
New Debt
When a stochastic growth rate is introduced,
g  t   N  g  t  ,  g2  t  
g  s  ds
g  s  ds 
g  s  ds 




 r  t   g  t   A  o  e 0   n t  p t  Cov  r t  , A  o  e 0   E  t  e 0 





E  d  t    
n t 
n t 
t
t
t
g  s  ds 

If Cov  r  t  , A  o  e 0
 is positive, d  t  in the previous analysis


is overestimated.
t
42
The higher of the Cov., the higher possibility to stop the cash dividends.
43
Conclusion - 1
• We derive an optimal payout ratio using an exponential utility
function to derive the stochastic dynamic dividend policy
model.
- Different from M&M model, our model considers 1) partial
payout; 2)uncertainty (risks); 3) stochastic earnings.
• A negative relationship between the optimal dividend payout
ratio and the growth rate.
• The relationship between firm’s optimal payout ratio and its
risks depends on its growth rate relative to its ROA.
- high growth firms pay dividends due to the consideration of
flexibility and low growth firms pay dividends due to the
44
consideration of free cash flow problem.
Conclusion - 2
• We derive a dynamic model of optimal growth rate and payout
ratio which allows a firm to finance its new assets by retained
earnings, new debt, and new equity.
• The optimal growth rate follows a convergence processes, and
the target rate is firm’s expected ROE.
• The firm’s dividend payout is negatively associated with the
covariance between the firm’s rate of return on equity and the
firm’s growth rate.
• The firm tends to pay a dividend if its covariance between the
firm’s rate of return on equity and the firm’s growth rate is
lower.
45