Sustainable Growth Rate, Optimal Growth Rate, and Optimal Dividend Policy:

A Joint Optimization Approach

Hong-Yi Chen Rutgers UniversityManak C. Gupta Temple UniversityAlice C. Lee State Street Corp.Cheng-Few Lee Rutgers University

Outline

• Introduction and Motivation

• The Model

• Optimal Growth Rate

• Optimal Dividend Policy

• Stochastic Growth Rate and Specification Error on Expected Dividend

• Empirical Evidence

• Conclusion

2

Introduction & Motivation

Dividend Policy and Growth Rate

• Gordon (1962), Lintner (1964), Lerner and Carleton (1966), Modigliani and Miller (1961), Miller and Modigliani (1966)

- Relationships between optimal dividend policy and rate of return under no growth and under both internally and externally, financed growth assumptions.

• Higgins (1977, 1981, and 2008)

- Sustainable growth rate: assuming that a firm can use retained earnings and issue new debt to finance the growth opportunity of the firm.

• DeAngelo and DeAngelo (2006) - M&M (1961) irrelevance result is “irrelevant” because it only considers

payout policies that pay out all free cash flow.- Payout policy matters when partial payouts are allowed.

3

Introduction & Motivation

• Rozeff (1982)

- The optimal dividend payout is related to the fraction of insider holdings, the growth of the firm, and the firm’s beta coefficient.

• Benartzi et al. (1997), DeAngelo et al. (1996 and 2006), Grullon et al. (2002), Blau and Fuller (2008), Lee et al. (2011)....

Only focus on conducting their analyses at the equilibrium point, but they do not focus on analyzing the time path that leads to the equilibrium

Do not jointly consider investment decision and finance decision at the same time

=> Is there an optimal dividend policy for a firm under the imperfect market, the uncertainty of the investment, and the dynamic growth rate?

4

Introduction & Motivation1. We develop a fully dynamic model for determining the time optimal

growth and dividend policy under stochastic conditions.

2. We study the effects of the time-varying horizons, the degree of market perfection, and stochastic initial conditions in determining an optimal growth and dividend policy for the firm.

- A convergence process in the optimal growth rate.

3. When the stochastic growth rate is introduced, the expected return may suffer a model specification.

- A firm’s payout ratio is negatively correlated to the covariance between its profitability and its growth rate.

4. Empirical evidence support the mean-reverting process of a firm’s optimal growth rate and provide an alternative explanation of diminishing cash dividend payouts.

5

Model

(3)

t

og s ds

g t A o e

6

• The new investment at time t is

where

( )Y t D t n t p t LA t

dA tA t

dt

the total dollar dividend at time ;D t t

price per share at time ;p t t

degree of market perfections, 0 < l;

n the proceeds of new equity issued at time ;t P t t

= the debt to total assets ratioL

Retained Earnings New Equity New Debt

Model

• The model defined in the equation (3) is for the convenience purpose. If we want the company’s leverage ratio unchanged after the expansion of assets then we need to modify equation (3) as

we can obtain the growth rate as

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.

7

( ) 1 / ( )t

og s ds

A t g t A o e Y t D t n t p t D E Y t D t n t p t

1 /( )

1 1 1 1

ROE d n t p t Eg t

ROE d ROE d

Our Model

Higgins’ sustainable g

Model

Discount cash flow

The price per share can be expressed as PV of future dividends with a risk adjustment.

=> maximize p(o) by jointly determine g(t) and n(t).

8

0

ˆ (8)T ktp o d t e dt

0 0

21 2 22

0

1t t

T g s ds g s ds ktp o n t r t g t A o e a A o t n t e e dtn o

Future Dividends Risk Adj.

Optimal Growth Rate

Logistic Equation – Verhulst (1845) => a convergence process

9

*

2

2

(19)

1 1

rt

o

ort

o o

rg t

re

g

g r

g r g e

Mean-Reverting Process

10

Optimal Dividend Payout Ratio

where

• Assuming ,

- Wallingford (1972), Lee et al. (2011)11

*

0

*

2 2 * * 2 *

32 *

1

1 (29)2

tkt g s ds

D t g t

Y t r t

t t g t r g t t g te W

t r g t

*

0212 *

sT g u du ks

tW e s r g s ds

* *1 and g t g

*( )( )* 2

** 2

1 ( )1 1

( ) ( )

g k T teD t g tg

Y t r t g k t

Stochastic Growth Rate and Specification Error

When a stochastic growth rate is introduced,

12

( ) (3)t

og s dsdA t

A t g t A o e Y t D t n t p t LA tdt

2, gg t N g t t

Retained Earnings New Equity New Debt

Stochastic Growth Rate and Specification Error

13

0 0 0,

(32a)

t t tg s ds g s ds g s ds

r t g t A o e n t p t Cov r t A o e E t e

E d tn t n t

0If , is positive, in the previous analysis

is overestimated.

tg s ds

Cov r t A o e d t

Hypotheses Development

• Hypothesis 1: The firm’s growth rate follows a mean-reverting process.

H1a: There exists a target rate of the firm’s growth rate, and the target rate is the firm’s return on equity.

H1b: The firm partially adjusts its growth rate to the target rate.

H1c: The partial adjustment is fast in the early stage of the mean-reverting process.

• Hypothesis 2: 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.

H2a: The covariance between the firm’s rate of return on equity and the firm’s growth rate is one of the key determinants of the dividend payout policy.

14

Hypotheses Development

• Hypothesis 3: 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.

H3a: The firm tends to stop paying a dividend if its covariance between the firm’s rate of return on equity and the firm’s growth rate is higher.

H3b: The firm tends to start paying a dividend if its covariance between the firm’s rate of return on equity and the firm’s growth rate is lower.

15

Sample

• Stock price, stock returns, share codes, and exchange codes are CRPS. Firm information, such as total asset, sales, net income, and dividends payout , etc., is collected from COMPUSTAT.

• The sample period is from 1969 to 2011.

• Only common stocks (SHRCD = 10, 11) and firms listed in NYSE, AMEX, or NASDAQ (EXCE = 1, 2, 3, 31, 32, 33) are included.

• Utility firms and financial institutions (SICCD = 4900-4999, 6000-6999) are excluded.

• For the purpose of estimating their betas to obtain systematic risks, firm years in our sample should have at least 60 consecutively previous monthly returns.

16

Empirical Results – Mean Reverting Process

17

Partial Adjustment Model

18

Partial Adjustment Model

19

, , , , ,i t j t j i t i t i t jg ROE g e

j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7 j = 8 j = 9 j = 10

All Sample0.1299*** 0.1276*** 0.1230*** 0.1188*** 0.1082*** 0.0045 0.0008 0.0046 0.0081 0.0107*

(10.11) (10.83) (11.35) (11.27) (9.20) (0.89) (0.18) (0.76) (1.28) (1.91)

j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7 j = 8 j = 9 j = 10

P1 0.2199*** 0.2072*** 0.1865*** 0.1791*** 0.1683*** 0.0164* 0.0220*** 0.0252** 0.0346*** 0.0314***

(High Growth) (11.74) (11.59) (10.90) (10.22) (8.88) (1.91) (2.67) (2.37) (3.30) (3.03)

                   

P2 0.1838*** 0.1715*** 0.2003*** 0.2149*** 0.2018*** 0.1067*** 0.0875*** 0.0596* 0.0498** 0.0342

(7.31) (6.43) (8.63) (7.99) (6.09) (3.52) (3.19) (1.94) (2.08) (1.17)

                   

P3 0.2939*** 0.3175*** 0.3234*** 0.3739*** 0.5283*** 0.3525*** 0.3624*** 0.2827*** 0.2451** 0.1456**

(4.30) (3.57) (4.08) (4.97) (6.90) (4.40) (4.14) (2.73) (2.52) (2.18)

                   

P4 -0.0654 -0.1236** -0.176*** -0.222*** -0.1798** -0.2304** -0.2085** -0.1443* -0.0230 -0.0871

(-1.08) (-2.18) (-2.86) (-4.35) (-2.25) (-2.49) (-2.28) (-1.77) (-0.28) (-0.68)

                   

P5 0.0574*** 0.0420** 0.0168 0.0116 0.0144 -0.0685** -0.0569* -0.0292 -0.0222 -0.0169

(2.99) (2.09) (0.85) (0.57) (0.74) (-2.71) (-1.94) (-0.97) (-0.68) (-0.60)

                   

P6 0.0621*** 0.0680*** 0.0659*** 0.0616*** 0.0558*** 0.0006 -0.0125 -0.0173 -0.0236 -0.0166

(Low Growth) (6.45) (6.31) (6.24) (4.72) (4.51) (0.05) (-1.04) (-1.40) (-1.57) (-1.09)

,ˆ

t j

20

21

22

Conclusion

• Given the uncertainty of ROE, the theoretical model shows the existence of an optimal growth rate and an optimal payout ratio.

• 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.

23